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Curricular Standardsfor

Mathematics

Kansas State Board of EducationRevised July 2003

Kansas Curricular Standards for Mathematics

Table of Contents

Kansas Curricular Standards for Mathematics Writing Committee.........................................................

Kansas Curricular Standards for Mathematics Writing Committee – Special Education........................

Introduction to the Kansas Curricular Standards for Mathematics........................................................

Mission Statement...................................................................................................................... Vision Statement........................................................................................................................ Purpose of this Document.......................................................................................................... Prior History ............................................................................................................................ The Revision Process................................................................................................................

Format of the Kansas Curricular Standards..........................................................................................

Standards, Benchmarks and Indicators..................................................................................... Grade Levels Where Benchmarks and Indicators Have Been Developed................................ Special Note Regarding Seventh and Tenth Grade Benchmarks and Indicators...................... Definitions Which Will Be Helpful In Understanding the Document...........................................

Uses of this Document...........................................................................................................................

Development of Local Curriculum and Assessments................................................................ Development of the Kansas Mathematics Assessments...........................................................

Grade Levels to be Assessed........................................................................................ Kansas Assessment Scores to be Reported.................................................................. The Use of Calculators on Kansas Mathematics Assessments..................................... Prioritization of Indicators to be Assessed.....................................................................

Supplemental Documents......................................................................................................................

One Page Summary of the Standards and Benchmarks.......................................................................

Curricular Standards by Standard and Benchmark...............................................................................

Number and ComputationNumber Sense............................................................................................................... Number Systems and Their Properties...........................................................................Estimation........................................................................................................................Computation....................................................................................................................

AlgebraPatterns...........................................................................................................................Variables, Equations, and Inequalities............................................................................Functions.........................................................................................................................Models.............................................................................................................................

continued on next page

Kansas Curricular Standards for Mathematics – July 2003

Curricular Standards by Standard and Benchmark (continued from previous page)

GeometryGeometric Figures and Their Properties.........................................................................Measurement and Estimation..........................................................................................Transformational Geometry.............................................................................................Geometry from an Algebraic Perspective.......................................................................

DataProbability........................................................................................................................Statistics..........................................................................................................................

Assessment Framework.........................................................................................................................

Appendix 1 - Information on Mathematical Communication, Reasoning, and Problem Solving.....................................................................................

Appendix 2 – Curricular Standards Grade by Grade..............................................................................

Appendix 3 – Glossary of Terms.............................................................................................................

Appendix 4 – National Standards in Personal Finance..........................................................................

Appendix 5 – Selected Bibliography.......................................................................................................

Appendix 6 – Helpful Resources.............................................................................................................

Appendix 7 – Template for Pattern Blocks.............................................................................................

Appendix 8 – Scope and Sequence.......................................................................................................


The Kansas Curricular Standards for Mathematics Writing Committee

The writing committee would like to thank the more than 1,600 individuals and/or groups who submitted written responses to the first and sixth working drafts of this document as well as the more than 100 persons who provided input at work sessions and at public meetings held throughout the state. The committee thoughtfully read and considered each of the responses received and felt this input was invaluable to the development of this document.

In addition, the committee would like to thank the teachers, school administrators, parents, and community members who have and will continue to work toward improving mathematics education in Kansas.

Co-Chair: Betsy Wiens, Auburn Washburn Public Schools (Topeka), USD 437

Co-Chair: George Abel, Emporia Public Schools, USD 253

Bill FaFlick, Wichita Public Schools, USD 259

Michelle Flaming, ESSDACK (Hutchinson)

Pat Foster, Oskaloosa Public Schools, USD 341

Ann Goodman, Garden City Public Schools, USD 457

Margie Hill, Blue Valley Public Schools (Overland Park), USD 229

Nancy McDonnell, Prairie View Public Schools (LaCygne), USD 362

Sarah Meadows, Topeka Public Schools, USD 501

Jo Musselwhite, Salina Public Schools, USD 306

Cheryl Murra, Leavenworth Public Schools, USD 453

Joan Purkey, Newman University (Wichita)

Judy Roitman, University of Kansas (Lawrence)

Teresa San Martin, Maize Public Schools, USD 266

Allen Sylvester, Geary County Public Schools (Junction City), USD 475

Ethel Edwards, KSDE – School Improvement and Accreditation

Canda Mueller Engheta, KSDE – Planning and Research

Lynnett Wright, KSDE – Student Support Services


The Kansas Curricular Standards for Mathematics Writing Committee – Special Education

Chris Butler, Victoria Public Schools, USD 432

Melissa Donaldson, Columbus Public Schools, USD 493

Debra Esau, SCKSEC (Pratt)

Larry Finn, Kansas State School for the Deaf (Olathe)

Linda Hickey, Blue Valley Public Schools (Overland Park), USD 229

Marjorie Hill, Blue Valley Public Schools (Overland Park), USD 229

Sharon Laverentz, Perry Public Schools, USD 343

Charlene Lueck, Geary County Public Schools (Junction City), USD 475

Sheila R. Nigh, Wichita Public Schools, USD 259

Sally Richardson, Wichita Public Schools, USD 259

Cecilia Rijfkogel, Garden City Public Schools, USD 475

Deb Stephenson, Blue Valley Public Schools (Overland Park), USD 229

K. Nadine Voth, Hesston Public Schools, USD 460

Mary Kay Woods, Three Lakes Educational Cooperative (Quenemo)

Ethel Edwards, KSDE – School Improvement and Accreditation

Scott Smith, KSDE – School Improvement and Accreditation

Lynnett Wright, KSDE – Student Support Services


Introduction to the Kansas Curricular Standards for Mathematics

Mission Statement

The mission of Kansas mathematics education is for all Kansas students to learn mathematical content and skills that are used to solve a variety of problems.

Vision Statement

The vision of mathematics education in Kansas is to work toward the following:

Kansas mathematics education will be recognized as one of the premier programs in the United States.

Kansas mathematics education will be equally effective for all students, irrespective of gender, race, or socioeconomic background.

Kansas families will broadly recognize the importance of and be encouraged to participate actively in their child(ren)’s mathematics learning.

Technology will be a fundamental part of mathematics teaching and learning.

The Purpose of this Document

The standards, benchmarks, and indicators in this document have been created to assist Kansas educators in developing local curricula and assessments, as well as to serve as the basis for the development of the state assessments in mathematics. The committee strove to recommend high, yet reasonable expectations for all students. High, yet reasonable expectations for all students are components of fairness in education. All students includes: those who choose to attend college, those who choose technical preparation, those who will enter the workforce, those from various socio-economic backgrounds, those who have been identified as gifted in the area of mathematics, those who have been identified with learning disabilities, those who have previously been successful with mathematics, and those who have struggled with mathematics sometime in the past.

Students may need additional support both within and outside the regular classroom to meet those expectations. Teachers should be given the professional development and resources necessary to enable them to help all students strive to meet or exceed these expectations . This may seem a daunting task, but the alternative is not acceptable.

Prior History

The Kansas State Board of Education developed and adopted the Kansas Mathematics Improvement Program in 1989. As part of that plan, Kansas Mathematics Curriculum Standards were developed by Kansas educators in 1990. A new Kansas mathematics assessment program began as a statewide pilot in 1991. In 1992, the Kansas legislature directed that further work be done in the development of the mathematics curriculum standards, so that benchmarks would be established at a minimum of three grade levels. In 1993, the State Board adopted revised standards for mathematics. The 1993 Mathematics Curriculum Standards then served as the basis for the Kansas Mathematics Assessment program through the end of the 1998-99 school year. In August 1997, the State Board directed that additional work be done in each of the curriculum areas. The mathematics writing committee convened in October 1997


and worked through February 1999. At the March 1999 meeting of the Kansas State Board of Education, the Kansas Curricular Standards for Mathematics were approved. These standards served as the basis for the Kansas Mathematics Assessments that were administered for the first time during the spring of 2000. The Standards approved in March 1999 serve(d) as the basis for the Kansas Mathematics Assessments through the spring of 2005.

The Revision Process

At the May 2002 meeting of the Kansas State Board of Education, the State Board directed that academic standards committees composed of stakeholders from throughout Kansas were to be convened for the curriculum areas defined by state law, and, at this time, only reading, writing, and mathematics. The mathematics committee was charged to:

1) review the current standards document.2) review modified and extended standards for inclusion in the document.3) review the format to ensure usability.4) determine the level of specificity of skills assessed.5) recommend essential indicators to be assessed.6) review previous assessment results and review past KSDE studies as they related to

student achievement.

The mathematics writing committee began its work in June 2002. Members worked more than thirty days on revising the standards. Members of the committee reviewed various research articles, examples of mathematics curriculum standards from other states, and reviews of the current Kansas Mathematics Curriculum Standards by various individuals and organizations. Because earlier editions of the Kansas Mathematics Curriculum Standards had been used to guide curriculum, instruction, and assessment throughout the state since 1993, the committee felt it was very important to build upon existing efforts to improve mathematics education. The committee unanimously elected to use the 1999 Standards document as the basis of their work.

At their initial meeting, the writing committee reached consensus on the following revisions for the first draft of the revised Kansas Curricular Standards for Mathematics:

1) definitions of curriculum standards remain the same.2) greater clarity and specificity to the benchmarks and indicators by grade level so

teachers know what they should teach and what students should learn (grades K-8 and high school).

3) format and language revisions that make the standards and the benchmarks easier for non-mathematics educators to understand, as well as aligning the formatting of the mathematics standards document with standards documents from the other curriculum areas.

4) expectations for students were to be high, yet reasonable to achieve.5) prioritize the state curriculum indicators to be assessed by the state assessments

and provide advice regarding the assessment of the state curricular standards.

During meetings throughout the summer of 2002, members of the mathematics revision committee and subcommittees worked to bring specificity and clarity to the benchmarks in the current document by revising the benchmarks and indicators outlined in the 1999 Kansas Mathematics Curriculum Standards. In August 2002, the first working draft was mailed to schools throughout the state to obtain feedback on the draft. The committee continued to work


during the fall and winter months reviewing and discussing the feedback from schools, revising grade level expectations, learning about and discussing various assessment issues, and brainstorming possible assessment priorities. By February of 2003, a sixth working draft was mailed to educators, community members, and professional organizations. Public meetings were held in March throughout the state to receive public input on the sixth draft. Summaries of the public meetings and written feedback were compiled for the committee. At the same time, the sixth draft was submitted to the State Board’s designated outside reviewer, Mid-Continent Regional Educational Laboratory, for review. In March and April, the writing committee met to make revisions based on recommendations from McREL’s review and to further prioritize indicators that would be used as the basis for assessment items on the revised Kansas Mathematics Assessments. In May, the committee met to refine the document. Grade level expectations were revised and further prioritization of benchmarks and indicators to be used in developing the Kansas Mathematics Assessments were evaluated. The tenth working draft was prepared for submission to the State Board and discussed at their June meeting.

After the June 2003 meeting of the State Board, subcommittees met to refine their work including additional examples, completing the Financial Literacy connection, and designing a glossary. In addition, during June and July, committee members were facilitating Summer Mathematics Standards Academies across the state; nine such academies were held. The committee's recommendations, as presented in the eleventh draft, were acted upon and approved at the July 2003 meeting of the Kansas State Board of Education . Kansas Mathematics Assessments, based on these standards, are to be administered for the first time during the spring of 2006.

To be completed for standards document:


Format of the Kansas Curricular Standards for Mathematics

Uses of this Document

Supplemental Documents

One Page Summary of the Standards and Benchmarks

Curricular Standards by Standard and Benchmark

Assessment Framework


Standard 1: Number and Computation KINDERGARTEN

Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a variety of situations.

Benchmark 1: Number Sense – The student demonstrates number sense for whole numbers, fractions, and money using concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. establishes a one-to-one correspondence with whole numbers from

0 through 20 using concrete objects and identifies, states, and writes the appropriate cardinal number (2.4.K1a) ($).

2. compares and orders whole numbers from 0 through 20 using concrete objects (2.4.K1a) ($).

3. recognizes a whole, a half, and parts of a whole using concrete objects (2.4.K1a,c) ($), e.g., half a pizza, part of a cookie, or the whole school.

4. identifies positions as first and last (2.4.K1a). 5. identifies pennies and dimes and states the value of the coins using

money models (2.4.K1d) ($).

The student... 1. solves real-world problems using equivalent representations and

concrete objects to compare and order whole numbers from 0 through 10 (2.4.A1a) ($).

K-9January 31, 2004

N – Noncalculator($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.

Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in number sense.

When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that include an awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the effects of operations, and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person predicts with some accuracy the result of an operation and consistently chooses appropriate measurement units. This “friendliness with numbers” goes far beyond mere memorization of computational algorithms and number facts; it implies an ability to use numbers flexibly, to choose the most appropriate representation of a number for a given circumstance, and to recognize when operations have been correctly performed. (Number Sense and Operations: Addenda Series, Grades K-6, NCTM, 1993)

Mathematical models such as concrete objects, pictures, number lines, or unifix cubes are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.

The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.

Standard 1: Number and Computation KINDERGARTENK-10

January 31, 2004N – Noncalculator($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.


Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole numbers with a special emphasis on place value in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. reads and writes whole numbers from 0 through 20 in numerical

form ($).2. represents whole numbers from 0 through 20 using place value

models (2.4.K1b) ($), e.g., ten frames, unifix cubes, straws bundled in 10s, or base ten blocks.

3. counts (2.4.K1a) ($):a. whole numbers from 0 through 20,b. whole numbers from 10 to 0 backwards,c. subsets of whole numbers from 0 through 20.

4. groups objects by 5s and by 10s (2.4.K1a).5. uses the concept of the zero property of addition (additive identity)

with whole numbers from 0 through 20 and demonstrates its meaning using concrete objects (2.4.K1a) ($), e.g., 4 apples and no (zero) other apples are 4 apples.

The student...1. solves real-world problems with whole numbers from 0 through 20

using place value models (2.4.A1b) ($), e.g., group the class into tens, count by tens; then continue counting by ones to find the total.

2. counts forwards and backwards from a specific whole number using a number line from 0 through 10 (2.4.A1a).

K-11January 31, 2004


Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:

Property of a number: 8 is divisible by 2. Property of a geometric shape: Each of the four sides of a square is of the same length. Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x. Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c. Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.






Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in avariety of situations.

Benchmark 3: Estimation – The student uses computational estimation with whole numbers in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. determines if a group of 20 concrete objects or less has more, less,

or about the same number of concrete objects as a second set of the same kind of objects (2.4.K1a).

The student...1. compares two randomly arranged groups of 10 concrete objects or

less and states the comparison using the terms: more, less, about the same (2.4.A1a).



Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark” computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer.

Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational methods including mental math, paper and pencil, and appropriate technology. However, being able to compute does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students develop a range of estimation strategies. (Principles and Standards for School Mathematics, NCTM, 2000)

In order for students to become more familiar with estimation, estimation should be introduced with examples where rounded or estimated numbers are used emphasizing real-world examples where only estimation is required, e.g., about how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible.





Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers using concrete objects in a variety of situations.



Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. adds and subtracts using whole numbers from 0 through 10 and

various mathematical models (2.4.K1a) ($), e.g., concrete objects, number lines, or unifix cubes.

2. uses repeated addition (multiplication) with whole numbers to find the sum when given the number of groups (three or less) and given the same number of concrete objects in each group (five or less) (2.4.K1a), e.g., two nests with three eggs in each nest means 3 + 3 = 6 or 2 groups of 3 makes 6.

3. uses repeated subtraction (division) with whole numbers when given the total number of concrete objects in each group to find the number of groups (2.4.K1a), e.g., there are 9 pencils. If each student gets 2 pencils, how many students get pencils? 9 – 2 – 2 – 2 – 2 or 9 minus 2 four times means four students get 2 pencils each and there is 1 pencil left over. or There are eight cookies to be shared equally among four people, how many cookies will each person receive?

The student...1. solves one-step real-world addition or subtraction problems with

whole numbers from 0 through 10 using concrete objects in various groupings and explains reasoning (2.4.A1a) ($), e.g., seven apples are in a basket and five students each take an apple; how many apples are left in the basket?

Teacher Notes: The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, 24 ÷ 2 are all representations for the standard representation of 12.





Standard 2: Algebra KINDERGARTEN

Standard 2: Algebra – The student uses algebraic concepts and procedures in a variety of situations.

Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships in patterns using concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. uses concrete objects, drawings, and other representations to work

with types of patterns (2.4.K1a):a. repeating patterns, e.g., an AB pattern is like red-blue, red-blue,

…; an ABC pattern is like dog-horse-pig, dog-horse-pig, …; or an AAB pattern is like Δ-Δ-Ο, Δ-Δ-Ο, …;

b. growing (extending) patterns, e.g., 5, 6, 7, … is an example of a pattern that adds one to the previous number to continue the pattern.

2. uses these attributes to generate patterns:a. whole numbers (2.4.K1a), e.g., 2, 4, 6, …;b. geometric shapes with one attribute change (2.4.K1e), e.g., Δ,

Ο, Δ, Ο, Δ, Ο, …;c. things related to daily life (2.4.K1a), e.g., breakfast, lunch, and

dinner.3. identifies and continues a pattern presented in various formats

including numeric (list or table), visual (picture, table, or graph), verbal (oral description), and kinesthetic (action) (2.4.K1a) ($).

4. generates (2.4.K1a): a. repeating patterns for the AB pattern, the ABC pattern, and the

AAB pattern; b. growing (extending) patterns that add 1, 2, or 10 to continue the

pattern.5. classifies and sorts concrete objects by similar attributes (2.4.K1a)

($).

The student...1. generalizes the following patterns using pictorial, and/or oral

descriptions including the use of concrete objects:a. repeating patterns for the AB pattern, the ABC pattern, and the

AAB pattern (2.4.A1a) ($);b. patterns using geometric shapes with one attribute change

(2.4.A1c).2. recognizes multiple representations of the AB pattern (2.4.A1a),

e.g., big- little, big-little, big-little, ... and 1-2, 1-2, 1-2, ..., or AB, AB, AB, ....

3. uses concrete objects to model a whole number pattern (2.4.A1a):a. counting by ones: ,, , …;b. counting by twos: , , , …;c. counting by tens: , , , …



Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney, Macmillan Publishing Company, 1994)

This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain.







Benchmark 2: Variables, Equations, and Inequalities – The student solves addition equations using concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. finds the unknown sum using the basic facts with sums through 10

using concrete objects and pictures (2.4.K1a) ($), e.g., 5 marbles + 5 marbles = .

The student... 1. describes real-world problems using concrete objects and pictures

and the basic facts with sums through 10 (2.4.A1a) ($), e.g., given some marbles, Sue says: There are 3 red marbles and 3 blue marbles. Altogether, there are 6 marbles.

Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various symbols, including letters and geometric shapes, to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ.

Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.






Benchmark 3: Functions – The student recognizes and describes whole number relationships using concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. locates whole numbers from 0 through 20 on a number line

(2.4.K1a).

The student...1. represents and describes mathematical relationships for whole

numbers from 0 through 10 using concrete objects, pictures, and oral descriptions (2.4.A1a) ($).

Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set. A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line, an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane.







Benchmark 4: Models – The student uses mathematical models including concrete objects to represent, show, and communicate mathematical relationships in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. knows, explains, and uses mathematical models to represent

mathematical concepts, procedures, and relationships. Mathematical models include:a. process models (concrete objects, pictures, number lines, unifix

cubes, measurement tools, or calendars) to model computational procedures and mathematical relationships, to compare and order numerical quantities, and to represent fractional parts (1.1.K1-4, 1.2.K3-5, 1.3.K1, 1.4.K1-3, 2.1.K1, 2.1.K2a, 2.1.K2c, 2.1.K3-5 2.2.K1, 2.3.K1, 3.1.K2, 3.2.K1-3, 3.3.K1-2, 3.4.K1-2) ($);

b. place value models (ten frames, unifix cubes, bundles of straws, or base ten blocks) to represent numerical quantities (1.2.K2) ($);

c. fraction models (fraction strips or pattern blocks) to represent numerical quantities (1.1.K3) ($);

d. money models (base ten blocks or coins) to represent numerical quantities (1.1.K5) ($);

e. two-dimensional geometric models (geoboards, dot paper, or attribute blocks), three-dimensional geometric models (solids), and real-world objects to compare size and to model attributes of geometric shapes (2.1.K1a, 3.1.K3);

f. two-dimensional geometric models (spinners), three-dimensional geometric models (number cubes), and concrete objects to model probability (4.1.K1-2) ($);

g. graphs using concrete objects, pictographs, and frequency tables to organize and display data (4.2.K1-3) ($).

2. uses concrete objects, pictures, drawings, diagrams, or dramatizations to show the relationship between two or more things ($).

The student...1. recognizes that various mathematical models can be used to

represent the same problem situation. Mathematical models include: a. process models (concrete objects, pictures, number lines,

unifix cubes, measurement tools, or calendars) to model computational procedures and mathematical relationships, to compare and order numerical quantities, and to model problem situations (1.1.A1, 1.2.A2, 1.3.A1, 1.4.A1, 2.1.A1a, 2.1.A2-3, 2.2.A1, 2.3.A1, 3.1.A3, 3.2.A1-2, 3.3.A1-2, 3.4.A1) ($);

b. place value models (ten frames, unifix cubes, bundles of straws, or base ten blocks) to represent numerical quantities (1.2.A1) ($);

c. two-dimensional geometric models (geoboards, dot paper, or attribute blocks), three-dimensional geometric models (solids), and real-world objects to compare size and to model attributes of geometric shapes (3.1.A1-2);

d. two-dimensional geometric models (spinners), three-dimensional geometric models (number cubes), and concrete objects to model probability (4.1.A1);

e. graphs using concrete objects, pictographs, and frequency tables to organize and display data (4.1.A1, 4.2.A1) ($).



Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item being assessed.

The mathematical modeling process involves:a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through

some sort of mathematical model,b. performing manipulations and mathematical procedures within the mathematical model,c. interpreting the results of the manipulations within the mathematical model, d. using these results to make inferences about the original real-world situation.

The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, they begin to develop an understanding of the advantages and disadvantages of the various representations/models. For example, comparing the number of boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can be represented by pictures of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color tiles).

Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).

For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item being assessed.




Standard 3: Geometry KINDERGARTEN

Standard 3: Geometry – The student uses geometric concepts and procedures in a variety of situations.

Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and their attributes using concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. recognizes circles, squares, rectangles, triangles, and ellipses

(ovals) (plane figures/ two-dimensional figures) (2.4.K1e). 2. recognizes and investigates attributes of circles, squares,

rectangles, triangles, and ellipses using concrete objects, drawings, and/or appropriate technology (2.4.K1a,e).

3. sorts cubes, rectangular prisms, cylinders, cones, and spheres (solids/three-dimensional figures) by their attributes using concrete objects (2.4.K1e).

The student...1. demonstrates how several plane figures (circles, squares,

rectangles, triangles, ellipses) can be combined to make a new shape (2.4.A1c).

2. sorts by one attribute real-world geometric shapes that are representations of the solids (cubes, rectangular prisms, cylinders, cones, spheres) (2.4.A1c), e.g., boxes can be sorted as rectangular prisms, cans can be sorted as cylinders, some ice cream cones can be sorted as cones, and some balls can be sorted as spheres.

3. recognizes (2.4.A1a):a. circles, squares, rectangles, triangles, and ellipses (plane

figures) within a picture;b. cubes, rectangular prisms, cylinders, cones, and spheres

(solids) within a picture.



Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (one-dimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to as three-dimensional.

From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example:








Benchmark 2: Measurement and Estimation – The student estimates and measures using standard and nonstandard units of measure with concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. uses whole number approximations (estimations) for length using

nonstandard units of measure (2.4.K1a) ($), e.g., the classroom door is about two kindergartners high or this paper is about two pencils long.

2. compares two measurements using these attributes (2.4.K1a) ($): a. longer, shorter (length); b. taller, shorter (height); c. heavier, lighter (weight).d. hotter, colder (temperature).

3. reads and tells time at the hour using analog and digital clocks (2.4.K1a).

The student...1. compares and orders concrete objects by length or weight

(2.4.A1a) ($). 2. locates and names concrete objects that are about the same length

or weight as a given concrete object (2.4.A1a) ($).

Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” The process of learning to measure at the early grades focuses on identifying what property (length, weight) is to be measured and to make comparisons. Estimation in measurement is defined as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps students develop basic properties of measurement, and it gives students a tool to determine whether a given measurement is reasonable.







Benchmark 3: Transformational Geometry – The student develops the foundation for spatial sense using concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. describes the spatial relationship between two concrete objects

using appropriate vocabulary (2.4.K1a), e.g., behind, above, below, on, or under.

2. identifies two like objects or shapes from a set of four objects or shapes (2.4.K1a).

The student...1. shows two concrete objects or shapes are congruent by physically

fitting one object or shape on top of the other (2.4.A1a).2. follows directions to move concrete objects from one location to

another using appropriate vocabulary (2.4.A1a), e.g., up, down, behind, or above.

Teacher Notes: Transformational geometry is another way to investigate geometric figures by moving every point in a plane figure to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph paper, mirrors or other reflective surfaces, or appropriate technology.







Benchmark 4: Geometry From An Algebraic Perspective – The student identifies one or more points on a number line in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. locates and plots whole numbers from 0 through 20 on a horizontal

number line (2.4.K1a).2. counts forwards and backwards from a given whole number from 0

through 10 on a number line (2.4.K1a).

The student...1. solves real-world problems involving counting whole numbers from

0 through 20 using a number line (2.4.A1a) ($), e.g., if Bill has 8 pieces of candy and his dad gives him 4 more pieces, how many pieces of candy does he have now?

Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line, and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane.





Standard 4: Data KINDERGARTEN

Standard 4: Data – The student uses concepts and procedures of data analysis in a variety of situations.

Benchmark 1: Probability – The student applies the concepts of probability using concrete objects in a variety ofsituations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. recognizes whether an event is impossible or possible (2.4.K1f) ($),

e.g., the possibility of a person having ten heads is impossible, while the possibility of a person having red hair is possible.

2. recognizes and states whether a simple event in an experiment or simulation including the use of concrete objects can have more than one outcome (2.4.K1a,f).

The student...1. conducts an experiment or simulation with a simple event and

records the results in a graph using concrete objects or frequency tables (tally marks) (2.4.A1a,d-e).

Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. In the early grades, students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Beginning probability experiences should be addressed through the use of concrete objects, coins, and geometric models (spinners or number cubes).


The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels – grades 4, 8, and 12 – and are grouped into four major categories – Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.



Standard 4: Data KINDERGARTEN


Benchmark 2: Statistics – The student collects, records, and explains numerical (whole numbers) and non-numerical data sets including the use of concrete objects in a variety of situations.

Kindergarten Knowledge Base Indicators Kindergarten Application IndicatorsThe student...1. records numerical (quantitative) and non-numerical (qualitative)

data including concrete objects, graphs, and tables using these data displays (2.4.K1a,g) ($): a. graphs using concrete objects, b. pictographs with a whole symbol or picture representing one

(no partial symbols or pictures),c. frequency tables (tally marks).

2. collects data related to familiar everyday experiences by counting and tallying (2.4.K1a,g) ($).

3. determines the mode (most) after sorting by one attribute (2.4.K1a,g) ($), e.g., color, shape, or size.

The student...1. communicates the results of data collection from graphs using

concrete objects and frequency tables (2.4.A1e) ($), e.g., there are sixteen kindergartners. Using themselves as concrete objects, the six students wearing tennis shoes line up in a row. The ten students wearing sandals line up in a row. The kindergartners become the bar graph. Then someone says: There are less kids wearing tennis shoes than kids wearing sandals.



Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are used to tell a story. Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate. Graphs take many forms: pictographs and graphs using concrete objects compare discrete data, frequency tables show how many times a certain piece of data occurs using tally marks to record the data, circle graphs (pie charts) model parts of a whole, and line graphs show change over time. The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.





Standard 1: Number and Computation FIRST GRADE



First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. knows, explains, and represents whole numbers from 0 through 100

using concrete objects (2.4.K1a) ($).2. compares and orders ($):

a. whole numbers from 0 through 100 using concrete objects (2.4.K1a),

b. fractions with like denominators (halves and fourths) using concrete objects (2.4.K1a,c).

3. recognizes a whole, a half, and a fourth and represents equal parts of a whole (halves, fourths) using concrete objects, pictures, diagrams, fraction strips, or pattern blocks (2.4.K1a,c) ($).

4. identifies and uses ordinal numbers first (1st) through tenth (10th) (2.4.K1a).

5. identifies coins (pennies, nickels, dimes, quarters) and currency ($1, $5, $10) and states the value of each coin and each type of currency using money models (2.4.K1d) ($).

6. recognizes and counts a like group of coins (pennies, nickels, dimes) (2.4.K1d) ($).

The student...1. solves real-world problems using equivalent representations and

concrete objects to compare and order whole numbers from 0 through 50 (2.4.A1a) ($).

2. determines whether or not numerical values using whole numbers from 0 through 50 are reasonable (2.4.A1a) ($), e.g., when asked if 40 dictionaries will fit inside the student’s desk, the student answers no and explains why.

3. demonstrates that smaller whole numbers are within larger whole numbers using whole numbers from 0 to 30 (2.4.A1a) ($), e.g., if there are five pigs in a pen, there are also three pigs in the pen.

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Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole numbers with a special emphasis on place value and recognizes, applies, and explains the concepts of properties as they relate to whole numbers in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. reads and writes whole numbers from 0 through 100 in numerical

form ($). 2. represents whole numbers from 0 through 100 using various

groupings and place value models (place value mats, hundred charts, or base ten blocks) emphasizing ones, tens, and hundreds (2.4.K1b) ($), e.g., how many groups of tens are there in 32 or how many groups of tens and ones in 62?

3. counts subsets of whole numbers from 0 through 100 both forwards and backwards (2.4.K1a) ($).

4. writes in words whole numbers from 0 through 10.5. identifies the place value of the digits in whole numbers from 0

through 100 (2.4.K1b) ($).6. identifies any whole number from 0 through 30 as even or odd

(2.4.K1a).7. uses the concepts of these properties with whole numbers from 0

through 100 and demonstrates their meaning using concrete objects (2.4.K1a) ($):a. commutative property of addition, e.g., 3 + 2 = 2 + 3,b. zero property of addition (additive identity), e.g., 4 + 0 = 4.

The student...1. solves real-world problems with whole numbers from 0 through 50

using place value models (place value mats, hundred charts, or base ten blocks) and the concepts of these properties to explain reasoning (2.4.A1a-b) ($):a. commutative property of addition, e.g., group 5 students into a

group of 3 and a group of 2, add to find the total; then reverse the order of the students to show that 2 + 3 still equals 5;

b. zero property of addition, e.g., have students lay out 11 crayons, tell them to add zero (crayons). Then ask: How many crayons are there?

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Benchmark 3: Estimation – The student uses computational estimation with whole numbers in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. estimates whole number quantities from 0 through 100 using

various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a) ($).

2. estimates to check whether or not results of whole number quantities from 0 through 100 are reasonable (2.4.K1a) ($).

The student...1. adjusts original whole number estimate of a real-world problem

using whole numbers from 0 through 50 based on additional information (a frame of reference) (2.4.A1a) ($), e.g., an estimate is made about the number of tennis balls in a shoebox; about half of the tennis balls are removed from the box and counted. With this additional information, an adjustment of the original estimate is made.

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Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)

Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. Students develop mental math skills easier when they are taught specific strategies and specific strategies based on understanding also need to be developed for estimation. An estimation strategy for the early grades is front-end; the focus is on the “front-end” or the leftmost digit because these digits are the most significant for forming an estimate. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., about how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible.



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Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers using concrete objects in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. computes with efficiency and accuracy using various computational

methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a) ($).

2. N states and uses with efficiency and accuracy basic addition facts with sums from 0 through 10 and corresponding subtraction facts ($).

3. skip counts by 2s, 5s, and 10s through 50 (2.4.K1a). 4. uses repeated addition (multiplication) with whole numbers to find the

sum when given the number of groups (ten or less) and given the same number of concrete objects in each group (ten or less) (2.4.K1a), e.g., three plates of cookies with 10 cookies on each plate means 10 + 10 +10 = 30 cookies.

5. uses repeated subtraction (division) with whole numbers when given the total number of concrete objects in each group to find the number of groups (2.4.K1a), e.g., there are 9 pencils. If each student gets 2 pencils, how many students get pencils? 9 – 2 – 2 – 2 – 2 or 9 minus 2 four times means four students get 2 pencils each and there is 1 pencil left over. or There are 30 pieces of candy to put equally into five bowls, how many pieces of candy will be in each bowl? 30 – 5 – 5 – 5 – 5 – 5 – 5 means there are six in each bowl.

6. performs and explains these computational procedures (2.4.K1a-b):a. adds whole numbers with sums through 99 without regrouping

using concrete objects, e.g., 42 straws (bundled in 10s) + 21


various groupings of two-digit whole numbers without regrouping (2.4.A1a-b) ($), e.g., Jo has 48 crayons and 16 markers in her desk. How many more crayons does she have than markers? This problem could be solved using base 10 models or a number line or by saying 48 – 10 = 38 and 38 – 6 = 32.

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straws (bundled in 10s) = 63 straws (bundled in 10s);b. subtracts two-digit whole numbers without regrouping using

concrete objects, e.g., 63 cubes – 21 cubes = 42 cubes.7. shows that addition and subtraction are inverse operations using

concrete objects (2.4.K1a) ($).

8. reads and writes horizontally and vertically the same addition expression, e.g., 5 + 4 is the same as 4

+ 5.

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Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations and by counting on for addition and counting up for subtraction and unknown-addend situations. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable, in other words, using mental math to estimate to see if the answer makes sense. Proficiency in mental math contributes to a better understanding of place value, mathematical operations, and basic number properties and an increased skill in estimation. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies for the early grades include counting on, counting back, counting up, doubling numbers (doubles), and making ten. Other mental math strategies for the early grades include skip counting and using number patterns. One way to improve the understanding of numbers and operations is to encourage children to develop computational procedures that are meaningful to them. Invented procedures promote the idea of mathematics as a meaningful activity. This is not to suggest that algorithm invention is the only way to promote student understanding or that children should be discouraged from using a standard algorithm if that is their choice. Different problems are best solved by different methods. For example, solving 7000 – 25 by counting down by tens and fives, yet solving 41 – 25 by adding up, or decomposing, the numbers. Another example is adding 52 + 45; a student may count, “52, 62, 72, 82, 92, plus 5 is 97.” (“Invented Strategies Can Develop Meaningful Mathematical Procedures” by William M. Carroll and Denise Porter, Teaching Children Mathematics, March 1997, pp. 370-374)

Regrouping refers to the reorganization of objects. In computation, regrouping is based on a “partitioning to multiples of ten” strategy. For example, 46 + 7 could be solved by partitioning 46 into 40 and 6, then 40 + (6 + 7) = 40 + 13 (and then 13 is partitioned into 10 and 3) which then becomes (40 + 10) + 3 which becomes 50 + 3 = 53 or 7 could be partitioned as 4 and 3, then 46 + 4 (bridging through 10) = 50 and 50 + 3 = 53. Before algorithmic procedures are taught, an understanding of “what happens” must occur. For this to occur, instruction should involve the use of structured manipulatives. To emphasize the role of the base ten-numeration system in algorithms, some form of expanded notation is recommended. During instruction, each child should have a set of manipulatives to work with rather than sit and watch demonstrations by the teacher. Some additional computational strategies include doubles plus one or two (i.e., 6 + 8, 6 and 6 are 12, so the answer must be 2 more or 14), compensation (i.e., 9 + 7, if one is taken away from 9, it leaves 8; then that one is given to 7 to make 8, then 8 + 8 = 16), subtracting through ten (i.e., 13 – 5, 13 take away 3 is 10, then take 2 more away from 10 and that is 8), and nine is one less than ten (i.e., 9 + 6, 10 and 6 are 16, and 1 less than 16 makes 15). (Teaching Mathematics in Grades K-8: Research Based Methods, ed. Thomas R. Post, Allyn and Bacon, 1988)

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Standard 2: Algebra FIRST GRADE



First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. uses concrete objects, drawings, and other representations to work

with types of patterns (2.4.K1a): c. repeating patterns, e.g., an AB pattern is like 1-2, 1-2, …; an

ABC pattern is like dog-horse-pig, dog-horse-pig, …; an AAB pattern is like Δ-Δ-Ο, Δ-Δ-Ο, …;

d. growing (extending) patterns, e.g., 1, 2, 3, … 2. uses the following attributes to generate patterns:

a. counting numbers related to number theory (2.4.K1.a), e.g., evens, odds, or skip counting by 2s, 5s, or 10s;

b. whole numbers that increase (2.4.K1a) ($), e.g., 11, 21, 31, ... or like 2, 4, 6, …;

c. geometric shapes (2.4.K1f), e.g., ▲, ▀, ◊, ▲, ▀, ◊, …; d. measurements (2.4.K1a), e.g., counting by inches or feet;e. the calendar (2.4.K1a), e.g., January, February, March, …; f. money and time (2.4.K1d) ($), e.g., 10¢, 20¢, 30¢, … or 1:00,

1:30, 2:00, ...;g. things related to daily life (2.4.K1a), e.g., seasons, temperature,

or weather;h. things related to size, shape, color, texture, or movement

(2.4.K1a); e.g., tall-short, tall-short, tall-short, …; or snapping fingers, clapping hands, or stomping feet (kinesthetic patterns).

3. identifies and continues a pattern presented in various formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written (2.4.K1a) ($).

4. generates (2.4.K1a):a. repeating patterns for the AB pattern, the ABC pattern, and the

AAB pattern;

The student...1. generalizes the following patterns using pictorial, oral, and/or written

descriptions including the use of concrete objects: a. whole number patterns (2.4.A1a) ($);b. patterns using geometric shapes (2.4.A1c);c. calendar patterns (2.4.A1a);d. patterns using size, shape, color, texture, or movement

(2.4.A1a).2. recognizes multiple representations of the same pattern (2.4.A1a), e.g.,

the AB pattern could be represented by clap, snap, clap, snap, … or red, green, red, green, … or square, circle, square, circle, ….

3. uses concrete objects to model a whole number pattern (2.4.A1a):d. counting by ones: ,, , …e. counting by twos: , , , …f. counting by fives: xxxxx, xxxxx xxxxx xxxxx, xxxxx xxxxx, …g. counting by tens: , , , …

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b. growing patterns that add 1, 2, 5, or 10.


This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain.

Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest common factor and least common multiple.



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Benchmark 2: Variable, Equations, and Inequalities – The student solves addition and subtraction equations using concrete objects in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. explains and uses symbols to represent unknown whole number

quantities from 0 through 20 (2.4.K1a).2. finds the unknown sum or difference of the basic facts using

concrete objects (2.4.K1a) ($), e.g., 12 dominoes – 5 dominoes = Δ dominoes or Δ cubes = 2 cubes + 4 cubes.

3. describes and compares two whole numbers from 0 through 100 using the terms: is equal to, is less than, is greater than (2.4.K1a-b) ($).

The student...1. represents real-world problems using concrete objects, pictures, oral

descriptions, and symbols and the basic addition and subtraction facts with one operation and one unknown (2.4. A1a) ($), e.g., given some marbles, Sue says: 3 red marbles and 3 blue marbles equal 6 marbles. Sue also shows and writes the problem and solution: 3 + 3 = or RRR + BBB = , 3 + 3 = 6.

2. generates and solves problem situations using the basic facts to find the unknown sum or difference with concrete objects (2.4.A1a), e.g., a student generates this problem: I have 6 marbles. My sister has 4. How many do we have altogether? The student shows 6 + 4 = , and 6 + 4 = 10.

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Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various symbols including letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ.






First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. plots whole numbers from 0 through 100 on segments of a number

line (2.4.K1a).2. states mathematical relationships between whole numbers from 0

through 50 using various methods including mental math, paper and pencil, and concrete objects (2.4.K1a) ($), e.g., every time a hand is added to the set, five more fingers are added to the total.

3. states numerical relationships for whole numbers from 0 through 50 in a horizontal or vertical function table (input/output machine, T-table) (2.4.K1e) ($), e.g.,

The student...1. represents and describes mathematical relationships for whole

numbers from 0 through 50 using concrete objects, pictures, oral descriptions, and symbols (2.4.A1a) ($).

2. recognizes numerical patterns (counting by 2s, 5s, and 10s) through 50 using a hundred chart (2.4.A1a).

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Number of bicycles 1 2 3 4 5 …Total number of wheels 2 4 6 8 10 …

The student states: For every bicycle added, you add two more wheels.

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Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line, an example of one-to-one correspondence (a relation). There are many kinds of relations – one-to-many, many-to-one, many-to-many, and one-to-one. A function is a special kind of relation.

Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set (an ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according to some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines, and T-tables may be used interchangeably and serve the same purpose.

Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the sequence – the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3. The pattern could be described as add 3 meaning that 3 must be added to any term after the first to find the next. This is an example of a recursive pattern. In a recursive pattern, after one or more consecutive terms are given, each successive term in the sequence is obtained from the previous term(s).In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This is an example of an explicit pattern. In an explicit pattern, each term in the sequence is defined by the term’s number in the sequence.



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First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. knows, explains, and uses mathematical models to represent

mathematical concepts, procedures, and relationships. Mathematical models include:a. process models (concrete objects, pictures, diagrams, number

lines, unifix cubes, hundred charts, measurement tools, or calendars) to model computational procedures and mathematical relationships, to compare and order numerical quantities, and to represent fractional parts (1.1.K1-4, 1.2.K3, 1.2.K6-7, 1.3.K1-2, 1.4.K1, 1.4.K2-7, 2.1.K1, 2.1.K1d-h, 2.1.K2a-b, 2.2.K3-4, 2.3.K1-2, 3.2.K1-6, 3.3.K1-3, 3.4.K1-3 4.2.K3-4) ($);

b. place value models (place value mats, hundred charts, or base ten blocks) to compare, order, and represent numerical quantities and to model computational procedures (1.2.K2, 1.2.K5, 1.4.K6, 2.2.K3) ($);

c. fraction models (fraction strips or pattern blocks) to compare, order, and represent numerical quantities (1.1.K2-3) ($);

d. money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K5-6, 2.1.K2f) ($);

e. function tables (input/output machines, T-tables) to model numerical relationships (2.3.K3) ($);

f. two-dimensional geometric models (geoboards, dot paper, pattern blocks, tangrams, or attribute blocks), three-dimensional geometric models (solids), and real-world objects to compare size and to model attributes of geometric shapes (2.1.K1c, 3.1.K1-3);

g. two-dimensional geometric models (spinners), three-dimensional geometric models (number cubes), and concrete objects to model probability (4.1.K1-2) ($);

The student…1. recognizes that various mathematical models can be used to

represent the same problem situation. Mathematical models include:f. process models (concrete objects, pictures, diagrams, number

lines, unifix cubes, measurement tools, or calendars) to model computational procedures and mathematical relationships, to compare and order numerical quantities, and to model problem situations (1.1.A1-3, 1.2.A1, 1.3.A1, 1.4.A1, 2.1.A1a, 2.1.A1c-d, 2.1.A2-3, 2.2.A1-2, 2.3.A1-2, 3.2.A1-3, 3.3.A1-2, 3.4.A1, 4.2.A2) ($);

g. place value models (place value mats, hundred charts, or base ten blocks) to compare, order, and represent numerical quantities and to model computational procedures (1.2.A1, 1.4.A1) ($);

h. two-dimensional geometric models (geoboards, dot paper, pattern blocks, tangrams, or attribute blocks), three-dimensional geometric models (solids), and real-world objects to compare size and to model attributes of geometric shapes (2.1.A1b, 3.1.A1-2);

d. two-dimensional geometric models (spinners), three-dimensional geometric models (number cubes), and concrete objects to model probability (4.1.A1) ($);

e. graphs using concrete objects, pictographs, frequency tables, and horizontal and vertical bar graphs to organize, display, and explain data (4.1.A1, 4.2.A1) ($).

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h. graphs using concrete objects, pictographs, frequency tables, horizontal and vertical bar graphs, and Venn diagrams or other pictorial displays to organize, display, and explain data (4.1.A1, 4.2.A1-2) ($);

i. Venn diagrams to sort data (4.2.K4).2. uses concrete objects, pictures, diagrams, drawings, or

dramatizations to show the relationship between two or more things ($).


