Orange School District

MathematicsGrades 3-5

Curriculum Guide 2010 Edition

APPROVED ON:__________________________

BOARD OF EDUCATION

Patricia A. ArthurPresident

Arthur GriffaVice-President

MembersStephanie Brown Rev. Reginald T. Jackson Maxine G. Johnson

Eunice Y. Mitchell David Wright

SUPERINTENDENT OF SCHOOLSRonald Lee

DEPUTYSUPERINTENDENT

ADMINISTRATIVE ASSISTANTTO THE SUPERINTENDENT

Dr. Paula HowardCurriculum and Instructional Services

Belinda Scott-SmileyOperations/Human Resources

BUSINESS ADMINISTRATOR

DIRECTORSBarbara L. Clark, Special Services

Candace Goldstein, Special ProgramsCandace Wallace, Curriculum & Testing

Curriculum ContributorsCandace Wallace

Ron NelkinMaria Pankin

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James DeLoatchMeghan Barrios

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BOARD OF EDUCATION 6

PHILOSOPHY 4VISION 4PURPOSE 5MATH FOCUS POINTS 6Blueprint 10

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Mathematics Curriculum GuideGrades 3-5

Philosophy

The philosophy upon which the Mathematics Curriculum Guide is to encourage and support the enjoyment of learning mathematics, as a way to make sense of the world in students’ everyday lives. Mathematics is everywhere, from the practicalities of counting, to find easier ways of organizing numbers and data to model and represent daily life experiences. Mathematics involves other disciplines, and is a way in which ideas are communicated, such as in tables and graphs.

Mathematics is developmental by nature. Therefore it is important that should any concerns arise related to mathematics understanding, that this is communicated with the student’s teacher as soon as possible. There are varied approaches used to teach and learn mathematics, which is referred to as a balanced mathematics approach. This includes traditional algorithms to approaching the study of mathematics that have been used for many years, along with newer and varied approaches, to provide multiple representations to model solving a problem.

The study of mathematics provides pathways to higher level thinking skills. As students learn mathematics, specialized terminology assist their development. This enables students to not only learn mathematics in a routine way, but to enable them to become problem solvers in novel situations, able to draw on a repertoire of skills and approaches.

We hope these beliefs will assist students to develop their understanding to use mathematics to make meaning, as well as to promote their critical thinking and development as lifelong learners. The goals are to promote problem-solving, and communication, to foster an understanding of the world, that has a conceptual foundation in the study of mathematics.

Vision

In Orange, we recognize that each student is unique and that the purpose of education is to enable every student to acquire the learning skills necessary to compete in the global community. It is essential that we provide a rigorous, high-quality Mathematics curriculum that allows each student’s talents and abilities to be developed to their full potential.

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Purpose

The Curriculum Guide was prepared by teachers and administrators with input from consultants who have expertise in Mathematics. Students and parents are welcome to read, review, and ask questions about the curriculum, to understand what they and their children are learning.

The Mathematics Curriculum Guide is based on an alignment with the New Jersey Core Content Curriculum Standards, and the Common Core State Standards which are a national set of shared standards which adopted by over 30 states. It is also based on national standards shared through the National Council of Teachers of Mathematics, which develops agreed upon content at each grade level, referred to as Curriculum Focal Points.

Content was designed with a student development perspective across each grade, as well as a vertical articulation, with spirals learning upward, based on the foundation that is developed. Mathematics is developmental in nature, so that it is important for students and parents to address any learning needs with the student’s teacher as soon as possible.

A curriculum must include assessment, as evidence of learning. Assessments are both formative and summative, and may include culminating projects related to a topic or unit, as well as traditional and standardized testing.

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Math Focus Points

Grade 3

Numbers and Operations – Students develop an understanding of multiplication and division, and strategies for basic multiplication facts.

Review Grade 2 Goal: Add and subtract numbers of up to 3-digit numbers Understand the meaning of multiplication (e.g. equal groups, seeing

different models, repeated addition, proportional relationship) Understand how multiplication and addition differ Know where multiplication is used Know how the product increases when the multiplier increases by 1 Understand