The mathematical modeling process involves:a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through some

sort of mathematical model,b. performing manipulations and mathematical procedures within the mathematical model,c. interpreting the results of the manipulations within the mathematical model, d. using these results to make inferences about the original real-world situation.

The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, they begin to develop an understanding of the advantages and disadvantages of the various representations/models. For example, comparing the number of boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can be represented by pictures of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color tiles).



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1-49January 31, 2004


Standard 3: Geometry FIRST GRADE


Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and describes their attributes using concrete objects in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...4. recognizes and draws circles, squares, rectangles, triangles, and

ellipses (ovals) (plane figures/two-dimensional figures) (2.4.K1f). 2. recognizes and investigates attributes of circles, squares,

rectangles, triangles, and ellipses (plane figures) using concrete objects, drawings, and appropriate technology (2.4.K1f).

3. recognizes cubes, rectangular prisms, cylinders, cones, and spheres (solids/three-dimensional figures) (2.4.K1f).

The student...1. demonstrates how (2.4.A1c):

a. a geometric shape made of several plane figures (circles, squares, rectangles, triangles, ellipses) can be separated to make two or more different plane figures;

b. several plane figures (circles, squares, rectangles, triangles, ellipses) can be combined to make a new geometric shape;

c. several solids (cubes, rectangular prisms, cylinders, cones, spheres) can be combined to make a new geometric shape.

2. sorts plane figures and solids (circles, squares, rectangles, triangles, ellipses, cubes, rectangular prisms, cylinders, cones, spheres) by a given attribute (2.4.A1c).

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First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. uses whole number approximations (estimations) for length and

weight using nonstandard units of measure (2.4.K1a) ($), e.g., the width of the chalkboard is about 10 erasers long or the weight of one encyclopedia is about five picture books.

2. compares two measurements using these attributes (2.4.K1a) ($): a. longer, shorter (length);b. taller, shorter (height); c. heavier, lighter (weight);d. hotter, colder (temperature).

3. reads and tells time at the hour and half-hour using analog and digital clocks (2.4.K1a).

4. selects appropriate measuring tools for length, weight, volume, and temperature for a given situation (2.4.K1a) ($).

5. measures length and weight to the nearest whole unit using nonstandard units (2.4.K1a) ($).

6. states the number of days in a week and months in a year (2.4.K1a).

The student...1. compares and orders concrete objects by length or weight (2.4.A1a)

($).2. compares the weight of two concrete objects using a balance

(2.4.A1a).3. locates and names concrete objects that are about the same length,

weight, or volume as a given concrete object (2.4.A1a) ($).

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Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” The process of learning to measure at the early grades focuses on identifying what property (length, weight) is to be measured and to make comparisons. Estimation in measurement is defined as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.



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Benchmark 3: Transformational Geometry – The student develops the foundation for spatial sense using concrete objects in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. describes the spatial relationship between two concrete objects

using appropriate vocabulary (2.4.K1a), e.g., behind, above, below, on, under, beside, or in front of.

2. recognizes that changing an object's position or orientation does not change the name, size, or shape of the object (2.4.K1a).

3. describes movement of concrete objects using appropriate vocabulary (2.4.K1a), e.g., right, left, up, or down.


fitting one object or shape on top of the other (2.4.A1a).2. gives and follows directions to move concrete objects from one

location to another using appropriate vocabulary (2.4.A1a), e.g., right, left, up, down, behind, or above.




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First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. locates and plots whole numbers from 0 through 100 on a segment

of a number line (horizontal/vertical) (2.4.K1a), e.g., using a segment of a number line from 45 to 60 to locate the whole number 50.

2. describes a given whole number from 0 to 100 as coming before or after another number on a number line (2.4.K1a).

3. uses a number line to model addition and counting using whole numbers from 0 to 100 (2.4.K1a).

The student...1. solves real-world problems involving counting and adding whole

numbers from 0 to 50 using a number line (2.4.A1a) ($), e.g., Nancy has 23¢. She finds 18¢ more in her pocket. How much money does she now have?

Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line, and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane.



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Standard 4: Data FIRST GRADE


Benchmark 1: Probability – The student applies the concepts of probability using concrete objects in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. recognizes whether an outcome of a simple event in an experiment

or simulation is impossible, possible, or certain (2.4.K1g) ($).2. recognizes and states whether a simple event in an experiment or

simulation including the use of concrete objects can have more than one outcome (2.4.K1g).

The student...1. makes a prediction about a simple event in an experiment or

simulation, conducts the experiment or simulation, and records the results in a graph using concrete objects, a pictograph with a symbol or picture representing only one, or a bar graph (2.4.A1d-e).

Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. In the early grades, students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Beginning probability experiences should be addressed through the use of concrete objects, coins, and geometric models (spinners or number cubes).



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Standard 4: Data FIRST GRADE


Benchmark 2: Statistics – The student collects, displays, and explains numerical (whole numbers) and non-numerical data sets including the use of concrete objects in a variety of situations.

First Grade Knowledge Base Indicators First Grade Application IndicatorsThe student...1. displays and reads numerical (quantitative) and non-numerical

(qualitative) data in a clear, organized, and accurate manner including a title, labels, and whole number intervals using these data displays (2.4.K1h) ($):a. graphs using concrete objects,b. pictographs with a whole symbol or picture representing one

(no partial symbols or pictures),c. frequency tables (tally marks),d. horizontal and vertical bar graphs, e. Venn diagrams or other pictorial displays, e.g., glyphs.

2. collects data using different techniques (observations or interviews) and explains the results (2.4.K1h) ($).

3. identifies the minimum (lowest) and maximum (highest) values in a data set (2.4.K1a) ($).

4. determines the mode (most) after sorting by one attribute (2.4.K1a,i) ($).

5. sorts and records qualitative (non-numerical, categorical) data sets using one attribute (2.4.K1a) ($), e.g., color, shape, or size.

The student...1. communicates the results of data collection and answers questions

(identifying more, less, fewer, greater than, or less than) based on information from (2.4.A1e) ($): a. graphs using concrete objects, b. a pictograph with a whole symbol or picture representing only

one (no partial symbols or pictures),c. a horizontal or vertical bar graph.

2. determines categories from which data could be gathered (2.4.A1a) ($), e.g., categories could include shoe size, height, favorite candy bar, or number of pockets in clothing.

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Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are used to tell a story. Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate. Graphs take many forms: pictographs and graphs using concrete objects compare discrete data, frequency tables show how many times a certain piece of data occurs using tally marks to record the data, circle graphs (pie charts) model parts of a whole, line graphs show change over time, and Venn diagrams show relationships between sets of objects.

The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.



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Standard 1: Number and Computation SECOND GRADE



Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student… 1. ■ knows, explains, and represents whole numbers from 0 through

1,000 using concrete objects (2.4.K1a) ($).2. compares and orders:

a. whole numbers from 0 through 1,000 using concrete objects (2.4.K1a) ($);

b. fractions greater than or equal to zero with like denominators (halves, fourths, thirds, eighths) using concrete objects (2.4.K1a,c).

3. uses addition and subtraction to show equivalent representations for whole numbers from 0 through 100 (2.4.K1a-b), e.g., 8 – 5 = 2 + 1 or 20 + 40 = 70 – 10.

4. identifies and uses ordinal positions from first (1st) through twentieth (20th) (2.4.K1a).

5. ▲identifies coins, states their values, and determines the total value to $1.00 of a mixed group of coins using pennies, nickels, dimes, quarters, and half-dollars (2.4.K1d) ($).

6. counts a like combination of currency ($1, $5, $10, $20) to $100 (2.4.K1d) ($).

The student...2. solves real-world problems using equivalent representations and

concrete objects to ($):a. compare and order whole numbers from 0 through 1,000

(2.4.A1b), e.g., using base ten blocks, represent the students in each class in the school; represent the numbers using digits (24) and compare and order in different ways;

b. add and subtract whole numbers from 0 through 100 (2.4.A1b), e.g., using base ten blocks, represent the number of students in each class in the school; find the total of all students in grades K, 1, and 2 and the total of all of the students in grades 3, 4, and 5 and then subtract to find the difference between the primary and intermediate grades;

c. compare and order a mixed group of coins to $1.00 (2.4.A1c), e.g., use actual coins to show 2 different amounts; students write: 47¢ is more than 31¢;

d. find equivalent values of coins to $1.00 without mixing coins (2.4.A1c), e.g., 50 pennies = 2 quarters, 5 dimes = 2 quarters, or 10 nickels = 2 quarters.

2. determines whether or not numerical values that involve whole numbers from 0 through 1,000 are reasonable (2.4.A1a-b) ($), e.g., if there are 26 children, plus 10 more children, is it reasonable to say there are 50 children?

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▲ – Assessed Indicator■ – Assessed Indicator on the Optional Response Assessment N – Noncalculator ($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.



Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.


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Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole numbers with a special emphasis on place value and recognizes, uses, and explains the concepts of properties as they relate to whole numbers in a variety of situations.

Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. reads and writes ($):

a. whole numbers from 0 through 1,000 in numerical form, e.g., 942 is read as nine hundred forty-two and is written in numerical form as 942;

b. whole numbers from 0 through 100 in words, e.g., 76 is read as seventy-six and is written in words as seventy-six.

c. whole numbers from 0 through 1,000 in numerical form when presented in word form, e.g., nine hundred forty-six is read as nine hundred forty-six and is written as 946.

2. ▲ represents whole numbers from 0 through 1,000 using various groupings and place value models emphasizing 1s, 10s, and 100s; explains the groups; and states the value of the digit in ones place, tens place, and hundreds place (2.4.K1b) ($), e.g., in 385, the 3 represents 3 hundreds, 30 tens, or 300 ones; the 8 represents 8 tens or 80 ones; and the 5 represents 5 ones.

3. counts subsets of whole numbers from 0 through 1,000 forwards and backwards (2.4.K1a) ($), e.g., 311, 312, …, 320; or 210, 209, …, 204.

4. ▲ identifies the place value of the digits in whole numbers from 0 through 1,000 (2.4.K1b) ($).

5. identifies any whole number from 0 through 100 as even or odd (2.4.K1a).

uses the concepts of these properties with whole numbers from 0 through 100 and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($):a. commutative property of addition, e.g., 5 + 6 = 6 + 5;

The student...1. solves real-world problems with whole numbers from 0 through

100 using place value models and the concepts of these properties to explain reasoning (2.4.A1a-b) ($):a. commutative property of addition, e.g., group 17 students into

a 9 and an 8, add to find the total, then reverse the students to show 8 + 9 still equals 17;

b. zero property of addition, e.g., have students lay out 22 crayons, tell them to add zero (crayons). How many crayons? 22 + 0 = 22.

2. performs various computational procedures with whole numbers from 0 through 100 using these properties and explains how they were used (2.4.A1b):a. commutative property of addition (5 + 6 = 6 + 5), e.g., given

6 + 5, the student says: I know that the answer is 11 because 5 + 6 is 11 and the order you add them in does not matter;

b. zero property of addition (17 + 0 = 0 + 17), e.g., given 17 + 0, the student says: I know that the answer is 17 because adding 0 does not change the answer (sum).

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b. zero property of addition (additive identity), e.g., 4 + 0 = 4;c. associative property of addition, e.g., (3 + 2) + 4 = 3 + (2 + 4);d. symmetric property of equality applied to basic addition and

subtraction facts, e.g., 10 = 2 + 8 is the same as 2 + 8 = 10 or 7 = 10 – 3 is the same as 10 – 3 = 7.





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Benchmark 3: Estimation – The student uses computational estimation with whole numbers and money in a variety

of situations.Second Grade Knowledge Base Indicators Second Grade Application Indicators

The student...1. estimates whole number quantities from 0 through 1,000 and

monetary amounts through $50 using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.Ka-b,d) ($).

2. uses various estimation strategies to estimate whole number quantities from 0 through 1,000 (2.4.K1a) ($).

The student...1. adjusts original whole number estimate of a real-world problem using

numbers from 0 through 1,000 based on additional information (a frame of reference) (2.4.A1a) ($), e.g., given a pint container and told the number of marbles it has in it, the student would estimate the number of marbles in a quart container.

2. estimates to check whether or not the result of a real-world problem using whole numbers from 0 through 1,000 and monetary amounts through $50 is reasonable and makes predictions based on the information (2.4.A1a-c) ($), e.g., in the lunchroom, good behavior that day can earn the class an extra 5 minutes of recess. Is it reasonable to think you can earn an hour of extra recess in one week? After answering the first question, then ask: About how many days would it take?

3. selects a reasonable magnitude from three given quantities, a one-digit numeral, a two-digit numeral, and a three-digit numeral (5, 50, 500) based on a familiar problem situation and explains the reasonableness of the selection (2.4.A1a), e.g., could the basket of fruit on the kitchen table hold 7 apples, 70 apples, or 700 apples? The student chooses 7 apples because apples are about the size of baseballs and 7 will fit in a basket on the kitchen table.

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Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. Students develop mental math and estimation easier when they are taught specific strategies. An estimation strategy for the early grades is front-end. The focus is on the “front-end” or the leftmost digit because these digits are the most significant for forming an estimate. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible. In addition to front-end estimation and rounding, another estimation strategy is compatible “nice” numbers.


The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the

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Appendix.

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Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers and money using concrete objects in a variety of situations.

Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. computes with efficiency and accuracy using various computational


2. N states and uses with efficiency and accuracy basic addition facts with sums from 0 through 20 and corresponding subtraction facts (2.4.K1a) ($).

3. skip counts by 2s, 5s, and 10s through 100 and skip counts by 3s through 36 (2.4.K1a).

4. uses repeated addition (multiplication) with whole numbers to find the sum when given the number of groups (ten or less) and given the same number of concrete objects in each group (twenty or less) (2.4.K1a) ($), e.g., five classes of 15 students visit the zoo; 15 + 15 + 15 + 15 + 15 = 75.

5. uses repeated subtraction (division) with whole numbers when given the total number of concrete objects in each group to find the number of groups (2.4.K1a) ($), e.g., there are 25 cookies. If each student gets 3 cookies, how many students get cookies? 25 – 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3 or 25 minus 3 eight times means eight students get 3 cookies each and there is 1 cookie left over.

6. fair shares/measures out (divides) a total amount through 100 concrete objects into equal groups (2.4.K1a-b), e.g., fair sharing 48 eggs into four groups resulting in four groups of 12 eggs or measuring out 48 eggs with 12 eggs in each group resulting in four groups of 12 eggs.


various groupings of ($):a. two-digit whole numbers with regrouping (24A1a-b), e.g., for the

food drive, the class collected 64 cans (cylinders) and 28 boxes (rectangular prisms). How many did they collect in all? This problem could be solved with base 10 models, or by saying 64 + 20 = 84 and 84 + 8 = 92 or 60 + 20 = 80 and 4 + 8 = 12 and 80 + 12 = 92 or with the traditional algorithm;

b. monetary amounts to 99¢ with regrouping (2.4.A1a-c), e.g., an extra carton of milk costs 25¢. If three students want an extra carton, how much money should the teacher collect? The student could solve by using coins (q + q + q or d + d + n + d + d + n + d + d + n) or by counting by 25s or by drawing or using base 10 models or with the traditional algorithm.

2. generates a family of basic addition and subtraction facts given one fact/equation (2.4.A1a), e.g., given 9 + 8 = 17; the other facts are 8 + 9 = 17, 17 – 8 = 9, and 17 – 9 = 8.

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7. ▲ N performs and explains these computational procedures:a. ■ adds and subtracts three-digit whole numbers with and without

regrouping including the use of concrete objects (2.4.K1a-b), b. adds and subtracts monetary amounts through 99¢ using cent

notation (25¢ + 52¢) and money models (2.4.K1a-b,d) ($). 8. ▲N identifies basic addition and subtraction fact families (facts with

sums from 0 through 20 and corresponding subtraction facts) (2.4.K1a).

9. reads and writes horizontally and vertically the same addition or subtraction expression e.g., 6 – 3 is the same as 6.

-3Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations and by counting on for addition and counting up for subtraction and unknown-addend situations. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number — calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable, in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies for the early grades include counting on, counting back, counting up, doubling numbers (doubles), and making ten. Other mental math strategies for the early grades include skip counting and using number patterns.

One way to improve the understanding of numbers and operations is to encourage children to develop computational procedures that are meaningful to them. Invented procedures promote the idea of mathematics as a meaningful activity. This is not to suggest that algorithm invention is the only way to promote student understanding or that children should be discouraged from using a standard algorithm if that is their choice. Different problems are best solved by different methods. For example, solving 7000 – 25 by counting down by tens and fives, yet solving 41 – 25 by adding up, or decomposing, the numbers. Another example is adding 52 + 45; a student may count, “52, 62, 72, 82, 92, plus 5 is 97.” (“Invented Strategies Can Develop Meaningful Mathematical Procedures”) by William M. Carroll and Denise Porter, Teaching Children Mathematics, March 1997, pp. 370-374)

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Regrouping refers to the reorganization of objects. In computation, regrouping is based on a “partitioning to multiples of ten” strategy. For example, 46 + 7 could be solved by partitioning 46 into 40 and 6, then 40 + (6 + 7) = 40 + 13 (and then 13 is partitioned into 10 and 3) which then becomes (40 + 10) + 3 becomes 50 + 3 = 53 or 7 could be partitioned as 4 and 3, then 46 + 4 (bridging through 10) = 50 and 50 + 3 = 53. Before algorithmic procedures are taught, an understanding of “what happens” must occur. For this to occur, instruction should involve the use of structured manipulatives. To emphasize the role of the base ten numeration system in algorithms, some form of expanded notation is recommended. During instruction, each child should have a set of manipulatives to work with rather than sit and watch demonstrations by the teacher. Some additional computational strategies include doubles plus one or two (i.e., 6 + 8, 6 and 6 are 12, so the answer must be 2 more or 14), compensation (i.e., 9 + 7, if one is taken away from 9, it leaves 8; then that one is given to 7 to make 8, then 8 + 8 = 16), subtracting through ten (i.e., 13 – 5, 13 take away 3 is 10, then take 2 more away from 10 and that is 8), and nine is one less than ten (i.e., 9 + 6, 10 and 6 are 16, and 1 less than 16 makes 15). (Teaching Mathematics in Grades K-8: Research Based Methods, ed. Thomas R. Post, Allyn and Bacon, 1988)



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Standard 2: Algebra SECOND GRADE



Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. uses concrete objects, drawings, and other representations to work

with types of patterns (2.4.K1a): a. repeating patterns, e.g., an AB pattern is like left-right, left-right,

…; an ABC pattern is like dog-horse-pig, dog-horse-pig, …; an AAB pattern is like ↑↑→, ↑↑→, …;

b. growing (extending) patterns, e.g., 7, 9, 11, … where the rule could be add 2 or the odd numbers beginning with 7.

2. uses the following attributes to generate patterns:b. counting numbers related to number theory (2.4.K1a), e.g.,

evens, odds, or skip counting by 3s, or 4s; b. whole numbers that increase or decrease (2.4.K1a) ($), e.g., 11,

22, 33, ... or 98, 88, 78, …; c. geometric shapes (2.4.K1f), e.g., Δ-Ο-Ο, Δ-Ο-Ο, …;d. measurements (2.4.K1a), e.g., 1”, 3”, 5”, … or 5 lbs, 10 lbs, 15

lbs, …;e. the calendar (2.4.K1a), e.g., Sunday, Monday, Tuesday, …;f. money and time (2.4.K1a,d) ($), e.g., $5, $10, $15, ... or 1:15,

1:30, 1:45, …;g. things related to daily life (2.4.K1a), e.g., seasons, temperature,

or weather;h. things related to size, shape, color, texture, or movement

(2.4.K1a), e.g., ,, , … ; or snapping fingers, clapping hands, or stomping feet or over, under, or behind using a bean bag toss (kinesthetic patterns).

3. ▲ ■ identifies and continues a pattern presented in various formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written (2.4.K1a) ($).

The student...1. generalizes these patterns using a written description:

a. whole number patterns(2.4.A1a) ($);b. patterns using geometric shapes (2.4.A1d);c. calendar patterns (2.4.A1a);d. money and time patterns (2.4.A1a,c) ($);e. patterns using size, shape, color, texture, or movement

(2.4.A1a); recognizes multiple representations of the same pattern (2.4.A1a),

e.g., the ABB pattern could be represented by clap, snap, snap, … or red, blue, blue, … or square, circle, circle, ….

uses concrete objects to model a whole number patterns (2.4.A1a), e.g., counting by twos: , , , …;counting by fives: xxxxx, xxxxx xxxxx xxxxx, xxxxx xxxxx, …;counting by tens: , ,

, …; counting by twenty-fives:

…

generates (2.4.K1a):

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, , ,

repeating patterns, e.g., 1-2, 1-2, 1-2, … where the elements repeat; growing (extending) patterns, e.g., 1, 4, 7, …where the rule is add 3.

Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney, Macmillan Publishing Company, 1994) This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain. Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest common factor and least common multiple.



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Benchmark 2: Variables, Equations, and Inequalities – The student uses symbols and whole numbers to solve addition and subtraction equations using concrete objects in a variety of situations.

Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. explains and uses symbols to represent unknown whole number

quantities from 0 through 100 (2.4.K1a).2. finds the sum or difference in one-step equations with : ($)

a. whole numbers from 0 through 99 (2.4.K1a-b), e.g., 32 + 19 = Δ or Δ = 79 – 46;

b. up to two different coins (2.4.K1d), e.g., nickel + penny = Δ¢. 3. finds unknown addend or subtrahend using basic addition and

subtraction facts (fact family) (2.4.K1a) ($), e.g., 12 = Δ + 7 or 12 – Δ = 7.

4. describes and compares two whole numbers from 0 through 1,000 using the terms: is equal to, is less than, is greater than (2.4.K1a-b) ($).

The student...1. represents real-world problems using symbols and whole numbers

from 0 through 30 with one operation (addition, subtraction) and one unknown (2.4.A1a) ($), e.g., when asked to give the total number of students in class today, the students write: 14 boys and 9 girls = students.

2. generates (2.4.A1a) ($):a. addition or subtraction equations to match a given real-world

problem with one operation and one unknown using whole numbers from 0 through 99, e.g., a boy has 45 stickers. How many more stickers does he need to have 80 stickers? This is represented by 45 + n = 80 or 80 – 45 = n.

b. a real-world problem to match a given addition or subtraction equation with one operation using the basic facts, e.g., the student is given the addition equation, 9 + = 17 and writes this problem situation: You have 9¢ and a piece of candy costs 17¢. How much more money do you need to buy the candy?

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Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various symbols including letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ.

Mathematical models such as concrete objects, pictures, diagrams, number lines, unifix cubes, hundred charts, or base ten blocks are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.]


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Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. states mathematical relationships between whole numbers from 0

through 100 using various methods including mental math, paper and pencil, and concrete objects (2.4.K1a) ($), e.g., every time a dog is added to the pack, 2 more ears are added to the total.

2. finds the values and determines the rule that involve addition or subtraction of whole numbers from 0 through 100 using a horizontal or vertical function table (input/output machine, T-table) (2.4.K1e), e.g., after looking at the function table,different students might respond that the rule is In + 2 equals Out, the rule is N + 2, or the rule is plus 2.

3. generalizes numerical patterns using whole numbers from 0 through 100 with one operation (addition, subtraction) by stating the rule using words, e.g., if a set of numbers is 2, 4, 6, 8,10, …; the rule is add two.

The student...1. represents and describes mathematical relationships between

whole numbers from 0 through 100 using concrete objects, pictures, oral descriptions, and symbols (2.4.A1a) ($).

2. finds the rule, states the rule, and extends numerical patterns with whole numbers from 0 through 100 (2.4.A1a), e.g., given 1, 3, 5, 7, 9 and continues with 11, 13, 15, 17, … recognizing that the pattern could be the odd numbers.

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In Out9 112 4

13 1542 4457 596 ?

72 ?N ?

..

Teacher Notes: Functions are relationships or rules in which each member of one set is paired with one, and only one, member of another set (an ordered pair). The concept of function can be introduced using function machines. Any number put in the machine will be changed according to some rule. A record of inputs and corresponding outputs can be maintained in a two-column format. Function tables, input/output machines, and T-tables may be used interchangeably and serve the same purpose.

Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the sequence — the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3. The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is explained by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of statements that explains how each successive term in the sequence is obtained from the previous term(s).

In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the number of the term — the 5th term is 52, the 8th term is 82, ….

Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.



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Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. knows, explains, and uses mathematical models to represent


lines, unifix cubes, hundred charts, or measurement tools) to model computational procedures and mathematical relationships, to compare and order numerical quantities, and to represent fractional parts (1.1.K1-4, 1.2.K3, 1.2.K5-6, 1.3.K1-2, 1.4.K1-8, 2.1.K1, 2.2K1, 2.1K1a-b, 2.1K1d-h, 2.1.K3-4, 2.2.K2a, 2.2.K3-4, 2.3.K1, 3.2.K1-5, 3.3.K1, 3.4.K1-3, 4.2.K3-5) ($);

b. place value models (place value mats, hundred charts, or base ten blocks) to compare, order, and represent numerical quantities and to model computational procedures (1.1.K3, 1.2.K2, 1.2.K4, 1.3.K1, 1.4.K6-7, 1.4.K7a, 2.2.K2a, 2.2.K4) ($);

c. fraction models (fraction strips or pattern blocks) to compare, order, and represent numerical quantities (1.1.K2b) ($);

d.money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K5-6, 1.3.K1, 1.4.K7b, 2.1.K1f, 2.2.K2b) ($);

e. function tables (input/output machines, T-tables) to model numerical relationships (2.3.K2) ($);

f. two-dimensional geometric models (geoboards, dot paper, pattern blocks, tangrams, or attribute blocks) to model perimeter and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model attributes of geometric shapes (2.1.K2c, 3.1.K1-6, 3.3.K2-3);

The student...1. recognizes that various mathematical models can be used to

represent the same problem situation. Mathematical models include:a. process models (concrete objects, pictures, diagrams,

number lines, unifix cubes, hundred charts, or measurement tools) to model computational procedures and mathematical relationships, to compare and order numerical quantities, and to model problem situations (1.1.A1a-b, 1.1.A2, 1.2.A1-2, 1.3.A1, 1.4.A1-2, 2.1.A1a, 2.1.A1c-e, 2.2.A1-2, 2.3.A1-2, , 3.2.A1-4, 3.3.A1-2, 3.4.A1, 4.2.A2) ($);

b. place value models (place value mats, hundred charts, or base ten blocks) to compare, order, and represent numerical quantities and to model computational procedures (1.1.A1a-b, 1.1.A2, 1.2.A1-2, 1.3.A2, 1.4.A1a) ($);

c. money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1c-d, 1.3.A2, 1.4.A1b, 2.1.A1d) ($);

d. two-dimensional geometric models (geoboards, dot paper, pattern blocks, tangrams, or attribute blocks) to model perimeter and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model attributes of geometric shapes (2.1.A1b, 3.1.A1-3);

e. two-dimensional geometric models (spinners), three- dimensional geometric models (number cubes), and process models (concrete objects) to model probability (4.1.A1) ($);

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g. two-dimensional geometric models (spinners), three-dimensional geometric models (number cubes), and process models (concrete objects) to model probability (4.1.K1-2) ($);

h. graphs using concrete objects, representational objects, or abstract representations, pictographs, frequency tables, horizontal and vertical bar graphs, Venn diagrams or other pictorial displays, and line plots to organize and display data (4.1.K2, 4.2.K1, 4.2.K2) ($);

i. Venn diagrams to sort data.

f. graphs using concrete objects, representational objects, or abstract representations, pictographs, horizontal and vertical bar graphs (4.1.A1, 4.2.A1-4) ($).

2. selects a mathematical model that is more useful than other mathematical models in a given situation.

2. creates a mathematical model to show the relationship between two or more things, e.g., using pattern blocks, a whole (1) can be represented using

a (1/1) or

two (2/2) or

three (3/3) or

six (6/6).

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The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, e.g., comparing the number of boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can be represented by pictures of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color tiles). As students work with the different representations of the same situation, they begin to develop an understanding of the advantages and disadvantages of the various representations/models.

Many mathematical models are listed in this benchmark. The indicator lists some of the mathematical models that could be used to teach a concept. Each indicator in this benchmark is linked to other indicators in other benchmarks; those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3. In addition, the indicator in the other benchmarks identifies, in parentheses, the Models’ indicator. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models).


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Standard 3: Geometry SECOND GRADE


Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and describes their properties using concrete objects in a variety of situations.

Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. recognizes and investigates properties of circles, squares,

rectangles, triangles, and ellipses (ovals) (plane figures/two-dimensional shapes) using concrete objects, drawings, and appropriate technology (2.4.K1f).

2. ■ recognizes, draws, and describes circles, squares, rectangles, triangles, ellipses (ovals) (plane figures) (2.4.K1f).

3. recognizes cubes, rectangular prisms, cylinders, cones, and spheres (solids/three-dimensional figures) (2.4.K1f).

4. recognizes the square, triangle, rhombus, hexagon, parallelogram, and trapezoid from a pattern block set (2.4.K1f).

5. compares geometric shapes (circles, squares, rectangles, triangles, ellipses) to one another (2.4.K1f).

6. recognizes whether a shape has a line of symmetry (2.4.K1f).

The student...1. solves real-world problems by applying the properties of plane

figures (circles, squares, rectangles, triangles, ellipses) (2.4.A1d), e.g., which shape could be used to completely cover the lid of a pencil box with no overlapping?

2. demonstrates how (2.4.A1d):a. ▲ plane figures (circles, squares, rectangles, triangles, ellipses)

can be combined or separated to make a new shape;b. solids (cubes, rectangular solids, cylinders, cones, spheres) can

be combined or separated to make a new shape.3. identifies the plane figures (circles, squares, rectangles, triangles,

ellipses) used to form a composite figure (2.4.A1d).

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Pattern blocks are a collection of six geometric shapes in six colors. Each set contains 250 pieces — 50 green triangles, 25 orange squares, 50 blue rhombi, 50 tan rhombi, 50 red trapezoids, and 25 yellow hexagons. The blue rhombus and the tan rhombus also are parallelograms, and the orange square is a parallelogram. The blocks are designed so that their sides are all the same length with the exception that the trapezoid has one side twice as long. This feature allows the blocks to be nested together and encourages the exploration of relationships among the shapes. Activities with pattern blocks help students explore patterns, functions, fractions, congruence, similarity, symmetry, perimeter, area, and graphing. A pattern block template can be found in the Appendix.



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Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. uses whole number approximations (estimations) for length, weight,

and volume using standard and nonstandard units of measure (2.4.K1a) ($), e.g., the height of the classroom door is 14 chalkboard erasers laid end to end or 7 feet high or an apple weighs about 42 unifix cubes.

2. ▲ reads and tells time by five-minute intervals using analog and digital clocks (2.4.K1a).

3. selects and uses appropriate measurement tools and units of measure for length, weight, volume, and temperature for a given situation (2.4.K1a) ($).

4. measures (2.4.K1a) ($):a. ▲ length to the nearest inch or foot and to the nearest whole

unit of a nonstandard unit;b. weight to the nearest nonstandard unit;c. volume to the nearest cup, pint, quart, or gallon; d. temperature to the nearest degree.

5. states (2.4.K1a):a. the number of minutes in an hour,b. the number of days in each month.

The student...1. compares the weights of more than two concrete objects using a

balance (2.4.A1a) ($). 2. solves real-world problems by applying appropriate measurements

(2.4.A1a): a. length to the nearest inch or foot, e.g., a cookie is almost how

many inches wide?b. length to the nearest whole unit of a nonstandard unit, e.g., how

many paper clips long is a candy bar? 3. estimates to check whether or not measurements or calculations for

length in real-world problems are reasonable (2.4.A1a) ($), e.g., is it reasonable to say that you measured your thumb and it is 2 feet long?

4. adjusts original measurement or estimation for length and weight in real-world problems based on additional information (a frame of reference) (2.4.A1a), e.g., I estimated that the stapler is 20 paperclips long. Then I lay out 4 paper clips next to the stapler. I realize that since I am half done, my estimate is too high; so I adjust my estimate to 8 paper clips.

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Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” The process of learning to measure at the early grades focuses on identifying what property (length, weight, volume) is to be measured and to make comparisons. Estimation in measurement is defined as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps students develop basic properties of measurement, and it gives students a tool to determine whether a given measurement is reasonable.



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Benchmark 3: Transformational Geometry – The student recognizes and shows one transformation on simple shapes and concrete objects in a variety of situations.

Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. knows and uses the cardinal points (north, south, east, west)

(2.4.K1a).2. recognizes that changing an object's position or orientation

including whether the object is nearer or farther away does not change the name, size, or shape of the object (2.4.K1f).

3. recognizes when a shape has undergone one transformation (flip/reflection, turn/rotation, slide/translation) (2.4.K1f).


fitting one shape or object on top of the other (2.4.A1a).2. follows directions to move objects from one location to another

using appropriate vocabulary and the cardinal points (north, south, east, west) (2.4.A1a).




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Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...

1. locates and plots whole numbers from 0 through 1,000 on a segment of a number line (horizontal/vertical) (2.4.K1a), e.g., using a segment of a number line from 800 to 820 to locate the whole number 805.

2. represents the distance between two whole numbers from 0 through 1,000 on a segment of a number line (2.4.K1a).

3. uses a segment of a number line to model addition and subtraction using whole numbers from 0 through 1,000 (2.4.K1a), e.g., 333 + n = 349 or 333 + 16 = n or 400 – n = 352 or 400 – 48 = n.

The student...1. solves real-world problems involving counting, adding, and

subtracting whole numbers from 0 through 1,000 using a segment of a number line (2.4.A1a) ($), e.g., Adam had collected 894 marbles. He lost nine marbles. How many does he have now? Using the number line, Adam shows how he solved the problem.

Teacher Notes: A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line, and is an example of one-to-one correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a number line is a precursor to absolute value.



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Standard 4: Data SECOND GRADE


Benchmark 1: Probability – The student applies the concepts of probability using concrete objects in a variety of situations.

Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. recognizes any outcome of a simple event in an experiment or

simulation as impossible, possible, certain, likely, or unlikely (2.4.K1g) ($).

2. lists some of the possible outcomes of a simple event in an experiment or simulation including the use of concrete objects (2.4.K1g-h).

The student...1. makes a prediction about a simple event in an experiment or

simulation; conducts the experiment or simulation including the use of concrete objects; records the results in a chart, table, or graph; and makes an accurate statement about the results (2.4.A1e-f).

Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be addressed through the use of concrete objects, coins, and geometric models (spinners or number cubes).



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Standard 4: Data SECOND GRADE


Benchmark 2: Statistics – The student collects, organizes, displays, and explains numerical (whole numbers) and non-numerical data sets including the use of concrete objects in a variety of situations.

Second Grade Knowledge Base Indicators Second Grade Application IndicatorsThe student...1. organizes, displays, and reads numerical (quantitative) and non-

numerical (qualitative) data in a clear, organized, and accurate manner including a title, labels, categories, and whole number intervals using these data displays (2.4.K1h) ($): a. ▲ graphs using concrete objects;b. ▲ pictographs with a whole symbol or picture representing one,

two, or ten (no partial symbols or pictures);c. ▲ ■ frequency tables (tally marks);d. ▲ horizontal and vertical bar graphs; e. Venn diagrams or other pictorial displays, e.g., glyphs; f. line plots.

2. collects data using different techniques (observations, interviews, or surveys) and explains the results (2.4.K1h) ($).

3. identifies the minimum (lowest) and maximum (highest) values in a whole number data set (2.4.K1a) ($).

4. finds the range for a data set using two-digit whole numbers (2.4.K1a) ($).

5. finds the mode (most) for a data set using concrete objects that include (2.4.K1a) ($):a. quantitative/numerical data (whole numbers through 100); b. qualitative/non-numerical data (category that occurs most

often).

The student...3. communicates the results of data collection and answers questions

based on information from (2.4.A1f) ($):a. graphs using concrete objects, b. pictographs with a whole symbol or picture representing one (no

partial symbols or pictures),c. horizontal and vertical bar graphs.

2. determines categories from which data could be gathered (2.4.A1f) ($), e.g., categories could include shoe size, height, favorite candy bar, or number of pockets in clothing.

3. recognizes that the same data set can be displayed in various formats including the use of concrete objects (2.4.A1f) ($).

4. recognizes appropriate conclusions from data collected (2.4.A1f) ($).

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Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are used to tell a story. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data. Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate. Graphs take many forms: bar graphs and pictographs compare discrete data, frequency tables show how many times a certain piece of data occurs using tally marks to record the data, circle graphs (pie charts) model parts of a whole, line graphs show change over time, Venn diagrams show relationships between sets of objects, and line plots show frequency of data on a number line.

The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.



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Standard 1: Number and Computation THIRD GRADE


Benchmark 1: Number Sense – The student demonstrates number sense for whole numbers, fractions, decimals, and money using concrete objects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. knows, explains, and represents ($):

a. whole numbers from 0 through 10,000 (2.4.K1a-b) b. fractions greater than or equal to zero (halves, fourths, thirds,

eighths, tenths, sixteenths) (2.4.K1c) ($); c. decimals greater than or equal to zero through tenths place

(2.4.K1c).2. compares and orders:

a. ▲ ■ whole numbers from 0 through 10,000 with and without the use of concrete objects (2.4.K1a-b) ($);

b. fractions greater than or equal to zero with like denominators (halves, fourths, thirds, eighths, tenths, sixteenths) using concrete objects (2.4.K1a,c);

c. decimals greater than or equal to zero through tenths place using concrete objects (2.4.K1a-c).

3. ▲ knows, explains, and uses equivalent representations including the use of mathematical models for:a. addition and subtraction of whole numbers from 0 through 1,000

(2.4.K1a-b) ($), e.g., 144 + 236 = 300 + 80 █ █ █

▌▌▌▌ ▌▌▌ ▪▪▪▪ ▪▪▪▪▪▪

The student…1. solves real-world problems using equivalent representations

and concrete objects to ($):a. compare and order whole numbers from 0 through 5,000

(2.4.A1a-b), e.g., using base ten blocks, represent the total school attendance for a week; then represent the numbers using digits and compare and order in different ways;

b. add and subtract whole numbers from 0 through 1,000 and when used as monetary amounts (2.4.A1a,d) ($), e.g., use real money to show at least 2 ways to represent $10.42; then subtract the cost of a book purchases at the school’s book fair from $10.42 (the amount you have earned and can spend).

2. determines whether or not solutions to real-world problems that involve the following are reasonable ($).a. whole numbers from 0 through 1,000 (2.4.A1a-b), e.g., a

student says that there are 1,000 students in grade 3 at her school, is this reasonable?

b. fractions greater than or equal to zero (halves, fourths, thirds, eighths, tenths, sixteenths) (2.4.A1a,c); e.g., you ate ½ of a sandwich and a friend ate ¾ of the same sandwich; is this reasonable?

c. decimals greater than or equal to zero when used as monetary amounts (2.4.A1d), e.g., a pack of chewing gum costs what amount - $62 $.75 9¢ $75.00 750¢? Is this reasonable?;

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▲– Assessed Indicator on the Objective Assessment ■ – Assessed Indicator on the Optional Constructed Response Assessment N – Noncalculator ($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.

100

100

one hundred

$100 $10 $10$100 $10 $10$100 $10 $10

$10 $10

b. multiplication using the basic facts through the 5s and the multiplication facts of the 10s (2.4.K1a), e.g., 3 x 2 can be represented as 4 + 2 or as an array, X X X

X X X;

3. determines the amount of change owed through $100.00 (2.4.A1d), e.g., school supplies cost $12.37. What was the amount of change received after giving the clerk $20.00? To solve, $20.00 – $12.37 = $7.63 (the change).

c. addition and subtraction of money (2.4.K1d) ($), e.g., three halfdollars equals 50¢ + 50¢ + 50¢ or 50¢ + 100¢.

4. ▲N determines the value of mixed coins and bills with a total value of $50 or less (2.1.K1d) ($).





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Appendix.

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Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of whole numbers with a special emphasis on place value and recognizes, uses, and explains the concepts of properties as they relate to whole numbers, fractions, decimals, and money in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. identifies, reads, and writes numbers using numerals and words from

tenths place through ten thousands place (2.4.K1a-b) ($), e.g., sixty-four thousand, three hundred eighty and five tenths is written in numerical form as 64,380.5.

2. identifies, models, reads, and writes numbers using expanded form from tenths place through ten thousands place (2.4.K1b), e.g., 56,277.3 = (5 x 10,000) + (6 x 1,000) + (2 x 100) + (7 x 10) + (7 x 1) + (3 x .1) = 50,000 + 6,000 + 200 + 70 + 7 + .3.

3. classifies various subsets of numbers as whole numbers, fractions (including mixed numbers), or decimals (2.4.K1a-c, 2.4.K1i)

4. identifies the place value of various digits from tenths to one hundred thousands place (2.4.K1b) ($).

5. identifies any whole number through 1,000 as even or odd (2.4.K1a).6. uses the concepts of these properties with whole numbers from 0

through 100 and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($): a. commutative properties of addition and multiplication, e.g., 7 + 8

= 8 + 7 or 3 x 6 = 6 x 3;b. zero property of addition (additive identity), e.g., 4 + 0 = 4; c. property of one for multiplication (multiplicative identity), 1 x 3 = 3; d. associative property of addition, e.g., (3 + 2) + 4 = 3 + (2 + 4);e. symmetric property of equality applied to addition and

multiplication, e.g., 100 = 20 + 80 is the same as 20 + 80 = 100 and 3 x 4 = 12 is the same as 12 = 3 x 4;

The student…1. solves real-world problems with whole numbers from 0 through

100 using place value models, money, and the concepts of these properties to explain reasoning (2.4.A1a-b,d) ($):a. commutative property of addition, e.g., a student has a dime, a

nickel, and a quarter to purchase a pencil; he/she totals the amount of the coins to see whether or not there is enough money; the student could count the quarter, nickel, and dime as 25¢ + 5¢ + 10¢ or as 25¢ + 10¢ + 5¢ because adding in any order does not change the sum;

b. zero property of addition, e.g., a student has 6 marbles in one pocket and none in the other, so all together there are: 6 + 0 = 6;

c. associative property of addition, e.g., a student has two dimes and a quarter; there are 2 ways to group the coins to find the total: 10¢ (dime) + 10¢ (dime) = 20¢, then add the quarter, 20¢ + 25¢ (quarter) = 45¢ or 10¢ (dime) + 25¢ (quarter) = 35¢, then add the other dime to 35¢ and 35¢ + 10¢ = 45¢ or (D + D) + Q = D + (D + Q) using coins or money models.

2. performs various computational procedures with whole numbers from 0 through 100 using the concepts of these properties and explains how they were used (2.4.A1a-b): a. commutative property of multiplication, e.g., given 4 x 6, the

student says: I know that 4 x 6 is 24 because I know 6 x 4 is 24 and multiplying in any order gets the same answer;

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f. zero property of multiplication, e.g., 9 x 0 = 0 or 0 x 32 = 0.7. divides whole numbers from 0 through 99,999 into groups of 10,000s;

1,000s; 100s; 10s, and 1s using base ten models (2.4.K1b).

b. zero property of multiplication without computing, e.g., 7 x 3 x 4 x 0 x 5 = □, the student says: I know the answer (product) is zero because no matter how many factors you have, when you multiply with a 0, the product is zero;

c. associative property of addition, e.g., 9 + 8 could be solved as 1 + (8 + 8) or (1 + 8) + 8, the student says: I don’t know 9 + 8, but I know my doubles (8 + 8), so I made the 9 into 1 + 8 and added 8 + 8 and then added 1 more to make 17.





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Benchmark 3: Estimation – The student uses computational estimation with whole numbers, fractions, and money in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. estimates whole numbers quantities from 0 through 1,000; fractions

(halves, fourths); and monetary amounts through $500 using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a-d) ($).

2. uses various estimation strategies to estimate using whole number quantities from 0 through 1,000 and explains the process used (2.4.K1a) ($) e.g., 362 rounded to the nearest ten is 360 and 362 rounded to the nearest hundred is 400. Using front-end estimation, 362 is about 300 or 400 depending on the context of the problem. Using a “nice” number, 362 is about 350 because of the benchmark number – 350, since 350 is the halfway point between 300 and 400.

3. recognizes and explains the difference between an exact and an approximate answer (2.4.K1a), e.g., when asked how many students are in a classroom, an exact answer could be 24. Whereas, an approximate answer could be 20 since 24 could be rounded down to the nearest ten (underestimated) or rounded up to 30 (overestimated).

The student…1. adjusts original whole number estimate of a real-world problem

using numbers from 0 through 1,000 based on additional information (a frame of reference) (2.4.A1a) ($), e.g., if given a pint container and told the number of marbles it has in it, the student would estimate the number of marbles in a quart container.

2. estimates to check whether or not the result of a real-world problem using whole numbers from 0 through 1,000 and monetary amounts through $500 is reasonable and makes predictions based on the information (2.4.A1a-b,d) ($), e.g., at the movies, you bought popcorn for $2.35 and a soda for $2.50; and then paid $4.50 for a ticket. Is it reasonable to say you spent $10? How much will you need to save to go to the movies once a week for the next month?

3. selects a reasonable magnitude from three given quantities based on a familiar problem situation and explains the reasonableness of the results (2.4.A1a), e.g., about how many students are in my class today – 2, 20, 200?

4. determines if a real-world problem with whole numbers from 0 through 1,000 calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.A1a) ($).

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Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible. In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation strategies include front-end with adjustment, compatible “nice” numbers, clustering, or special numbers.



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money including the use of concrete objects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. computes with efficiency and accuracy using various computational


2. N states and uses with efficiency and accuracy the multiplication facts through the 5s and the multiplication facts of the 10s and corresponding division facts (2.4.K1a) ($).

3. skip counts (multiples) by 2s, 3s, 4s, 5s, and 10s (2.4.K1a).4. N performs and explains these computational procedures:

a. adds and subtracts whole numbers from 0 through 10,000 (2.4.K1a-b);

b. multiplies whole numbers when one factor is 5 or less and the other factor is a multiple of 10 through 1,000 with or without the use of concrete objects (2.4.K1a-b), e.g., 400 x 3 = 120 or 70 x 5 = 350;

c. adds and subtracts monetary amounts using dollar and cents notation through $500.00 (2.4.K1d) ($), e.g., $47.07 + $356.96 = $404.03.

5. fair shares/measures out (divides) a total amount through 100 concrete objects into equal groups (2.4.K1a-b), e.g., fair sharing 52 pieces of candy with 8 friends resulting in eight groups of 6 with four pieces left over or measuring out into groups of eight 52 pieces of candy with four pieces left over.

6. explains the relationship between addition and subtraction (2.4.K1a-b) ($).

7. ▲■ N identifies multiplication and division fact families through the 5s and the multiplication and division fact families of the 10s (2.4.K1a), e.g., when given 6 x = 18, the student recognizes the remaining members of the fact family.

The student…1. ▲N solves one-step real-world addition or subtraction problems

with ($):a. whole numbers from 0 through 10,000 (2.4.A1a-b), e.g., for

the food drive, the school collected 564 cans (cylinders) and 297 boxes (rectangular prisms). How many items did they collect in all? This problem could be solved with base 10 models: by adding 500 + 200 (700), 60 + 90 (150), and 4 + 7 (11), so 700+ 150 + 11= 861; by adding 564 + 300 (864) and 297 is 3 less than 300, so 864 – 3 = 861; or by using the traditional algorithm;

b. monetary amounts using dollar and cents notation through $500.00 (2.4.A1a-b,d), e.g., you are shopping for a new bicycle; at The Bike Store, the bike you want is $189.69 and at Sports for All, it is $162.89. How much will you save by buying the bike at Sports for All?

2. N generates a family of multiplication and division facts through the 5s (2.4.A1a), e.g., if the student writes 5 x 9 = 45, the remaining facts generated are: 9 x 5 = 45, 45 ÷ 5 = 9, 45 ÷ 9 = 5.

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Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers and

8. reads and writes horizontally, vertically, and with different operational symbols the same addition, subtraction, multiplication, or division expression, e.g., 4 • 6 is the same as 4 x 6 or 4(6) or 6 and 10

divided by 2 is the same as 10 ÷ 2 or 10 x 4 2. Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations and by counting on for addition and counting up for subtraction and unknown-addend situations. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling numbers (doubles), making ten, and compatible numbers.


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3-97January 31, 2004


Standard 2: Algebra THIRD GRADE



Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…4. uses concrete objects, drawings, and other representations to work

with types of patterns (2.4.K1a): e. repeating patterns, e.g., an AB pattern is like 1-2, 1-2, …; an

ABC pattern is like dog-horse-pig, dog-horse-pig, …; an AAB pattern is like ↑↑→, ↑↑→, …;

f. growing patterns, e.g., 1, 4, 7, 10, …5. uses these attributes to generate patterns:

c. counting numbers related to number theory (2.4.K1a), e.g., evens, odds, or multiples through the 5s;

whole numbers that increase or decrease (2.4.K1a) ($),e.g., 3, 6, 9, …; 20, 15, 10, …;

geometric shapes including one attribute change (2.4.K1f), e.g., --Δ-▲, --Δ-▲, --Δ-▲,… where the pattern is filled-in square, square, triangle, filled-in triangle, …; or when using attribute blocks the change is size only, then shape only, … such as

…;measurements (2.4.K1a), e.g., 1 ft, 2 ft, 3 ft, …; 3 lbs, 6 lbs, 9 lbs; or

2 cups, 4 cups, 6 cups, …; e. money and time (2.4.K1a,d) ($), e.g., $.25, $.50, $.75, … or 1:05

p.m., 1:10 p.m., 1:15 p.m., …;f. things related to daily life (2.4.K1a), e.g., water cycle, food

cycle, or life cycle;

The student…1. generalizes the following patterns using a written description:

a. counting numbers related to number theory (2.4.A1a);b. whole number patterns (2.4.A1a) ($),c. patterns using geometric shapes (2.4.A1f),d. measurement patterns (2.4.A1a),e. money and time patterns (2.4.A1a,d) ($),f. patterns using size, shape, color, texture, or movement

(2.4.A1a).2. ▲ recognizes multiple representations of the same pattern (2.4.A1a)

e.g., the ABC pattern could be represented by clap, snap, stomp, …; red, green, yellow, …; tricycle, bicycle, unicycle, …; or 3, 2, 1, …

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g. things related to size, shape, color, texture, or movement (2.4.K1a), e.g., red-green, red-green, red-green, …; snapping fingers; clapping hands; stomping feet; or tossing a bean bag over the head, under the leg, and behind the back (kinesthetic patterns).

3. identifies, states, and continues a pattern presented in various formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written (2.4.K1a) ($).

4. generates: a. repeating patterns (2.4.K1a), b. growing (extending) patterns (2.4.K1a),c. patterns using function tables (input/output machines, T-tables)

(2.4.K1e).

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Teacher Notes: Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney, Macmillan Publishing Company, 1994) This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain.

Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest common factor and least common multiple.



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Benchmark 2: Variables, Equations, and Inequalities – The student uses symbols and whole numbers to solve equations including the use of concrete objects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. explains and uses symbols to represent unknown whole number

quantities from 0 through 1,000 (2.4.K1a)2. finds the sum or difference in one-step equations with ($):

a. whole numbers from 0 through 99 (2.4.K1a) e.g., 89 = 76 + y or y – 23 = 32;

b. monetary values through a dollar (2.4.K1d), e.g., 25¢ + 10¢ + 5¢ = n.

3. finds the unknown in the multiplication and division fact families through the 5s and the 10s (2.4.K1a), e.g., 3 • = 4 • 6.

4. compares two whole numbers from 0 through 1,000 using the equality and inequality symbols (=, <, >) and their corresponding meanings (is equal to, is less than, is greater than) (2.4.K1a-b) ($).

The student…1. represents real-world problems using symbols with one

operation and one unknown that (2.4.A1a) ($):a. adds or subtracts using whole numbers from 0 through 99, e.g.,

when asked to represent the number of 3rd graders in a school, students write: 21 + 18 + 19 = ;

b. multiplies or divides using the basic facts through the 5s and the basic facts of the 10s, e.g., juice comes in packs of 4. How many packs are needed for 32 third-graders? Students could write: 32 ÷ 4 = J.

2. generates one-step equations to solve real-world problems with one unknown and a whole number solution that (2.4.A1a) ($):a. adds or subtracts using the basic fact families, e.g., when asked

to generate a simple equation, a student says: I have 5 dogs and 2 fish. How many pets do I have? This is represented by 5 + 2 = P and to solve for P, add 5 and 2, P = 7.

b. multiplies or divides using the basic facts through the 5s and the basic facts of the 10s, e.g., Tom has a sticker book and each page holds 5 stickers. If the same number of stickers is placed on each page, the book will hold 30 stickers. How many pages are in his book? This is represented by 5 x S = 30 or 30 ÷ 5 = S.

3. generates (2.4.A1a) ($):a. a real-world problem with one operation that matches a given

addition equation or subtraction equation using whole numbers from 0 through 99, e.g., given the subtraction equation, 69 – G = 37, the problem could be written: You have 69 guppies and give away some to a friend and have 37 left. How many guppies did you give away?

b. a real-world problem with one operation that matches a given

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multiplication equation or division equation using basic facts through the 5s and the basic facts of the 10s, e.g., the problem could be: I have 25 pictures and glue 5 pictures on each page of my album. How many pages will I need to use? The equation: 25/5 = ▲.

c. number comparison statements using equality and inequality symbols (=, <, >) for whole numbers from 0 through 100, measurement, and money $, e.g.4 ft 4 in > 4 ft 2 in.

Teacher Notes: Understanding the concept of variable is fundamental to algebra. In the early grades, students use various symbols, including letters and geometric shapes, to represent unknown quantities that do or do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 x 4 = Δ.



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Benchmark 3: Functions – The student recognizes and describes whole number relationships using concreteobjects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. states mathematical relationships between whole numbers from 0

through 200 using various methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a) ($), e.g., every time a quarter is added to the amount; 25¢ is added to the total.

2. finds the values and determines the rule with one operation (addition, subtraction) of whole numbers from 0 through 200 using a horizontal or vertical function table (input/output machine, T-table) (2.4.K1e), e.g., using this input/output machine, different student responses might be that the rule is Input minus

10 equals Output, the rule is N – 10, or the rule is subtract 10.

3. ▲ generalizes numerical patterns using whole numbers from 0 through 200 with one operation (addition, subtraction) by stating the rule using words, e.g., if the sequence is 30, 50, 70, 90, …; in words, the rule is add twenty to the number before.

4. uses a function table (input/output machine, T-table) to identify and plot ordered pairs in the first quadrant of a coordinate plane (2.4.K1a,e).

The student…1. represents and describes mathematical relationships between whole

numbers from 0 through 100 using concrete objects, pictures, written descriptions, symbols, equations, tables, and graphs (2.4.A1a) ($).

1. finds the rule, states the rule using words, and extends numerical patterns with whole numbers from 0 through 100 (2.4.A1a,e), e.g., at school each student must check out three library books. After the tenth student has checked out, how many total books will have been checked out? A solution using a function table might be:

Number of Students 1 2 5 10Total Number Of Books 3 6 15 ?

The rule could be that for every student, add three books or multiply the number of children by three to get the total number of books. Other solutions might be using a pattern to count by three ten times - 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 - or skip count by three ten times.

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Input Output92 82

156 14613 3

113 103? 59

106 ?? ?N ?

..


Function concepts should be developed from growing patterns. Each term in a number sequence is related to its position in the sequence – the functional relationship. The pattern – 4, 7, 10, 13, 16, 19, and so on – is an arithmetic sequence with a difference of 3. The pattern could be described as add 3 meaning that 3 must be added to the previous term to find the next. This pattern is explained by using the recursive definition for a sequence. The recursive definition for a sequence is a statement or a set of statements that explains how each successive term in the sequence is obtained from the previous term(s).

In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the number of the term – the 5th term is 52, the 8th term is 82, …




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Benchmark 4: Models – The student develops and uses mathematical models including the use of concrete objects to represent and show mathematical relationships in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. knows, explains, and uses mathematical models to represent

mathematical concepts, procedures, and relationships. Mathematical models include:a. process models (concrete objects, pictures, number lines,

coordinate planes/grids, hundred charts, measurement tools, multiplication arrays, or division sets) to model computational procedures and mathematical relationships (1.2.K1, 1.2.K.1a, 1.2.K2 1.2.K3, 1.2.K5-6, 1.3.K1-3, 1.4.K1-3, 1.4.K1a-b, 1.4.K5-7, 2.1.K1, 2.1.K2a, 2.1.K2d-g, 2.1.K3, 2.1.K4a-b, 2.2.K1, 2.2.K2, 2.2.K3-4, 2.3.K1, 2.3.K4, 3.2.K1-4, 3.3.K1, 3.4.K1-3, K.2.K3) ($);

b. place value models (place value mats, hundred charts, base ten blocks or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures (1.1.K1c, 1.1.K2a, 1.1.K2c, 1.1.K3a, 1.2.K1-4, 1.2.K7, 1.3.K1, 1.4.K4a-b, 1.4.K5-6, 2.2.K4) ($);

c. fraction models (fraction strips or pattern blocks) and decimal models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K1b, 1.1.K2b-c, 1.2.K3, 1.3.K1) ($);

d. money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1K3c, 1.1.K4, 1.3.K1, 1.4.K4c, 2.1.K2e, 2.2.K2b) ($);

e. function tables (input/output machines, T-tables) to find numerical relationships (2.1.K4c, 2.3.K2, 2.3.K4) ($);


represent the same problem situation. Mathematical models include:a. process models (concrete objects, pictures, number lines,

coordinate planes/grids, hundred charts, measurement tools, multiplication arrays, or division sets) to model computational procedures and mathematical relationships and to model problem situations (1.2.A1, 1.2.A2a-b, 1.3.A1-4, 1.4.A1a-b, 1.4.A2, 2.1.A1a-b, 2.1.A1d-f, 2.1.A2, 2.2.A1-3, 2.2.A3a-c, 2.3.A1-2, 3.2.A1-3, 3.3.A1-2, 3.4.A1, 4.2.A2) ($);

a. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures (1.1.A1a, 1.1.A2a, 1.2.A1-2, 1.3.A2, 1.4.A1a-b) ($);

b. fraction models (fraction strips or pattern blocks) and decimal models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A2b) ($);

c. money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1b, 1.1.A2c, 1.2.A1, 1.3.A2, 1.4.A1b, 2.1.A1e, 2.2.A3c) ($);

d. function tables (input/output machines, T-tables) to model numerical relationships (2.3.A2) ($);

e. two-dimensional geometric models (geoboards, dot paper, pattern blocks, or tangrams) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model attributes of geometric shapes (2.1.A1c, 3.1.A1-3);

f. two-dimensional geometric models (geoboards, dot paper, g. two-dimensional geometric models (spinners), three-dimensional 3-107

January 31, 2004▲– Assessed Indicator on the Objective Assessment ■ – Assessed Indicator on the Optional Constructed Response Assessment N – Noncalculator ($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.

pattern blocks, or tangrams) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model attributes of geometric shapes (2.1.K2c, 3.1.K1-6, 3.2.K5, 3.3.K2);

g. two-dimensional geometric models (spinners), three-dimensional models (number cubes), and process models (concrete objects) to model probability (4.1.K1-2) ($);

h. graphs using concrete objects, representational objects, or abstract representations, pictographs, frequency tables, horizontal and vertical bar graphs, Venn diagrams or other pictorial displays, line plots, charts, and tables to organize and display data (2.3.K4, 4.1.K2, 4.2.K1a-d, 4.2.K1f-g, 4.2.K2) ($);

i. Venn diagrams to sort data and show relationships (1.2.K3).

models (number cubes), and process models (concrete objects) to model probability (4.1.A1-2) ($);

h. graphs using concrete objects, representational objects, or abstract representations pictographs, frequency tables, horizontal and vertical bar graphs, Venn diagrams or other pictorial displays, line plots, charts and tables to organize and display data (4.1.A1-2, 4.2.A1a-d, 4.2.A1f-g, 4.2.A3) ($);

i. Venn diagrams to sort data and show relationships.2. selects a mathematical model that is more useful than other

mathematical models in a given situation.

2. creates a mathematical model to show the relationship between two or more things, e.g., using pattern blocks, a whole (1) can be represented as

a (1/1) or

two (2/2) or

three (3/3) or

six (6/6).

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Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item being assessed.The mathematical modeling process involves:

a. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through some sort of mathematical model,

b. performing manipulations and mathematical procedures within the mathematical model,c. interpreting the results of the manipulations within the mathematical model, d. using these results to make inferences about the original real-world situation.

The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, e.g., comparing the number of boys and girls in the classroom can be represented by lining them up in two different lines. The same situation also can be represented by pictures of the children (pictograph), a bar graph, or by using two different colors of the same manipulative (unifix cubes or color tiles). As students work with the different representations of the same situation, they begin to develop an understanding of the advantages and disadvantages of the various representations/models.



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Standard 3: Geometry THIRD GRADE


Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and investigates their properties using concrete objects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. recognizes and investigates properties of plane figures (circles,

squares, rectangles, triangles, ellipses, rhombi, octagons) using concrete objects, drawings, and appropriate technology (2.4.K1f).

2. recognizes, draws, and describes plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, octagons) (2.4.K1f).

3. ■ recognizes the solids (cubes, rectangular prisms, cylinders, cones, spheres) (2.4.K1f).

4. ▲ recognizes and describes the square, triangle, rhombus, hexagon, parallelogram, and trapezoid from a pattern block set (2.4.K1f).

5. recognizes and describes a quadrilateral as any four-sided figure (2.4.K1f).

6. determines if geometric shapes and real-world objects contain line(s) of symmetry and draws the line(s) of symmetry if the line(s) exist(s) (2.4.K1f).

The student…1. solves real-world problems by applying properties of plane figures

(circles, squares, rectangles, triangles, ellipses) to (2.4.A1f), e.g., the teacher asked each student to draw a rectangle. A student draws a square. Did the student follow directions? Why or why not?

2. demonstrates how (2.4.A1f):a. plane figures (circles, squares, rectangles, triangles, ellipses,

rhombi, hexagons, trapezoids) can be combined to make a new shape;

b. solids (cubes, rectangular prisms, cylinders, cones, spheres) can be combined to make a new shape.

3. identifies the plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, hexagons, trapezoids) used to form a composite figure (2.4.A1f).

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Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to describe their properties and their relationships to other shapes. The term geometry comes from two Greek words meaning “earth measure.” The fundamental concepts in geometry are point (no dimension), line (one-dimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to as three-dimensional.




Pattern blocks are a collection of six geometric shapes in six colors. Each set contains 250 pieces – 50 green triangles, 25 orange squares, 50 blue rhombi, 50 tan rhombi, 50 red trapezoids, and 25 yellow hexagons. The blue rhombus and the tan rhombus also are parallelograms, and the orange square is a parallelogram. The blocks are designed so that their sides are all the same length with the exception that the trapezoid has one side twice as long. This feature allows the blocks to be nested together and encourages the exploration of relationships among the shapes. Activities with pattern blocks help students explore patterns, functions, fractions, congruence, similarity, symmetry, perimeter, area, and graphing. A template for a pattern block set can be found in the Appendix.


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Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. uses whole number approximations (estimations) for length, width,

weight, volume, temperature, time, and perimeter using standard and nonstandard units of measure (2.4.K1a) ($).

2. ▲ reads and tells time to the minute using analog and digital clocks (2.4.K1a).

3. selects, explains the selection of, and uses measurement tools, units of measure, and degree of accuracy appropriate for a given situation to measure (2.4.K1a) ($):a. length width, and height to the nearest half inch, inch, foot, and

yard; and to the nearest whole unit of nonstandard unit;b. length, width, and height to the nearest centimeter and meter;c. weight to the nearest whole unit of a nonstandard unit;d. volume to the nearest cup, pint, quart, and gallon; e. volume to the nearest liter; f. temperature to the nearest degree.

4. states (2.4.K1a):a. the number of hours in a day and days in a year;b. the number of inches in a foot, inches in a yard, and feet in a

yard;c. the number of centimeters in a meter;d. the number of cups in a pint, pints in a quart, and quarts in a

gallon.5. finds the perimeter of squares, rectangles, and triangles given the

measures of all the sides (2.4.K1f).

The student…1. solves real-world problems by applying appropriate

measurements: a. ▲ length to the nearest inch, foot, or yard, e.g., Jill has a piece

of rope that is 36 inches long and Bob has a piece that is 15 inches long. If they put their pieces together, how long would the piece of rope be?

b. ▲ length to the nearest centimeter or meter, e.g., a new pencil is about how many centimeters long?

c. length to the nearest whole unit of a nonstandard unit, e.g., how many paper clips long is a hot dog?

d. temperature to the nearest degree, e.g., what would the temperature outside be if it was a good day for swimming?

e. ▲ number of days in a week, e.g., if school started 37 weeks ago, how many days of school have passed?

2. estimates to check whether or not measurements or calculations for length, temperature, and time in real-world problems are reasonable (2.4.A1a) ($), e.g., after finding the range of temperature over a two-week period, determine whether or not the answer is reasonable.

3. adjusts original measurement or estimation for length, weight, temperature, and time in real-world problems based on additional information (a frame of reference) (2.4.A1a) ($), e.g., the class estimates that the class gerbil weighs as much as a box of 24 crayons. The gerbil is placed on one side of the pan balance and a box of 16 crayons is placed on the other side. The pan balance barely moves. Should the estimate of the gerbil’s weight be adjusted?

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Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps students develop basic properties of measurement and it gives students a tool to determine whether a given measurement is reasonable.



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Benchmark 3: Transformational Geometry – The student recognizes and performs one transformation on simple shapes or concrete objects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. knows and uses cardinal points (north, south, east, west) and

intermediate points (northeast, southeast, northwest, southwest) (2.4.K1a).

2. recognizes and performs one transformation (reflection/flip, rotation/turn, and translation/slide) on a two-dimensional figure (2.4.K1f).

The student…1. recognizes real-world transformations (reflection/flip, rotation/turn,

and translation/slide) (2.4.A1a), e.g., tiles in a ceiling, bricks in a sidewalk, or steps on a playground slide.

2. gives and uses directions to move from one location to another on a map and follows directions including the use of cardinal and intermediate points (2.4.A1a).




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Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number lineand the first quadrant of a coordinate plane in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. uses a number line (horizontal/vertical) to model the basic

multiplication facts through the 5s and the multiplication facts of the 10s (2.4.K1a).

2. identifies points on a coordinate plane (coordinate grid) using (2.4.K1a):a. two positive whole numbers,b. a letter and a positive whole number.

3. identifies points as ordered pairs in the first quadrant of a coordinate plane (coordinate grid) (2.4.K1a).

The student…1. solves real-world problems using coordinate planes (coordinate

grids) and map grids that have positive whole number and letter coordinates (2.4.A1a), e.g., identifying locations on a map or giving and following directions to move from one location to another.

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A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the coordinate plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the y-coordinate. Since the pair is always named in order (first x, then y), it is called an ordered pair.



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Standard 4: Data THIRD GRADE


Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions and to make predictions and decisions including the use of concrete objects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. recognizes any outcome of a simple event in an experiment or

simulation as impossible, possible, certain, likely, unlikely, or equally likely (2.4.K1g) ($).

2. ▲ ■ lists some of the possible outcomes of a simple event in an experiment or simulation including the use of concrete objects (2.4.K1g-h).

The student…1. makes predictions about a simple event in an experiment or

simulation; conducts the experiment or simulation including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions about the event (2.4.A1g-h).

2. compares what should happen (theoretical probability/expected results) with what did happen (experimental probability/empirical results) in an experiment or simulation with a simple event (2.4.A1g).

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Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be addressed through the use of concrete objects, coins, and geometric models (spinners, number cubes, or dartboards). Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments. Some examples of uses of probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win every game?



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Standard 4: Data THIRD GRADE


Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (whole numbers) and non-numerical data sets including the use of concrete objects in a variety of situations.

Third Grade Knowledge Base Indicators Third Grade Application IndicatorsThe student…1. organizes, displays, and reads numerical (quantitative) and non-

numerical (qualitative) data in a clear, organized, and accurate manner including a title, labels, categories, and whole number intervals using these data displays (2.4.K1h) ($):a. graphs using concrete objects;b. pictographs with a whole symbol or picture representing one,

two, five, ten, twenty-five, or one-hundred (no partial symbols or pictures);

c. frequency tables (tally marks); d. horizontal and vertical bar graphs;e. Venn diagrams or other pictorial displays, e.g., glyphs; f. line plots;g. charts and tables.

2. collects data using different techniques (observations, polls, surveys, or interviews) and explains the results (2.4.K1h) ($).

3. ▲ finds these statistical measures of a data set with less than ten data points using whole numbers from 0 through 1,000 (2.4.K1a) ($):a. minimum and maximum data values,b. range,c. mode (uni-modal only),d. median when data set has an odd number of data points.

The student…1. interprets and uses data to make reasonable inferences and

predictions, answer questions, and make decisions from these data displays (2.4.A1h) ($): a. graphs using concrete objects;b. pictographs with a whole symbol or picture representing one,

two, five, ten, twenty-five, or one-hundred (no partial symbols or pictures);

c. frequency tables (tally marks);d. horizontal and vertical bar graphs;e. Venn diagrams or other pictorial displays;f. line plots;g. charts and tables.

2. uses these statistical measures with a data set of less than ten data points using whole numbers from 0 through 1,000 to make reasonable inferences and predictions, answer questions, and make decisions (2.4.A1a) ($):a. minimum and maximum data values,b. range,c. mode,d. median when data set has an odd number of data points.

3. recognizes that the same data set can be displayed in various formats including the use of concrete objects (2.4.A1h) ($).

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Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships, are used to tell a story, and are an important part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data set. Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data display is accurate.

Graphs take many forms:– bar graphs and pictographs compare discrete data, – frequency tables show how many times a certain piece of data occurs, – circle graphs (pie charts) model parts of a whole, – line graphs show change over time, – Venn diagrams show relationships between sets of objects, and– line plots show frequency of data on a number line.

An important aspect of data is its center. The measures of central tendency (averages) of a data set are mean, median, and mode. Conceptual understanding of mean, median, and mode is developed through the use of concrete objects that represent the data values.



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Standard 1: Number and Computation FOURTH GRADE


mixed numbers), decimals, and money including the use of concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. knows, explains, and uses equivalent representations for ($):

a. whole numbers from 0 through 100,000 (2.4.K1a-b); b. fractions greater than or equal to zero (halves, fourths, thirds,

eighths, tenths, twelfths, sixteenths, hundredths) including mixed numbers (2.4.K1c);

c. decimals greater than or equal to zero through hundredths place and when used as monetary amounts (2.4.K1c-d) ($), e.g., 7¢ = $.07 = 7/100 of a dollar or a hundreds grid with 7 sections colored or .1 = 1/10 = .

2. compares and orders:a. whole numbers from 0 through 100,000 (2.4.K1a-b) ($);b. fractions greater than or equal to zero (halves, fourths, thirds,

eighths, tenths, twelfths, sixteenths, hundredths) including mixed numbers with a special emphasis on concrete objects (2.4.K1c);

c. decimals greater than or equal to zero through hundredths place and when used as monetary amounts (2.4.K1c-d) ($).

The student…1. solves real-world problems using equivalent representations and

concrete objects to ($): a. compare and order whole numbers from 0 through 100,000

(2.4.A1a-b); e.g., using base ten blocks, represent the attendance at the circus over a three day stay; then represent the numbers using digits and compare and order in different ways;

b. add and subtract whole numbers from 0 through 10,000 and decimals when used as monetary amounts (2.4.A1a-d), e.g., use real money to show at least 2 ways to represent $142.78, then subtract the cost of a pair of tennis shoes;

c. multiply a one-digit whole number by a two-digit whole number (2.4.A1a-b), e.g., use base ten blocks to represent 24 x 5 to find the total number of hours in 5 days, or use repeated addition 24 + 24 + 24 + 24 + 25 to solve, or use the algorithm.

2. determines whether or not solutions to real-world problems that involve the following are reasonable ($):a. whole numbers from 0 through 10,000 (2.4.A1a-b), e.g., a

student says that there are 1,000 students in grade 4 at her school, is this reasonable?

b. fractions greater than or equal to zero (halves, fourths, thirds, eighths, tenths, sixteenths) (2.4.A1c), e.g., you ate ½ of a sandwich and a friend ate ¾ of the same sandwich; is this reasonable?

c. decimals greater than or equal to zero when used as monetary amounts (2.4.A1c-d), e.g., a pack of chewing gum costs what amount - $62 $.75 9¢ 75.00 750¢? Is this reasonable?

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▲– Assessed Indicator■ – Assessed Indicator on the Optional Constructed Response Assessment N – Noncalculator ($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.

Benchmark 1: Number Sense – The student demonstrates number sense for whole numbers, fractions (including





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Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of wholenumbers with a special emphasis on place value; recognizes, uses, and explains the concepts of

properties as they relate to whole numbers; and extends these properties to fractions (including mixed numbers), decimals, and money.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. ▲ identifies, models, reads, and writes numbers using numerals,

words, and expanded notation from hundredths place through one-hundred thousands place (2.4.K1a-b) ($), e.g., four hundred sixty-two thousand, two hundred eighty-four and fifty hundredths = 462,284.50 or 462,284.50 = (4 x 100,000) + (6 x 10,000) + (2 x 1,000) + (2 x 100) + (8 x 10) + (4 x 1) + (5 x .1) + (0 x .01) = 400,000 + 60,000 + 2,000 + 200 + 80 + 4 + .5 + .00.

2. classifies various subsets of numbers as whole numbers, fractions (including mixed numbers), or decimals (2.4.K1b-c, 2.4.K1i).

3. identifies the place value of various digits from hundredths place through one hundred thousands place (2.4.K1b) ($).

4. identifies any whole number as even or odd (2.4.K1a).5. uses the concepts of these properties with the whole number system

and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($): a. ▲ commutative properties of addition and multiplication, e.g.,

12 + 18 = 18 + 12 and 8 x 9 = 9 x 8; b. ▲ zero property of addition (additive identity) and property of one

for multiplication (multiplicative identity), e.g., 24 + 0 = 24 and 75 x 1 = 75;c. ▲ associative properties of addition and multiplication, e.g.,

4 + (2 + 3) = (4 + 2) + 3 and 2 x (3 x 4) = (2 x 3) x 4;d. ▲ symmetric property of equality applied to addition and

multiplication, e.g., 100 = 20 + 80 is the same as 20 + 80 = 100 and 21 = 7 x 3 is the same as 3 x 7 = 21;

e. zero property of multiplication, e.g., 9 x 0 = 0 or 0 x 112 = 0;


10,000 using place value models; money; and the concepts of these properties to explain reasoning (2.4.A1a-b,d) ($):

a. commutative properties of addition and multiplication, e.g., a student has a $5, a $10, and a $20 bill; a student totals the amount to see how much can be spent shopping for school supplies. The student says: Because you can add in any order, I can rearrange the money and count $20, $10, and $5 for $20 + $10 + $5. Another student has 4 $5 bills. The student is asked the amount. The student says: I don’t know 4 x 5 but I know 5 x4 is $20, since multiplication can be done in any order.

b. zero property of addition, e.g., a student has 6 marbles in one pocket and none in the other pocket. How many marbles altogether?

c. property of one for multiplication, e.g., there are 24 students in our class, each student should have one math book; so I compute 24 x 1 = 24. Multiplying times 1 does not change the product because it is one group of 24.

d. associative properties of addition and multiplication, e.g., a student has two dimes and a quarter. Using coins or money models, there are at least 2 ways to group the coins to find the total. One way is 10¢ (dime) + 10¢ (dime) = 20¢, then add the quarter, so 20¢ + 25¢ (quarter) = 45¢. Another way 10¢ (dime) + 25¢ (quarter) = 35¢, then add the other dime to 35¢ so 35¢ + 10¢ = 45¢ This models that (D + D) + Q =

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D + (D + Q).

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f. distributive property, e.g., 6(7 + 3) = (6 • 7) + (6 • 3). e. zero property of multiplication, e.g., in science, you are observing a snail. The snail does not move over a 4-hour period. To figure its total movement, you say 4 x 0 = 0.

2. performs various computational procedures with whole numbers from 0 through 10,000 using the concepts of the following properties; extends the properties to fractions (halves, fourths, thirds, eighths, tenths, sixteenths) including mixed numbers, and decimals through hundredths place; and explains how the properties were used (2.4.A1a-c): a. commutative property of addition and multiplication, e.g., 5 + 6

= 6 + 5, the student says: I know that 5 + 6 = 11 and adding in any order still gets the answer, so 6 + 5 is the same as 5 + 6. 4 x 6 = 6 x 4, the student says: I know that 4 x 6 = 24 and multiplying in any order still gets the answer, so 4 x 6 is the same as 6 x 4.

b. zero property of multiplication without computing, e.g., 158 x 0 = 0; the student says: I know the answer (product) is zero because no matter how many factors you have, when you multiply with a 0, the product is zero.

c. associative property of addition, e.g., 9 + 8 could be solved as 1 + (8 + 8) or (1 + 8) + 8, the student says: I don’t know 9 + 8, but I know my doubles of 8 + 8, so I made the 9 into 1 + 8 and then added 1 more to make 17.

3. states the reason for using whole numbers, fractions, mixed numbers, or decimals when solving a given real-world problem (2.4.A1a-d).

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Teacher Notes: From the Mathematics Dictionary and Handbook (Nichols Schwartz Publishing, 1999), property as a mathematical term means a characteristic (an attribute) of a number, geometric shape, mathematical operation, equation, or inequality. To give an example: Property of a number: 8 is divisible by 2. Property of a geometric shape: Each of the four sides of a square is of the same length. Property of an operation: Addition is commutative. For all numbers x and y, x + y = y + x. Property of an equation: For all numbers a, b, and c, if a = b, then a + c = b + c. Property of an inequality: For all numbers a, b, and c, if a > b, then a – c > b – c.



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Benchmark 3: Estimation – The student uses computational estimation with whole numbers, fractions (including mixed numbers) and money in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. estimates whole number quantities from 0 through 10,000;

fractions (halves, fourths, thirds); and monetary amounts through $1,000 using various computational methods including mental math, paper and pencil, concrete materials, and appropriate technology (2.4.K1a-d) ($).

2. uses various estimation strategies and explains how they are used when estimating whole numbers quantities from 0 through 10,000; fractions [(halves, fourths, thirds) including mixed numbers)]; and monetary amounts through $1,000 (2.4.K1a-d) ($).

3. recognizes and explains the difference between an exact and an approximate answer (2.4.K1a), e.g., when asked how many desks are in the room, the student gives an estimate of about 30 and then counts the desks and indicates an exact answer is 28 desks.

4. selects from an appropriate range of estimation strategies and determines if the estimate is an overestimate or underestimate, (2.4.K1a).

The student…1. adjusts original whole number estimates of a real-world problem using

numbers from 0 through 10,000 based on additional information (a frame of reference) (2.4.A1a) ($), e.g., if given a small jar and told the number of pieces of candy it has in it, the student would adjust his/her original estimate of the number of pieces of candy in a larger jar.

2. estimates to check whether or not the result of a real-world problem using whole numbers from 0 through 10,000, fractions (including mixed numbers), and monetary amounts is reasonable and makes predictions based on the information (2.4.A1a-d) ($), e.g., at the movies, you bought popcorn for $2.35, a soda for $2.50, and paid $4.50 for the ticket. Is it reasonable to say you spent $10? How much will you need to save to go to the movies once a week for the next month?

3. selects a reasonable magnitude from three given quantities based on a familiar problem situation and explains the reasonableness of selection (2.4.A1a), e.g., about how many new pencils will fit in your pencil box? Is it about 25, about 50, or about 100? The answer will depend on the size of your pencil box.

4. determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.A1a) ($).

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Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000).




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Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers, fractions, and money including the use of concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. computes with efficiency and accuracy using various computational

methods including mental math, paper and pencil, concrete materials, and appropriate technology (2.4.K1a) ($).

2. N states and uses with efficiency and accuracy multiplication facts from 1 x 1 through 12 x 12 and corresponding division facts (2.4.K1a) ($).

3. N performs and explains these computational procedures ($): a. adds and subtracts whole numbers from 0 through 100,000 and

when used as monetary amounts (2.4.K1a-b,d);b. multiplies through a three-digit whole number by a two-

digit whole number (2.4.K1a-b);c. multiplies whole dollar monetary amounts (through three-digits)

by a one- or two-digit whole number (2.4.K1d), e.g., $45 x 16;d. multiplies monetary amounts less than $100.00 by whole

numbers less than ten (2.4.K1d), e.g., $14.12 x 7;e. divides through a two-digit whole number by a one-digit whole

number with a one-digit whole number quotient with or without a remainder (2.4.K1a-b), e.g., 47 ÷ 5 = 9 r 2;

f. adds and subtracts fractions greater than or equal to zero with like denominators (2.4.K1c);

g. figures correct change through $20.00 (2.4.K1d).4. identifies multiplication and division fact families (2.4.K1a). 5. reads and writes horizontally, vertically, and with different

operational symbols the same addition, subtraction, multiplication, or division expression, e.g., 6 • 4 is the same as 6 x 4 is the same as 4 and 6(4) or 10 divided by 2 is the same as 10 ÷ 2 or 10. x 6

2

The student…1. ▲N solves one- and two-step real-world problems with one or two

operations using these computational procedures ($):a. adds and subtracts whole numbers from 0 through 10,000 and

when used as monetary amounts (2.4.A1a-b,d), e.g., Lee buys a bicycle for $139, a helmet for $29, and a reflector for $6. He paid for it with a $200 check from his grandparents. How much will he have left from the $200 check?

b. multiplies through a two-digit whole number by a two-digit whole number (2.4.A1a-b), e.g., at school, there are 22 students in each classroom. If there are 24 classes, how many students are in the classrooms?

c. multiplies whole dollar monetary amounts (up through three-digit) by a one- or two-digit whole number (2.4.A1a-b,d), e.g., 112 third and fourth graders are planning a field trip. The cost per student is $9.00. How much will the trip cost?

d. multiplies monetary amounts less than $100 by whole numbers less than ten (2.4.A1a-d), e.g., at the book fair, a student buys 8 books on animals for $2.69 each. How much did the student pay for the books?

e. ■ figures correct change through $20.00 (2.4.A1a-d), e.g., buying a 65¢ drink, paying for it with a $1 bill, and then figuring the amount of change.

2. generates a family of multiplication and division facts given one equation/fact (2.4.A1b), e.g., given 8 x 9 = 72, the other facts are 9 x 8 = 72, 72 ÷ 8 = 9, and 72 ÷ 9 = 8.

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6. ▲N shows the relationship between these operations with the basic fact families (addition facts with sums from 0 through 20 and corresponding subtraction facts, multiplication facts from 1 x 1 through 12 x 12 and corresponding division facts) including the use of mathematical models (2.4.K1a) ($): a. addition and subtraction, b. addition and multiplication,c. multiplication and division,d. subtraction and division.

7. finds factors and multiples of whole numbers from 1 through 100 (2.4.K1a).

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Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations and by counting on for addition and counting up for subtraction and unknown-addend situations. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling, repeated doubling, halving, making tens, dividing with tens, thinking money, and using compatible “nice” numbers.




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Standard 2: Algebra FOURTH GRADE


patterns using concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. uses concrete objects, drawings, and other representations to work

with types of patterns(2.4.K1a): a. repeating patterns, e.g., an AB pattern is like 1-2, 1-2, …; an

ABC pattern is like dog-horse-pig, dog-horse-pig, …; an AAB pattern is like ↑↑→, ↑↑→, …;

b. growing patterns e.g., 2, 5, 11, 20, …2. uses these attributes to generate patterns:

a. counting numbers related to number theory (2.4.K1a), e.g., multiples and factors through 12 or multiplying by 10, 100, or 1,000;

b. whole numbers that increase or decrease (2.4.K1a) ($) , e.g., 20, 15, 10, …;

c. geometric shapes including one or two attributes changes (2.4.K1f), e.g.,

… when the next shape has one more side; or when both color and shape change at the same time such as

…,d. measurements (2.4.K1a), e.g., 3 ft., 6 ft., 9 ft., …;e. money and time (2.4.K1a,d) ($), e.g., $.25, $.50, $.75, … or 1:05

p.m., 1:10 p.m., 1:15 p.m., …; f. things related to daily life (2.4.K1a), e.g., water cycle, food cycle,

or life cycle;g. things related to size, shape, color, texture, or movement

(2.4.K1a), e.g., rough, smooth, rough, smooth, rough, smooth, or

The student…1. generalizes these patterns using a written description:

a. counting numbers related to number theory (2.4.A1a),b. whole number patterns (2.4.A1a) ($), c. patterns using geometric shapes (2.4.A1f),d. measurement patterns (2.4.A1a),e. money and time patterns (2.4.A1a,d) ($),f. patterns using size, shape, color, texture, or movement

(2.4.A1a).2. recognizes multiple representations of the same pattern (2.4.A1a),

e.g., skip counting by 5s to 60; whole number multiples of 5 through 60; the multiplication table of 5 given the numerical pattern of 5, 10, 15, …, 60; relating the concept of five minute time intervals to each of the numerals on a clock giving the pattern of 5, 10, 15, …, 60; one nickel, two nickels, three nickels, …; the number of fingers on twelve hands; recognizing that all of these representations are the same general pattern.

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Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships in

clapping hands (kinesthetic patterns).3. identifies, states and continues a pattern presented in visual various

formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written (2.4.K1a) ($).

4. generates:a. a pattern (repeating, growing) (2.4.K1a);

a pattern using a function table (input/output machines, T-tables) (2.4.K1e).


This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, whole numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain. Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest common factor and least common multiple.



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solve equations including the use of concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. explains and uses variables and symbols to represent unknown

whole number quantities from 0 through 1,000 (2.4.K1a). 2. ▲ solves one-step equations using whole numbers with one variable

and a whole number solution that: a. find the unknown in a multiplication or division equation based

on the multiplication facts from 1 x 1 through 12 x 12 and corresponding division facts (2.4.K1a), e.g., 60 = 10 x n;

b. find the unknown in a money equation using multiplication and division based upon the facts and addition and subtraction with values through $10 (2.4.K1d) ($), e.g., 8 quarters + 10 dimes = y dollars;

c. find the unknown in a time equation involving whole minutes, hours, days, and weeks with values through 200 (2.4.K1a), e.g., 180 minutes = y hours.

3. compares two whole numbers from 0 through 10,000 using the equality and inequality symbols (=, ≠, <, >) and their corresponding meanings (is equal to, is not equal to, is less than, is greater than) (2.4.K1b) ($).

4. reads and writes whole number equations and inequalities using mathematical vocabulary and notation, e.g., 15 = 3 x 5 is the same as fifteen equals three times five or 4,564 > 1,000 is the same as four thousand, five hundred sixty-four is greater than one thousand.

The student…1. represents real-world problems using variables and symbols with

unknown whole number quantities from 0 through 1,000 (2.4.A1a) ($), e.g., How many weeks in twenty-eight days? can be represented by n x 7 = 28 or n = 28 ÷ 7.

2. generates one-step equations to solve real-world problems with one unknown (represented by a variable or symbol) and a whole number solution that (2.4.A1a) ($): a. add or subtract whole numbers from 0 through 1,000; e.g.,

Homer, Kansas has 832 nonfiction books in its library. Homer, Idaho has 652 nonfiction books in its library. How many fewer books nonfiction books are in Homer, Idaho’s library? 832 - 652 = B;

b. multiply or divide using the basic facts, e.g., Tom has a sticker book and each page holds 5 stickers. If the same number of stickers is placed on each page, the book will hold 30 stickers. How many pages are in his book? This is represented by 5 x S = 30 or 30 ÷ 5 = S.

3. generates (2.4.A1a) ($):a. real-world problems with one operation to match a given

addition, subtraction, multiplication, or division equation using whole numbers through 99, e.g., given 12 x 3 = Y, the student writes: I was sick for 3 days, when I got back I had 3 pages of homework. There are 12 problems on each page. How many total problems must I work?

b. number comparison statements using equality and inequality symbols (=, <, >) with whole numbers, measurement, and money, e.g., 1 ft < 15 in or 10 quarters > $2.

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Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, and whole numbers to

Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters and geometric shapes, to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ or 3 + s = 7. Various symbols or letters should be used interchangeably in equations.



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concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. states mathematical relationships between whole numbers from 0

through 1,000 using various methods including mental math, paper and pencil, concrete materials, and appropriate technology (2.4.K1a) ($).

2. ▲ finds the values, determines the rule, and states the rule using symbolic notation with one operation of whole numbers from 0 through 200 using a horizontal or vertical function table (input/output machine, T-table) (2.4.K1e), e.g., using the function table, find the rule, the rule is N • 4.

N ?1 45 202 83 ?4 ?? 24

3. generalizes numerical patterns using whole numbers from 0 through 200 with one operation by stating the rule using words, e.g., if the pattern is 46, 68,90, 112, 134, …; in words, the rule is add 22 to the number before.

4. uses a function table (input/output machine, T-table) to identify, plot, and label the ordered pairs in the first quadrant of a coordinate plane (2.4.K1a,e).

The student…1. ▲ represents and describes mathematical relationships between

whole numbers from 0 through 1,000 using concrete objects, pictures, written descriptions, symbols, equations, tables, and graphs (2.4.A1a) ($).

2. finds the rule, states the rule, and extends numerical patterns using real-world applications using whole numbers from 0 through 200 (2.4.A1a,e), e.g., the teacher must order supplies for field day. For every 12 students, one red rubber ball is needed. If 6 balls are ordered, how many students will be able to play? A solution using a function table might be:

The rule is divide the number of students by 12 or for each group of12 students, another ball is added. Other solutions might be using a pattern to count by 12 six times – 12, 24, 36, 48, 60, 72 or to skip count by 12 for each ball ordered.

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Benchmark 3: Functions – The student recognizes and describes whole number relationships including the use of

Number of Students

Number of Balls

12 124 236 348 460 572 6N N ÷ 12



In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the number of the term – the 5 th term is 52, the 8th term is 82, …




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to represent and explain mathematical relationships in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. knows, explains, and uses mathematical models to represent


lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures, mathematical relationships, and equations (1.1.K1a, 1.1.K2a, 1.2.K1, 1.2.K4-5, 1.3.K1-4, 1.4.K1-2, 1.4.K3a-b, 1.4.K3e, 1.4.K4, 1.4.K6-7, 2.1.K1, 2.1.K.1a-b, 2.1.K2d-g, 2.1.K3, 2.1.K4a, 2.2.K1, 2.2.K2a, 2.2.K3-4, 2.3.K1, 2.3.K4, 3.2.K1-4, 3.3.K1-2, 3.4.K1-4, 4.2.K3) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures (1.1.K1a, 1.1.K2a, 1.2.K1-3, 1.3.K1-2, 1.4.K3a-b, 1.4.K3e, 2.2.K4) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K1b-c, 1.1.K2b-c, 1.2.K2, 1.3.K1-2, 1.4.K1f) ($);

d. money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K1c, 1.2.K1c, 1.3.K1-2, 1.4.K3a, 1.4.K3a, 1.4.K3c-d, 1.4.K3g, 2.1.K2e, 2.2.K2b) ($);

e. function tables (input/output machines, T-tables) to model numerical and algebraic relationships (2.1.K4b, 2.3.K2, 2.3.K4, 3.4.K4) ($);


represent the same problem situation. Mathematical models include:a. process models (concrete objects, pictures, diagrams, number

lines, coordinate planes/grids, hundred charts, measurement tools, multiplication arrays, or division sets) to model computational procedures, mathematical relationships, and problem situations (1.1.A1, 1.1.A2a, 1.2.A1-3, 1.3.A1-4, 1.4.A1, 2.1.A1a-b, 2.1.A1d-f, 2.1.A2, 2.2.A1-3, 2.3.A1-2, 3.2.A1a-g, 3.2.A2-3, 3.3.A1-2, 3.4.A1-2, 4.2.A2) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unfix cubes) to model problem situations (1.1A1, 1.1.A2a, 1.2.A1-3, 1.3.A2, 1.4.A1) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1b, 1.1.A2b-c, 1.2.A2-3, 1.3.A2, 1.4.A1d-e) ($);

d. money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1b, 1.1.A2c, 1.2.A1, 1.2.A3, 1.3.A1, 1.4.A1a, 1.4.A1c-e, 2.1.A1e) ($);

e. function tables (input/output machines, T-tables) to model numerical and algebraic relationships (2.3.A2) ($);

f. two-dimensional geometric models (geoboards, dot paper, pattern blocks, or tangrams) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model properties of geometric shapes (2.1.A1c, 3.1.A1-2, 3.2.A1h, 3.3.A3);

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Benchmark 4: Models – The student develops and uses mathematical models including the use of concrete objects

f. two-dimensional geometric models (geoboards, dot paper, pattern blocks, or tangrams) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model properties of geometric shapes (2.1.K2c, 2.1.K1e, 3.1.K1-6, 3.2.K5, 3.3.K3);

g. two-dimensional geometric models (spinners), three-dimensional models (number cubes), and process models (concrete objects) to model probability (4.1.K1-3) ($);

h. graphs using concrete objects, pictographs, frequency tables, horizontal and vertical bar graphs, line graphs, circle graphs, Venn diagrams, line plots, charts, and tables to organize and display data (4.1.K2, 4.2.K1-2) ($);

i. Venn diagrams to sort data and show relationships (1.2.K2).

2. creates a mathematical model to show the relationship between two or more things, e.g., using pattern blocks, a whole (1) can be represented as

a (1/1) or

two (2/2) or

three (3/3) or

six (6 (6/6).

g. two-dimensional geometric models (spinners), three-dimensional geometric models (number cubes), and process models (concrete objects) to model probability (4.1.A1-3) ($);

h. graphs using concrete objects, pictographs, frequency tables, horizontal and vertical bar graphs, line graphs, Venn diagrams, line plots, charts, and tables to organize, display, explain, and interpret data (4.1.A2, 4.2.A1, 4.2.A3-4) ($);

i. Venn diagrams to sort data and show relationships.2. selects a mathematical model and explains why some

mathematical models are more useful than other mathematical models in certain situations.

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The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, students begin to develop an understanding of the advantages and disadvantages of the various representations/models.



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Standard 3: Geometry FOURTH GRADE


Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and investigatestheir properties including the use of concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. recognizes and investigates properties of plane figures (circles,

squares, rectangles, triangles, ellipses, rhombi, octagons, hexagons, pentagons) using concrete objects, drawings, and appropriate technology (2.4.K1f).

2. recognizes, draws, and describes plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, octagons, hexagons, pentagons) (2.4.K1f).

3. describes the solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) using the terms faces, edges, and vertices (corners) (2.4.K1f).

4. recognizes and describes the square, triangle, rhombus, hexagon, parallelogram, and trapezoid from a pattern block set (2.4.K1f).

5. recognizes (2.4.k1f):a. squares, rectangles, rhombi, parallelograms, trapezoids as

special quadrilaterals; b. similar and congruent figures; c. points, lines (intersecting, parallel, perpendicular), line segments,

and rays.6. determines if geometric shapes and real-world objects contain line(s)

of symmetry and draws the line(s) of symmetry if the line(s) exist(s) (2.4.K1f).

The student…1. solves real-world problems by applying the properties of (2.4.A1f):

a. plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, parallelograms, hexagons) and lines of symmetry, e.g., print your name or the school’s name in all capital letters. Identify the lines of symmetry in each letter.

b. solids (cubes, rectangular prisms, cylinders, cones, spheres), e.g., you want to design something to store school supplies. Which of the solids could you use for storage? Why did you select that solid?

2. ▲ ■ identifies the plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, octagons, hexagons, pentagons, trapezoids) used to form a composite figure (2.4.A1f).

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Pattern blocks are a collection of six geometric shapes in six colors. Each set contains 250 pieces – 50 green triangles, 25 orange squares, 50 blue rhombi, 50 tan rhombi, 50 red trapezoids, and 25 yellow hexagons. The blue rhombus and the tan rhombus also are parallelograms, and the orange square is a parallelogram. The blocks are designed so that their sides are all the same length with the exception that the trapezoid has one side twice as long. This feature allows the blocks to be nested together and encourages the exploration of relationships among the shapes. Activities with pattern blocks help students explore patterns, functions, fractions, congruence, similarity, symmetry, perimeter, area, and graphing. A pattern block template can be found in the Appendix.



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units of measure including the use of concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. uses whole number approximations (estimations) for length, width,

weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure (2.4.K1a) ($).

2. ▲ selects, explains the selection of, and uses measurement tools, units of measure, and degree of accuracy appropriate for a given situation to measure (2.4.K1a) ($): a. length, width, and height to the nearest fourth of an inch or to

the nearest centimeter; b. volume to the nearest cup, pint, quart, or gallon; to the nearest

liter; or to the nearest whole unit of a nonstandard unit; c. weight to the nearest ounce or pound or to the nearest whole

unit of a nonstandard unit of measure;d. temperature to the nearest degree; e. time including elapsed time.

3. states:a. the number of weeks in a year;b. the number of ounces in a pound;c. the number of milliliters in a liter, grams in a kilogram, and

meters in a kilometer; d. the number of items in a dozen.

4. converts (2.4.K1a):a. within the customary system: inches and feet, feet and yards,

inches and yards, cups and pints, pints and quarts, quarts and gallons;

b. within the metric system: centimeters and meters. 5. finds(2.4.K1f):

a. the perimeter of two-dimensional figures given the measures of all the sides.

b. the area of squares and rectangles using concrete objects.

The student…1. solves real-world problems by applying appropriate measurements:

a. length to the nearest fourth of an inch (2.4.A1a), e.g., how much longer is the math textbook than the science textbook?

b. length to the nearest centimeter (2.4.A1a), e.g., a new pencil is about how many centimeters long?

c. temperature to the nearest degree (2.4.A1a), e.g., what would the temperature outside be if it was a good day for sledding?

d. weight to the nearest whole unit (pounds, grams, nonstandard unit) (2.4.A1a), e.g., Brendan went to the store and bought 2 packages of hamburger for a meatloaf. One of the hamburger packages weighed 1 lb. and 8 ozs. The other packages weighed 1 lb. and 7 ozs. What is the combined weight (to the nearest pound) of the two packages of hamburger?

e. time including elapsed time (2.4.A1a), e.g., Joy went to the mall at 10:00 a.m. She shopped until 4:15 p.m. How long did she shop at the mall?

f. months in a year (2.4.A1a), e.g., if it takes 208 weeks to get a college degree, and Susan has completed one year, how many more weeks does she have to complete to get her degree?

g. minutes in an hour (2.4.A1a), e.g., Bob has spent 240 minutes working on a project for Science. How many hours has he worked on the project?

h. perimeter of squares, rectangles, and triangles (2.4.A1f), e.g., a triangle has 3 equal sides of 32 inches. What is the perimeter of the triangle?

2. ▲ estimates to check whether or not measurements and calculations for length, width, weight, volume, temperature, time, and perimeter in real-world problems are reasonable (2.4.A1a) ($), e.g., which is the most reasonable weight for your scissors – 2 ounces, 2 pounds, 20 ounces, or 20 pounds? A teacher measures one side of

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Benchmark 2: Measurement and Estimation – The student estimates and measures using standard and nonstandard

a square desktop at 2 feet. Which of the following perimeters is reasonable for the desktop – 2 feet, 4 square feet, 6 square feet, or 8 feet?

3. adjusts original measurement or estimation for length, width, weight, volume, temperature, time, and perimeter in real-world problems based on additional information (a frame of reference) (2.4.A1a) ($), e.g., your class has a large jar and a small jar. You estimate it will take 5 small jars of liquid to fill the large jar. After you pour the contents of 2 small jars in, the large jar is more that half full. Should you need to adjust your estimate?




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shapes or concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. describes a transformation using cardinal points or positional

directions (2.4.K1a), e.g., go north three blocks and then west four blocks or move the triangle three units to the right and two units up.

2. ▲ ■ recognizes, performs, and describes one transformation (reflection/flip, rotation/turn, translation/slide) on a two-dimensional figure or concrete object (2.4.K1a).

3. recognizes three-dimensional figures (rectangular prisms, cylinders) and concrete objects from various perspectives (top, bottom, sides, corners) (2.4.K1f).

The student…1. recognizes real-world transformations (reflection/flip, rotation/turn,

translation/slide) (2.4.A1a).2. gives and uses cardinal points or positional directions to move from

one location to another on a map or grid (2.4.A1a).3. describes the properties of geometric shapes or concrete objects

that stay the same and the properties that change when a transformation is performed (2.4.A1f).




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Benchmark 3: Transformational Geometry – The student recognizes and performs one transformation on simple



and the first quadrant of a coordinate plane in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. uses a number line (horizontal/vertical) to model whole number

multiplication facts from 1 x 1 through 12 x 12 and corresponding division facts (2.4.K1a).

2. uses points in the first quadrant of a coordinate plane (coordinate grid) to identify locations (2.4.K1a).

3. ▲ ■ identifies and plots points as whole number ordered pairs in the first quadrant of a coordinate plane (coordinate grid) (2.4.K1a).

4. organizes whole number data using a T-table and plots the ordered pairs in the first quadrant of a coordinate plane (coordinate grid) (2.4.K1a,e).

The student…1. solves real-world problems that involve distance and location using

coordinate planes (coordinate grids) and map grids with positive whole number and letter coordinates (2.4.A1a), e.g., identifying locations and giving and following directions to move from one location to another.

2. solves real-world problems by plotting whole number ordered pairs in the first quadrant of a coordinate plane (coordinate grid) (2.4.A1a) ($), e.g., given that each movie ticket cost $5, the student graphs the number of tickets bought and the total cost of tickets to attend a movie.

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Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line





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Standard 4: Data FOURTH GRADE



Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. recognizes that the probability of an impossible event is zero and

that the probability of a certain event is one (2.4.K1g) ($). 2. lists all possible outcomes of a simple event in an experiment or

simulation including the use of concrete objects (2.4.K1g-h). 3. recognizes and states the probability of a simple event in an

experiment or simulation (2.4.K1g), e.g., when a coin is flipped, the probability of landing heads up is ½ and the probability of landing tails up is ½. This can be read as one out of two or one half.

The student…1. makes predictions about a simple event in an experiment or

simulation; conducts an experiment or simulation including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions about the event(2.4.A1g-h).

2. uses the results from a completed experiment or simulation of a simple event to make predictions in a variety of real-world problems (2.4.A1g-h), e.g., the manufacturer of Crunchy Flakes puts a prize in 20 out of every 100 boxes. What is the probability that a shopper will find a prize in a box of Crunchy Flakes, if they purchase 10 boxes?

3. compares what should happen (theoretical probability/expected results) with what did happen (empirical probability/experimental results) in an experiment or simulation with a simple event(2.4.A1g).

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The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories) Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.

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Standard 4: Data FOURTH GRADE


Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (whole numbers) and non-numerical data sets including the use of concrete objects in a variety of situations.

Fourth Grade Knowledge Base Indicators Fourth Grade Application IndicatorsThe student…1. ▲ ■ organizes, displays, and reads numerical (quantitative) and

non-numerical (qualitative) data in a clear, organized, and accurate manner including a title, labels, categories, and whole number intervals using these data displays (2.4.K1h) ($):a. graphs using concrete objects, (for testing, does not have to use

concrete objects in items);b. pictographs with a symbol or picture representing one, two, five,

ten, twenty-five, or one-hundred including partial symbols when the symbol represents an even amount;

c. frequency tables (tally marks); d. horizontal and vertical bar graphs; e. Venn diagrams or other pictorial displays, e.g., glyphs; f. line plots; g. charts and tables; h. line graphs; i. circle graphs.

2. collects data using different techniques (observations, polls, surveys, interviews, or random sampling) and explains the results (2.4.K1h) ($).

3. identifies, explains, and calculates or finds these statistical measures of a data set with less than ten whole number data points using whole numbers from 0 through 1,000 (2.4.K1a) ($):a. minimum and maximum values,b. range,c. mode,d. median when data set has an odd number of data points, e. mean when data set has a whole number mean.

The student…1. interprets and uses data to make reasonable inferences and

predictions, answer questions, and make decisions from these data displays (2.4.A1h) ($): a. graphs using concrete objects; b. pictographs with a symbol or picture representing one, two,

five, ten, twenty-five, or one-hundred including partial symbols when the symbol represents an even amount;

c. frequency tables (tally marks);d. horizontal and vertical bar graphs;e. Venn diagrams or other pictorial displays;f. line plots; g. charts and tables;h. line graphs.

2. ▲ uses these statistical measures of a data set using whole numbers from 0 through 1,000 with less than ten whole number data points to make reasonable inferences and predictions, answer questions, and make decisions (2.4.A1a) ($): a. minimum and maximum values,b. range, c. mode, d. median when the data set has an odd number of data points, e. mean when the data set has a whole number mean.

3. recognizes that the same data set can be displayed in various formats including the use of concrete objects (2.4.A1h) ($).

4. recognizes and explains the effects of scale and interval changes on graphs of whole number data sets (2.4.A1h).

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Graphs take many forms: – bar graphs and pictographs compare discrete data, – frequency tables show how many times a certain piece of data occurs, – circle graphs (pie charts) model parts of a whole, – line graphs show change over time, – Venn diagrams show relationships between sets of objects, and– line plots show frequency of data on a number line.




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Standard 1: Number and Computation FIFTH GRADE

Number and Computation – The student uses numerical and computational concepts and procedures in a variety of situations.

Benchmark 1: Number Sense – The student demonstrates number sense for integers, fractions, decimals, and money in a variety of situations.

The student…1. ▲N knows, explains, and uses equivalent representations for ($):

a. whole numbers from 0 through 1,000,000 (2.4.K1a-b); b. fractions greater than or equal to zero (including mixed

numbers) (2.4.K1c); c. decimals greater than or equal to zero through hundredths

place and when used as monetary amounts (2.4.K1c).2. compares and orders (2.4.K1a-c) ($) :

a. integers,b. fractions greater than or equal to zero (including mixed

numbers),c. decimals greater than or equal to zero through hundredths

place.3. explains the numerical relationships (relative magnitude) between

whole numbers, fractions greater than or equal to zero (including mixed numbers), and decimals greater than or equal to zero through hundredths place (2.4.K1a-c).

4. knows equivalent percents and decimals for one whole, one-half, one-fourth, three-fourths, and one tenth through nine tenths (2.4.K1c), e.g., 1 = 100% = 1.0, 3/4 = 75% = .75, 3/10 = 30% = .3.

5. identifies integers and gives real-world problems where integers are used (2.4.K1a), e.g., making a T-table of the temperature each hour over a twelve hour period in which the temperature at the beginning is 10 degrees and then decreases 2 degrees per hour.

The student…1. solves real-world problems using equivalent representations and

concrete objects to ($): a. compare and order (2.4.A1a-d) –

i. whole numbers from 0 through 1,000,000; e.g., using base ten blocks, represent the attendance at the circus over a three day stay; then represent the numbers using digits and compare and order in different ways;

ii. fractions greater than or equal to zero (including mixed numbers), e.g., Frank ate 2 ½ pizzas, Tara ate 9/4 of the pizza. Frank says he ate more. Is he correct? Use a model to explain. With drawings and shadings, student shows amount of pizza eaten by Frank and the amount eaten by Tara.

iii. decimals greater than or equal to zero to hundredths place, e.g., uses decimal squares, money (dimes as tenths, pennies as hundredths), the correct amount of hundred chart filled in, or a number line to show that .42 is less than .59.

iv. integers, e.g., plot winter temperature for a very cold region for a week (use Internet data); represent on a thermometer, number line, and with integers;

b. add and subtract whole numbers from 0 through 100,000 and decimals when used as monetary amounts (2.4.A1a,c), e.g., use real money to show at least 2 ways to represent $846.00, then subtract the cost of a new computer setup;

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▲– Assessed Indicator on the Objective Assessment ■ – Assessed Indicator on the Optional Constructed Response AssessmentN – Noncalculator($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.

Fifth Grade Knowledge Base Indicators Fifth Grade Application Indicators

c. multiply through a two-digit whole number by a two-digit whole number (2.4.A1a-b), e.g., George charges $23 for mowing a lawn. How much will he make after he mows 3 lawns? Represent the $23 with money models - 2 $10 bills and 3 $1 bills and repeat that 3 times or represent the $23 using base ten blocks or 23 x 3 or 23 + 23 + 23.;

d. divide through a four-digit whole number by a two-digit whole number (2.4.A1a-b), e.g., the Boy Scout troop collected cans and held bake sales for a year and earned $492.60. The money will be divided evenly among the 12 troop members to buy new uniforms. Represent each boy’s share of the money at least 2 ways - traditional division; use 4 hundreds, 9 tens, 2 ones, and 6 dimes to act out the situation; or use base ten blocks to act it out.

2. determines whether or not solutions to real-world problems that involve the following are reasonable ($): a. whole numbers from 0 through 100,000 (2.4.A1a-b), e.g., the

football is placed on your own 10-yard line with 90 yards to go for a touchdown. After the first down, your team gains 7 yards. On the second down, your team loses 4 yards. Is it reasonable for the football to be placed on the 3-yard line for the beginning of the third down?

b. fractions greater than or equal to zero (including mixed numbers) (2.4.A1c), e.g., explain if it is reasonable to say that a dog is ½ boxer, ¼ bulldog, ¼ collie, and ¼ rotweiler;

3. decimals greater than or equal to zero through hundredths place (2.4.A1c), e.g., five people a ate pizza. Is it reasonable to say that each person ate .3 of the pizza?

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Teacher Notes: Number sense refers to one’s ability to reason with numbers and to work with numbers in a flexible way. The ability to compute mentally, to estimate based on understanding of number relationships and magnitudes, and to judge reasonableness of answers are all involved in number sense. When we say that someone has good number sense, we mean that he or she possesses a variety of abilities and understandings that include an awareness of the relationships between numbers, an ability to represent numbers in a variety of ways, a knowledge of the effects of operations, and an ability to interpret and use numbers in real-world counting and measurement situations. Such a person predicts with some accuracy the result of an operation and consistently chooses appropriate measurement units. This “friendliness with numbers” goes far beyond mere memorization of computational algorithms and number facts; it implies an ability to use numbers flexibly, to choose the most appropriate representation of a number for a given circumstance, and to recognize when operations have been correctly performed. (Number Sense and Operations: Addenda Series, Grades K-6, NCTM, 1993)

At this grade level, rational numbers include positive numbers, large numbers (one million), and small numbers (one-hundredth). Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close, or about the same? For example, using the numbers 219, 264, and 457, answer questions such as –

Which two are closest? Why? Which is closest to 300? To 250? About how far apart are 219 and 500? 5,000? If these are ‘big numbers,’ what are small numbers? Numbers about the same? Numbers that make these seem

small? (Elementary and Middle School Mathematics, John A. Van de Walle, Addison Wesley Longman, Inc., 1998)

Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.


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Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the whole number system; recognizes, uses, and explains the concepts of properties as they relate to the whole number

system; and extends these properties to integers, fractions (including mixed numbers), and decimals.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. classifies subsets of numbers as integers, whole number, fractions

(including mixed numbers), or decimals (2.4.K1a-c, 2.4.K1k).2. identifies prime and composite numbers from 0 through 50.3. uses the concepts of these properties with whole numbers,

integers, fractions greater than or equal to zero (including mixed numbers), and decimals greater than or equal to zero and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($): a. commutative properties of addition and multiplication, e.g.,

43 + 34 = 34 + 43 and 12 x 15 = 15 x 12;b. associative properties of addition and multiplication, e.g.,

4 + (3 + 5) = (4 + 3) + 5;c. zero property of addition (additive identity) and property of one

for multiplication (multiplicative identity), e.g., 342 + 0 = 342 and 576 x 1 = 576;

d. symmetric property of equality, e.g., 35 = 11 + 24 is the same as 11 + 24 = 35;

e. zero property of multiplication, e.g., 438,223 x 0 = 0;f. distributive property, e.g., 7(3 + 5) = 7(3) + 7(5); g. substitution property, e.g., if a = 3 and a = b, then b = 3.

4. recognizes Roman Numerals that are used for dates, on clock faces, and in outlines.

5. recognizes the need for integers, e.g., with temperature, below zero is negative and above zero is positive; in finances, money in your pocket is positive and money owed someone is negative.


100,000 and decimals through hundredths using place value models; money; and the concepts of these properties to explain reasoning (2.4.A1a-c,e) ($):a. commutative and associative properties of addition and

multiplication, e.g., lay out a $5, $10 and $20 bills. Ask for the total of the money. The student says: Because you can add in any order (commutative) I can rearrange the money and count $20, $10 and $5 for $20 + $10 + $5 or Lay out 4 $5

bills. The student is asked the amount. The student says: I don’t know what 4 x 5 is, but I know 5 x 4 is $20 and since multiplication can be done in any order, then it is $20.

b. zero property of addition, e.g., have students lay out 6 dimes. Tell them to add zero. How many dimes? 6 + 0 = 6

c. property of one for multiplication, e.g., there are 24 students in our class. I want one math book per student, so I compute 24 x 1= 24. Multiplying times 1 does not change the product because it is one group of 24.

d. symmetric property of equality, e.g., Pat knows he has $56. He has 2 twenty-dollar bills in his wallet. How much does he have at home in his bank? This can be represented as –

56 = (2 x 20) + , so ( 2 x 20) + = 56 ▎ ▎5-155

January 31, 2004▲– Assessed Indicator on the Objective Assessment ■ – Assessed Indicator on the Optional Constructed Response AssessmentN – Noncalculator($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.

56 = 40 + , so 40 + = 56▎ ▎ 56 = 20 + 20 + 16, so 20 + 20 + 16 = 56.

e. zero property of multiplication, e.g., in science, you are observing a snail. The snail does not move over a 4-hour period. To figure its total movement, you say 4 x 0 = 0.

f. distributive property, e.g., Juan has 7 quarters and 7 dimes. What is the total amount of money he has? 7($.25 + $.10) = 7($.25) + 7($.10).

3. performs various computational procedures with whole numbers from 0 through 100,000 using the concepts of these properties; extends these properties to fractions greater than or equal to zero (including mixed numbers) and decimals greater than or equal to zero through hundredths place; and explains how the properties were used (2.4.A1a-c,e):a. commutative and associative properties of addition and

multiplication, e.g., given 4.2 x 10, the student says: I know that it is 42 because I know that 10 x 4.2 = 42, since you can multiply in any order and get the same answer. or The student says I don’t know what 9 + 8 is, but I know my doubles of 8 + 8, so I make the 9 into 1 + 8 and after adding 8 and 8, I add 1 more;

b. zero property of addition, e.g., given 47 + 917 + 0, the student says: I know that the answer is 964 because adding 0 does not change the answer (sum);

c. property of one for multiplication, e.g., $9.62 x 1. The student says: I know the product is still $9.62 because multiplication by one never changes the product. It is like if I had $9.62 in one pile, I would just have $9.62;

d. symmetric property of equality, e.g., given = ½ + ¼, the ▎student says: That is the same as ½ + ¼ because I must make both sides equal;

e. zero property of multiplication e.g., given .7 x 0, the student says: I know the answer (product) is zero because no matter how many factors you have, multiplying by 0, the product is 0;

f. distributive property, e.g., given 4 x 614, the student can explain that you can solve it (in your head?) by computing 4(600) +

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4(10) + 4(4), which is 2,400 + 40 + 4 = 2,444.4. states the reason for using integers, whole numbers, fractions

(including mixed numbers), or decimals when solving a given real-world problem (2.4.A1a-c) ($).





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Number and Computation – The student uses numerical and computational concepts and procedures in avariety of situations.

Benchmark 3: Estimation – The student uses computational estimation with whole numbers, fractions, decimals, and money in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student:1. estimates whole numbers quantities from 0 through 100,000;

fractions greater than or equal to zero (including mixed numbers); decimals greater than or equal to zero through hundredths place; and monetary amounts to $10,000 using various computational methods including mental math, paper and pencil, concrete materials, and appropriate technology (2.4.K1a-c) ($).

2. ▲N uses various estimation strategies to estimate whole number quantities from 0 through 100,000; fractions greater than or equal to zero (including mixed numbers); decimals greater than or equal to zero through hundredths place; and monetary amounts to $10,000 and explains how various strategies are used (2.4.K1a-c) ($).

3. recognizes and explains the difference between an exact and an approximate answer (2.4.K1a-c).

4. explains the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer (2.4.K1a).

The student:1. adjusts original estimate using whole numbers from 0 through

100,000 of a real-world problem based on additional information (a frame of reference) (2.4.A1a) ($), e.g., given a large container of marbles, estimate the quantity of marbles. Then, using a smaller container filled with marbles, count the number of marbles in the smaller container and adjust your original estimate.

2. estimates to check whether or not the result of a real-world problem using whole numbers from 0 through 100,000; fractions greater than or equal to zero (including mixed numbers); decimals greater than or equal to zero to tenths place; and monetary amounts to $10,000 is reasonable and makes predictions based on the information (2.4.A1a-c) ($), e.g., at your birthday party, you ate 4 ½ pepperoni pizzas, 3 ¼ cheese pizzas, and 2 ¾ sausage pizzas. On the bill they charged you for 10 pizzas. Is that reasonable? If pizzas cost $6.99

each, about how much should you save for your next birthday party?3. selects a reasonable magnitude from given quantities based on a

real-world problem using whole numbers from 0 through 100,000 and explains the reasonableness of selection (2.4.A1a), e.g., about how many tulips can fit in the flower vase, 2, 10, or 25? The student chooses ten and explains that the vase at home is a jelly jar and either two or ten will fit, but ten looks prettier.

4. ▲ ■ determines if a real-world problem calls for an exact or approximate answer using whole numbers from 0 through 100,000 and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete materials, and appropriate technology (2.4.A1a) ($).

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Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind the strategy to make a valid estimation is important. [For example, an invented approach for adding 37 + 89 could involve arriving at the solution by adding the tens, adding the ones, and then adding the partial sums to get the total.] (Principles and Standards for School Mathematics, NCTM, 2000)



The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in

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Number and Computation – The student uses numerical and computational concepts and procedures in avariety of situations.

Benchmark 4: Computation – The student models, performs, and explains computation with whole numbers, fractions including mixed numbers, and decimals including the use of concrete objects in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. computes with efficiency and accuracy using various

computational methods including mental math, paper and pencil, concrete materials, and appropriate technology (2.4.K1a).

2. performs and explains these computational procedures: a. N divides whole numbers through a 2-digit divisor and a 4-digit

dividend with the remainder as a whole number or a fraction using paper and pencil (2.4.K1a-b), e.g., 7452 ÷ 24 = 310 r 12 or 310 ½;

b. divides whole numbers beyond a 2-digit divisor and a 4-digit dividend using appropriate technology (2.4.K1a-b), e.g., 73,368 ÷ 36 = 2,038;

c. N adds and subtracts decimals from thousands place through hundredths place (2.4.K1c);

d. N multiplies decimals up to three digits by two digits from hundreds place through hundredths place (2.4.K1c);

e. N adds and subtracts fractions (like and unlike denominators) greater than or equal to zero (including mixed numbers) without regrouping and without expressing answers in simplest form with special emphasis on manipulatives, drawings, and models; (2.4.K1c);

f. N multiplies and divides by 10; 100; 1,000; or single-digit multiples of each (2.4.K1a-b), e.g., 20 • 300 or 4,400 ÷ 500.

3. reads and writes horizontally, vertically, and with different operational symbols the same addition, subtraction, multiplication, or division expression, e.g., 6 • 4 is the same as 6 x 4 is the same as 6(4) and 6 or 10 divided by 2 is the same as 10 ÷ 2 or 10

The student…1. ▲N solves one- and two-step real-world problems using these

computational procedures ($) (For the purpose of assessment, two-step could include any combination of a, b, c, d, e, or f.): a. adds and subtracts whole numbers from 0 through 100,000

(2.4.A1a-b); e.g., Lee buys a bike for $139, a helmet for $29 and a reflector for $6. How much of his $200 check from his grandparents will he have left?

b. multiplies through a four-digit whole number by a two-digit whole number (2.4.A1a-b), e.g., at the amusement park, Monday’s attendance was 4,414 people. Tuesday’s attendance was 3,042 people. If the cost per person is $23, how much money was collected on those days?

c. multiplies monetary amounts up to $1,000 by a one- or two-digit whole number (2.4.A1c), e.g., what is the cost of 4 items each priced at $3.49?

d. divides whole numbers through a 2-digit divisor and a 4-digit dividend with the remainder as a whole number or a fraction (2.4.A1a-c);

a. adds and subtracts decimals from thousands place through hundredths place when used as monetary amounts (2.4.A1a-c) (The set of decimal numbers includes whole numbers.), e.g., at the track meet, Peter ran the 100 meter dash in 12.3 seconds. Tanner ran the same race in 12.19 seconds. How much faster was Tanner?

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x 4 2 .

4. ▲N identifies, explains, and finds the greatest common factor and least common multiple of two or more whole numbers through the basic multiplication facts from 1 x 1 through 12 x 12 (2.4.K1d).

b. ■ multiplies and divides by 10; 100; and 1,000 and single digit multiples of each (10, 20, 30, …; 100, 200, 300, …; 1,000; 2,000; 3,000; …) (2.4.A1a-b), e.g., Matti has 1,590 stamps to place in her stamp album. 30 stamps fit on a page. What is the minimum number of pages she needs in her album?

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Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction, addition and multiplication, multiplication and division, and subtraction and division. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. [For example, an invented approach for adding 37 + 89 could involve arriving at the solution by adding the tens, adding the ones, and then adding the partial sums to get the total.] (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling, repeated doubling, halving, making tens, dividing with tens, thinking money, and using compatible “nice” numbers.


Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of growing up in a complex society. It is not only necessary to estimate appropriate answers accurately when required; it is also important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure. Technology does not replace the need to learn basic facts, to compute mentally, or to do reasonable paper-and-pencil computation. However, dividing a 5-digit number by a 2-digit number is appropriate with the exception of dividing by 10, 100, or 1,000 and simple multiples of each.

The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending

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and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.

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Standard 2: Algebra FIFTH GRADE

Algebra – The student uses algebraic concepts and procedures in a variety of situations.

Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains relationships inpatterns in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…6. uses concrete objects, drawings, and other representations to work

with these types of patterns(2.4.K1a): a. repeating patterns, e.g., 9, 10, 11, 9, 10, 11, …; b. growing patterns, e.g., 20, 30, 28, 38, 36, … where the rule is

add 10, then subtract 2; or 2, 5, 8, … as an example of an arithmetic sequence – each term after the first is found by adding the same number to the preceding term.

7. uses these attributes to generate patterns:a. counting numbers related to number theory (2.4.K1a), e.g.,

multiples or perfect squares; b. whole numbers (2.4.K1a) ($), e.g., 10; 100; 1,000; 10,000;

100,000; … (powers of ten); a. geometric shapes through two attribute changes (2.4.K1g), e.g.,

… when the next shape has one more side; or when both the color and the shape change at the same time;

c. measurements (2.4.K1a), e.g., 3 m, 6 m, 9 m, …;a. things related to daily life (2.4.K1a), e.g., sports scores,

longitude and latitude, elections, eras, or appropriate topics across the curriculum;

b. things related to size, shape, color, texture, or movement (2.4.K1a), e.g., square dancing moves (kinesthetic patterns)

The student…1. generalizes these patterns using a written description:

a. numerical patterns (2.4.K1a) ($),b. patterns using geometric shapes through two attribute changes

(2.4.A1a,g),c. measurement patterns (2.4.A1a),d. patterns related to daily life (2.4.A1a)

2. recognizes multiple representations of the same pattern (2.4.A1a) ($), e.g., 10; 100; 1,000; …– represented as 10; 10 x 10; 10 x 10 x 10; …;

– represented as a rod, a flat, a cube, … using base ten blocks; or – represented by a $10 bill; a $100 bill; a $1,000 bill; ….

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3. identifies, states, and continues a pattern presented in various formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written (2.4.K1a) ($).

4. generates:a. a pattern (repeating, growing) (2.4.K1a).b. a pattern using a function table (input/output machines, T-tables)

(2.4.K1g).


This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, positive rational numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain. Number theory is the study of the properties of the counting (natural) numbers, their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, greatest common factor and least common multiple, and sequences.



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Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, whole numbers, andalgebraic expressions in one variable to solve linear equations in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. ▲ explains and uses variables and symbols to represent unknown

whole number quantities from 0 through 1,000 and variable relationships (2.4.K1a)

2. ▲N solves one-step linear equations with one variable and a whole number solution using addition and subtraction with whole numbers from 0 through 100 and multiplication with the basic facts (2.4.K1a,e) ($), e.g., 3y = 12, 45 = 17 + q, or r – 42 = 36.

3. explains and uses equality and inequality symbols (=, , <, ≤, >, ≥) and corresponding meanings (is equal to, is not equal to, is less than, is less than or equal to, is greater than, is greater than or equal to) with whole numbers from 0 to 100,000 (2.4.K1a-b) ($).

4. recognizes ratio as a comparison of part-to-part and part-to-whole relationships (2.4.K1a), e.g., the relationship between the number of boys and the number of girls (part-to-part) or the relationship between the number of girls to the total number of students in the classroom (part-to-whole).

The student…1. represents real-world problems using variables, symbols, and one-

step equations with unknown whole number quantities from 0 through 1,000 (2.4.A1a,e) ($); e.g., Your parents say you must read 5 minutes each and every day of the next year. How many minutes will you read? This is represented by 365 x 5 = M.

2. generates one-step linear equations to solve real-world problems with whole numbers from 0 through 1,000 with one unknown and a whole number solution using addition, subtraction, multiplication, and division (2.4.A1a,e) ($), e.g., Ninety-six items are being shared with four people. How much does each person receive? becomes 96 ÷ 4 = n.

3. generates (2.4.A1a,e) ($):a. a real-world problem with one operation to match a

given addition, subtraction, multiplication, or division equation using whole numbers from 0 through 1,000 (2.4.A1a), e.g., given 95 ÷ 5 = x students write: There are 95 kids at camp who need to be divided into teams of 5. How many teams will there be?

b. number comparison statements using equality and inequality symbols (=, <, >) with whole numbers, measurement, and money e.g., 1 ft < 15 in or 10 quarters > $2.

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Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ or 3 + s = 7. Various symbols or letters should be used interchangeably in equations.



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Benchmark 3: Functions – The student recognizes, describes, and examines whole number relationships in avariety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. states mathematical relationships between whole numbers from 0

through 10,000 using various methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a) ($).

2. finds the values, determines the rule, and states the rule using symbolic notation with one operation of whole numbers from 0 through 10,000 using a vertical or horizontal function table (input/output machine, T-table) (2.4.K1f), e.g., using the function table, fill in the values and find the rule, the rule is N • 80.

N 4 9 11 ? 2 7 ?? 320 720 880 640 ? ? 800

3. generalizes numerical patterns using whole numbers from 0 through 5,000 up to two operations by stating the rule using words, e.g., If the sequence is 2400, 1200, 600, 300, 150, …; in words, the rule could be split the number in half or divide the previous number by 2 or if the sequence is 4, 11, 25, 53, 109, …; in words, the rule could be double the number and add 3 to get the next number or multiply the number by 2 and add 3.

4. ▲ ■ uses a function table (input/output machine, T-table) to identify, plot, and label whole number ordered pairs in the first quadrant of a coordinate plane (2.4.K1a,f).

5. plots and locates points for integers (positive and negative whole numbers) on a horizontal number line and vertical number line (2.4.K1a).

6. describes whole number relationships using letters and symbols.

The student…1. represents and describes mathematical relationships between whole

numbers from 0 through 5,000 using written and oral descriptions, tables, graphs, and symbolic notation (2.4.A1a) ($).

2. finds the rule, states the rule, and extends numerical patterns using real-world problems with whole numbers from 0 through 5,000 (2.4.A1a,f) ($), e.g., the class sells cookies at lunch recess to raise money for a field trip. The goal is to sell 3,000 cookies at 25¢ each. A student notices that every 4th day, a new case of cookies has to be opened. Each case holds 450 cookies. If the class keeps selling cookies at the same rate, how many days will it take to sell 3,000 cookies? A student’s answer might be: 28 days because that will be 150 overthe goal or on day 27 until 3,000 cookies are sold.

3. translates between verbal, numerical, and graphical representations including the use of concrete objects to describe mathematical relationships (2.4.A1a,k), e.g., when the temperature is 20º and then it drops 2º an hour for 12 hours, the result is a negative number; the student could model this using a vertical number line.

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Day # of Cookies Sold4 4508 900

12 135016 180020 225024 270028 3150







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Benchmark 4: Models – The student develops and uses mathematical models including the use of concrete objectsto represent and explain mathematical relationships in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. knows, explains, and uses mathematical models to represent


lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures and mathematical relationships and to solve equations (1.1.K1a, 1.1K1c, 1.1.K2, 1.1.K3, 1.1.K5, 1.2.K1, 1.2.K3, 1.3.K1-4, 1.4.K1, 1.4.K2a-b, 1.4.K.2f, 2.1.K1, 2.1.K2a-b, 2.1.K2d-h, 2.1.K2, 2.2.K1-4, 2.3.K1, 2.3.K4-5, 3.1.K1-6, 3.2.K1-4, 3.3.K1-2, 3.4.K1-4, 4.2.K3) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures (1.1.K1a, 1.1.K2, 1.1.K4, 1.2.K1, 1.3.K1-3, 1.4.K2a-b, 1.4.K2f, 2.2.K3) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K1b, 1.1.K2-4, 1.2.K1, 1.3.K1-3, 1.4.K2c-e, 4.1.K4) ($);

d. factor trees to find least common multiple and greatest common factor (1.2.K2, 1.4.K4);

e. equations and inequalities to model numerical relationships (2.2.K2) ($);

f. function tables (input/output machines, T-tables) to model numerical and algebraic relationships (2.1.K1c, 2.1.K1j, 3.1.K1-8, 3.2.K7-8, 3.3.K1-3) ($);



lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures, mathematical relationships, and problem situations and to solve equations (1.1.A1, 1.1.A2a, 1.2.A1-3, 1.3.A1-4, 1.4.A1a-b, 1.4.A1d-f, 2.1.A1a, 2.1.A1c-d, 2.1.A2, 2.2.A1-3, 2.3.A1-3, 3.2.A1a-f, 3.2.A2-4, 3.3.A1, 3.4.A1-2. 4.2.A2) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations (1.1.A1, 1.1.A2a, 1.2.A1-3, 1.3.A2, 1.4.A1a-b, 1.4.A1f, 1.4.A3a-e, 2.2.K3) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1a-b, 1.1.A2b-c, 1.2.A1-3, 1.3.A2, 1.4.A1c-e) ($);

d. factor trees to find least common multiple and greatest common factor;

e. equations and inequalities to model numerical relationships (2.1.A1-2, 2.2.A1-3) ($);

f. function tables (input/output machines, T-tables) to model numerical and algebraic relationships (2.3.A2, 3.2.A1g-h, 3.3.A3) ($);

g. two-dimensional geometric models (geoboards or dot paper) to model perimeter, area, and properties of geometric shapes and three-dimensional models (nets or solids) and real-world objects

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to compare size and to model volume and properties of geometric shapes (2.1.A1b, 3.1.A1-2, 3.2.A4, 4.1.A1-3);

g. two-dimensional geometric models (geoboards or dot paper) to model perimeter, area, and properties of geometric shapes and three-dimensional models (nets or solids) and real-world objects to compare size and to model volume and properties of geometric shapes (2.1.K2c, 2.1.K4b, 3.2.K5, 3.3.K3, 4.1.K2);

h. tree diagrams to organize attributes through three different sets and determine the number of possible combinations (4.1.K2, 4.2.K1a-d, 4.2.K1f-I; 4.2.K2, 4.2);

i. two- and three-dimensional geometric models (spinners or number cubes) and process models (concrete objects, pictures, diagrams, or coins) to model probability (4.1.K1-3, 4.2.K1e, 4.2.K2) ($) ;

j. graphs using concrete objects, pictographs, frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, line plots, charts, tables, and single stem-and-leaf plots to organize and display data (4.1.K2, 4.2.K1-2) ($) ;

k. Venn diagrams to sort data and show relationships.2. creates mathematical models to show the relationship between two

or more things, e.g., using trapezoids to represent numerical quantities –

.5 or 1/2

1 or 1.0

1.5 or 1 1/2

2 or 2.0

h. scale drawings to model large and small real-world objects (3.3.A2);

i. tree diagrams to organize attributes through three different sets and determine the number of possible combinations;

j. two- and three-dimensional geometric models (spinners or number cubes) and process models (concrete objects, pictures, diagrams, or coins) to model probability (4.1.A1-3, 4.2.A1) ($);

k. graphs using concrete objects, pictographs, frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, line plots, charts, and tables to organize, display, explain, and interpret data (2.3.A3; 4.1.A1-2, 4.2.A1, 4.2.A3-4) ($);

l. Venn diagrams to sort data and show relationships.2. selects a mathematical model and explains why some

mathematical models are more useful than other mathematical models in certain situations.

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Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item being assessed. The mathematical modeling process involves:



The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, they begin to develop an understanding of the advantages and disadvantages of the various representations/models.



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Standard 3: Geometry FIFTH GRADE

Geometry – The student uses geometric concepts and procedures in a variety of situations.

Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric shapes and comparestheir properties in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. recognizes and investigates properties of plane figures and solids

using concrete objects, drawings, and appropriate technology (2.4.K1g).

2. recognizes and describes (2.4.K1g):a. regular polygons having up to and including ten sides;b. similar and congruent figures.

3. ▲ recognizes and describes the solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms, rectangular pyramids, triangular pyramids) using the terms faces, edges, and vertices (corners) (2.4.K1g).

4. determines if geometric shapes and real-world objects contain line(s) of symmetry and draws the line(s) of symmetry if the line(s) exist(s) (2.4.K1g).

5. recognizes, draws, and describes (2.4.K1g):a. points, lines, line segments, and rays;b. angles as right, obtuse, or acute.

6. recognizes and describes the difference between intersecting, parallel, and perpendicular lines (2.4.K1g).

7. identifies circumference, radius, and diameter of a circle (2.4.K1g).

The student…1. solves real-world problems by applying the properties of (2.4.A1g):

a. ▲ plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, parallelograms, hexagons, pentagons) and the line(s) of symmetry; e.g., twins are having a birthday party. The rectangular birthday cake is to be cut into two pieces of equal size and with the same shape. How would the cake be cut? Would the cut be a line of symmetry? How would you know?

b. solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases; e.g., ribbon is to be glued on all of the edges of a cube. If one edge measures 5 inches, how much ribbon is needed? If a letter was placed on each face, how many letters would be needed?

c. intersecting, parallel, and perpendicular lines; e.g., relate these terms to maps of city streets, bus routes, or walking paths. Which street is parallel to the street where the school is located?

2. identifies the plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, octagons, pentagons, hexagons, trapezoids, parallelograms) used to form a composite figure (2.4.A1g).

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Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses measurement formulas ina variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. determines and uses whole number approximations (estimations)

for length, width, weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure (2.4.K1a) ($).

2. selects, explains the selection of, and uses measurement tools, units of measure, and degree of accuracy appropriate for a given situation to measure length, width, weight, volume, temperature, time, perimeter, and area using (2.4.K1a) ($):a. customary units of measure to the nearest fourth and eighth

inch, b. metric units of measure to the nearest centimeter,c. nonstandard units of measure to the nearest whole unit, d. time including elapsed time.

3. states the number of feet and yards in a mile (2.4.K1a).4. converts (2.4.K1a):

a. ▲ ■ within the customary system: inches and feet, feet and yards, inches and yards, cups and pints, pints and quarts, quarts and gallons, pounds and ounces;

b. within the metric system: centimeters and meters, meters and kilometers, milliliters and liters, grams and kilograms.

5. knows and uses perimeter and area formulas for squares and rectangles (2.4.K1g).

The student…1. solves real-world problems by applying appropriate measurements

and measurement formulas ($):a. ▲ length to the nearest eighth of an inch or to the nearest

centimeter (2.4.A1a), e.g., in science, we are studying butterflies. What is the wingspan of each of the butterflies studied to the nearest eighth of an inch?

b. temperature to the nearest degree (2.4.A1a), e.g., what would the temperature be if it was a good day for swimming?

c. ▲ weight to the nearest whole unit (pounds, grams, nonstandard units) (2.4.A1a), e.g., if you bought 200 bricks (each one weighed 5 pounds), how much would the whole load of bricks weigh?

d. time including elapsed time (2.4.A1a), e.g., Bob left Wichita at 10:45 a.m. He arrived in Kansas city at 1:30. How long did it take Bob to travel to Kansas City?

e. hours in a day, days in a week, and days and weeks in a year (2.4.A1a), e.g., John spent 59 days in New York City. How many weeks did he stay in New York City?

f. ▲ months in a year and minutes in an hour (2.4.A1a), e.g., it took Susan 180 minutes to complete her homework assignment. How many hours did she spend doing homework?

g. ▲ perimeter of squares, rectangles, and triangles (2.4.A1g), e.g., Mark wants to put up a fence up in his rectangle-shaped back yard. If his yard measures 18 feet by 36 feet, how many feet of fence will he need to go around his yard?

h. ▲ area of squares and rectangles (2.4.A1g), e.g., a farmer's square-shaped field is 35 feet on each side. How many square feet does he have to plow?

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2. solves real-world problems that involve conversions within the same measurement system: inches and feet, feet and yards, inches and yards, cups and pints, pints and quarts, quarts and gallons, centimeters and meters (2.4.A1a), e.g., you estimate that each person will chew 6 inches of bubblegum tape. If each package has 9 feet of bubblegum tape, how many people will get gum from that package?

3. estimates to check whether or not measurements or calculations for length, weight, temperature, time, perimeter, and area in real-world problems are reasonable (2.4.A1a) ($), e.g. is it reasonable to say you need 30 mL of water to fill a fish tank or would you need 30 L of water to fill the fish tank?

4. adjusts original measurement or estimation for length, width, weight, volume, temperature, time, and perimeter in real-world problems based on additional information (a frame of reference) (2.4.A1a,g) ($), e.g., after estimating the outside temperature to be 75º F, you find out that yesterday’s high temperature at 3 p.m. was 62º. Should you adjust your estimate? Why or why not?




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Benchmark 3: Transformational Geometry – The student recognizes and performs transformations on geometric shapes including the use of concrete objects in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. recognizes and performs through two transformations (reflection,

rotation, translation) on a two-dimensional figure (2.4.K1a).2. recognizes when an object is reduced or enlarged (2.4.K1a). 3. ▲ recognizes three-dimensional figures (rectangular prisms,

cylinders, cones, spheres, triangular prisms, rectangular pyramids) from various perspectives (top, bottom, side, corners) (2.4.K1g).

The student…1. describes and draws a two-dimensional figure after performing one

transformation (reflection, rotation, translation) (2.4.A1a).2. makes scale drawings of two-dimensional figures using a simple

scale and grid paper (2.4.A1h), e.g., using the scale 1 cm = 3 m, the student makes a scale drawing of the classroom.



The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories – Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.

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Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line and the first quadrant of a coordinate plane in a variety of situations.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. locates and plots points on a number line (vertical/horizontal) using

integers (positive and negative whole numbers) (2.4.K1a).2. explains mathematical relationships between whole numbers,

fractions, and decimals and where they appear on a number line (2.4.K1a).

3. identifies and plots points as ordered pairs in the first quadrant of a coordinate plane (coordinate grid) (2.4.K1a).

4. organizes whole number data using a T-table and plots the ordered pairs in the first quadrant of a coordinate plane (coordinate grid) (2.4.K1a,f).

The student…1. solves real-world problems that involve distance and location using

coordinate planes (coordinate grids) and map grids with positive whole number and letter coordinates (2.4.A1a), e.g., identifying locations and giving and following directions to move from one location to another.

2. solves real-world problems by plotting ordered pairs in the first quadrant of a coordinate plane (coordinate grid) (2.4.A1a) ($) , e.g., graph daily the cumulative number of recess minutes in a 5-day school week.

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Standard 4: Data FIFTH GRADE

Data – The student uses concepts and procedures of data analysis in a variety of situations.


Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. recognizes that all probabilities range from zero (impossible) through

one (certain) (2.4.K1i) ($).2. lists all possible outcomes of a simple event in an experiment or

simulation in an organized manner including the use of concrete objects (2.4.K1g-j).

3. recognizes a simple event in an experiment or simulation where the probabilities of all outcomes are equal (2.4.K1i).

4. represents the probability of a simple event in an experiment or simulation using fractions (2.4.K1c).

The student...1. ■ conducts an experiment or simulation with a simple event

including the use of concrete materials; records the results in a chart, table, or graph; uses the results to draw conclusions about the event; and makes predictions about future events (2.4.A1j-k).

2. uses the results from a completed experiment or simulation of a simple event to make predictions in a variety of real-world situations (2.4.A1j-k), e.g., the manufacturer of Crunchy Flakes puts a prize in 20 out of every 100 boxes. What is the probability that a shopper will find a prize in a box of Crunchy Flakes, if they purchase 10 boxes?

3. compares what should happen (theoretical probability/expected results) with what did happen (empirical probability/experimental results) in an experiment or simulation with a simple event (2.4.A1j).

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Standard 4: Data FIFTH GRADE


Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (rational numbers) and non-numerical data sets in a variety of situations with a special emphasis on measures of central tendency.

Fifth Grade Knowledge Base Indicators Fifth Grade Application IndicatorsThe student…1. organizes, displays, and reads numerical (quantitative) and non-

numerical (qualitative) data in a clear, organized, and accurate manner including a title, labels, categories, and whole number and decimal intervals using these data displays (2.4.K1j) ($):a. graphs using concrete objects,b. pictographs,c. frequency tables,d. bar and line graphs,e. Venn diagrams and other pictorial displays, e.g., glyphs,f. line plots,g. charts and tables,h. circle graphs, i. single stem-and-leaf plots.

2. collects data using different techniques (observations, polls, tallying, interviews, surveys, or random sampling) and explains the results (2.4.K1j) ($).

3. ▲ identifies, explains, and calculates or finds these statistical measures of a whole number data set of up to twenty whole number data points from 0 through 1,000 (2.4.K1a) ($): a. minimum and maximum values,b. range,c. mode (no-, uni-, bi-),d. median (including answers expressed as a decimal or a fraction

without reducing to simplest form), e. mean (including answers expressed as a decimal or a fraction

without reducing to simplest form).

The student…1. ▲ interprets and uses data to make reasonable inferences,

predictions, and decisions, and to develop convincing arguments from these data displays (2.4.A1k) ($):a. graphs using concrete materials,b. pictographs,c. frequency tables, d. bar and line graphs,e. Venn diagrams and other pictorial displays,f. line plots,g. charts and tables, h. circle graphs.

2. uses these statistical measures of a whole number data set to make reasonable inferences and predictions, answer questions, and make decisions (2.4.A1a) ($): a. minimum and maximum values,b. range,c. mode,d. median, e. mean when the data set has a whole number mean.

3. recognizes that the same data set can be displayed in various formats and discusses why a particular format may be more appropriate than another (2.4.A1k) ($).

4. recognizes and explains the effects of scale and interval changes on graphs of whole number data sets (2.4.A1k).

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Graphs take many forms: – bar graphs and pictographs compare discrete data, – frequency tables show how many times a certain piece of data occurs, – circle graphs (pie charts) model parts of a whole, – line graphs show change over time, – Venn diagrams show relationships between sets of objects, – line plots show frequency of data on a number line, and– single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side.




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Standard 1: Number and Computation SIXTH GRADE


Benchmark 1: Number Sense – The student demonstrates number sense for rational numbers and simple algebraic expressions in one variable in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. knows, explains, and uses equivalent representations for rational

numbers expressed as fractions, terminating decimals, and percents; positive rational number bases with whole number exponents; time; and money (2.4.K1a-c) ($).

2. ▲ compares and orders (2.4.K1a-c) ($): a. integers; b. fractions greater than or equal to zero,c. decimals greater than or equal to zero through thousandths

place.3. explains the relative magnitude between whole numbers, fractions

greater than or equal to zero, and decimals greater than or equal to zero (2.4.K1a-c).

4. ▲ N knows and explains numerical relationships between percents, decimals, and fractions between 0 and 1 (2.4.K1a,c), e.g., recognizing that percent means out of a 100, so 60% means 60 out of 100, 60% as a decimal is .60, and 60% as a fraction is 60/100.

5. uses equivalent representations for the same simple algebraic expression with understood coefficients of 1 (2.4.K1a), e.g., when students are developing their own formula for the perimeter of a square, they combine s + s + s + s to make 4s.

The student…1. generates and/or solves real-world problems using

equivalent representations of (2.4.A1a-c) ($):a. integers, e.g., the basketball team made 15 out of 25

free throws this season. Express their free throw shooting as a fraction and as a decimal.

b. fractions greater than or equal to zero, e.g., the basketball team made 15 out of 25 free throws this season, express their free throw shooting as a fraction.

c. decimals greater than or equal to zero through thousandths place (2.4.1a), e.g., the basketball team made 15 out of 25 free throws this season, express their free throw shooting as a decimal.

2. determines whether or not solutions to real-world problems that involve the following are reasonable ($).a. integers (2.4.A1a), e.g., the football is placed on your own 10-

yard line with 90 yards to go for a touchdown. After the first down, your team gains 7 yards. On the second down, your team loses 4 yards; and on the third down your team gains 2 yards. Is it reasonable for the football to be placed on the 5 yard line for the beginning of the fourth down? Why or why not?

b. fractions greater than or equal to zero (2.4.A1c), e.g., Gary, Tom, and their parents are selling greeting cards. Gary receives 1/3 of the profit and Tom receives 1/4 of the profit. Is it reasonable that together they received 2/7 of the profits?

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Why or why not?

c. decimals greater than or equal to zero through thousandths place (2.4.A1c), e.g., the beginning bank balance is $250.40 A deposit of $175, a withdrawal of $198, and a $2 service charge are made. The checkbook balance reads $127.40. Is this a reasonable balance? Why or why not?

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At this grade level, rational numbers include positive and negative numbers, large numbers (one million), and small numbers (one-thousandth). Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close, or about the same? For example, using the numbers 219, 264, and 457, answer questions such as –




The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories) Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.Standard 1: Number and Computation SIXTH GRADE

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Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the rationalnumber system and the irrational number pi; recognizes, uses, and describes their properties; andextends these properties to algebraic expressions in one variable.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. classifies subsets of the rational number system as counting (natural)

numbers, whole numbers, integers, fractions (including mixed numbers), or decimals (2.4.K1a,c,k).

2. identifies prime and composite numbers and explains their meaning (2.4.K1d).

3. uses and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($): a. commutative and associative properties of addition and

multiplication (commutative – changing the order of the numbers does not change the solution; associative – changing the grouping of the numbers does not change the solution);

b. identity properties for addition and multiplication (additive identity – zero added to any number is equal to that number; multiplicative identity – one multiplied by any number is equal to that number);

c. symmetric property of equality, e.g., 24 x 72 = 1,728 is the same as 1,728 = 24 x 72;

d. zero property of multiplication (any number multiplied by zero is zero);

e. distributive property (distributing multiplication or division over addition or subtraction), e.g., 26(9 + 15) = 26(9) + 26(15);

f. substitution property (one name of a number can be substituted for another name of the same number), e.g., if a = 3 and a + 2 = b, then 3 + 2 = b;

g. addition property of equality (adding the same number to each side of an equation results in an equivalent equation – an equation with the same solution), e.g., if a = b, then a + 3 = b + 3;

The student…1. generates and/or solves real-world problems with rational numbers

using the concepts of these properties to explain reasoning (2.4.A1a-c,e) ($):a. commutative and associative properties for addition and

multiplication, e.g., at a delivery stop, Sylvia pulls out a flat of eggs. The flat has 5 columns and 6 rows of eggs. Show two ways to find the number of eggs: 5 • 6 = 30 or 6 • 5 = 30.

b. additive and multiplicative identities, e.g., the outside temperature was T degrees during the day. The temperature rose 5 degrees and by the next morning it had dropped 5 degrees.

c. symmetric property of equality, e.g., Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $15 = $10 + $5 is the same as $10 + $5 = $15

d. distributive property, e.g., trim is used around the outside edges of a bulletin board with dimensions 3 ft by 5 ft. Show two different ways to solve this problem: 2(3 + 5) = 16 or 2 • 3 + 2 • 5 = 6 + 10 = 16. Then explain why the answers are the same.

e. substitution property, e.g., V = IR [Ohm’s Law: voltage (V) = current (I) x resistance (R)] If the current is 5 amps (I = 5) and the resistance is 4 ohms (R = 4), what is the voltage?

f. addition property of equality, e.g., Bob and Sue each read the same number of books. During the week, they each read 5

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more books. Compare the number of books each read: b= number of books Bob read, s= number of books Sue read, so b + 5 = s + 5..

h. multiplication property of equality (for any equation, if the same number is multiplied to each side of that equation, then the new statement describes an equation equivalent to the original), e.g., if a= b, then a x 7 = b x 7;

additive inverse property (every number has a value known as its additive inverse and when the original number is added to that additive inverse, the answer is zero), e.g., +5 + (-5) = 0.

4. recognizes and explains the need for integers, e.g., with temperature, below zero is negative and above zero is positive; in finances, money in your pocket is positive and money owed someone is negative.

5. recognizes that the irrational number pi can be represented by an approximate rational value, e.g., 22/7 or 3.14.

g. multiplication property of equality, e.g., Jane watches television half as much as Tom. Jane watches T.V. for 3 hours. How long does Tom watch television? Let T = number of hours Tom watches TV. 3 = ½T, so 2•3 + 2•½T.

h. additive inverse property, e.g., at the shopping mall, you are at ground level when you take the elevator down 5 floors. Describe how to get to ground level.

2. analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals, or the irrational number pi and its rational approximations in solving a given real-world problem (2.4.A1a-c) ($), e.g., in the store everything is 50% off. When calculating the discount, which representation of 50% would you use and why?

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Benchmark 3: Estimation – The student uses computational estimation with rational numbers and the irrational number pi in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. estimates quantities with combinations of rational numbers and/or

the irrational number pi using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a-c) ($).

2. uses various estimation strategies and explains how they were used to estimate rational number quantities or the irrational number pi (2.4.K1a-c) ($)

3. recognizes and explains the difference between an exact and an approximate answer (2.4.K1a-c).

4. determines the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer and its potential impact on the result (2.4.K1a).

The student…1. adjusts original rational number estimate of a real-world problem

based on additional information (a frame of reference) (2.4.A1a) ($), e.g., given a large container of marbles, estimate the quantity of marbles. Then, using a smaller container filled with marbles, count the number of marbles in the smaller container and adjust your original estimate.

2. ▲N estimates to check whether or not the result of a real-world problem using rational numbers is reasonable and makes predictions based on the information (2.4.A1a) ($), e.g., a class of 28 students has a goal of reading 1,000 books during the school year. If each student reads 13 books each month, will the class reach their goal?

3. selects a reasonable magnitude from given quantities based on a real-world problem and explains the reasonableness of the selection (2.4.A1a), e.g., length of a classroom in meters – 1-3 meters, 5-8 meters, 10-15 meters.

4. determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, or appropriate technology (2.4.A1a) ($), e.g., Kathy buys items at the grocery store priced at: $32.56, $12.83, $6.99, 5 for $12.49 each. She has $120 with her to pay for the groceries. To decide if she can pay for her items, does she need an exact or an approximate answer?

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Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible. In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation strategies include front-end with adjustment, compatible “nice” (friendly) numbers, clustering, special numbers, or truncation.



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Benchmark 4: Computation – The student models, performs, and explains computation with positive rational numbers and integers in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. computes with efficiency and accuracy using various

computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a).

2. performs and explains these computational procedures:a. ▲N divides whole numbers through a two-digit divisor and a

four-digit dividend and expresses the remainder as a whole number, fraction, or decimal (2.4.K1a-b), e.g., 7452 ÷ 24 = 310 r 12, 310 12/24, 310 ½, or 310.5;

b. N adds and subtracts decimals from millions place through thousandths place (2.4.K1c);

c. N multiplies and divides a four-digit number by a two-digit number using numbers from thousands place through hundredths place (2.4.K1a-b), e.g., 4,350 ÷ 1.2 = 3,625;

d. N multiplies and divides using numbers from thousands place through thousandths place by 10; 100; 1,000; .1; .01; .001; or single-digit multiples of each (2.4.K1a-c); e.g., 54.2 ÷ .002 or 54.3 x 300;

e. N adds integers (2.4.K1a); e.g., +6 + –7 = –1 f. ▲N adds, subtracts, and multiplies fractions (including mixed

numbers) expressing answers in simplest form (2.4.K1c); e.g., 5¼ • 1/3 = 21/4 • 1/3 = 7/4 or 1¾

g. N finds the root of perfect whole number squares (2.4.K1a);h. N uses basic order of operations (multiplication and division in

order from left to right, then addition and subtraction in order from left to right) with whole numbers;

The student…1. generates and/or solves one- and two-step real-world problems with

rational numbers using these computational procedures ($): a. division with whole numbers (2.4.A1b), e.g., the perimeter of a

square is 128 feet. What is the length of its side?b. ▲ ■ addition, subtraction, multiplication, and division of

decimals through hundredths place (2.4.A1a-c), e.g., on a recent trip, Marion drove 25.8 miles from Allen to Barber, then 15.2 miles from Barber to Chase, then 14.9 miles from Chase to Douglas. When Marion had completed half of her drive from Allen to Douglas how many miles did she drive?

c. addition, subtraction, and multiplication of fractions (including mixed numbers) (2.4.A1c), e.g., the student council is having a contest between classes. On the average, each student takes 3 1/3 minutes for the relay. How much time is needed for a class of 24 to run the relay?

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i. adds, subtracts multiplies, and divides rational numbers using concrete objects.

3. recognizes, describes, and uses different representations to express

the same computational procedures, e.g., 3/4 = 3 ÷ 4 = .

4. identifies, explains, and finds the prime factorization of whole numbers (2.4.K1d).

5. finds prime factors, greatest common factor, multiples, and the least common multiple (2.4.K1d).

6. finds a whole number percent (between 0 and 100) of a whole number (2.4.K1a,c) ($), e.g., 12% of 40 is what number?

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Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in single-digit addition and subtraction by developing relationships within addition and subtraction combinations [and by developing relationships with multiplication and division combinations]. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling, repeated doubling, halving, making tens, multiplying by powers of tens, dividing with tens, thinking money, and using compatible “nice” numbers.


Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of growing up in a complex society. It is not only necessary to estimate appropriate answers accurately when required, but also it is also important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure. Technology does not replace the need to learn basic facts, to compute mentally, or to do reasonable paper-and-pencil computation. However, dividing a 5-digit number by a 2-digit number is appropriate with the exception of dividing by 10, 100, or 1,000 and simple multiples of each.

The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked

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Standard 2: Algebra SIXTH GRADE


Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule of a pattern in variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application Indicators

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The student…1. identifies, states, and continues a pattern presented in various

formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes include:a. counting numbers including perfect squares, and factors and

multiples (number theory) (2.4.K1a);b. positive rational numbers limited to two operations (addition,

subtraction, multiplication, division) including arithmetic sequences (a sequence of numbers in which the difference of two consecutive numbers is the same) (2.4.K1a);

c. geometric figures through two attribute changes (2.4.K1g);d. measurements (2.4.K1a);e. things related to daily life (2.4.K1a) ($), e.g., time (a full moon

every 28 days), tide, calendar, traffic, or appropriate topics across the curriculum.

2. generates a pattern (repeating, growing) (2.4.K1a).3. extends a pattern when given a rule of one or two simultaneous

operational changes (addition, subtraction, multiplication, division) between consecutive terms (2.4.K1a), e.g., find the next three numbers in a pattern that starts with 3, where you double and add 1 to get the next number; the next three numbers are 7, 15, and 31.

4. ▲ states the rule to find the next number of a pattern with one operational change (addition, subtraction, multiplication, division) to move between consecutive terms (2.4.K1a), e.g., given 4, 8, and 16, double the number to get the next term, multiply the term by 2 to get the next term, or add the number to itself for the next term.

The student… 1. recognizes the same general pattern presented in different

representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1a,k), e.g., you are selling cookies by the box. Each box costs $3. You have $2 to begin your sales. This can be written as a pattern that begins with 2 and adds three each time, as a table or graph.

2. recognizes multiple representations of the same pattern (2.4.A1a)

($), e.g., 1, 10; 100; 1,000; 10,000… – represented as 1; 10; 10 x 10; 10 x 10 x 10; 10 x 10 x 10 x 10; …; – represented as 100; 101; 102; 103; 104; …; – represented as a unit; a rod; a flat; a cube; … using base ten

blocks; or – represented as a $1 bill; a $10 bill; a $100 bill ; a $1,000 bill; ….

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X Y0 22 83 114 14

Teacher Notes: A fundamental component in the development of classification, number, and problem solving skills is inventing, discovering, and describing patterns. Patterns pervade all of mathematics and much of nature. All patterns are either repeating or growing or a variation of either or both. Translating a pattern from one medium to another to find two patterns that are alike, even though they are made with different materials, is important so students can see the relationships that are critical to repeating patterns. With growing patterns, students not only extend patterns, but also look for a generalization or an algebraic relationship that will tell them what the pattern will be at any point along the way.

Working with patterns is an important process in the development of mathematical thinking. Patterns can be based on geometric attributes (shapes, regions, angles); measurement attributes (color, texture, length, weight, volume, number); relational attributes (proportion, sequence, functions); and affective attributes (values, likes, dislikes, familiarity, heritage, culture). (Learning to Teach Mathematics, Randall J. Souviney, Macmillan Publishing Company, 1994)

This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain. Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, greatest common factor and least common multiple, and sequences.



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Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, positive rational numbers, and algebraic expressions in one variable to solve linear equations and inequalities in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. explains and uses variables and/or symbols to represent unknown

quantities and variable relationships (2.4.K1a), e.g., x < 2. 2. uses equivalent representations for the same simple algebraic

expression with understood coefficients of 1 (2.4.K1a), e.g., when students are developing their own formula for the perimeter of a square they combine s + s + s + s to make 4s.

3. solves (2.4.K1a,e) ($):a. one-step linear equations (addition, subtraction, multiplication,

division) with one variable and whole number solutions, e.g., 2 x = 8 or x + 7 = 12

b. one-step linear inequalities(addition, subtraction) in one variable with whole numbers, e.g., x – 5 < 12;

4. explains and uses equality and inequality symbols (=, , <, ≤, >, ≥) and corresponding meanings (is equal to, is not equal to, is less than, is less than or equal to, is greater than, is greater than or equal to) to represent mathematical relationships with positive rational numbers (2.4.K1a-b) ($).

5. knows and uses the relationship between ratios, proportions, and percents and finds the missing term in simple proportions where the missing term is a whole number (2.4.K1a,c), e.g., 1 = x, 2 = 4,

1 = x_. 2 4 3 x 4 1006. finds the value of algebraic expressions using whole numbers

(2.4.Ka), e.g., If x =3, then 5x = 5(3).

The student… 1. represents real-world problems using variables and symbols to

(2.4.A1a,e) ($):a. write algebraic or numerical expressions or one-step equations

(addition, subtraction, multiplication, division) with whole number solutions, e.g., John has three times as much money as his sister. If M is the amount of money his sister has, what is the expression that represents the amount of money that John has? The expression would be written as 3M.

b. ▲ ■ write and/or solve one-step equations (addition, subtraction, multiplication, and division), e.g., a player scored three more points today than yesterday. Today, the player scored 17 points. How many points were scored yesterday? Write an equation to represent this problem. Let Y = number of points scored yesterday. The equation would be written as y + 3 = 17. The answer is y = 14.

4. generates real-world problems that represent simple expressions or one-step linear equations (addition, subtraction, multiplication, division) with whole number solutions (2.2.A1a,e), ($) e.g., write a problem situation that represents the expression x + 10. The problem could be: How old will a person be ten years from now if x represents the person’s current age?

5. explains the mathematical reasoning that was used to solve a real-world problem using a one-step equation (addition, subtraction, multiplication, division) (2.2.A1a,e) ($), e.g., use the equation form y + 3 = 17. Solve by subtracting 3 from both sides to get y = 14.

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Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + 4 = Δ or 3 • s = 15 where a triangle and s are used as variables. Various symbols or letters should be used interchangeably in equations.



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Benchmark 3: Functions – The student recognizes, describes, and analyzes linear relationships in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. recognizes linear relationships using various methods including

mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology (2.4.K1a).

2. finds the values and determines the rule with one operation using a function table (input/output machine, T-table) (2.4.K1f).

3. generalizes numerical patterns up to two operations by stating the rule using words (2.4.K1a), e.g., If the sequence is 2400, 1200, 600, 300, 150, …, what is the rule? In words, the rule could be split the previous number in half or divide the previous number before by 2.

4. uses a given function table (input/output machine, T-table) to identify, plot, and label the ordered pairs using the four quadrants of a coordinate plane (2.4.K1a,f).

The student…1. represents a variety of mathematical relationships using written and

oral descriptions of the rule, tables, graphs, and when possible, symbolic notation (2.4.A1f,k) ($), e.g., linear patterns and graphs can be used to represent time and distance situations. Pretend you are in a car traveling from home at 50 miles per hour. Then, represent the nth term. 50n meaning 50 times the number of hours traveling equals the distance away from home.

Time | Distance 0 | 0 1 | 50 2 | 100 . | . . | . . | . n | 50n 2. interprets and describes the mathematical relationships of

numerical, tabular, and graphical representations (2.4.Alf,k).

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In the pattern 1, 4, 9, 16, 25,.., 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the number of the term – the 5th term is 52, the 8th term is 82, …




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Benchmark 4: Models – The student generates and uses mathematical models to represent and justify mathematical relationships in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. knows, explains, and uses mathematical models to represent


lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures and mathematical relationships and to solve equations (1.1.K1-5, 1.2.K1, 1.3.K1-4, 1.4.K1, 1.4.K2a, 1.4.K2c-e, 1.4.K2g, 1.4.K2i, 1.4.K6, 2.1.K1a-b, 2.1.K1d-e, 2.1.K2-4, 2.2.K1-6, 2.3.K1, 2.3.K3-4, 3.2.K1-4, 3.2.K8, 3.3.K1-4, 3.4.K1-3, 4.2.K4) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures (1.1.K1-4, 1.2.K1, 1.3.K1-3, 1.4.K2b, 1.4.K2c-d, 2.2K4) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K1-4, 1.2.K1, 1.3.K1-3, 1.4.K2b, 1.4.K2d, 1.4.K2f, 1.4.K6, 2.2.K5, 4.1.K4, 4.2.K4) ($):

d. factor trees to find least common multiple and greatest common factor (1.4.K4-5);

e. equations and inequalities to model numerical relationships (2.2.K3,) ($);

f. function tables (input/output machines, T-tables) to model numerical and algebraic relationships (2.3.K2, 2.3.K4) ($);



lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures and mathematical relationships, to represent problem situations, and to solve equations (1.1.A1, 1.1.A1a, 1.2.A1-2, 1.3.A1-4, 1.4.A1a-b, 2.1.A1-2, 2.1.A1-3, 3.2.A1a, 3.2.A1c, 3.2.A2, 3.3.A1-2, 3.4.A1-2, 4.2.A1) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations (1.1.A1, 1.2.A1-2, 2.2.A3) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1, 1.1.A2b-c, 1.2.A1-2, 1.4.A1b-c) ($);

d. factor trees to find least common multiple and greatest common factor;

e. equations and inequalities to model numerical relationships (2.2.A1-3) ($);

f. function tables (input/output machines, T-tables) to model numerical and algebraic relationships (2.3.A1-2) ($);

g. two-dimensional geometric models (geoboards or dot paper) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (nets or solids) and real-world objects to model volume and to identify attributes (faces, edges, vertices, bases) of geometric shapes (3.1.A1-3. 3.2.A1b, 3.4.A2);

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g. two-dimensional geometric models (geoboards or dot paper) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (nets or solids) and real-world objects to model volume and to identify attributes (faces, edges, vertices, bases) of geometric shapes (2.1.K1c, 3.1.K1-5, 3.1.K7-10, 3.2.K7, 3.3.K1-4);

h. tree diagrams to organize attributes and determine the number of possible combinations (4.1.K2);

i. graphs using concrete objects, two- and three-dimensional geometric models (spinners or number cubes) and process models (concrete objects, pictures, diagrams, or coins) to model probability (4.1.K1-4) ($).

j. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, line plots, charts, tables, single stem-and-leaf plots, and scatter plots to organize and display data (4.2.K1-3) ($);

k. Venn diagrams to sort data and to show relationships (1.2.K1). 2. uses one or more mathematical models to show the relationship

between two or more things.

h. scale drawings to model large and small real-world objects (3.4.A2);

i. tree diagrams to organize attributes and determine the number of possible combinations;

j. two- and three-dimensional geometric models (spinners or number cubes) and process models (concrete objects, pictures, diagrams, or coins) to model probability (4.1.A1-3) ($);

k. graphs using concrete objects, frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, line plots, charts, tables, and single stem-and-leaf plots to organize, display, explain, and interpret data (2.1.A1, 2.3.A1-2, 4.1.A1-2, 4.2.A1-3) ($);

l. Venn diagrams to sort data and to show relationships. 2. selects a mathematical model and justifies why some

mathematical models are more accurate than other mathematical models in certain situations.

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The use of mathematical models is necessary for conceptual understanding. The ways in which mathematical ideas are represented is fundamental to how students understand and use those ideas. As students begin to use multiple representations of the same situation, they begin to develop an understanding of the advantages and disadvantages of various representations/models.



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Standard 3: Geometry SIXTH GRADE


Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric figures and compares their properties in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student… 1. recognizes and compares properties of plane figures and solids

using concrete objects, constructions, drawings, and appropriate technology (2.4.K1g).

2. recognizes and names regular and irregular polygons through 10 sides including all special types of quadrilaterals: squares, rectangles, parallelograms, rhombi, trapezoids, kites (2.4.K1g).

3. names and describes the solids [prisms (rectangular and triangular), cylinders, cones, spheres, and pyramids (rectangular and triangular)] using the terms faces, edges, vertices, and bases (2.4.K1g).

4. recognizes all existing lines of symmetry in two-dimensional figures (2.4.K1g).

5. recognizes and describes the attributes of similar and congruent figures (2.4.K1g).

6. recognizes and uses symbols for angle (find symbol for), line(↔), line segment (―), ray (→), parallel (||), and perpendicular ().

7. ▲ classifies (2.4.K1g):a. angles as right, obtuse, acute, or straight; b. triangles as right, obtuse, acute, scalene, isosceles, or

equilateral.8. identifies and defines circumference, radius, and diameter of circles

and semicircles.9. recognize that the sum of the angles of a triangle equals 180°

(2.4.K1g). 10. determines the radius or diameter of a circle given one or the other.

The student…2. solves real-world problems by applying the properties of (2.4.A1g):

a. plane figures (regular polygons through 10 sides, circles, and semicircles) and the line(s) of symmetry, e.g., twins are having a birthday party. The rectangular birthday cake is to be cut into two equal sizes of the same shape. How would you cut the cake?

b. solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases, e.g., lace is to be glued on all of the edges of a cube. If one edge measures 34 cm, how much lace is needed?

c. intersecting, parallel, and perpendicular lines, e.g., railroad tracks form what type of lines? Two roads are perpendicular, what is the angle between them?

2. decomposes geometric figures made from (2.4.A1g):a. regular and irregular polygons through 10 sides, circles, and

semicircles, e.g., draw a picture of a house (rectangular base) with a roof (triangle) and a chimney on the side of the roof (trapezoid). Identify the three geometrical figures.

b. nets (two-dimensional shapes that can be folded into three-dimensional figures), e.g., the cardboard net that becomes a shoebox.

3. composes geometric figures made from (2.4.A1g):a. regular and irregular polygons through 10 sides, circles,

and semicircles;b. nets (two-dimensional shapes that can be folded into three-

dimensional figures).

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Teacher Notes: Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (one-dimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional. Solids are referred to as three-dimensional. The base, in terms of geometry, generally refers to the side on which a figure rests. Therefore, depending on the orientation of the solid, the base changes.



The application of the Knowledge Indicators from the Geometry Benchmark, Geometric Figures and Their Properties are most often applied within the context of the other Geometry Benchmarks — Measurement and Estimation, Transformational Geometry, and Geometry From an Algebraic Perspective — rather than in isolation.



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Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses measurement formulas in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student… 1. determines and uses whole number approximations (estimations) for

length, width, weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure (2.4.K1a) ($).

2. selects, explains the selection of, and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, area, and angle measurements (2.4.K1a) ($).

3. converts (2.4.K1a):a. within the customary system, e.g., converting feet to inches,

inches to feet, gallons to pints, pints to gallons, ounces to pounds, or pounds to ounces;

b. ▲ within the metric system using the prefixes: kilo, hecto, deka, deci, centi, and milli; e.g., converting millimeters to meters, meters to millimeters, liters to kiloliters, kiloliters to liters, milligrams to grams, or grams to milligrams.

4. uses customary units of measure to the nearest sixteenth of an inch and metric units of measure to the nearest millimeter (2.4.K1a).

5. recognizes and states perimeter and area formulas for squares, rectangles, and triangles (2.4.K1g). a. uses given measurement formulas to find perimeter and area of:

squares and rectangles,b. figures derived from squares and/or rectangles.

6. describes the composition of the metric system (2.4.K1a):a. meter, liter, and gram (root measures);b. kilo, hecto, deka, deci, centi, and milli (prefixes).

7. finds the volume of rectangular prisms using concrete objects (2.4.K1g).

The student…1. solves real-world problems by applying these measurement

formulas ($):a. ▲ perimeter of polygons using the same unit of measurement

(2.4.A1a,g), e.g., measures the length of fence around a yard;b. ▲ ■ area of squares, rectangles, and triangles using the same

unit of measurement (2.4.A1g), e.g., finds the area of a room for carpeting;

c. conversions within the metric system (2.4.A1a), e.g., your school is having a balloon launch. Each student needs 40 centimeters of string, and there are 42 students. How many meters of string are needed?

2. estimates to check whether or not measurements and calculations for length, width, weight, volume, temperature, time, perimeter, and area in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) ($), e.g., students estimate, in feet, the height of a bookcase in their classroom. Then a student who is about 5 feet tall stands beside it. The students then adjust the estimate.

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8. estimates an approximate value of the irrational number pi (2.4.K1a).


Using concrete objects (mathematical models) to conduct experiments to compare the circumference and the radius/diameter of a circle is important in developing students’ understanding of pi, its approximate value, and the circumference formula. The circumference formula will be emphasized in the seventh grade.



.

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Benchmark 3: Transformational Geometry – The student recognizes and performs transformations on two- and three-dimensional geometric figures in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. ▲ ■ identifies, describes, and performs one or two transformations

(reflection, rotation, translation) on a two-dimensional figure (2.4.K1a).

2. reduces (contracts/shrinks) and enlarges (magnifies/grows) simple shapes with simple scale factors (2.4.K1a), e.g., tripling or halving.

3. recognizes three-dimensional figures from various perspectives (top, bottom, sides, corners) (2.4.K1a).

4. recognizes which figures will tessellate (2.4.K1a).

The student… 1. describes a transformation of a given two-dimensional figure that

moves it from its initial placement (preimage) to its final placement (image) (2.4.A1a).

2. makes a scale drawing of a two-dimensional figure using a simple scale (2.4.A1a), e.g., using the scale 1 cm = 30 m, the student makes a scale drawing of the school.

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Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do not change the figure itself. Other transformations like reduction (contraction/shrinking) or enlargement (magnification/growing) change the size of a figure, but not the shape (congruence vs. similarity).


The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and ) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.



Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line and a coordinate plane in a variety of situations.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. uses a number line (horizontal/vertical) to order integers and positive

rational numbers (in both fractional and decimal form) (2.4.K1a). 2. organizes integer data using a T-table and plots the ordered pairs in

all four quadrants of a coordinate plane (coordinate grid) (2.4.K1a).3. ▲ uses all four quadrants of the coordinate plane to (2.4.K1a):

a. identify the ordered pairs of integer values on a given graph;b. plot the ordered pairs of integer values.

The student…1. represents, generates, and/or solves real-world problems using a

number line with integer values (2.4.A1a) ($), e.g., the difference between –2 degrees and 10 degrees on a thermometer is 12 degrees (units); similarly, the distance between –2 to +10 on a number line is 12 units.

2. represents and/or generates real-world problems using a coordinate plane with integer values to find (2.4.A1a,g-h):a. the perimeter of squares and rectangles, e.g., Alice made a

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scale drawing of her classroom and put it on a coordinate plane marked off in feet. The rectangular table in the back of the room was described by the points (8,9), (8,12), (14,12) and (14,9). Now Alice wants to put a skirting around the outer edge of the table. Using the drawing, find the amount of skirting she will need.

b. the area of triangles, squares, and rectangles, e.g., a scale drawing of a flower garden is found in a book with the coordinates of the four corners being (9,5), (9,13), (18,13) and (18,5). The scale is marked off in meters. How many square meters is the flower garden?





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Standard 4: Data SIXTH GRADE



Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. recognizes that all probabilities range from zero (impossible) through

one (certain) and can be written as a fraction, decimal, or a percent (2.4.K1i) ($) , e.g., when you flip a coin, the probability of the coin landing on heads (or tails) is ½, .5, or 50%. The probability of flipping a head on a two-headed coin is 1. The probability of flipping a tail on a two-headed coin is 0.

2. ▲ ■ lists all possible outcomes of an experiment or simulation with a compound event composed of two independent events in a clear and organized way (2.4.K1h-j), e.g., use a tree diagram or list to find all the possible color combinations of pant and shirt ensembles, if there are 3 shirts (red, green, blue) and 2 pairs of pants (black and brown).

3. recognizes whether an outcome in a compound event in an experiment or simulation is impossible, certain, likely, unlikely, or equally likely (2.4.K1i).

4. ▲ represents the probability of a simple event in an experiment or simulation using fractions and decimals (2.4.K1c,i), e.g., the probability of rolling an even number on a single number cube is represented by ½ or .5.

The student…1. conducts an experiment or simulation with a compound event

composed of two independent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions about the events and make predictions about future events (2.4.A1j-k).

2. analyzes the results of a given experiment or simulation of a compound event composed of two independent events to draw conclusions and make predictions in a variety of real-world situations (2.4.A1j-k), e.g., given the equal likelihood that a customer will order a pizza with either thick or thin crust, and an equal probability that a single topping of beef, pepperoni, or sausage will be selected –1) What is the probability that a pizza ordered will be thin crust with

beef topping? 2) Given sales of 30 pizzas on a Friday night, how many would the

manager expect to be thin crust with beef topping? 3. compares what should happen (theoretical probability/expected

results) with what did happen (empirical probability/experimental results) in an experiment or simulation with a compound event composed of two independent events (2.4.A1j).

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Teacher Notes: Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be addressed through the use of concrete objects (process models); spinners, number cubes, or dartboards (geometric models); and coins (money models). Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments. Some examples of uses of probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win every game?



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Standard 4: Data SIXTH GRADE


Benchmark 2: Statistics – The student collects, organizes, displays, and explains numerical (rational numbers) and non-numerical data sets in a variety of situations with a special emphasis on measures of central tendency.

Sixth Grade Knowledge Base Indicators Sixth Grade Application IndicatorsThe student…1. organizes, displays, and reads quantitative (numerical) and qualitative

(non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays (2.4.K1j) ($):a. graphs using concrete objects;b. frequency tables and line plots;c. bar, line, and circle graphs;d. Venn diagrams or other pictorial displays;e. charts and tables;f. single stem-and-leaf plots; g. scatter plots;

2. selects and justifies the choice of data collection techniques (observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, or purposeful sampling) in a given situation (2.4.K1j).

3. uses sampling to collect data and describe the results (2.4.K1j) ($).4. determines mean, median, mode, and range for (2.4.K1a,c) ($):

a. a whole number data set,b. a decimal data set with decimals greater than or equal to

zero.

The student…1. uses data analysis (mean, median, mode, range) of a whole

number data set or a decimal data set with decimals greater than or equal to zero to make reasonable inferences, predictions, and decisions and to develop convincing arguments from these data displays (2.4.A1k) ($): a. graphs using concrete objects;b. frequency tables and line plots;c. bar, line, and circle graphs;d. Venn diagrams or other pictorial displays;e. charts and tables; f. single stem-and-leaf plots.

2. explains advantages and disadvantages of various data displays for a given data set (2.4.A1k) ($).

3. recognizes and explains the effects of scale and/or interval changes on graphs of whole number data sets (2.4.A1k).

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Graphs take many forms: – bar graphs and pictographs compare discrete data, – frequency tables show how many times a certain piece of data occurs, – circle graphs (pie charts) model parts of a whole, – line graphs show change over time, – Venn diagrams show relationships between sets of objects, – line plots show frequency of data on a number line, – single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side, and– scatter plots show the relationship between two quantities.




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Standard 1: Number and Computation SEVENTH GRADE


Benchmark 1: Number Sense – The student demonstrates number sense for rational numbers, the irrational numberpi, and simple algebraic expressions in one variable in a variety of situations.

Seventh Grade Knowledge Base Indicators Seventh Grade Application IndicatorsThe student…1. knows, explains, and uses equivalent representations for rational

numbers and simple algebraic expressions including integers, fractions, decimals, percents, and ratios; integer bases with whole number exponents; positive rational numbers written in scientific notation with positive integer exponents; time; and money (2.4.K1a-c) ($), e.g., 253,000 is equivalent to 2.53 x 105 or x + 5x is equivalent to 6x.

2. compares and orders rational numbers and the irrational number pi (2.4.K1a) ($).

3. explains the relative magnitude between rational numbers and between rational numbers and the irrational number pi (2.4.K1a).

4. knows and explains what happens to the product or quotient when (2.4.K1a):a. a whole number is multiplied or divided by a rational number

greater than zero and less than one, b. a whole number is multiplied or divided by a rational number

greater than one, c. a rational number (excluding zero) is multiplied or divided by

zero. 5. explains and determines the absolute value of rational numbers

(2.4.K1a).

The student…1. generates and/or solves real-world problems using (2.4.A1a) ($):

a. ▲equivalent representations of rational numbers and simple algebraic expressions, e.g., you are in the mountains. Wilson Mountain has an altitude of 5.28 x 103 feet. Rush Mountain is 4,300 feet tall. How much higher is Wilson Mountain than Rush Mountain?

b. fraction and decimal approximations of the irrational number pi, e.g., Mary measured the distance around her 48-inch diameter circular table to be 150 inches. Using this information, approximate pi as a fraction and as a decimal.

2. determines whether or not solutions to real-world problems using rational numbers, the irrational number pi, and simple algebraic expressions are reasonable (2.4.A1a) ($), e.g., a sweater that cost $15 is marked 1/3 off. The cashier charged $12. Is this reasonable?

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The student with number sense will look at a problem holistically before confronting the details of the problem. The student will look for relationships among the numbers and operations and will consider the context in which the question was posed. Students with number sense will choose or even invent a method that takes advantage of their own understanding of the relationships between numbers and between numbers and operations, and they will seek the most efficient representation for the given task. Number sense can also be recognized in the students' use of benchmarks to judge number magnitude (e.g., 2/5 of 49 is less that half of 49), to recognize unreasonable results for calculations, and to employ non-standard algorithms for mental computation and estimation. (Developing Number Sense: Addenda Series, Grades 5-8, NCTM, 1991)

At this grade level, rational numbers include positive and negative numbers and very large numbers [ten million (107)] and very small numbers [hundred-thousandth (10-5)]. Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close, or about the same? For example, using the numbers 219, 264, and 457, answer questions such as





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Standard 1: Number and Computation – The student uses numerical and computational concepts and procedures in a

variety of situations.

Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the rational number system and the irrational number pi; recognizes, uses, and describes their properties; and extends these properties to algebraic expressions in one variable.

Seventh Grade Knowledge Base Indicators Seventh Grade Application IndicatorsThe student…4. knows and explains the relationships between natural (counting)

numbers, whole numbers, integers, and rational numbers using mathematical models (2.4.K1a,k), e.g., number lines or Venn diagrams.

5. classifies a given rational number as a member of various subsets of the rational number system (2.4.K1a,k), e.g., - 7 is a rational number and an integer.

6. names, uses, and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($): a. commutative properties of addition and multiplication (changing

the order of the numbers does not change the solution); b. associative properties of addition and multiplication (changing the

grouping of the numbers does not change the solution); c. distributive property [distributing multiplication or division over

addition or subtraction, e.g., 2(4 – 1) = 2(4) – 2(1) = 8 – 2 = 6];d. substitution property (one name of a number can be substituted

for another name of the same number), e.g., if a = 2, then 3a = 3 x 2 = 6.

7. uses and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($): a. identity properties for addition and multiplication (additive identity

– zero added to any number is equal to that number;


and the irrational number pi using the concepts of these properties to explain reasoning (2.4.K1a) ($):a. commutative and associative properties of addition and

multiplication, e.g., at a delivery stop, Sylvia pulls out a flat of eggs. The flat has 5 columns and 6 rows of eggs. Express how to find the number of eggs in 2 ways.

b. distributive property, e.g., trim is used around the outside edges of a bulletin board with dimensions 3 ft by 5 ft. Explain two different methods of solving this problem.

c. substitution property, e.g., V = IR [Ohm’s Law: voltage (V) = current (I) x resistance (R)] If the current is 5 amps (I = 5) and the resistance is 4 ohms (R = 4), what is the voltage?

d. symmetric property of equality, e.g., Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $15 = $10 + $5 is the same as $10 + $5 = $15.

e. additive and multiplicative identities, e.g., Bob and Sue each read the same number of books. During the week, they each read 5 more books. Compare the number of books each read. Let b=the number of books Bob read and s=the number of books Sue read, then b+5=s+5

f. zero property of multiplication, e.g., Jenny was thinking of two numbers. Jenny said that the product of the two numbers was

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multiplicative identity – one multiplied by any number is equal to that number);

b. symmetric property of equality (if 7 + 2x = 9 then 9 = 7 + 2x);c. zero property of multiplication (any number multiplied by zero is

zero);d. addition and multiplication properties of equality

(adding/multiplying the same number to each side of an equation results in an equivalent equation);

e. additive and multiplicative inverse properties. (Every number has a value known as its additive inverse and when the original number is added to that additive inverse, the answer is zero, e.g., +5 + –5 = 0. Every number except 0 has a value known as its multiplicative inverse and when the original number multiplied by its inverse, the answer will be 1, e.g., 8 x 1/8 =1.)

5. recognizes that the irrational number pi can be represented by approximate rational values, e.g., 22/7 or 3.14.

0. What could you deduct from this statement? Explain your reasoning.

g. addition and multiplication properties of equality, e.g., the total price (P) of a car, including tax (T), is $14, 685. 33. If the tax is $785.42, what is the sale price of the car (S)?

h. additive and multiplicative inverse properties, e.g., if 5 candy bars cost $1.00, what does one candy bar cost? Explain your reasoning.

2. analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals, or the irrational number pi and its rational approximations in solving a given real-world problem (2.4.K1a, e.g., in the store everything is 25% off. When calculating the discount, which representation of 25% would you use and why?





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Benchmark 3: Estimation – The student uses computational estimation with rational numbers and the irrational

number pi in a variety of situations.

Seventh Grade Knowledge Base Indicators Seventh Grade Application Indicators

The student…1. estimates quantities with combinations of rational numbers and/or

the irrational number pi using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a) ($).

2. N uses various estimation strategies and explains how they were used to estimate rational number quantities and the irrational number pi (2.4.K1a) ($).

3. recognizes and explains the difference between an exact and approximate answer (2.4.K1a).

4. determines the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer and its potential impact on the result (24.K1a).

5. knows and explains why the fraction (22/7) or decimal (3.14) representation of the irrational number pi is an approximate value (2.4.K1c).


based on additional information (a frame of reference) (2.4.A1a) ($), e.g., estimate the weight of a bookshelf of books. Then weigh one book and adjust your estimate.

2. estimates to check whether or not the result of a real-world problem using rational numbers, the irrational number pi, and/or simple algebraic expressions is reasonable and makes predictions based on the information (2.4.A1a), e.g., a goat is staked out in a pasture with a rope that is 7 feet long. The goat needs 200 square feet of grass to graze. Does the goat have enough pasture? If not, how long should the rope be?

3. determines a reasonable range for the estimation of a quantity given a real-world problem and explains the reasonableness of the range (2.4.A1a), e.g., how long will it take your teacher to walk two miles? The range could be 25-35 minutes.

4. determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.A1a) ($), e.g., Kathy buys items at the grocery store priced at $32.56, $12.83, $6.99, and 5 for $12.49 each. She has $120 with her to pay for the groceries. To decide if she can pay for her items, does she need an exact or an approximate answer?

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Teacher Notes: Estimate, as a verb, means to make an educated guess based on information in a problem or to give an answer close to the exact number. Estimation is used when an exact answer is not needed, as in many real-life situations for which “ballpark” computations are acceptable. Good number sense enables one to estimate a quantity, estimate a measure, or estimate an answer. Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. However, being able to compute does not automatically lead to an ability to estimate or judge reasonableness of answers. Frequent modeling by the teacher helps students develop a range of estimation strategies. Students should be encouraged to frequently explain their thinking as they estimate. As with exact computation, sharing estimation strategies allows students access to others’ thinking and provides opportunities for class discussion. Identifying the estimation strategy by name is not critical; however, explaining the thinking behind the strategy to make a valid estimation is important. (Principles and Standards for School Mathematics, NCTM, 2000)

Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible. In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation strategies include front-end with adjustment, compatible “nice” numbers, clustering, special numbers, or truncation.



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Benchmark 4: Computation – The student models, performs, and explains computation with rational numbers, the

irrational number pi, and first-degree algebraic expressions in one variable in a variety of situations.

Seventh Grade Knowledge Base Indicators Seventh Grade Application IndicatorsThe student…1. computes with efficiency and accuracy using various computational

methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a-c) ($).

2. performs and explains these computational procedures (2.4.K1a): a. ▲N adds and subtracts decimals from ten millions place

through hundred thousandths place;b. ▲N multiplies and divides a four-digit number by a two-digit

number using numbers from thousands place through thousandths place;

c. ▲N multiplies and divides using numbers from thousands place through thousandths place by 10; 100; 1,000; .1; .01; .001; or single-digit multiples of each, e.g., 54.2 ÷ .002 or 54.3 x 300;

d. ▲N adds, subtracts, multiplies, and divides fractions and expresses answers in simplest form;

e. N adds, subtracts, multiplies, and divides integers;f. N uses order of operations (evaluates within grouping

symbols, evaluates powers to the second or third power, multiplies or divides in order from left to right, then adds or subtracts in order from left to right) using whole numbers;

g. simplifies positive rational numbers raised to positive whole number powers;

h. combines like terms of a first degree algebraic expression.recognizes, describes, and uses different ways to express computational

procedures, e.g., 5 – 2 = 5 + (–2) or 49 x 23 = (40 x 23) + (9 x 23) or 49 x 23 = (49 x 20) + (49 x 3) or 49 x 23 = (50 x 23) – 23.

finds prime factors, greatest common factor, multiples, and the least

The student…1. generates and/or solves one- and two-step real-world problems

using these computational procedures and mathematical concepts (2.4.A1a) ($):a. ■ addition, subtraction, multiplication, and division of rational

numbers with a special emphasis on fractions and expressing answers in simplest form, e.g., at the candy store, you buy ¾ of a pound of peppermints and ½ of a pound of licorice. The cost per pound for each kind of candy is $3.00. What is the total cost of the candy purchased?

b. addition, subtraction, multiplication, and division of rational numbers with a special emphasis on integers, e.g., the high temperatures for the week were: 4o, 10o, 1o, 0o, 7o, 3o, and –5o. What is the mean temperature for the week?

c. first degree algebraic expressions in one variable, e.g., Jenny rents 3 videos plus $20 of other merchandise. Barb rents 5 videos plus $15 of other merchandise. Represent the total purchases of Jenny and Barb using V as the price of a video rental.

d. percentages of rational numbers, e.g., if the sales tax is 5.5%, what is the sales tax on an item that costs $36?

e. approximation of the irrational number pi, e.g., what is the approximate diameter of a 400-meter circular track?

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common multiple (2.4.K1d).

5. ▲ finds percentages of rational numbers (2.4.K1a,c) ($), e.g., 12.5% x $40.25 = n or 150% of 90 is what number? (For the purpose of assessment, percents will not be between 0 and 1.)

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Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction, addition and multiplication, multiplication and division, and subtraction and division. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling, repeated doubling, halving, making tens, multiplying by powers of ten, dividing with tens, thinking money, and using compatible “nice” numbers.




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Standard 2: Algebra SEVENTH GRADE


Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule of a

pattern in a variety of situations.


The student…1. identifies, states, and continues a pattern presented in various

formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes:a. ▲ counting numbers including perfect squares, cubes, and

factors and multiples (number theory) (2.4.K1a);b. ▲ positive rational numbers including arithmetic and geometric

sequences (arithmetic: sequence of numbers in which the difference of two consecutive numbers is the same, geometric: a sequence of numbers in which each succeeding term is obtained by multiplying the preceding term by the same number) (2.4.K1a), e.g., 2, ½, 1/8, 1/32, …;

c. geometric figures (2.4.K1f); d. measurements (2.4.K1a);e. things related to daily life (2.4.K1a) ($), e.g., tide, moon cycle, or

temperature.2. generates a pattern (2.4.K1a). 3. extends a pattern when given a rule of one or two simultaneous

changes (addition, subtraction, multiplication, division) between consecutive terms (2.4.K1a), e.g., find the next three numbers in a pattern that starts with 3, where you double and add 1 to get the next number; the next three numbers are 7, 15, and 31.

4. ▲ ■ states the rule to find the nth term of a pattern with one operational change (addition or subtraction) between consecutive terms (2.4.K1a), e.g., given 3, 5, 7, and 9; the nth term is 2n +1. (This is the explicit rule for the pattern.)

The student…1. generalizes a pattern by giving the nth term using symbolic notation

(2.4.A1a,f), e.g., given the following, the nth term is 2n.

1 person has 2 eyes 2 people have 4 eyes 3 people have 6 eyes 4 people have 8 eyes . . . . . . n people have 2n eyes

2. recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1a,f,j-k) ($).

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X Y1 22 43 64 8...

.

.

.n 2n

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In the pattern that begins with 3, 5, 7, and 9; the explicit rule is 2n +1 and the recursive rule is add 2 to the previous term. Patterns themselves are not explicit or recursive. The RULE for the pattern can be expressed explicitly or recursively and MOST patterns can be explained using either format especially IF that pattern reflects either an arithmetic sequence or geometric sequence.

This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain. Number theory is the study of the properties of the counting (natural) numbers, their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, greatest common factor and least common multiple, and sequences.


The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in

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Benchmark 2: Variable, Equations, and Inequalities – The student uses variables, symbols, rational numbers, and

simple algebraic expressions in one variable to solve linear equations and inequalities in a variety of

situations.


The student…1. knows and explains that a variable can represent a single quantity

that changes (2.4.K1a), e.g., daily temperature. 2. knows, explains, and uses equivalent representations for the same

simple algebraic expressions (2.4.K1a), e.g., x + y + 5x is the same as 6x + y.

3. shows and explains how changes in one variable affects other variables (2.4.A1a), e.g., changes in diameter affects circumference.

4. explains the difference between an equation and an expression.5. solves (2.4.K1a,e) ($):

a. one-step linear equations in one variable with positive rational coefficients and solutions, e.g., 7x = 28 or x + 3 = 7 or x = 5;

4 3b. two-step linear equations in one variable with counting number

coefficients and constants and positive rational solutions;c. one-step linear inequalities with counting numbers and one

variable, e.g., 3x > 12.6. explains and uses the equality and inequality symbols (=, , <, ≤, >,

≥) and corresponding meanings (is equal to, is not equal to, is less than, is less than or equal to, is greater than, is greater than or equal to) to represent mathematical relationships with rational numbers (2.4.K1a) ($).

7. ▲ knows the mathematical relationship between ratios, proportions, and percents and how to solve for a missing term in a proportion with positive rational number solutions and monomials (2.4.K1a,c) ($), e.g.,

5 = 2 6 x.

The student…1. ▲ represents real-world problems using variables and symbols to

write linear expressions, one- or two-step equations (2.4.A1e) ($), e.g., John has three times as much money as his sister. If M is the amount of money his sister has, what is the equality that represents the amount of money that John has? To represent the problem situation, J = 3M could be written.

2. solves real-world problems with one- or two-step linear equations in one variable with whole number coefficients and constants and positive rational solutions intuitively and analytically (2.4.A1e) ($), e.g., Kim has read 5 more than twice the number of pages as Hank. Kim has read 15 pages. How many pages has Hank read? To solve analytically, write 2h + 5 = 15. Then find the answer.

3. generates real-world problems that represent one- or two-step linear equations (2.4.A1e) ($), e.g., given the equation x + 10 = 30, the problem could be: Two items cost $30.00. If one item costs $10.00, what is the cost of the other item?

4. explains the mathematical reasoning that was used to solve a real-world problem using a one- or two-step linear equation (2.4.A1e) ($), e.g., Kim has read 5 more than twice the number of pages as Hank. Kim has read 15 pages. How many pages has Hank read? To solve, write 2h + 5 = 15. Let h=number of pages Hank read. Then to find the answer subtract 5 from both sides of the equation. 2h = 10, then divide both sides of the equation by 2, so h = 5.

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8. ▲ evaluates simple algebraic expressions using positive rational numbers (2.4.K1c) ($), e.g., if x = 3/2, y = 2, then 5xy + 2 = 5(3/2)(2) + 2 = 17.

Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + Δ = 4 or 3 • s = 15 where a triangle and s are used as variables. Various symbols or letters should be used interchangeably in equations.



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Benchmark 3: Functions – The student recognizes, describes, and analyzes constant and linear relationships in a



The student…1. recognizes constant and linear relationships using various methods

including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology (2.4.K1a,e-g) ($).

2. finds the values and determines the rule through two operations using a function table (input/output machine, T-table) (2.4.K1f).

3. demonstrates mathematical relationships using ordered pairs in all four quadrants of a coordinate plane (2.4.K1g).

4. describes and/or gives examples of mathematical relationships that remain constant (2.4.K1e-g) ($), e.g., you will get $10.00 to do a job, no matter how long it takes for you to do it.

The student…1. represents a variety of constant and linear relationships using

written or oral descriptions of the rule, tables, graphs, and when possible, symbolic notation (2.4.A13-g,k) ($), e.g., the relationship between cars and their wheels (written) becomes a table:

Cars | Wheels 1 | 4 (1,4) 2 | 8 (2,8) 10 | 40 (10,40) . | . . | . . | . n | 4n and then the ordered pairs of (1,4), (2,8), (10,40), and (n,4n) can be graphed. 2. interprets, describes, and analyzes the mathematical relationships

of numerical, tabular, and graphical representations, including translations between the representations (2.4.A1k) ($).

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Benchmark 4: Models – The student generates and uses mathematical models to represent and justify mathematical

relationships found in a variety of situations.

Seventh Grade Knowledge Base Indicators Seventh Grade Application IndicatorsThe student…1. knows, explains, and uses mathematical models to represent and

explain mathematical concepts, procedures, and relationships. Mathematical models include:

a. process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K1-5, 1.2.K1-4, 1.3.K1-4, 1.4.K1-2, 1.4.K5, 2.1.K1a-b, 2.1.K1e. 2.1.K2-4, 2.2.K1-3, 2.2.K5-6, 2.3.K1, 3.1.K9, 3.2.K1-3, 3.2.K9, 3.3.K1-4, 3.4.K1, 4.2.K4-6) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures (1.1.K1, 1.4.K2) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K1, 1.3.K5, 1,4,K2, 2.2.K7-8, 4.1.K3) ($):

d. factor trees to find least common multiple and greatest common factor and to model prime factorization (1.4.K4);

e. equations and inequalities to model numerical relationships (2.2.K5, 2.3.K1, 23.K4) ($);

f. function tables to model numerical and algebraic relationships (2.3.K1-2, 2.3.K4) ($);

g. coordinate planes to model relationships between ordered pairs and linear equations (2.3.K1, 2.3.K3-4, 3.4.K2-4) ($);


represent the same problem situation. Mathematical models include: a. process models (concrete objects, pictures, diagrams,

flowcharts, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations (1.1.A1, 1.2.A1-2, 1.3.A1-4, 1.4.A1, 2.1.A1-2, 3.1.A1, 3. 2.A1a, 3.2.A1d, 3.2.A1f, 3.2.A2, 3.3.A1. 4.2.A4) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations (1.1.A1a, 1.1.A2a, 1.2.A12, 1.3A1.2, 1.4.A3a-e, 2.2.A3) ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1b, 1.1.A2b, 1.2.A1-2, 1.3.A1-2) ($);

d. factor trees to find least common multiple and greatest common factor and to model prime factorization (1.4.K5)

e. equations and inequalities to model numerical relationships (2.2.A1-4, 2.3.A1, 3.2.A1e). ($)

f. function tables to model numerical and algebraic relationships (2.1.A1-2, 2.3.A1) ($);

g. coordinate planes to model relationships between ordered pairs and linear equations (2.3.A1, 3.4.A1) ($);

h. two- and three-dimensional geometric models (geoboards, dot paper, nets or solids) to model perimeter, area, volume, and surface area, and properties of two- and three-dimensional

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h. two- and three-dimensional geometric models (geoboards, dot paper, nets or solids) to model perimeter, area, volume, and surface area, and properties of two- and three-dimensional (2.1.K1c, 3.1.K1, 3.1.K3-8, 3.1.K10, 3.2.K1-2, 3.2.K4-8, 3.2.K10);

models (3.1.A2-3, 3.2.A1b-c, 3.2.A1e, 3.3.A2, 3.4.A1);i. scale drawings to model large and small real-world objects

(3.3.A3);

i. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.K1, 4.1.K4) ($);

j. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single stem-and-leaf plots, scatter plots, and box-and-whisker plots to organize and display data (4.2.K1) ($);

k. Venn diagrams to sort data and show relationships (1.2.K1-2).

j. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.A1) ($);

k. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single stem-and-leaf plots, scatter plots, and box-and-whisker plots to describe, interpret, and analyze data (2.1.A2, 2.3.A1-2, 4.1.A1-2, 4.2.A1-3) ($);

l. Venn diagrams to sort data and show relationships.2. selects a mathematical model and justifies why some

mathematical models are more accurate than other mathematical models in certain situations, e.g., recognizes that change over time is better represented through a line graph than through a table of ordered pairs.

3. uses the mathematical modeling process to make inferences about real-world situations when the mathematical model used to represent the situation is given.

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The mathematical modeling process involves:e. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through

some sort of mathematical model,f. performing manipulations and mathematical procedures within the mathematical model,g. interpreting the results of the manipulations within the mathematical model, h. using these results to make inferences about the original real-world situation.




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Standard 3: Geometry SEVENTH GRADE




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The student…1. recognizes and compares properties of two- and three-dimensional

figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology (2.4.K1h).

2. classifies regular and irregular polygons having through ten sides as convex or concave.

3. ▲ identifies angle and side properties of triangles and quadrilaterals (2.4.K1h):a. sum of the interior angles of any triangle is 180°;b. sum of the interior angles of any quadrilateral is 360°;c. parallelograms have opposite sides that are parallel and

congruent; d. rectangles have angles of 90°, opposite sides are congruent;e. rhombi have all sides the same length, opposite angles are

congruent;f. squares have angles of 90°, all sides congruent;g. trapezoids have one pair of opposite sides parallel and the other

pair of opposite sides are not parallel.4. identifies and describes (2.4.K1h):

a. the altitude and base of a rectangular prism and triangular prism, b. the radius and diameter of a cylinder.

5. identifies corresponding parts of similar and congruent triangles and quadrilaterals (2.4.K1h).

6. uses symbols for right angle within a figure ( ), parallel (||), perpendicular (), and triangle (Δ) to describe geometric figures(2.4.K1h).

7. classifies triangles as (2.4.K1h):a. scalene, isosceles, or equilateral;b. right, acute, obtuse, or equiangular.

The student…3. solves real-world problems by applying the properties of (2.4.A1a):

d. plane figures (regular and irregular polygons through 10 sides, circles, and semicircles) and the line(s) of symmetry; e.g., two guide wires are used to stabilize a tower. The wires with the ground form an isosceles triangle. The two base angles form 20º angles with the ground. What is the size of the vertex angle made where the wires meet on the tower?

e. solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases; e.g., lace is to be glued on all of the edges of a cube. If one edge measures 34 cm, how much lace is needed?

2. decomposes geometric figures made from (2.4.A1h):c. regular and irregular polygons through 10 sides, circles, and

semicircles;d. nets (two-dimensional shapes that can be folded into three-

dimensional figures), e.g., the cardboard net that becomes a shoebox;

e. prisms, pyramids, cylinders, cones, spheres, and hemispheres.

3. composes geometric figures made from (2.4.A1h):c. regular and irregular polygons through 10 sides, circles,

and semicircles;d. nets (two-dimensional shapes that can be folded into three-

dimensional figures);e. prisms, pyramids, cylinders, cones, spheres, and

hemispheres.

8. determines if a triangle can be constructed given sides of three different lengths(2.4.K1h).

9. generates a pattern for the sum of angles for 3-, 4-, 5-, … n-sides polygons (2.4.K1a).

10. describes the relationship between the diameter and the circumference of a circle (2.4.K1h).

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Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses measurement formulas in a



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The student… 1. determines and uses rational number approximations (estimations)

for length, width, weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure (2.4.K1a) ($).

2. selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, area, and angle measurements (2.4.K1a) ($).

3. converts within the customary system and within the metric system (2.4.K1a).

4. ▲ knows and uses perimeter and area formulas for circles, squares, rectangles, triangles, and parallelograms (2.4.K1h).

5. finds perimeter and area of two-dimensional composite figures of circles, squares, rectangles, and triangles (2.4.K1h).

6. ▲ uses given measurement formulas to find (2.4.K1h):a. surface area of cubes,b. volume of rectangular prisms.

7. finds surface area of rectangular prisms using concrete objects (2.4.K1h).

8. uses appropriate units to describe rate as a unit of measure (2.4.K1a), e.g., miles per hour.

9. finds missing angle measurements in triangles and quadrilaterals (2.4.K1h).

The student…1. solves real-world problems by ($):

a. converting within the customary and metric systems (2.4.A1a), e.g., James added 30 grams of sand to his model boat that weighed 2 kg. With the sand included, what is the total weight of his boat in kg?

b. finding perimeter and area of circles, squares, rectangles, triangles, and parallelograms (2.4.A1h), e.g., the dimensions of a room are 22’ x 12’. What is the total length of wallpaper border needed?

c. ▲ ■ finding perimeter and area of two-dimensional composite figures of squares, rectangles, and triangles (2.4.A1h), e.g., the front of a barn is rectangular in shape with a height of 10 feet and a width of 48 feet. Above the rectangle is a triangle that is 7 feet high with sides 25 feet long. What is the area of the front of the barn?

d. using appropriate units to describe rate as a unit of measure (2.4.A1a), e.g., a person traveled 20 miles in 10 minutes. What is the rate of travel? The answer could be 2 miles per minute or 120 miles per hour.

e. finding missing angle measurements in triangles and quadrilaterals (2.4.A1h), e.g., a fenced pasture is a quadrilateral with angles of 30o, 120o, and 90o degrees. What is the measure of the fourth angle?

f. applying various measurement techniques (selecting and using measurement tools, units of measure, and level of precision) to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, and area appropriate to a given situation (2.4.A1a).

2. estimates to check whether or not measurements or calculations for length, width, weight, volume, temperature, time, perimeter, and area in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) ($), e.g., students estimate the weight of their book in grams. Then the weight of their calculator is measured in grams. Students then adjust their estimate.

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Benchmark 3: Transformational Geometry – The student recognizes and performs transformations on two- and three-

dimensional geometric figures in a variety of situations.


The student…1. identifies, describes, and performs single and multiple

transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on a two-dimensional figure (2.4.K1a).

2. identifies three-dimensional figures from various perspectives (top, bottom, sides, corners) (2.4.K1a).

3. draws three-dimensional figures from various perspectives (top, bottom, sides, corners) (2.4.K1a).

4. generates a tessellation (2.4.K1a).

The student…1. describes the impact of transformations [reflection, rotation,

translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on the perimeter and area of squares and rectangles (2.4.A1a); e.g., when a square is rotated, the perimeter and area stays the same, however, when the length of the sides of a square are tripled, the perimeter triples, and the area is 9 times bigger.

2. investigates congruency and similarity of geometric figures using transformations (2.4.A1h).

3. ▲ ■ determines the actual dimensions and/or measurements of a two-dimensional figure represented in a scale drawing (2.4.A1i).

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Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do not change the figure itself. Other transformations like reduction (contraction/shrinking) or enlargement (magnification/growing) change the size of a figure, but not the shape (congruence vs. similarity).



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Benchmark 4: Geometry From An Algebraic Perspective – The student relates geometric concepts to a number line

and a coordinate plane in a variety of situations.


The student…1. finds the distance between the points on a number line by computing

the absolute value of their difference (2.4.K1a).2. uses all four quadrants of a coordinate plane to (2.4.K1g):

a. identify in which quadrant or on which axis a point lies when given the coordinates of a point,

b. plot points,c. identify points,d. list through five ordered pairs of a given line.

3. uses a given linear equation with whole number coefficients and constants and a whole number solution to find the ordered pairs, organize the ordered pairs using a T-table, and plot the ordered pairs on the coordinate plane (2.4.K1e-g).

4. examines characteristics of two-dimensional figures on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.A1g).

The student…1. represents and/or generates real-world problems using a coordinate

plane to find (2.4.A1g-h):a. perimeter of squares and rectangles; e.g., using a coordinate

plane Jack started at the school (1, 2), traveled to the post office (3 ½, 2), then went by the fire station (3 ½, 3), then visited the park (1, 3), and finally returned to the school. Determine the distance Jack traveled.

b. circumference (perimeter) of circles, e.g., Jane jogs on a circular track with a radius of 100 feet. How far would she jog in one lap?

c. area of circles, parallelograms, triangles, squares, and rectangles; e.g., if the sprinkler head is centered at (3, 4) and we know it reaches point (3, - 2) on the coordinate plane, determine the area being watered.

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Standard 4: Data SEVENTH GRADE


Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions, generate convincing

arguments, and make predictions and decisions including the use of concrete objects in a



The student…1. finds the probability of a compound event composed of two

independent events in an experiment or simulation (2.4.K1i) ($).2. explains and gives examples of simple or compound events in an

experiment or simulation having probability of zero or one. 3. uses a fraction, decimal, and percent to represent the probability of

(2.4.K1c):a. a simple event in an experiment or simulation;b. a compound event composed of two independent events in an

experiment or simulation.4. finds the probability of a simple event in an experiment or simulation

using geometric models (2.4.K1i), e.g., using spinners or dartboards, what is the probability of landing on 2? The answer is ¼, .25, or 25%.

The student…1. conducts an experiment or simulation with a compound event

composed of two independent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions and make predictions about future events (2.4.A1j-k).

2. analyzes the results of an experiment or simulation of a compound event composed of two independent events to draw conclusions, generate convincing arguments, and make predictions and decisions in a variety of real-world situations (2.4.A1j-k).

3. compares results of theoretical (expected) probability with empirical (experimental) probability in an experiment or situation with a compound event composed of two simple independent events and understands that the larger the sample size, the greater the likelihood that the experimental results will equal the theoretical probability(2.4.A1j).

4. makes predictions based on the theoretical probability of a simple event in an experiment or simulation (2.4.A1j).

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Standard 4: Data SEVENTH GRADE


Benchmark 2: Statistics – The student collects, organizes, displays, and explains numerical (rational numbers) and

non-numerical data sets in a variety of situations with a special emphasis on measures of central

tendency.


The student…1. ▲ organizes, displays, and reads quantitative (numerical) and

qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays (2.4.K1j) ($):a. frequency tables;b. bar, line, and circle graphs;c. Venn diagrams or other pictorial displays;d. charts and tables;e. stem-and-leaf plots (single);f. scatter plots; g. box-and-whiskers plots.

2. selects and justifies the choice of data collection techniques (observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, or purposeful sampling) in a given situation.

3. conducts experiments with sampling and describes the results.4. determines the measures of central tendency (mode, median, mean)

for a rational number data set (2.4.K1a) ($).5. identifies and determines the range and the quartiles of a rational

number data set (2.4.K1a) ($).6. identifies potential outliers within a set of data by inspection rather

than formal calculation (2.4.K1a) ($), e.g., consider the data set of 1, 100, 101, 120, 140, and 170; the outlier is 1.

The student…1. uses data analysis (mean, median, mode, range) of a rational

number data set to make reasonable inferences and predictions, to analyze decisions, and to develop convincing arguments from these data displays (2.4.A1k) ($):a. frequency tables;b. bar, line, and circle graphs;c. Venn diagrams or other pictorial displays;d. charts and tables;e. stem-and-leaf plots (single);f. scatter plots; g. box-and-whiskers plots.

2. explains advantages and disadvantages of various data displays for a given data set (2.4.A1k) ($).

3. ▲ recognizes and explains (2.4.A1k):a. ■ misleading representations of data;b. the effects of scale or interval changes on graphs of data sets.

1. determines and explains the advantages and disadvantages of using each measure of central tendency and the range to describe a data set (2.4.A1a) ($).

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Graphs take many forms: – bar graphs and pictographs compare discrete data, – frequency tables show how many times a certain piece of data occurs, – circle graphs (pie charts) model parts of a whole, – line graphs show change over time, – Venn diagrams show relationships between sets of objects, – line plots show frequency of data on a number line, – single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side, – scatter plots show the relationship between two quantities, and– box-and-whisker plots are visual representations of the five-number summary – the median, the upper and lower quartiles, and the least and greatest values in the distribution – therefore, the center, the spread, and the overall range are immediately evident by looking at the plot.

Two important aspects of data are its center and its spread. The mean, median, and mode are measures of central tendency (averages) that describe where data are centered. Each of these measures is a single number that describes the data. However, each does it slightly differently. The range describes the spread (dispersion) of data. The easiest way to measure spread is the range, the difference between the greatest and the least values in a data set. Quartiles are boundaries that break the data into fourths.



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Standard 1: Number and Computation EIGHTH GRADE


Benchmark 1: Number Sense – The student demonstrates number sense for real numbers and simple algebraic expressions in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. knows, explains, and uses equivalent representations for rational

numbers and simple algebraic expressions including integers, fractions, decimals, percents, and ratios; rational number bases with integer exponents; rational numbers written in scientific notation with integer exponents; time; and money (2.4.K1a) ($).

2. compares and orders rational numbers, the irrational number pi, and algebraic expressions (2.4.K1a) ($), e.g., which expression is greater –3n or 3n? It depends on the value of n. If n is positive, 3n is greater. If n is negative, -3n is greater. If n is zero, they are equal.

3. explains the relative magnitude between rational numbers, the irrational number pi, and algebraic expressions (2.4.K1a).

4. recognizes and describes irrational numbers (2.4.K1a), e.g., √ 2 is a non-repeating, non-terminating decimal; or (pi) is a non-terminating decimal.

5. ▲ knows and explains what happens to the product or quotient when (2.4.K1a):a. a positive number is multiplied or divided by a rational number

greater than zero and less than one, e.g., if 24 is divided by 1/3, will the answer be larger than 24 or smaller than 24? Explain.

b. a positive number is multiplied or divided by a rational number greater than one, C

c. a nonzero real number is multiplied or divided by zero, (For purposes of assessment, an explanation of division by zero will not be expected.)

6. explains and determines the absolute value of real numbers (2.4.K1a).

The student…1. generates and/or solves real-world problems using equivalent

representations of rational numbers and simple algebraic expressions (2.4.A1a) ($), e.g., a paper reports a company’s gross income as $1.2 billion and their total expenses as $30,450,000. What is the company’s net profit?

2. determines whether or not solutions to real-world problems using rational numbers, the irrational number pi, and simple algebraic expressions are reasonable (2.4.A1a) ($), e.g., the city park is putting a picket fence around their circular rose garden. The garden has a diameter of 7.5 meters. The planner wants to buy 20 meters of fencing. Is this reasonable?

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The student with number sense will look at a problem holistically before confronting the details of the problem. The student will look for relationships among the numbers and operations and will consider the context in which the question was posed. Students with number sense will choose or even invent a method that takes advantage of their own understanding of the relationships between numbers and between numbers and operations, and they will seek the most efficient representation for the given task. Number sense can also be recognized in the students' use of benchmarks to judge number magnitude (e.g., 2/5 of 49 is less that half of 49), to recognize unreasonable results for calculations, and to employ non-standard algorithms for mental computation and estimation. (Developing Number Sense: Addenda Series, Grades 5-8, NCTM, 1991)

If you think about division as undoing multiplication, you’ll see why we don’t divide by zero. EXAMPLE: Try to divide 245 ÷ 0. To divide, think of the related multiplication equation. 245 ÷ 0 = ? asks the same question as ? x 0 = 245. When you think about it this way, you can see that you’re really stuck because any number times zero is zero! So, mathematicians say that division by zero is undefined. Math On Call 226. Don’t try to divide by Zero.

At this grade level, real numbers include positive and negative numbers and very large numbers (one billion) and very small numbers (one-millionth). Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close, or about the same? For example, using the numbers 219, 264, and 457, answer questions such as:




The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at three grade levels, the indicators in the Kansas Curricular Standards for Mathematics that correlate with the National Standards in Personal Finance are indicated at each grade level with a ($). The National Standards in Personal Finance are included in the Appendix.8-260




Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the real number system; recognizes, applies, and explains their properties; and extends these

properties to algebraic expressions.

Eighth Grade Knowledge Base Indicators Eighth Grade Application Indicators

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The student…1. explains and illustrates the relationship between the subsets of the

real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] using mathematical models (2.4.K1a), e.g., number lines or Venn diagrams.

2. ▲ identifies all the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] to which a given number belongs (2.4.K1l). (For the purpose of assessment, irrational numbers will not be included.)

3. names, uses, and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($):a. commutative, associative, distributive, and substitution

properties [commutative: a + b = b + a and ab = ba; associative: a + (b + c) = (a + b) + c and a(bc) = (ab)c; distributive: a(b + c) = ab + ac; substitution: if a = 2, then 3a = 3 x 2 = 6];

b. identity properties for addition and multiplication and inverse properties of addition and multiplication (additive identity: a + 0 = a, multiplicative identity: a • 1 = a, additive inverse: +5 + –5 = 0, multiplicative inverse: 8 x 1/8 = 1);

c. symmetric property of equality, e.g., 7 + 2 = 9 has the same meaning as 9 = 7 + 2;

d. addition and multiplication properties of equalities, e.g., if a = b, then a + c = b + c;

e. addition property of inequalities, e.g., if a > b, then a + c > b + c; f. zero product property, e.g., if ab = 0, then a = 0 and/or b = 0.


using the concepts of these properties to explain reasoning (2.4.A1a) ($):a. ▲ commutative, associative, distributive, and substitution

properties; e.g., we need to place trim around the outside edges of a bulletin board with dimensions of 3 ft by 5 ft. Explain two different methods of solving this problem and why the answers are equivalent.

b. ▲ identity and inverse properties of addition and multiplication; e.g., I had $50. I went to the mall and spent $20 in one store, $25 at a second store and then $5 at the food court. To solve: [$50 – ($20 + $25 + $5) = $50 - $50 = 0]. Explain your reasoning.

c. symmetric property of equality; e.g., Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $15 = $10 + $5 is the same as $10 + $5 = $15

d. addition and multiplication properties of equality; e.g., the total price (P) of a car, including tax (T), is $14, 685. 33. If the tax is $785.42, what is the sale price of the car (S)?

e. zero product property, e.g., Jenny was thinking of two numbers. Jenny said that the product of the two numbers was 0. What could you deduct from this statement? Explain your reasoning

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2. analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), or decimals in solving a given real-world problem (2.4.A1a) ($), e.g., in the store everything is 33 1/3% off. When calculating the discount, which representation of 33 1/3% would you use and why?





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Benchmark 3: Estimation – The student uses computational estimation with real numbers in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. estimates real number quantities using various computational

methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a) ($).

2. uses various estimation strategies and explains how they were used to estimate real number quantities and simple algebraic expressions (2.4.K1a) ($).

3. knows and explains why a decimal representation of the irrational number pi is an approximate value (2.4.K1c).

4. knows and explains between which two consecutive integers an irrational number lies (2.4.K1a).


based on additional information (a frame of reference) (2.4.A1a) ($), e.g., estimate the height of a building from a picture. In another picture, a person is standing next to the building. By using the person as a frame of reference adjust your original estimate.

2. estimates to check whether or not the result of a real-world problem using rational numbers and/or simple algebraic expressions is reasonable and makes predictions based on the information (2.4.A1a) ($), e.g., you have a $4,000 debt on a credit card. You pay the minimum of $30 per month. Is it reasonable to pay off the debt in 10 years?

3. determines a reasonable range for the estimation of a quantity given a real-world problem and explains the reasonableness of the range (2.4.A1c) ($), e.g., determine the reasonable range for the weight of a book and explain why this range is reasonable.

4. determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental mathematics, paper and pencil, concrete objects, and/or appropriate technology (2.4.A1a) ($), e.g., do you need an exact or an approximate answer when calculating the area of the walls in a room to determine the number of rolls of wallpaper needed to paper the room?. An approximation is appropriate for the area but an exact answer is needed for the number of roles. What would you do if you were wallpapering 2 rooms?

5. explains the impact of estimation on the result of a real-world problem (underestimate, overestimate, range of estimates)

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(2.4.A1a) ($), e.g., you are estimating the total of three large purchases ($489, $553, and $92). If you rounded each to the nearest $10, would your estimate be slightly lower or higher than the actual amount spent? If you rounded each to the nearest $100, would your estimate be slightly lower or higher than the actual amount spent?

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Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible. In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation strategies include front-end with adjustment, compatible “nice” numbers, clustering, special numbers, or truncation.

Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, hundred charts, base ten blocks, or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator.


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Benchmark 4: Computation – The student models, performs, and explains computation with rational numbers, the irrational number pi, and algebraic expressions in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. computes with efficiency and accuracy using various computational


2. performs and explains these computational procedures with rational numbers (2.4.K1a):a. ▲N addition, subtraction, multiplication, and division of integers b. ▲N order of operations (evaluates within grouping symbols,

evaluates powers to the second or third power, multiplies or divides in order from left to right, then adds or subtracts in order from left to right);

c. approximation of roots of numbers using calculators;d. multiplication or division to find:

i. a percent of a number, e.g., what is 0.5% of 10?ii. percent of increase and decrease, e.g., if two coins are

removed from ten coins, what is the percent of decrease? iii. percent one number is of another number, e.g., what

percent of 80 is 120?iv. a number when a percent of the number is given, e.g.,

15% of what number is 30?e. addition of polynomials, e.g., (3x – 5) + (2x + 8).f. simplifies algebraic expressions in one variable by combining

like terms or using the distributive property (2.4.K1a), e.g., –3(x – 4) is the same as –3x + 12.

3. finds factors and common factors of simple monomial expressions (2.4.K1d), e.g., given the monomials 10m2n3 and 15a2mn2 some common factors would be 5m, 5mn2, and n2.

The student…1. ▲ generates and/or solves one- and two-step real-world

problems using computational procedures and mathematical concepts (2.4.A1a) with ($):a. ■ rational numbers, e.g., find the height of a triangular

garden given that the area to be covered is 400 square feet with a base of 12½ feet;

b. the irrational number pi as an approximation, e.g., before planting, a farmer plows a circular region that has an approximate area of 7,300 square feet. What is the radius of the circular region to the nearest tenth of a foot?

c. applications of percents, e.g., sales tax or discounts. (For the purpose of assessment, percents greater than or equal to 100% will NOT be used).

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Teacher Notes: Efficiency and accuracy means that students are able to compute single-digit numbers with fluency. Students increase their understanding and skill in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction, addition and multiplication, multiplication and division, and subtraction and division. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling, repeated doubling, halving, making tens, multiplying by powers of ten, dividing with tens, finding fractional parts, thinking money, and using compatible “nice” numbers.




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Standard 2: Algebra EIGHTH GRADE


Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule of a pattern from a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…2. identifies, states, and continues a pattern presented in various

formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes:f. counting numbers including perfect squares, cubes, and factors

and multiples with positive rational numbers (number theory) (2.4.K1a).

g. rational numbers including arithmetic and geometric sequences (arithmetic: sequence of numbers in which the difference of two consecutive numbers is the same, geometric: a sequence of numbers in which each succeeding term is obtained by multiplying the preceding term by the same number) (2.4.K1a), e.g., 1, 1, 3, …;

4 2 4c. geometric figures (2.4.K1h);d. measurements (2.4.K1a);e. things related to daily life ($);f. variables and simple expressions, e.g., 1 – x, 2 – x, 3 – x, 4 – x,

...; or x, x2, x3, ...2. generates and explains a pattern (2.4.K1a). 3. generates a pattern limited to two operations (addition, subtraction,

multiplication, division, exponents) when given the rule for the nth term (2.4.K1a), e.g., the nth term is n2+1, find the first 4 terms beginning with n = 1; the terms are 2, 5, 10, and 17.

4. states the rule to find the nth term of a pattern using explicit symbolic notation (2.4.K1a), e.g., given 2, 5, 8, 11, …; find the rule for the nth term, the rule is 3n –1.

The student…1. generalizes numerical patterns using algebra and then translates

between the equation, graph, and table of values resulting from the generalization (2.4.A1d-e,j) ($), e.g., water is billed at $1.00 per 1,000 gallons, plus a basic fee of $10 per month for being connected to the water district.

1,000 Gallons | Cost in a given month Graph these: 1 | $10 + 1*1.00 (1, 11) 2 | $10 + 2*.1.00 (2, 12) . | . . | . n | 10 + n*1.00 (n, 1.00n + 10)

where C = total cost and G = gallons used, C = 1.00 G + 10]

(Title/Labels needed)

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5. describes the pattern when given a table of linear values and plots the ordered pairs on a coordinate plane (2.4.K1f-g), e.g., in the table below, the pattern could be described as the x-coordinates are increasing by three, while the y-coordinates are increasing by 6, or the x is doubled and one is added to find the y.

2. recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1a,j) ($).




This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain.

Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, greatest common factor and least common multiple, and sequences.

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X 2 5 8 11Y 5 11 17 23



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Benchmark 2: Variable, Equations, and Inequalities – The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. identifies independent and dependent variables within a given

situation.2. simplifies algebraic expressions in one variable by combining like

terms or using the distributive property (2.4.K1a), e.g., –3(x – 4) is the same as – 3 x + 12.

3. solves (2.4.K1a,e) ($): a. ▲ one- and two-step linear equations in one variable with

rational number coefficients and constants intuitively and/or analytically;

b. one-step linear inequalities in one variable with rational number coefficients and constants intuitively, analytically, and graphically;

c. systems of given linear equations with whole number coefficients and constants graphically.

4. knows and describes the mathematical relationship between ratios, proportions, and percents and how to solve for a missing monomial or binomial term in a proportion (2.4.K1c), e.g., 2 = _ 1_

5 x + 2.5. represents and solves algebraically ($):

a. the number when a percent and a number are given, b. what percent one number is of another number,c. percent of increase or decrease, e.g., the price of a loaf of

bread is $2.00. With a coupon, the cost is $1.00. What is the percent of decrease?

6. evaluates formulas using substitution ($).

The student…1. represents real-world problems using (2.4.A1d) ($):

a. ▲ ■ variables, symbols, expressions, one- or two-step equations with rational number coefficients and constants, e.g., today John is 3.25 inches more than half his sister’s height. If J = John’s height, and S = his sister’s height, then J = 0.5S + 3.25.

b. one-step inequalities with rational number coefficients and constants, e.g., after Randy paid $38.50 for a watch, he did not have enough money to by a calculator for $5.50. Represent this situation with an inequality.

c. systems of linear equations with whole number coefficients and constants, e.g., two students collected the same amount of money for a walk-a-thon. One student received $5 per mile and a donation of $10, while the other student received $2 per mile and a donation of $20. How many miles did they walk?

2. solves real-world problems with two-step linear equations in one variable with rational number coefficients and constants and rational solutions intuitively, analytically, and graphically (2.4.A1e) e.g., Mike and Albert are friends, but Joe and Albert are not friends. Which of the following diagrams can be used to describe this situation? (Three dots labeled J, M, A: there is a line between J and M and line between M and A, but no line between J and A.)

3. generates real-world problems that represent (2.4.A1d) ($):a. one- or two-step linear equations, ($), e.g., given the equation 2x

+ 10 = 30, the problem could be I bought two shirts and a pair of pants for $10. How much was a shirt, if the total bill was $30?

b. one-step linear inequalities, e.g., write a real-world situation that represents the inequality x + 10 > 30. The problem could be: If you give me $10, I will have more than $30.

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4. explains the mathematical reasoning that was used to solve a real-world problem using one- or two-step linear equations and inequalities and discusses the advantages and disadvantages to various strategies that may have been used to solve the problem, (2.4.A1d) ($), e.g., given the inequality x + 10 > 30, subtract the same number from both sides of the inequality or graph as y1 = x + 10 and y = 30 and find on the graph where y1 is less than y2. The first method gives an exact answer; the second method is a visual representation and can be used to solve more difficult inequalities.

Teacher Notes: Understanding the concept of variable is fundamental to algebra. Students use various symbols, including letters and geometric shapes to represent unknown quantities that both do and do not vary. Quantities that are not given and do not vary are often referred to as unknowns or missing elements when they appear in equations, e.g., 2 + Δ = 4 or 3 • s = 15 where a triangle and s are used as variables. Various symbols or letters should be used interchangeably in equations.



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Benchmark 3: Functions – The student recognizes, describes, and analyzes constant, linear, and nonlinear relationships in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. recognizes and examines constant, linear, and nonlinear

relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology (2.4.K1a,e-g) ($).

2. knows and describes the difference between constant, linear, and nonlinear relationships (2.4.K1g).

3. explains the concepts of slope and x- and y-intercepts of a line (2.4.K1g).

4. recognizes and identifies the graphs of constant and linear functions (2.4.K1g) ($).

5. identifies ordered pairs from a graph, and/or plots ordered pairs using a variety of scales for the x- and y-axis (2.4.K1g).

The student…1. represents a variety of constant and linear relationships using written

or oral descriptions of the rule, tables, graphs, and symbolic notation (2.4.A1d-f) ($).

2. interprets, describes, and analyzes the mathematical relationships of numerical, tabular, and graphical representations (2.4.A1j) ($).

3. ▲ translates between the numerical, tabular, graphical, and symbolic representations of linear relationships with integer coefficients and constants (2.4.A1a), e.g., a fish tank is being filled with water with a 2-liter jug. There are already 5 liters of water in the fish tank. Therefore, you are showing how full the tank is as you empty 2-liter jugs of water into it. Y = 2x + 5 (symbolic) can be represented in a table (tabular) –

X 0 1 2 3Y 5 7 9 11

and as a graph (graphical) –

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In the Cartesian Coordinate System, the expression x = 3 cannot represent a function. When you graph all points where x = 3 you get a vertical line, so more than one y-value exists for the x-value x = 3. By definition, a function must have only one output value for any given input. For more information, consult a reference book under the topic, vertical line test.


The National Standards in Personal Finance identify what K-12 students should know and be able to do in personal finance; benchmarks are provided at three grade levels (grades 4, 8, and 12) and are grouped into four major categories: Income, Spending and Credit, Saving and Investing, and Money Management. Although the National Standards in Personal Finance are benchmarked at 8-276




Benchmark 4: Models – The student generates and uses mathematical models to represent and justify mathematical relationships found in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. knows, explains, and uses mathematical models to represent and

explain mathematical concepts, procedures, and relationships. Mathematical models include:

h. process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K1-6, 1.2.K1, 1.2.K3, 1.3.K1-2, 1.3.K4, 1.4.K1-2, 2.1.K1a-b, 2.1.K1d-e, 2.1.K2-4, 2.2.K2-3, 3.1.K9, 3.2.K1-4, 3.3.K1-4, 3.4.K4, 4.2.K4-5) ($);

i. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures ($);

j. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.3.K3, 2.3.K6) ($):

k. factor trees to model least common multiple, greatest common factor, and prime factorization (1.4.K3);

l. equations and inequalities to model numerical relationships (2.2.K3, (3.4.K2) ($);

m. function tables to model numerical and algebraic relationships (2.1.K5, 3.4.K2) ($) ;

n. coordinate planes to model relationships between ordered pairs and linear equations and inequalities (2.1.K5, 2.3.K1-5, 3.4.K2-3) ($);


represent the same problem situation. Mathematical models include: a. process models (concrete objects, pictures, diagrams,

flowcharts, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations (1.1.A1-2, 1.2.A1-2, 1.3.A1-5, 1.4.A1, 2.1.A1, 3.1.A1, 3.2.A1-2, 3.3.A1, 3.4.A1-2) ($);

b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations ($);

c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (3.2.A3) ($);

d. equations and inequalities to model numerical relationships ( 2.1.A2, 2.2.A1-2, 2.3.A1, 3.4.A2) ($);e. function tables to model numerical and algebraic relationships

(2.1.A2, 2.3.A2, 3.4.A2) ($);f. coordinate planes to model relationships between ordered pairs

and linear equations and inequalities (2.3.A1 3.4.A2) ($);g. two- and three-dimensional geometric models (geoboards, dot

paper, nets, or solids) and real-world objects to model perimeter, area, volume, surface area and properties of two- and three-dimensional figures (3.3.A3, 3.4.A2);

h. scale drawings to model large and small real-world objects (3.1.A1-2, 3.3.A4);

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o. two- and three-dimensional geometric models (geoboards, dot paper, nets, or solids) and real-world objects to model perimeter, area, volume, surface area, and properties of two-and three-dimensional figures (2.1.K1c, 3.1.K1-6, 3.1.K8, 3.1.K10, 3.2.K5, 3.3.K4-5);

p. scale drawings to model large and small real-world objects (3.3.K3-4);

q. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.K1-5) ($);

r. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, and histograms to organize and display data (4.2.K1, 4.2.K6) ($);

s. Venn diagrams to sort data and to show relationships (1.2.K2).

i. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.A1-4);

j. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, and histograms to describe, interpret, and analyze data (2.1.A1-2, 2.3.A2-3, 4.2.A1, 4.2.A3, 4.2.A1-3) ($);

k. Venn diagrams to sort data and to show relationships.2. ▲ determines if a given graphical, algebraic, or geometric model is

an accurate representation of a given real-world situation ($).3. uses the mathematical modeling process to analyze and make

inferences about real-world situations ($).

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The mathematical modeling process involves:i. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through

some sort of mathematical model,j. performing manipulations and mathematical procedures within the mathematical model,k. interpreting the results of the manipulations within the mathematical model, l. using these results to make inferences about the original real-world situation.




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Standard 3: Geometry EIGHTH GRADE



Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. recognizes and compares properties of two- and three-dimensional


2. discusses properties of triangles and quadrilaterals related to (2.4.K1h):h. sum of the interior angles of any triangle is 180°;i. sum of the interior angles of any quadrilateral is 360°;a. parallelograms have opposite sides that are parallel and

congruent, opposite angles are congruent;b. rectangles have angles of 90°, sides may or may not be equal;c. rhombi have all sides equal in length, angles may or may not be

equal;d. squares have angles of 90°, all sides congruent;e. trapezoids have one pair of opposite sides parallel and the other

pair of opposite sides are not parallel;f. kites have two distinct pairs of adjacent congruent sides.

3. recognizes and describes the rotational symmetries and line symmetries that exist in two-dimensional figures (2.4.K1h), e.g., draw a picture with a line of symmetry in it. Explain why it is a line of symmetry.

4. recognizes and uses properties of corresponding parts of similar and congruent triangles and quadrilaterals to find side or angle measures using standard notation for similarity (~) and congruence ( ) (2.4.K1h).

5. knows and describes Triangle Inequality Theorem to determine if a triangle exists (2.4.K1h).

The student…1. solves real-world problems by (2.4.A1a):

a. ▲ ■ using the properties of corresponding parts of similar and congruent figures, e.g., scale drawings, map reading, proportions, or indirect measurements.

b. applying the Pythagorean Theorem, e.g., indirect measurements, map reading/distance, or diagonals.

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6. ▲ uses the Pythagorean theorem to (2.4.K1h):a. determine if a triangle is a right triangle, b. find a missing side of a right triangle where the lengths of

all three sides are whole numbers.7. recognizes and compares the concepts of a point, line, and plane.8. describes the intersection of plane figures, e.g., two circles could

intersect at no point, one point, two points, or all points. 9. describes and explains angle relationships:

a. when two lines intersect including vertical and supplementary angles;

b. when formed by parallel lines cut by a transversal including corresponding, alternate interior, and alternate exterior angles.

10. recognizes and describes arcs and semicircles as parts of a circle and uses the standard notation for arc () and circle () (2.4.K1h).

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The application of the Knowledge Indicators from the Geometry Benchmark, Geometric Figures and Their Properties are most often applied within the context of the other Geometry Benchmarks - Measurement and Estimation, Transformational Geometry, and Geometry From an Algebraic Perspective - rather than in isolation.



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Benchmark 2: Measurement and Estimation – The student estimates, measures, and uses geometric formulas in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student… 1. determines and uses rational number approximations (estimations)

for length, width, weight, volume, temperature, time, perimeter, area, and surface area using standard and nonstandard units of measure (2.4.K1a) ($).

2. selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate real number representations for length, weight, volume, temperature, time, perimeter, area, surface area, and angle measurements (2.4.K1a) ($).

3. converts within the customary system and within the metric system.4. estimates the measure of a concrete object in one system given the

measure of that object in another system and the approximate conversion factor (2.4.K1a), e.g., a mile is about 2.2 kilometers; how far is 2 miles?

5. uses given measurement formulas to find (2.4.K1h): a. area of parallelograms and trapezoids;b. surface area of rectangular prisms, triangular prisms, and

cylinders;c. volume of rectangular prisms, triangular prisms, and cylinders.

6. recognizes how ratios and proportions can be used to measure inaccessible objects (2.4.K1c), e.g., using shadows to measure the height of a flagpole.

7. calculates rates of change, e.g., speed or population growth.

The student…1. solves real-world problems (2.4.A1a) by ($):

a. converting within the customary and the metric systems, e.g., James added 30 grams of sand to his model boat that weighed 2 kg before it sank. With the sand included, what is the total weight of his boat?

b. finding perimeter and area of circles, squares, rectangles, triangles, parallelograms, and trapezoids; e.g., Jane jogs on a circular track with a radius of 100 feet. How far would she jog in one lap?

c. finding the volume and surface area of rectangular prisms, e.g., how much paint would be needed to cover a box with dimensions of 3 feet by 4 feet by 5 feet?

2. estimates to check whether or not measurements or calculations for length, weight, volume, temperature, time, perimeter, area, and surface area in real world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) ($), e.g., to check your calculation in finding the area of the floor in the kitchen; you count how many foot-square tiles there are on the floor.

3. uses ratio and proportion to measure inaccessible objects (2.4.A1c), e.g., using the length of a shadow to measure the height of a flagpole.

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Benchmark 3: Transformational Geometry – The student recognizes and applies transformations on geometric figures in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. identifies, describes, and performs single and multiple

transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on a two-dimensional figure (2.4.K1a).

2. describes a reflection of a given two-dimensional figure that moves it from its initial placement (preimage) to its final placement (image) in the coordinate plane over the x- and y-axis (2.4.K1a,i).

3. draws (2.4.K1a):a. three-dimensional figures from a variety of perspectives (top,

bottom, sides, corners);b. a scale drawing of a two-dimensional figure;c. a two-dimensional drawing of a three-dimensional figure.

4. determines where and how an object or a shape can be tessellated using single or multiple transformations (2.4.K1a).

The student…1. generalizes the impact of transformations on the area and perimeter

of any two-dimensional geometric figure (2.4.A1a), e.g., enlarging by a factor of three triples the perimeter (circumference) and multiplies the area by a factor of nine.

2. describes and draws a two-dimensional figure after undergoing two specified transformations without using a concrete object.

3. investigates congruency, similarity, and symmetry of geometric figures using transformations (2.4.A1g).

4. uses a scale drawing to determine the actual dimensions and/or measurements of a two-dimensional figure represented in a scale drawing (2.4.A1h).

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Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do not change the figure itself. Other transformations, like reduction (contraction/shrinking) or enlargement (magnification/growing), change the size of a figure, but not the shape (congruence vs. similarity).



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Benchmark 4: Geometry from an Algebraic Perspective – The student uses an algebraic perspective to examine the geometry of two-dimensional figures in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. uses the coordinate plane to (2.4.K1a):

a. ▲ list several ordered pairs on the graph of a line and find the slope of the line;

b. ▲ recognize that ordered pairs that lie on the graph of an equation are solutions to that equation;

c. ▲ recognize that points that do not lie on the graph of an equation are not solutions to that equation;

d. ▲ determine the length of a side of a figure drawn on a coordinate plane with vertices having the same x- or y-coordinates;

e. solve simple systems of linear equations.2. uses a given linear equation with integer coefficients and constants

and an integer solution to find the ordered pairs, organizes the ordered pairs using a T-table, and plots the ordered pairs on a coordinate plane (2.4.K1e-g).

3. examines characteristics of two-dimensional figures on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.A1g).

The student…1. represents, generates, and/or solves distance problems (including

the use of the Pythagorean theorem, but not necessarily the distance formula) (2.4.A1a), e.g., a student lives five miles west and three miles north of school and another student lives 4 miles south and 7 miles east of school. What is the shortest distance between the students’ homes (as the crow flies)?

2. translates between the written, numeric, algebraic, and geometric representations of a real-world problem (2.4.A1a,d-g), e.g., given a situation: make a T-table, define the algebraic relationship, and graph the ordered pairs. The T-table can be represented as – as an algebraic relationship – 2 χ = 5,

X 0 1 2 3

and as a graph (graphical)

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Y 5 7 9 11





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Standard 4: Data EIGHTH GRADE


Benchmark 1: Probability – The student applies the concepts of probability to draw conclusions, generate convincing arguments, and make predictions and decisions including the use of concrete objects in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. knows and explains the difference between independent and

dependent events in an experiment, simulation, or situation (2.4.K1j) ($).

2. identifies situations with independent or dependent events in an experiment, simulation, or situation (2.4.K1j), e.g., there are three marbles in a bag. If you draw one marble and give it to your brother, and another marble and give it to your sister, are these independent events or dependent events?

3. ▲ finds the probability of a compound event composed of two independent events in an experiment, simulation, or situation (2.4.K1j), e.g., what is the probability of getting two heads, if you toss a dime and a quarter?

4. finds the probability of simple and/or compound events using geometric models (spinners or dartboards) (2.4.K1j).

5. finds the odds of a desired outcome in an experiment or simulation and expresses the answer as a ratio (2/3 or 2:3 or 2 to 3) (2.4.K1j).

6. describes the difference between probability and odds.

The student…1. conducts an experiment or simulation with independent or

dependent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions and make predictions about future events (2.4.A1i-j).

2. analyzes the results of an experiment or simulation of two independent events to generate convincing arguments, draw conclusions, and make predictions and decisions in a variety of real-world situations (2.4.A1i-j).

3. compares theoretical probability (expected results) with empirical probability (experimental results) in an experiment or simulation with a compound event composed of two independent events and understands that the larger the sample size, the greater the likelihood that the experimental results will equal the theoretical probability (2.4.A1i).

4. makes predictions based on the theoretical probability of (2.4.A1a,i): a. ▲ ■ a simple event in an experiment or simulation,b. compound events composed of two independent events in an

experiment or simulation.

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Odds is a ratio of favorable to unfavorable outcomes whereas probability is a ratio of favorable outcomes to all possible outcomes. Probability can be written as a fraction, a decimal, and a percent, whereas odds can only be written as a ratio, e.g., 2/3, 2:3, or 2 to 3.



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Standard 4: Data EIGHTH GRADE


Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (rational) and non-numerical data sets in a variety of situations.

Eighth Grade Knowledge Base Indicators Eighth Grade Application IndicatorsThe student…1. organizes, displays and reads quantitative (numerical) and

qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays (2.4.K1k) ($): a. frequency tables;b. bar, line, and circle graphs;c. Venn diagrams or other pictorial displays;d. charts and tables;e. stem-and-leaf plots (single and double);f. scatter plots;g. box-and-whiskers plots; h. histograms.

2. recognizes valid and invalid data collection and sampling techniques.

3. ▲ determines and explains the measures of central tendency (mode, median, mean) for a rational number data set (2.4.K1a).

4. determines and explains the range, quartiles, and interquartile range for a rational number data set (2.4.K1a).

5. explains the effects of outliers on the median, mean, and range of a rational number data set (2.4.K1a).

6. makes a scatter plot and draws a line that approximately represents the data, determines whether a correlation exists, and if that correlation is positive, negative, or that no correlation exists (2.4.K1k).

The student…1. uses data analysis (mean, median, mode, range) in real-world

problems with rational number data sets to compare and contrast two sets of data, to make accurate inferences and predictions, to analyze decisions, and to develop convincing arguments from these data displays (2.4.A1j) ($): a. frequency tables;b. bar, line, and circle graphs;c. Venn diagrams or other pictorial displays;d. charts and tables;e. stem-and-leaf plots (single and double);f. scatter plots;g. box-and-whiskers plots; h. histograms.

5. explains advantages and disadvantages of various data collection techniques (observations, surveys, or interviews), and sampling techniques (random sampling, samples of convenience, biased sampling, or purposeful sampling) in a given situation (2.4.A1j) ($).

6. recognizes and explains (2.4.A1j):a. misleading representations of data;b. the effects of scale or interval changes on graphs of data sets.

4. recognizes faulty arguments and common errors in data analysis.

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Graphs take many forms: – bar graphs and pictographs compare discrete data, – frequency tables show how many times a certain piece of data occurs, – circle graphs (pie charts) model parts of a whole, – line graphs show change over time, – Venn diagrams show relationships among sets of objects, – line plots show frequency of data on a number line, – single stem-and-leaf plots (closely related to line plots except that the number line is usually vertical and digits are used rather than x’s) show frequency distribution by arranging numbers (stems) on the left side of a vertical line with numbers (leaves) on the right side, – scatter plots show the relationship between two quantities, – box-and-whisker plots are visual representations of the five-number summary – the median, the upper and lower quartiles, and the least and greatest values in the distribution – therefore, the center, the spread, and the overall range are immediately evident by looking at the plot, and – histograms (closely related to stem-and-leaf plots) describe how data falls into different ranges.

Two important aspects of data are its center and its spread. The mean, median, and mode are measures of central tendency (averages) that describe where data are centered. Each of these measures is a single number that describes the data. However, each does it slightly differently. The range describes the spread (dispersion) of data. The easiest way to measure spread is the range, the difference between the greatest and the least values in a data set. Quartiles are boundaries that break the data into fourths.



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Benchmark 1: Number Sense – The student demonstrates number sense for real numbers and algebraic expressions in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. knows, explains, and uses equivalent representations for real

numbers and algebraic expressions including integers, fractions, decimals, percents, ratios; rational number bases with integer exponents; rational numbers written in scientific notation; absolute value; time; and money (2.4.K1a) ($), e.g., –4/2 = (–2); a(-2) b(3) = b3/a2.

2. compares and orders real numbers and/or algebraic expressions and explains the relative magnitude between them (2.4.K1a) ($), e.g., will (5n)2 always, sometimes, or never be larger than 5n? The student might respond with (5n)2 is greater than 5n if n > 1 and (5n)2 is smaller than 5 if o < n < 1.

3. knows and explains what happens to the product or quotient when a real number is multiplied or divided by (2.4.K1a):a. a rational number greater than zero and less than one, b. a rational number greater than one, c. a rational number less than zero.

The student…1. generates and/or solves real-world problems using equivalent

representations of real numbers and algebraic expressions (2.4.A1a) ($), e.g., a math classroom needs 30 books and 15 calculators. If B represents the cost of a book and C represents the cost of a calculator, generate two different expressions to represent the cost of books and calculators for 9 math classrooms.

2. determines whether or not solutions to real-world problems using real numbers and algebraic expressions are reasonable (2.4.A1a) ($), e.g., in January, a business gave its employees a 10% raise. The following year, due to the sluggish economy, the employees decided to take a 10% reduction in their salary. Is it reasonable to say they are now making the same wage they made prior to the 10% raise?

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▲– Assessed Indicator■ – Assessed Indicator on the Optional Response AssessmentN – Noncalculator($) – Financial Literacy THESE STANDARDS ARE ALIGNED ONLY TO THE ASSESSMENTS THAT WILL BEGIN DURING THE 2005-06 SCHOOL YEAR.

Standard 1: Number and Computation NINTH AND TENTH GRADES


At this grade level, real numbers include positive and negative numbers and very large numbers (one billion) and very small numbers (one-billionth). Relative magnitude refers to the size relationship one number has with another – is it much larger, much smaller, close, or about the same? For example, using the numbers 219, 264, and 457, answer questions such as





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Benchmark 2: Number Systems and Their Properties – The student demonstrates an understanding of the real number system; recognizes, applies, and explains their properties, and extends these properties to algebraic expressions.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. explains and illustrates the relationship between the subsets of the

real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] using mathematical models (2.4.K1a), e.g., number lines or Venn diagrams.

2. identifies all the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] to which a given number belongs (2.4.K1m).

3. ▲ names, uses, and describes these properties with the real number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($):a. commutative (a + b = b + a and ab = ba), associative [a + (b + c)

= (a + b) + c and a(bc) = (ab)c], distributive [a (b + c) = ab + ac], and substitution properties (if a = 2, then 3a = 3 x 2 = 6);

b. identity properties for addition and multiplication and inverse properties of addition and multiplication (additive identity: a + 0 = a, multiplicative identity: a • 1 = a, additive inverse: +5 + –5 = 0, multiplicative inverse: 8 x 1/8 = 1);

c. symmetric property of equality (if a = b, then b = a); d. addition and multiplication properties of equality (if a = b, then a

+ c = b + c and if a = b, then ac = bc) and inequalities (if a > b, then a + c > b + c and if a > b, and c > 0 then ac > bc);

e. zero product property (if ab = 0, then a = 0 and/or b = 0).4. uses and describes these properties with the real number system

(2.4.K1a) ($):a. transitive property (if a = b and b = c, then a = c),

The student…1. generates and/or solves real-world problems with real numbers

using the concepts of these properties to explain reasoning (2.4.A1a) ($):a. commutative, associative, distributive, and substitution

properties, e.g., the chorus is sponsoring a trip to an amusement park. They need to purchase 15 adult tickets at $6 each and 15 student tickets at $4 each. How much money will the chorus need for tickets? Solve this problem two ways.

b. identity and inverse properties of addition and multiplication, e.g., the purchase price (P) of a series EE Savings Bond is found by the formula ½ F = P where F is the face value of the bond. Use the formula to find the face value of a savings bond purchased for $500.

c. symmetric property of equality, e.g., Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $ addition and multiplication properties of equality, e.g., the total price for the purchase of three shirts in $62.54 including tax. If the tax is $3.89, what is the cost of one shirt, if all shirts cost the same?

d. addition and multiplication properties of equality, e.g., the total price for the purchase of three shirts is $62.54 including tax. If the tax is $3.89, what is the cost of one shirt?

T = 3s + t $62.54 = 3s + $3.89 - $3.89

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b. reflexive property (a = a). $62.54 - $3.89 = 3s $58.65 = 3s $58.65 = 3s = 3s ÷ 3 $19.55 = se. zero product property, e.g., Jenny was thinking of two numbers.

Jenny said that the product of the two numbers was 0. What could you deduct from this statement? Explain your reasoning.

2. analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals or irrational numbers and their rational approximations in solving a given real-world problem (2.4.A1a) ($), e.g., a store sells CDs for $12.99 each. Knowing that the sales tax is 7%, Marie estimates the cost of a CD plus tax to be $14.30. She selects nine CDs. The clerk tells Marie her bill is $157.18. How can Marie explain to the clerk she has been overcharged?





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Benchmark 3: Estimation – The student uses computational estimation with real numbers in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. estimates real number quantities using various

computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a) ($).

2. uses various estimation strategies and explains how they were used to estimate real number quantities and algebraic expressions (2.4.K1a) ($).

3. knows and explains why a decimal representation of an irrational number is an approximate value(2.4.K1a).

4. knows and explains between which two consecutive integers an irrational number lies (2.4.K1a).

The student…1. ▲ adjusts original rational number estimate of a real-world problem

based on additional information (a frame of reference) (2.4.A1a) ($), e.g., estimate how long it takes to walk from here to there; time how long it takes to take five steps and adjust your estimate.

2. estimates to check whether or not the result of a real-world problem using real numbers and/or algebraic expressions is reasonable and makes predictions based on the information (2.4.A1a) ($), e.g., if you have a $4,000 debt on a credit card and the minimum of $30 is paid per month, is it reasonable to pay off the debt in 10 years?

3. determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational strategies including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.A1a) ($), e.g., do you need an exact or an approximate answer in calculating the area of the walls to determine the number of rolls of wallpaper needed to paper a room? What would you do if you were wallpapering 2 rooms?

4. explains the impact of estimation on the result of a real-world problem (underestimate, overestimate, range of estimates) (2.4.A1a) ($), e.g., if the weight of 25 pieces of paper was measured as 530.6 grams, what would the weight of 2,000 pieces of paper equal to the nearest gram? If the student were to estimate the weight of one piece of paper as about 20 grams and then multiply this by 2,000 rather than multiply the weight of 25 pieces of paper by 80; the answer would differ by about 2,400 grams. In general, multiplying or dividing by a rounded number will cause greater discrepancies than rounding after

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multiplying or dividing.



Mental math and estimation are distinct but related mathematical skills. Proficiency in mental math contributes to increased skill in estimation. In order for students to become more familiar with estimation, teachers should introduce estimation with examples where rounded or estimated numbers are used. Emphasis should be placed on real-world examples where only estimation is required, e.g., About how many hours do you sleep a night? Using the language of estimation is important, so students begin to realize that a variety of estimates (answers) are possible. In addition, when students are taught specific estimation strategies, they develop mental math and estimation skills easier. Estimation strategies include front-end with adjustment, compatible “nice” numbers, clustering, special numbers, truncation, or simulation.



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Benchmark 4: Computation – The student models, performs, and explains computation with real numbers and polynomials in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. computes with efficiency and accuracy using various computational


2. performs and explains these computational procedures (2.4.K1a):a. N addition, subtraction, multiplication, and division using the

order of operations b. multiplication or division to find ($):

i. a percent of a number, e.g., what is 0.5% of 10?ii. percent of increase and decrease, e.g., a college raises its

tuition form $1,320 per year to $1,425 per year. What percent is the change in tuition?

iii. percent one number is of another number, e.g., 89 is what percent of 82?

iv. a number when a percent of the number is given, e.g., 80 is 32% of what number?

c. manipulation of variable quantities within an equation or inequality (2.4.K1d), e.g., 5x – 3y = 20 could be written as 5x – 20 = 3y or 5x(2x + 3) = 8 could be written as 8/(5x) = 2x + 3;

d. simplification of radical expressions (without rationalizing denominators) including square roots of perfect square monomials and cube roots of perfect cubic monomials;

e. simplification or evaluation of real numbers and algebraic monomial expressions raised to a whole number power and algebraic binomial expressions squared or cubed;

f. simplification of products and quotients of real number and algebraic monomial expressions using the properties of

The student…1. generates and/or solves multi-step real-world problems with real

numbers and algebraic expressions using computational procedures (addition, subtraction, multiplication, division, roots, and powers excluding logarithms), and mathematical concepts with ($): a. ▲ applications from business, chemistry, and physics that

involve addition, subtraction, multiplication, division, squares, and square roots when the formulae are given as part of the problem and variables are defined (2.4.A1a) ($), e.g., given F = ma, where F = force in newtons, m = mass in kilograms, a = acceleration in meters per second squared. Find the acceleration if a force of 20 newtons is applied to a mass of 3 kilograms.

b. ▲ volume and surface area given the measurement formulas of rectangular solids and cylinders (2.4.A1f), e.g., a silo has a diameter of 8 feet and a height of 20 feet. How many cubic feet of grain can it store?

c. probabilities (2.4.A1h), e.g., if the probability of getting a defective light bulb is 2%, and you buy 150 light bulbs, how many would you expect to be defective?

d. ▲ ■ application of percents (2.4.A1a), e.g., given the formula

A = P(1+ )nt, A = amount, P= principal, r = annual interest, n =

number of compounding periods per year, t = number of years. If $1,000 is placed in a savings account with a 6% annual interest rate and is compounded semiannually, how much money will be in the account at the end of 2 years?

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exponents;g. matrix addition ($), e.g., when computing (with one operation) a

building’s expenses (data) monthly, a matrix is created to include each of the different expenses; then at the end of the year, each type of expense for the building is totaled;

h. scalar-matrix multiplication ($), e.g., if a matrix is created with everyone’s salary in it, and everyone gets a 10% raise in pay; to find the new salary, the matrix would be multiplied by 1.1.

3. finds prime factors, greatest common factor, multiples, and the least common multiple of algebraic expressions (2.4.K1b).

e. simple exponential growth and decay (excluding logarithms) and economics (2.4.A1a) ($), e.g., a population of cells doubles every 20 years. If there are 20 cells to start with, how long will it take for there to be more than 150 cells? or If the radiation level is now 400 and it decays by ½ or its half-life is 8 hours, how long will it take for the radiation level to be below an acceptable level of 5?

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Teacher Notes: Efficiency and accuracy means that students are able to compute with fluency. Students increase their understanding and skill in addition, subtraction, multiplication, and division by understanding the relationships between addition and subtraction, addition and multiplication, multiplication and division, and subtraction and division. Students learn basic number combinations and develop strategies for computing that makes sense to them. Through class discussions, students can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and these invented approaches often can be used with efficiency and accuracy. (Principles and Standards for School Mathematics, NCTM, 2000)

The definition of computation is finding the standard representation for a number. For example, 6 + 6, 4 x 3, 17 – 5, and 24 ÷ 2 are all representations for the standard representation of 12. Mental math is mentally finding the standard representation for a number – calculating in your head instead of calculating using paper and pencil or technology. One of the main reasons for teaching mental math is to help students determine if a computed/calculated answer is reasonable; in other words, using mental math to estimate to see if the answer makes sense. Students develop mental math skills easier when they are taught specific strategies. Mental math strategies include counting on, doubling, repeated doubling, halving, making tens, multiplying by powers of tens, dividing with tens, finding fractional parts, thinking money, and using compatible “nice” numbers.

Mathematical models such as concrete objects, pictures, diagrams, Venn diagrams, number lines, or factor trees are necessary for conceptual understanding and should be used to explain computational procedures. If a mathematical model can be used to represent the concept, the indicator in the Models benchmark is identified in the parentheses. For example, (2.4.K1a) refers to Standard 2 (Algebra), Benchmark 4 (Models), and Knowledge Indicator 1a (process models). Then, the indicator in the Models benchmark lists some of the mathematical models that could be used to teach the concept. In addition, each indicator in the Models benchmark is linked back to the other indicators. Those indicators are identified in the parentheses. For example, process models are linked to 1.1.K3, 1.2.K6, 1.3.K1, … with 1.1.K3 referring to Standard 1 (Number and Computation), Benchmark 1 (Number Sense), and Knowledge Indicator 3.Technology is changing mathematics and its uses. The use of technology including calculators and computers is an important part of growing up in a complex society. It is not only necessary to estimate appropriate answers accurately when required, but also it is also important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure. Technology does not replace the need to learn basic facts, to compute mentally, or to do reasonable paper-and-pencil computation. However, dividing a 5-digit number by a 2-digit number is appropriate with the exception of dividing by 10, 100, or 1,000 and simple multiples of each.


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Standard 2: Algebra NINTH AND TENTH GRADES


Benchmark 1: Patterns – The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. identifies, states, and continues the following patterns using various

formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and writtena. arithmetic and geometric sequences using real numbers and/or

exponents (2.4.K1a); e.g., radioactive half-lives;b. patterns using geometric figures (2.4.K1h);c. algebraic patterns including consecutive number patterns or

equations of functions, e.g., n, n + 1, n + 2, ... or f(n) = 2n – 1 (2.4.K1c,e);

d. special patterns (2.4.K1a), e.g., Pascal’s triangle and the Fibonacci sequence.

2. generates and explains a pattern (2.4.K1f). 3. classify sequences as arithmetic, geometric, or neither. 4. defines (2.4.K1a):

a. a recursive or explicit formula for arithmetic sequences and finds any particular term,

b. a recursive or explicit formula for geometric sequences and finds any particular term.

The student…1. recognizes the same general pattern presented in different

representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1i) ($).

2. solves real-world problems with arithmetic or geometric sequences by using the explicit equation of the sequence (2.4.K1c) ($), e.g., an arithmetic sequence: A brick wall is 3 feet high and the owners want to build it higher. If the builders can lay 2 feet every hour, how long will it take to raise it to a height of 20 feet? or a geometric sequence: A savings program can double your money every 12 years. If you place $100 in the program, how many years will it take to have over $1,000?

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This process (working with patterns) can be used to develop or deepen understandings of important concepts in number theory, rational numbers, measurement, geometry, probability, and functions. Working with patterns provides opportunities for students to recognize, describe, extend, develop, and explain.

Number theory is the study of the properties of the counting numbers (positive integers), their relationships, ways to represent them, and patterns among them. Number theory includes the concepts of odd and even numbers, factors and multiples, primes and composites, and greatest common factor and least common multiple, and sequences.



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Benchmark 2: Variables, Equations, and Inequalities – The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. knows and explains the use of variables as parameters for a specific

variable situation (2.4.K1f), e.g., the m and b in y = mx + b or the h, k, and r in (x – h)2 + (y – k)2 = r2.

2. manipulates variable quantities within an equation or inequality (2.4.K1e), e.g., 5x – 3y = 20 could be written as 5x – 20 = 3y or 5x(2x + 3) = 8 could be written as 8/(5x) = 2x + 3.

3. solves (2.4.K1d) ($):a. N linear equations and inequalities both analytically and

graphically;b. quadratic equations with integer solutions (may be solved by

trial and error, graphing, quadratic formula, or factoring);c. ▲N systems of linear equations with two unknowns using

integer coefficients and constants;d. radical equations with no more than one inverse operation

around the radical expression; e. equations where the solution to a rational equation can be

simplified as a linear equation with a nonzero denominator, e.g., __3__ = __5__.

(x + 2) (x – 3) f. equations and inequalities with absolute value quantities

containing one variable with a special emphasis on using a number line and the concept of absolute value.

g. exponential equations with the same base without the aid of a calculator or computer, e.g., 3x + 2 = 35.

The student…1. represents real-world problems using variables, symbols,

expressions, equations, inequalities, and simple systems of linear equations (2.4.A1c-e) ($).

2. represents and/or solves real-world problems with (2.4.A1c) ($): a. ▲N linear equations and inequalities both analytically and

graphically, e.g., tickets for a school play are $5 for adults and $3 for students. You need to sell at least $65 in tickets. Give an inequality and a graph that represents this situation and three possible solutions.

b. quadratic equations with integer solutions (may be solved by trial and error, graphing, quadratic formula, or factoring), e.g., a fence is to be built onto an existing fence. The three sides will be built with 2,000 meters of fencing. To maximize the rectangular area, what should be the dimensions of the fence?

c. systems of linear equations with two unknowns, e.g., when comparing two cellular telephone plans, Plan A costs $10 per month and $.10 per minute and Plan B costs $12 per month and $.07 per minute. The problem is represented by Plan A = .10x + 10 and Plan B = .07x + 12 where x is the number of minutes.

d. radical equations with no more than one inverse operation around the radical expression, e.g., a square rug with an area of 200 square feet is 4 feet shorter than a room. What is the length of the room?

e. a rational equation where the solution can be simplified as a linear equation with a nonzero denominator, e.g., John is 2 feet taller than Fred. John’s shadow is 6 feet in length and Fred’s shadow is 4 feet in length. How tall is Fred?

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3. explains the mathematical reasoning that was used to solve a real-world problem using equations and inequalities and analyzes the advantages and disadvantages of various strategies that may have been used to solve the problem (2.4.A1c).

Teacher Notes: One of the most powerful ways we have of expressing the regularities found in mathematics is with variables. Variables enable us to use mathematical symbolism as a tool to think and help better understand mathematical ideas in the same way physical objects and drawings are used. But if variables are to be included with these other “thinker toys,” it is important to help students develop an understanding of the various ways they are used. A variable is a symbol that can stand for any one of a set of numbers or other objects. Although correct, this simple-sounding definition has a variety of interpretations, depending on how the variables are used.

Meanings of variables change with the way they are used. Usiskin (The ideas of Algebra, K-12, pp. 8-19, NCEM, 1988) identified three uses of variables that are commonly encountered in school mathematics:

1. As a specific unknown. In the early grades, this is the use found in equations such as 8 + = 12. Later, we see exercises such as this: If 3x + 2 = 4x – 1, solve for x.

2. As a pattern generalizer. Variables are used in statements that are true for all numbers. For example, a x b – b x a for all real numbers. Formulas such as A = L x W also show a pattern.

3. As quantities that vary in joint variation. Joint variation occurs when change in one variable determines a change in another. In y = 3x + 5, as x changes or varies, so does y. Formulas are also an example of joint variation. In C = 2πr, the radius, changes, so does C, the circumference. (Elementary and Middle School Mathematics, John A. Van de Walle, Addison Weslen Longman, Inc., 1988)



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Benchmark 3: Functions – The student analyzes functions in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. evaluates and analyzes functions using various methods including

mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.K1a,d-f).

2. matches equations and graphs of constant and linear functions and quadratic functions limited to y = ax2 + c (2.4.K1d,f).

3. determines whether a graph, list of ordered pairs, table of values, or rule represents a function (2.4.K1e-f).

4. determines x- and y-intercepts and maximum and minimum values of the portion of the graph that is shown on a coordinate plane (2.4.K1f).

5. identifies domain and range of: a. relationships given the graph or table (2.4.K1e-f), b. linear, constant, and quadratic functions given the equation(s)

(2.4.K1d).6. ▲ recognizes how changes in the constant and/or slope within a

linear function changes the appearance of a graph (2.4.K1f) ($).7. uses function notation.8. evaluates function(s) given a specific domain ($).9. describes the difference between independent and dependent

variables and identifies independent and dependent variables ($).

The student…1. translates between the numerical, graphical, and symbolic

representations of functions (2.4.A1c-e) ($).2. ▲ ■ interprets the meaning of the x- and y- intercepts, slope, and/or

points on and off the line on a graph in the context of a real-world situation (2.4.A1e) ($), e.g., the graph below represents a tank full of water being emptied. What does the y-intercept represent? What does the x-intercept represent? What is the rate at which it is emptying? What does the point (2, 25) represent in this situation? What does the point (2,30) represent in this situation?

The Water Tank

3. analyzes (2.4.A1c-e):a. the effects of parameter changes (scale changes or restricted

domains) on the appearance of a function’s graph,b. how changes in the constants and/or slope within a linear

function affects the appearance of a graph,c. how changes in the constants and/or coefficients within a

quadratic function in the form of y = ax2 + c affects the appearance of a graph.

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Hours

Gal

lons



In the pattern 1, 4, 9, 16, 25, …, 225; there is no common difference. This sequence is not arithmetic or geometric (no common ratio between geometric terms). Neither is it a combination of the two; however, there is a pattern and the missing terms between 25 and 225 can be found. To find the term value, square the number of the term. The next missing terms would be 36, 49, 64, 81, 100, 121, and 144. This pattern is explained by using the explicit formula for a sequence. The explicit formula for a sequence defines a rule for finding each term in the number sequence related to its position in the sequence. In other words, to find the term value, square the number of the term – the 5 th term is 52, the 8th term is 82, …


In the Cartesian Coordinate System, the expression x = 3 cannot represent a function. When you graph all points where x = 3 you get a vertical line, so more than one y-value exists for the x-value x = 3. By definition, a function must have only one output value for any given input. For more information, consult a reference book under the topic, vertical line test.



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Benchmark 4: Models – The student develops and uses mathematical models to represent and justify mathematical relationships found in a variety of situations involving tenth grade knowledge and skills.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. knows, explains, and uses mathematical models to represent and

explain mathematical concepts, procedures, and relationships. Mathematical models include:a. process models (concrete objects, pictures, diagrams, number

lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K1-3, 1.2.K1, 1.2.K3-4, 1.3.K1-4, 1.4.K1, 1.4.K2a-b, 2.1.K1a, 2.1.K1d, 2.1.K2, 2.2.K4, 2.3.K1, 3.2.K1-3, 3.2.K6, 3.3.K1-4, 4.2.K3-4) ($);

b. factor trees to model least common multiple, greatest common factor, and prime factorization (1.4.K3);

c. algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns (2.1.K1c);

d. equations and inequalities to model numerical and geometric relationships (1.4.K2c, 2.2.K3, 2.3.K1-2, 3.2.K7) ($);

e. function tables to model numerical and algebraic relationships (2.1.K1c, 2.2.K2, 2.3.K1, 2.3.K3, 2.3.K5) ($);

f. coordinate planes to model relationships between ordered pairs and equations and inequalities and linear and quadratic functions (2.2.K1, 2.3.K1-6, 3.4.K1-8) ($);

g. constructions to model geometric theorems and properties (3.1.K2, 3.1.K6);

h. two- and three-dimensional geometric models (geoboards, dot paper, coordinate plane, nets, or solids) and real-world objects to model perimeter, area, volume, and surface area, properties of two- and three-dimensional figures, and isometric views of


represent the same problem situation. Mathematical models include:a. process models (concrete objects, pictures, diagrams,

flowcharts, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations (1.1.K1, 1.2.A1-2, 1.3.A1-4, 1.4.A1a, 1.4A1d-e, 3.1.A1-3, 3.2.A1-3, 3.3.A2, 3.3.A4, 3.4.A2, 4.2.A1a-b) ($);

b. algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns;

c. equations and inequalities to model numerical and geometric relationships (2.1.A2, 2.2.A1-3, 2.3.A1) ($);

d. function tables to model numerical and algebraic relationships (2.3.A1, 2.3.A3, 3.4.A2) ($);

e. coordinate planes to model relationships between ordered pairs and equations and inequalities and linear and quadratic functions (2.2.A1, 2.3.A1-3, 3.4.A1-2, 3.4.A4) ($);

f. two- and three-dimensional geometric models (geoboards, dot paper, coordinate plane, nets, or solids) and real-world objects to model perimeter, area, volume, and surface area, properties of two- and three-dimensional figures and isometric views of three-dimensional figures (3.3.A1, 4.2.A1c);

g. scale drawings to model large and small real-world objects (3.3.A3, 3.4.A3);

h. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to

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three-dimensional figures (2.1.K1b, 3.1.K1-8, 3.2.K1, 3.2.K4-5, 3.3.K1-4);

model probability (1.4.A1c, 4.2.A1, 4.2.A3);

i. scale drawings to model large and small real-world objects; j. Pascal’s Triangle to model binomial expansion and probability;k. geometric models (spinners, targets, or number cubes), process

models (concrete objects, pictures, diagrams, or coins), and tree diagrams to model probability (4.1.K1-3);

l. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, histograms, and matrices to organize and display data (4.2.K1, 4.2.K5-6) ($);

m. Venn diagrams to sort data and show relationships (1.2.K2).

i. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, histograms, and matrices to describe, interpret, and analyze data (2.1.A1, 4.1.A1, 4.1.A3-4, 4.1.A6, 4.2.A1) ($);

j. Venn diagrams to sort data and show relationships.2. uses the mathematical modeling process to analyze and make

inferences about real-world situations ($).

Teacher Notes: For assessment purposes, the mathematical modeling process appropriate to the indicator may be included as part of the item being assessed. The mathematical modeling process involves:






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Standard 3: Geometry NINTH AND TENTH GRADES


Benchmark 1: Geometric Figures and Their Properties – The student recognizes geometric figures and compares and justifies their properties of geometric figures in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. recognizes and compares properties of two-and three-dimensional


2. discusses properties of regular polygons related to (2.4.K1g-h):a. angle measures,b. diagonals.

3. recognizes and describes the symmetries (point, line, plane) that exist in three-dimensional figures (2.4.K1h).

4. recognizes that similar figures have congruent angles, and their corresponding sides are proportional (2.4.K1h).

5. uses the Pythagorean Theorem to (2.4.K1h): a. determine if a triangle is a right triangle,b. find a missing side of a right triangle.

6. recognizes and describes (2.4.K1g-h): a. congruence of triangles using: Side-Side-Side (SSS), Angle-

Side-Angle (ASA), Side-Angle-Side (SAS), and Angle-Angle-Side (AAS);

b. the ratios of the sides in special right triangles: 30°-60°-90° and 45°-45°-90°.

7. recognizes, describes, and compares the relationships of the angles formed when parallel lines are cut by a transversal (2.4.K1h).

8. recognizes and identifies parts of a circle: arcs, chords, sectors of circles, secant and tangent lines, central and inscribed angles (2.4.K1h).

The student…1. solves real-world problems by (2.4.A1a):

a. using the properties of corresponding parts of similar and congruent figures, e.g., scale drawings, map reading, or proportions;

b. ▲ ■ applying the Pythagorean Theorem, e.g., when checking for square corners on concrete forms for a foundation, determine if a right angle is formed by using the Pythagorean Theorem;

c. using properties of parallel lines, e.g., street intersections.2. uses deductive reasoning to justify the relationships between the

sides of 30°-60°-90° and 45°-45°-90° triangles using the ratios of sides of similar triangles (2.4.A1a).

3. understands the concepts of and develops a formal or informal proof through understanding of the difference between a statement verified by proof (theorem) and a statement supported by examples (2.4.A1a).

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Teacher Notes: Geometry is the study of points, lines, angles, planes, and shapes and their relationships. Symbols and numbers are used to describe the properties of these points, lines, angles, planes, and shapes. The term geometry comes from two Greek words meaning “earth measure.” The fundamental concepts in geometry are point (no dimension), line (one-dimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional and solids are referred to as three-dimensional.






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Benchmark 2: Measurement and Estimation – The student estimates, measures and uses geometric formulas in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. determines and uses real number approximations (estimations) for

length, width, weight, volume, temperature, time, distance, perimeter, area, surface area, and angle measurement using standard and nonstandard units of measure (2.4.K1a) ($).

2. selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate real number representations for length, weight, volume, temperature, time, distance, area, surface area, mass, midpoint, and angle measurements (2.4.K1a) ($).

3. approximates conversions between customary and metric systems given the conversion unit or formula (2.4.K1a).

4. states, recognizes, and applies formulas for (2.4.K1h) ($):a. perimeter and area of squares, rectangle, and triangles;b. circumference and area of circles; volume of rectangular solids.

5. uses given measurement formulas to find perimeter, area, volume, and surface area of two- and three-dimensional figures (regular and irregular) (2.4.K1h).

6. recognizes and applies properties of corresponding parts of similar and congruent figures to find measurements of missing sides (2.4.K1a).

7. knows, explains, and uses ratios and proportions to describe rates of change (2.4.K1d) ($), e.g., miles per gallon, meters per second, calories per ounce, or rise over run.

The student…1. solves real-world problems by (2.4.A1a) ($):

a. converting within the customary and the metric systems, e.g., Marti and Ginger are making a huge batch of cookies and so they are multiplying their favorite recipe quite a few times. They find that they need 45 tablespoons of liquid. To the nearest ¼ of a cup, how many cups would be needed?

b. finding the perimeter and the area of circles, squares, rectangles, triangles, parallelograms, and trapezoids, e.g., a track is made up of a rectangle with dimensions 100 meters by 50 meters with semicircles at each end (having a diameter of 50 meters). What is the distance of one lap around the inside lane of the track?

c. finding the volume and the surface area of rectangular solids and cylinders, e.g., a car engine has 6 cylinders. Each cylinder has a height of 8.4 cm and a diameter of 8.8 cm. What is the total volume of the cylinders?

d. using the Pythagorean theorem, e.g., a baseball diamond is a square with 90 feet between each base. What is the approximate distance from home plate to second base?

e. using rates of change, e.g., the equation w = –52 + 1.6t can be used to approximate the wind chill temperatures for a wind speed of 40 mph. Find the wind chill temperature (w) when the actual temperature (t) is 32 degrees. What part of the equation represents the rate of change?

2. estimates to check whether or not measurements or calculations for length, weight, volume, temperature, time, distance, perimeter, area,

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surface area, and angle measurement in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) ($).

3. uses indirect measurements to measure inaccessible objects (2.4.A1a), e.g., you are standing next to the railroad tracks and a train passes. The number of cars in the train can be determined if you know how long it takes for one car to pass and the length of time the whole train takes to pass you.

Teacher Notes: The term geometry comes from two Greek words meaning “earth measure.” Measurement provides the tools required to apply geometric concepts in the real-world. Estimation in measurement is defined as making guesses as to the exact measurement of an object without using any type of measurement tool. Estimation helps students develop a relationship between the different sizes of units of measure. It helps students develop basic properties of measurement and it gives students a tool to determine if a given measurement is reasonable.



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Benchmark 3: Transformational Geometry – The student recognizes and applies transformations on two- and three- dimensional figures in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student…1. describes and performs single and multiple transformations

[refection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on two- and three-dimensional figures (2.4.K1a).

2. recognizes a three-dimensional figure created by rotating a simple two-dimensional figure around a fixed line (2.4.K1a), e.g., a rectangle rotated about one of its edges generates a cylinder; an isosceles triangle rotated about a fixed line that runs from the vertex to the midpoint of its base generates a cone.

3. generates a two-dimensional representation of a three-dimensional figure (2.4.K1a).

4. determines where and how an object or a shape can be tessellated using single or multiple transformations and creates a tessellation (2.4.K1a).

The student…1. ▲ analyzes the impact of transformations on the perimeter and area

of circles, rectangles, and triangles and volume of rectangular prisms and cylinders (2.4.A1f), e.g., reducing by a factor of ½ multiplies an area by a factor of ¼ and multiplies the volume by a factor of 1/8, whereas, rotating a geometric figure does not change perimeter or area.

2. describes and draws a simple three-dimensional shape after undergoing one specified transformation without using concrete objects to perform the transformation (2.4.A1a).

3. uses a variety of scales to view and analyze two- and three-dimensional figures (2.4.A1g).

4. analyzes and explains transformations using such things as sketches and coordinate systems (2.4.A1a).

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Teacher Notes: Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do not change the figure itself. Other transformations, like reduction (contraction/shrinking) or enlargement (magnification/growing), change the size of a figure, but not the shape (congruence vs. similarity).



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Benchmark 4: Geometry from an Algebraic Perspective – The student uses an algebraic perspective to analyze the geometry of two- and three-dimensional figures in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application IndicatorsThe student… 1. recognizes and examines two- and three-dimensional figures and

their attributes including the graphs of functions on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.K1f).

2. determines if a given point lies on the graph of a given line or parabola without graphing and justifies the answer (2.4.K1f).

3. calculates the slope of a line from a list of ordered pairs on the line and explains how the graph of the line is related to its slope (2.4.K1f).

4. ▲ finds and explains the relationship between the slopes of parallel and perpendicular lines (2.4.K1f), e.g., the equation of a line 2x + 3y = 12. The slope of this line is 2/3־ . What is the slope of a line perpendicular to this line?

5. uses the Pythagorean Theorem to find distance (may use the distance formula) (2.4.K1f).

6. ▲ recognizes the equation of a line and transforms the equation into slope-intercept form in order to identify the slope and y-intercept and uses this information to graph the line (2.4.K1f).

7. recognizes the equation y = ax2 + c as a parabola; represents and identifies characteristics of the parabola including opens upward or opens downward, steepness (wide/narrow), the vertex, maximum and minimum values, and line of symmetry; and sketches the graph of the parabola (2.4.K1f).

The student…1. represents, generates, and/or solves real-world problems that

involve distance and two-dimensional geometric figures including parabolas in the form ax2 + c (2.4.A1e), e.g., compare the heights of 2 different objects whose paths are represented h1(t) = 3t² + 1 and h2(t) = ½t² + 4 (where h represents the height in feet and t represents elapsed time in seconds) after 5 seconds.

2. translates between the written, numeric, algebraic, and geometric representations of a real-world problem (2.4.A1a-e) ($), e.g., given a situation, write a function rule, make a T-table of the algebraic relationship, and graph the order pairs.

3. recognizes and explains the effects of scale changes on the appearance of the graph of an equation involving a line or parabola (2.4.A1g).

4. analyzes how changes in the constants and/or leading coefficients within the equation of a line or parabola affects the appearance of the graph of the equation (2.4.A1e).

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8. explains the relationship between the solution(s) to systems of equations and systems of inequalities in two unknowns and their corresponding graphs (2.4.K1f), e.g., for equations, the lines intersect in either one point, no points, or infinite points; and for inequalities, all points in double-shaded areas are solutions for both inequalities.

Teacher Notes: A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the coordinate plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the y-coordinate. Since the pair is always named in order (first x, then y), it is called an ordered pair.



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Standard 4: Data NINTH AND TENTH GRADES


Benchmark 1: Probability – The student applies probability theory to draw conclusions, generate convincing arguments, make predictions and decisions, and analyze decisions including the use of concrete

objects in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application Indicators The student…1. finds the probability of two independent events in an experiment,

simulation, or situation (2.4.K1k) ($) .2. finds the conditional probability of two dependent events in an

experiment, simulation, or situation (2.4.K1k).3. ▲ explains the relationship between probability and odds and

computes one given the other (2.4.K1a,k).

The student…1. conducts an experiment or simulation with two dependent events;

records the results in charts, tables, or graphs; and uses the results to generate convincing arguments, draw conclusions and make predictions (2.4.A1h-i).

2. uses theoretical or empirical probability of a simple or compound event composed of two or more simple, independent events to make predictions and analyze decisions about real-world situations including:a. work in economics, quality control, genetics, meteorology, and

other areas of science (2.4.A1a); b. games (2.4.A1a); c. situations involving geometric models, e.g., spinners or

dartboards (2.4.A1f).3. compares theoretical probability (expected results) with empirical

probability (experimental results) of two independent and/or dependent events and understands that the larger the sample size, the greater the likelihood that experimental results will match theoretical probability (2.4.A1h).

4. uses conditional probabilities of two dependent events in an experiment, simulation, or situation to make predictions and analyze decisions.

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Whereas probability is a ratio of favorable outcomes to all possible outcomes, odds is a ratio of favorable to unfavorable outcomes. If the odds of rolling an even number on a number cube is 1:1 (1/1 or 1 to 1), then the probability is 1/2 (.5 or 50%).



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Standard 4: Data NINTH AND TENTH GRADES


Benchmark 2: Statistics – The student collects, organizes, displays, explains, and interprets numerical (rational) and non-numerical data sets in a variety of situations.

Ninth and Tenth Grades Knowledge Base Indicators Ninth and Tenth Grades Application Indicators

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The student…1. organizes, displays, and reads quantitative (numerical) and

qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays (2.4.K1l):a. frequency tables and line plots;b. bar, line, and circle graphs;c. Venn diagrams or other pictorial displays;d. charts and tables;e. stem-and-leaf plots (single and double);f. scatter plots;g. box-and-whiskers plots; h. histograms.

2. explains how the reader’s bias, measurement errors, and display distortions can affect the interpretation of data.

3. calculates and explains the meaning of range, quartiles and interquartile range for a real number data set (2.4.K1a).

4. ▲ explains the effects of outliers on the measures of central tendency (mean, median, mode) and range and interquartile range of a real number data set (2.4.K1a).

5. ▲ approximates a line of best fit given a scatter plot and makes predictions using the graph or the equation of that line (2.4.K1k).

6. compares and contrasts the dispersion of two given sets of data in terms of range and the shape of the distribution including (2.4.K1k):a. symmetrical (including normal), b. skew (left or right),c. bimodal, d. uniform (rectangular).

The student…1. ▲ uses data analysis (mean, median, mode, range, quartile,

interquartile range) in real-world problems with rational number data sets to compare and contrast two sets of data, to make accurate inferences and predictions, to analyze decisions, and to develop convincing arguments from these data displays (2.4.A1i) ($): a. ■ frequency tables and line plots;b. bar, line, and circle graphs;c. Venn diagrams or other pictorial displays;d. charts and tables;e. stem-and-leaf plots (single and double);f. scatter plotsg. box-and-whiskers plots;h. histograms.

2. determines and describes appropriate data collection techniques (observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, biased sampling, census of total population, or purposeful sampling) in a given situation.

3. uses changes in scales, intervals, and categories to help support a particular interpretation of the data (2.4.A1i).

4. determines and explains the advantages and disadvantages of using each measure of central tendency and the range to describe a data set (2.4.K1i).

5. analyzes the effects of: a. outliers on the mean, median, and range of a real number data

set; b. changes within a real number data set on mean, median, mode,

range, quartiles, and interquartile range.6. approximates a line of best fit given a scatter plot, makes

predictions, and analyzes decisions using the equation of that line (2.4.A1i).

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Teacher Notes: Graphs (data displays) are pictorial representations of mathematical relationships and are an important part of statistics. When a graph is made, the axes and the scale (numbers running along a side of the graph) are chosen for a reason. The difference between numbers from one grid line to another is the interval. The interval will depend on the lowest and highest values in the data. Emphasizing the importance of using equal-sized pictures or intervals is critical to ensuring that the data is displayed accurately.

Graphs take many forms: – bar graphs compare discrete data;– frequency tables show how many times a certain piece of data occurs; – circle graphs (pie charts) model parts of a whole;– line graphs show change over time;– Venn diagrams show relationships between sets of objects; – line plots show frequency of data on a number line;

stem-and-leaf plots (closely related to line plots) show frequency distribution by arranging numbers (stems) on the left side of a vertical line (single stem-and-leaf) with numbers (leaves) on the right side;

– scatter plots show the relationship between two quantities;box-and-whisker plots are visual representations of the five-number summary – the median, the upper and lower quartiles, and the least and

greatest values in the distribution – therefore, the center, the spread, and the overall range are immediately evident by looking at the plot and;

– histograms (closely related to stem-and-leaf plots) describe how data falls into different ranges.

Two important aspects of data are its center and its spread. The mean, median, and mode are measures of central tendency (averages) that describe where data are centered. Each of these measures is a single number that describes the data. However, each does it slightly differently. The range describes the spread (dispersion) of data. The easiest way to measure spread is the range, the difference between the greatest and the least values in a data set. Quartiles are boundaries that break the data into fourths. The interquartile range is the difference between the upper quartile and the lower quartile and is considered a measure of dispersion.


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Glossary

A

absolute value – a number’s distance from zero on the number line (The value is always positive.)

acute angle – an angle that measures less than 90

addend – one of a set of numbers to be added

addition problem – 8 (addend)+ 4 (addend)

12 (sum)

additive identity- 0 is the additive identity, because for any real number a: a + 0 = a

additive inverse – the opposite of a number; when a number is added to its additive inverse, the sum is 0. The additive inverse of –7 is: +7 because –7 + +7 = 0

algebra – a strand of mathematics in which variables are used to express general rules about numbers, number relationships, and operations

algebraic expression – a group of numbers, symbols, and variables that express an operation or a series of operations (3x is an expression for three times some number.)

algorithm – a systematic way for carrying out computations

altitude- a line segment which measures the height of a figure.

angle – a figure formed by two rays having a common endpoint

approximation – an amount or estimate that is nearly exact

area – the measure of the region inside a closed plane figure, measured in square units

arithmetic sequence – a sequence of numbers in which the difference of two consecutive numbers is the same

array – a rectangular arrangement of objects with equal amounts in each column and row

Associative Property – for any real numbers a, b, and c: (a + b) + c = a + (b + c); and for any real numbers a, b, and c: (ab)c = a(bc)

Kansas Curricular Standards for MathematicsJanuary 2004

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attribute – a characteristic of a shape or set of data

axis – one of the reference lines (perpendicular number lines) on a coordinate graph

Bbalance (bank) – the amount of money in a bank account

balance (scale) – a tool used for comparing the mass or weight of objects

bar graph – a way of organizing data in horizontal or vertical bars

base (polygon) – the side of a polygon that contains one end of the altitude OR a special side of a two-dimensional object.

base (solid figure) – a special face of a three-dimensional object

benchmark number – a number used in making comparisons or judgments

binomial – the sum or difference of two monomials

bisect – to divide into two equal parts

box-and-whisker plot – a representation of a five-number summary that displays the high and low values, the median, and the quartiles of a set of data, but does not display any other specific values (In the graph, a rectangle (“box”) represents the middle 50% of the data set and the line segments (“whiskers”) at both ends of the rectangle represent the remainder of the data.)

C

calendar – a tool to keep track of the date

capacity – a measurement of the amount that an object can hold

cardinal number – a whole number that answers the question “How many objects in a set?”

Cartesian plane – a two-dimensional coordinate grid

Celsius – a temperature scale where 0 is the freezing point of water, and 100 is the boiling point of water

center (of a circle) – the point inside a circle that is the same distance from every point on the circle

centimeter -- a metric unit of length equal to 1/100 of a meter


326

central tendency – a number somewhere in the middle of the data set, or a number with a lot of data clustered around it (Mean, median, and mode are measures of central tendency (averages).)

certain (event) – an event with a probability of one

chart – a sheet of information arranged in lists, pictures, tables, or diagrams

chord – a line segment connecting any two points on a circle

circle – a closed plane figure with every point the same distance from the center

circumference – the perimeter of a circle, the distance around the outside of the circle

clustering – an estimation strategy which groups numbers with relatively the same estimation value (7,225 + 6,734 + 7,123 + 6,642 to 7,000 + 7,000 + 7,000 + 7,000)

coefficient – in algebra, the numerical factor of a term (In 3x, the coefficient of x is 3.)

coins – metal money

Commutative Property – for any real numbers a and b: a + b = b + a; and for any real numbers a and b: ab = ba

compass – a tool for drawing circles

compatible numbers – numbers that easily “fit together” in computation and are easy to manipulate mentally

composite figure – a figure made up of several different geometric shapes

composite number – a number that has more than itself and 1 as factors (6 has factors of 1, 2, 3, and 6.)

compound event – an event composed of at least two simple events

compound interest – interest paid on the accumulated amount, including previous interest

compute – to find a numerical result, usually by adding, subtracting, multiplying, or dividing

cone – a three-dimensional figure with a circular base and a vertex (like an ice cream cone)

congruency – having the same size and shape


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congruent – figures that are exact “duplicates” of each other; having the same size and shape

coordinate grid/plane – a two-dimensional system in which the coordinates of every point can be expressed as an ordered pair that describes the distance of that point from the x-axis and y-axis (Cartesian plane)

coordinates – a pair of numbers that give the location of a point on a grid/plane or graph

cube – a three-dimensional figure having six congruent, square faces

cube number – a number raised to the third power

cup – a unit of liquid measure; 8 ounces = 1 cup, 2 cups = 1 pint

customary system of measurement – a system of measurement commonly used in the United States (inches, cups, pounds, etc.)

cylinder – a three-dimensional figure with parallel, congruent circular bases

D

data – information, especially numerical, usually organized for analysis

data displays – representations such as frequency tables, histograms, line plots, bar graphs, line graphs, circle graphs, Venn diagrams, charts, and tables

decimal – a name for a fractional number expressed with a decimal point such as .27

decompose (figures) – breaking apart a composite figure into the basic shapes from which it was made

denominator – the number or expression below the line in fraction, the equal parts in the whole

dependent event – an event whose outcome is affected by the outcome of another event

dependent variable – in a function, a variable whose value is determined by the value of the related independent variable

diagonal – a line segment that joins two non-consecutive vertices of a polygon

diameter – a line segment that has its endpoints on a circle (chord) and passes through the center of the circle

difference – the result of subtracting one number from another

digit – one of the ten numerals in our numeration system: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9


328

digital root – the result of adding digits in a number until only one digit remains

discount – a reduction applied to a given price

discrete – discontinuous, made up of distinct parts

Distributive Property – for all real numbers a, b, and c: a(b + c) = ab + ac and a(b - c) = ab - ac

dividend (financial) - payment to stockholders, usually quarterly, that represents their part of the company’s profits

dividend (mathematical) – the number that is being divided by another number

divisibility – a number n is divisible by a number k if the quotient n/k is a whole number

divisibility rules – .Rules for determining which numbers are divisible by a given numberSome common divisibility rules include:

Two: A number is divisible by 2 if the ones digit is 0,2,4,6, or 8Three: A number is divisible by 3 if the sum of the digits is divisible by 3Four: A number is divisible by 4 if the number formed by the last two digits is divisible by 4Five: A number is divisible by 5 if the last digit is 0 or 5Six: A number is divisible by 6 if the number is divisible by 2 and 3Seven: No easy rule existsEight: A number is divisible by 8 if the number formed by the last 3 digits is divisible by 8Nine: A number is divisible by 9 if the digital root is 9

division - the operation of making equal groups to find how many in each group or how many groups

division problem – 12 (quotient) (divisor) 4 48 (dividend)

48 ÷ 12 = 4

48/12 = 4

divisor – the number by which the dividend is divided

double – twice as much


329

E

ellipse – a closed plane curve generated by a point moving in such a way that the sums of its distances from two fixed points is a constant

empirical results – results based on experiments

equation – a mathematical sentence that states that two expressions are equal; 2 + 3 = 1 + 4

equilateral – all sides congruent

equivalent – numbers or variable expressions that have the same value (5 is equivalent to 2 + 3)

estimation strategies – Front-end with Adjustment$1.26 4.79 Add whole number part (left most digit.)

.99 1 + 4 + 0 + 1 + 2 = 81.37 Adjust by looking at decimal parts.2 .58 .26 + .79 Estimate is 1

.99 Estimate is 1.37 + .58 Estimate is 1 3

Therefore, the estimate is 8 + 3 = 11.

Rounding$4.78

2.93 Round number to nearest whole number.1.25

+ 3.12 5 + 3 + 1 + 3Therefore, the estimate is 12.

Special Numbers

Examples of special numbers are 1; 10; 100; 1,000; 1/2; 0.

Estimate by rounding to one of the special numbers.

67 x 102 Estimate 67 x 100 or 6,700 5,421/9.87 Estimate 5,421/10 or 542.1 4/7+12/13 Estimate 1 + ½ or 1½ 4,805 x 11/21 Estimate 4800 x ½ or 2,400

Clustering


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Clustering can be used as an estimation strategy when you have a group of numbers that cluster around a common value.72256734 This set of numbers clusters around 7,000.6829 A good estimate would be 7000 x 6 or 42,000.72957101

+ 6642

Compatible “nice” NumbersCompatible (nice) numbers are numbers that can easily “fit together” and are easy to manipulate mentally. Take a global look at all numbers, look for numbers that can be paired for easy computation.

2749 27 + 81 = 10038 49 + 56 = 10056 38 + 65 = 10081

+ 65 Therefore, the estimate is 300.

TruncationTruncation is the process of ignoring all digits to the right of a chosen place value. An example of using truncation as an estimation technique is saying gasoline sells for $1.13 per gallon when the actual price is $1.139 ($1.139).

evaluate (an algebraic expression) – substitute the values given for the variables and solve using the order of operations

even number – a number that can be divided by 2 with no remainder (2, 4, 6, 8, 10, ...)

expanded form – the way to write numbers that shows the place values of each digit

explicit rule for a pattern – statement that explains how each term in the number sequence is related to its position in the sequence

exponent – a numeral telling how many times a number is to be multiplied by itself

expression – a variable or combination of variables, numbers, and symbols that represents a mathematical relationship

F

face - a plane figure that serves as one side of a solid figure; a flat surface on a solid figure

fact family – a group of addition/subtraction or multiplication/division facts that use the same set of numbers; 3+5=8, 5+3=8, 8-3=5, 8-5=3


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factor (of a number) – a number that divides evenly into a given number

factor (in multiplication) – one of two or more numbers that are multiplied to get an answer (in 3 x 4 = 12, both the 3 and the 4 are known as factors of 12)

Fahrenheit – a temperature scale where the freezing point of water is 32 and the boiling point of water is 212

first quadrant – the upper right fourth of a plane

flip – a mirror-image of a figure on the opposite side of a line; ( the size and shape of the figure stay the same); also called reflection

foot – a customary unit of length; 12 inches = 1 foot

formulate – to express as a formula; to work out and state in an orderly, exact way

fraction – a part of a whole; the name for a fractional number written in the form a/b (4/9)

numerator – the number or expression above the line in fraction (number of equal parts being described) 4 (numerator)7 (denominator)

denominator – the number or expression below the line in fraction (total number of equal parts)

improper fraction – a fraction with a numerator larger than a denominator (7/3)

proper fraction – a fraction with a numerator smaller than a denominator (3/7)

mixed number – writing a number greater than one with an integer part and a fractional part

lowest terms – a fraction written so that the numerator and denominator have 1 as the only common factor (4/7 is in lowest terms but 8/14 is not since 2 will go into both the numerator and the denominator, i.e., 2 is a common factor.)

reduce or simplify – to write a fraction in lowest terms depending on the context, simplest form does not always mean a mixed number

frame of reference – adjusting estimation based on additional information

frequency – the number of times an event occurs

front-end with adjustment – rounding to the nearest whole number such as ($4.96 + $5.88 to $5.00 + $6.00)

function – a special set of ordered pairs; one output value for each input value

function table – a table that assigns exactly one output value for each input value


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G

gallon – a customary unit of liquid measure; 4 quarts = 1 gallon

geometric sequence – a sequence of terms in which the ratio of two succeeding terms is the same

geometry – a strand of mathematics dealing with figures and their parts

glyph – a simple pictorial shorthand used to display data and details of a subject

gram - a metric unit of mass

graphical displays – (list below and define/also in document)bar graphcircle graphline plothistogramstem and leaf

greater than – more than, the symbol used is >

greatest common factor – the greatest number that is a common factor of two or more numbers (6 is the GCF of 12 and 18 because it is the largest number that goes into both 12 and 18.)

grid – a set of horizontal and vertical lines spaced uniformly used to graph points

gross income – income before any expenses and deductions are subtracted

growing pattern – a sequence in which each element is larger than the previous one.

H

hexagon – a six-sided plane figure

histogram – a graph in which the labels for the bars are numerical intervals

I

impossible event – an event with a probability of zero


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improper fraction – a fraction with a numerator larger than a denominator

inch – a customary unit of length; 1/12 of a foot

independent events – two or more events in which the outcome of the one event does not affect the outcome of the other events

inequality (algebraic) – an algebraic statement that a quantity is greater than or less than another quantity, such as 7t + 5 < 40

inequality (numerical) -- a numerical statement that a quantity is greater than or less than another quantity; 7 + 5 < 40

integer – any member of the set of whole numbers and their opposites (... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...)

interest – an amount of money paid for the use of money

interquartile range – the difference between the upper (third) and lower (first) quartiles in a set of data

intersecting lines – 2 or more lines that cross each other at one or more points

intersection – the point at which two lines meet

irrational number – a real number that is not rational; A number that can be written as a non-repeating infinite decimal

isosceles (triangle) – at least two congruent sides

isosceles (trapezoid) – non-parallel opposite sides are congruent

K

kilometer – a metric unit of length equal to 1000 meters

kilogram – a metric unit of mass equal to 1000 grams

kite – a quadrilateral in which there are two pairs of consecutive (adjacent) congruent sides

L

least common multiple – the smallest number that is a common multiple of two or more numbers (24 is the LCM of 6 and 8 because it is the smallest number that both 6 and 8 will go into evenly)

length – the distance from one point to another; one dimension of a 2- or 3-dimensional figure


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less than – fewer than, the symbol used is <

line – a set of points that form a straight path extending infinitely in both directions

linear equation – an equation that results in a straight line when graphed

linear function – a function of the form f(x) = ax + b where a and b are constants

linear inequality – an inequality whose boundary in the coordinate plane is a line

liter -- a metric unit of volume

M

mass – a measurement that tells how much matter an object has

matrix – a rectangular array of data

mean – one of the measures of central tendency (average); the sum of numbers in a set of data divided by the number of pieces of data

measures of central tendency – measures of central tendency (average) include mean, median, mode

median – one measure of central tendency (average); the middle number in a set of numbers, determined by arranging them in order from lowest to highest and counting to the middle; in the case of an even number data set, add the two middle numbers and divide by two

meter – a metric unit of length

meter stick – a measurement tool used for measuring meters

metric units – a system of measurement based on the decimal system (meters, centimeters, etc.)

mile – a unit of length (5,280 feet = 1 mile)

milliliter -- a metric unit of volume equal to 1/1000 of a liter

milligram – a metric unit of mass equal to 1/1000 of a gram

millimeter – a metric unit of length equal to 1/1000 of a meter

minuend – the name given to the number from which another number is to be subtracted


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mixed number – writing a number greater than one with an integer part and a fractional part

mode – a measure of central tendency (average); the number found most frequently in a set of data; there can be no mode, one mode, or more than one mode

model – a representation of a situation that includes process models, place value models, fraction and mixed number models, decimal and money models, two- and three-dimensional models, equations, inequalities, and graphs; models can be mathematical representations of situations, or non-mathematical models of mathematical situations

money – anything that is generally accepted as a medium of exchange to buy goods and services; serves as a standard of value; and has a store of value; gold, silver, shells, tobacco, beads, and paper have all been used as money

monomial – an expression that can be a constant, a variable, or a product of a constant and one or more variables such as 5 (a constant), x (a variable), -3z (a product of a constant and a variable), 6xy (a product of a constant and more than one variable), or x ^3 ( a variable to a power)

multiple – the product of a given whole number and any other whole number

multiplicand – a name given to the number that is being multiplied by another number

multiplication problem – 5 (multiplicand or factor)x 4 (multiplier or factor)

20 (product)

5 X 4 = 20

5 · 4 = 20

5(4) = 20

Multiplication Property of One – for any real number a: a • 1 =1; 1 • a = 1

Multiplication Property of Zero – for any real number a: a • 0 =0; 0 • a = 0

Multiplicative Identity – 1 is the multiplicative identity, because for any real number a: a • 1 = a

Multiplicative Inverse – the reciprocal of a number (When a number is multiplied by its multiplicative inverse, the product is always one.) The multiplicative inverse of 4 is ¼, because 4 · ¼ = 1

Multiplicative Property of Equality – for any real numbers a and b: if a = b, then ac = bc


336

multiplier – the number by which the multiplicand is being multiplied

N

neighbor – the whole number that is adjacent to another whole number (The neighbors of 6 are 5 and 7.)

net – a pattern to be cut and folded without overlapping to make a solid shape

net income – amount left over after the expenses and deductions have been subtracted from the total

nice numbers – numbers that are easy to think about and work with (100, 500, and 750 are easier to work with than 94, 517, and 762; other multiplies of 100 are very “nice” numbers as are multiples of 10.)

nonstandard unit – measurement unit other than customary or metric, such as paperclip, penny, thumbnail

number – a mathematical idea contained in a set; a number can be named in word form (six), standard form (22), or expanded form [(3 x 1,000) + (0 x 100) + (4 x 10) + (6 x 1)]

numeral – the symbolic representation of a number

numerator – the number above the line in fraction; the number of equal parts being described

O

obtuse – an angle that measures more than 90 and less than 180

octagon – an eight-sided plane figure

odds – the ratio of favorable to unfavorable outcomes, whereas probability is a ratio of favorable outcomes to all possible outcomes. If the odds of rolling an even number on a number cube is 1:1 (1/1 or 1 to 1), then the probability is ½ (.5 or 50%)

odds (against an event) – the ratio of the number of ways an event cannot occur to the number of ways that an event can occur

odds (for an event) – the ratio of the number of ways an event can occur to the number of ways that an event cannot occur

ordered pair – a pair of numbers that give the coordinates of points on a grid in this order (horizontal coordinate, vertical coordinate)

order of operations – the agreed-upon order of evaluating numerical expressions (First, work inside parentheses, then do powers. Then do multiplications or divisions, from left to right. Then do additions or subtractions, from left to right.)

ordinal number – a number used to name the position in an ordered arrangement


337

ounce – a measurement of weight;16 ounces = 1 pound

outcome – a possible result of an experiment

outlier – a piece of numerical data that is much smaller or larger than the rest of the data in a set.

P

parabola –the graph of a quadratic equation of the form y= ax^2+bx+c

parallel lines – lines in the same plane that do not intersect

parallelogram – a four-sided plane figure with parallel opposite sides

pattern – a repeated sequence, either repeating or growing

pattern blocks – a collection of six geometric shapes in color (green triangle, orange square, blue rhombus, tan rhombus, red trapezoid, yellow hexagon), also see Appendix 7 for a pattern block template

pentagon – a five-sided plane figure

percent – a comparison of a number with 100 (43 compared to 100 is 43%.)

perimeter – the measurement of the distance around the outside of a figure

perpendicular – lines in the same plane that intersect at right angles

pi – the constant ratio of the diameter of a circle to its circumference

pictograph – a visual display of data that uses pictures to represent amounts

pint – a customary unit of liquid measure; 2 cups = 1 pint

place value – the value assigned to a digit due to its position in a numeral

plot – to place points on a coordinate grid/plane

point – a location in space represented by a dot


338

poll – a survey of public opinion concerning a particular subject

polygon – a simple closed plane figure having line segments as sides

polynomial – a sum and/or difference of monomials

pound – a customary measurement of weight;16 ounces =1 pound

precision of measurement – the smallest unit of measurement used in measuring

prediction – an educated guess of the number of occurrences in a probability situation; something that is guessed in advance, based on known facts.

prime factorization – an expression showing a whole number as the product of prime factors (24 = 4 x 6 is not a prime factorization since 4 can be broken down into 2 x 2 and 6 can be broken into 2 x 3. The correct prime factorization of 24 is 2 x 2 x 2 x 3.)

prime number – a positive integer that can be divided by only 1 and itself (2, 3, 5, 7, 11...)

probability (of an event) – the likelihood that an event will occur, the ratio of the number of favorable outcomes to the number of equally likely possible outcomes; a number from 0 to 1 that measures the likelihood that an event will occur.

product – the answer in a multiplication problem

profit – income minus expenses, usually stated in dollars. (Extent to which people or organizations are better off at the end of a period than at the beginning of that period.)

proper fraction – a fraction with a numerator smaller than a denominator

property – a characteristic of a number, geometric figure, mathematical operation, equation, or inequality

proportion – a statement of equality between two ratios (For example, 3:6 = 1:2, read: 3 is to 6 as 1 is to 2, can also be written as two equal fractions 3/6 = 1/2.)

protractor – a tool for measuring and drawing angles

purposeful sampling – a process used to select a sample; sampling from a deliberately limited population

pyramid – a three-dimensional figure whose base is a polygon and all other faces are triangles that share a common vertex.

Pythagorean Theorem – in a right triangle, the square of the length of the hypotenuse = the sum of the squares of the legs


339

Q

quadrant – one of the four regions of the coordinate plane formed by the x-axis and y-axis

quadratic equation – a polynomial equation with a variable to the second degree (power of 2) of the form y=ax2 + bx + c where a cannot be zero

quadratic function – a function whose value is given by a quadratic polynomial

quadrilateral – a four-sided polygon

quart – a customary unit of liquid measure; 2 pints = 1 quart

quarter – a coin with a value of $.25, one-fourth of something

quotient – the answer in a division problem

R

radius – a line segment having one endpoint in the center of the circle and another on the circle

range – in a collection of numbers, the difference between the maximum value and the minimum value;, the length of an interval

ratio – a comparison of two numbers that can be expressed as a fraction (3/4), or with a colon (3:4), or in words (3 to 4)

rational number – a number than can be written as the ratio of two integers

ray – a portion of a line extending from one endpoint in one direction indefinitely

real number system – the combined set of rational and irrational numbers

real numbers – rational and irrational numbers

rectangle – a four-sided plane figure with 90° angles and opposite sides are parallel

recursive rule for a pattern– statement or a set of statements that explains how each successive term in the sequence is obtained from the previous term(s)

reflection – a mirror-image of a figure on the opposite side of a line; (the size and shape of the figure stay the same); also called flip

regrouping – the process of renaming one place value to another; The number 1 in 10 represents ten ones


340

relative magnitude – the size relationship one number has with another number (is it much larger, much smaller, close, or about the same)

remainder – the number left over when things are divided into equal shares

repeating pattern – a pattern made up of repeating a segment, e.g., 235235235235235…

revenue – business’ receipts from the sale or rental of goods or services; money from other sources such as dividends and interest

rhombus – a quadrilateral that has four congruent sides

right angle – an angle that measures 90

right triangle – a triangle that has a 90angle

Roman numerals – symbols used in the ancient Roman number system

rotation – turning a figure about a point, which serves as the center of rotation; (does not change size or shape); also called turn

rounding – approximating a number to a specific place value so it is easier to work with; round 678 to the nearest ten: 680

ruler – a measuring tool used to determine length

S

sales tax rate – tax specified as a percent of the price of an item

scalar – a single number used with multiplication of a given data such as a vector or a matrix

scale – an arrangement of numbers in some order at uniform intervals

scalene triangle – a triangle having no sides equal

segment – a part of a line defined by two endpoints

similarity – the property of geometric figures having angles of the same size

simple event – an outcome which cannot be broken down further; rolling a number cube and getting 6 is a simple event, rolling two number cubes and getting a sum of six is a compound event

simulation – a model of an experiment that might be impractical to carry out


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simultaneous change – two or more changes occurring at the same time

slide – a transformation that slides a figure a given distance in a given direction; (shape and size do not change); also called a translation

slope – the slant of a line given as the ratio of the rate of change in y (rise) with respect to a change in x (run)

sphere – a three-dimensional figure formed by a set of points equidistant from a center point; a round ball

square – a quadrilateral with four right angles and four congruent sides

square number – the number of dots in a square array; a number that is the result of multiplying an integer by itself

standard form – the numeric representation of written or pictorial forms of a number concept, e.g., one hundred thirty-two is represented by 132

standard unit of measure – measurement unit such as customary or metric

stem-and-leaf plot – a method of organizing data from least to greatest using the digits of the greatest place value to group data; shows the greatest, least, and median values in a set of data

subtraction problem – 28 (minuend)- 7 (subtrahend)

21 (difference)

28 – 7 = 21

subtrahend – the number or the term to be subtracted

sum – the result of adding two or more numbers

Symmetric Property – for any real numbers a and b, if a = b, then b = a

Symmetry (line of) – a line that divides a figure into two congruent halves that are mirror images of each other; e.g. diameter of a circle, diagonal of a square

systems of linear equations – two or more related linear equations for which you can seek a common solution; The system may have no common solutions, one common solution, or many common solutions.


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systems of linear inequalities - two or more related linear inequalities for which you can seek a common solution; The system may have no common solutions, one common solution, or many common solutions

T

T-chart/T-table – a chart/table of ordered pairs; input-output table; function table

temperature – the measurement of hotness or coldness

term – a number, variable, product or quotient in an expression

tessellate – to arrange an area in a repeating geometric pattern

thermometer – a measurement tool used to measure temperature in Fahrenheit or Celsius

three-dimensional – a figure in space having height, length, and depth

ton (customary) – a customary unit of weight equal to 2,000 pounds ton (metric) -- a metric unit of mass equal to 1000 kilograms

transformation – a movement or change of a figure in two- or three-dimensional space. Standard transformations include rotations, reflections, and translations (do not change size of a figure) and reduction and enlargement (change size of a figure)

transitive – a law of equality which says if a = b, and b = c, then a = c

translation – a transformation that moves every point on a figure a given distance in a given direction; (shape and size do not change); also called a slide

transversal – a line that intersects two or more other lines

trapezoid – a quadrilateral with exactly one pair of parallel sides

tree diagram – a connected, branching graph used to diagram probabilities or factors

triangle – a polygon with three sides

truncate – to make numbers with many digits easier to read and use by ignoring all digits to the right of a chosen place

turn - rotating a figure about a point, which serves as the center of rotation; (does not change size or shape); also called rotation


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two-dimensional – a plane figure having only the dimensions of width and length

U

unit – a precisely fixed quantity used to measure; one’s place

unit price – cost for a small unit of measure, such as an ounce or pound, used to compare the costs of a product in different-sized packages

V

variable – a symbol that represents an unstated numerical value in a number sentence; In the equation 6 + a = 10, “a” is the variable which stands for 4 and a^2 + b^2 = C^2 where the variable’s values depend on the specific situation

vector – a line segment that has direction and distance

Venn diagram – a drawing with circles that shows relationships among sets of data

vertex – a common point at which two line segments, lines, or rays meet

volume – the measure of space enclosed by a figure

W

weight – the measure of how heavy an object is

whole number – the set of numbers {0, 1, 2, 3, ...}

width – the distance from one point to another; one dimension of a 2- or 3-dimensional figure

withdrawal – subtracting money from an account

X

x-axis – the horizontal axis in a coordinate graph

x-intercept -- the value of x at the point where a line or graph of a function intersects the x-axisY

y-axis – the vertical axis in a coordinate graph


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yard – a customary unit of length;3 feet = 1 yard

yardstick – a measurement tool used to measure yards, feet, and inches

y-intercept – the value of y at the point where a line or graph of a function intersects the y-axis

Z

zero product property – for any real numbers a and b, if ab = 0, then a = 0 or b = 0

Zero Property – for any real number a, a · 0 = 0 and 0 · a = 0


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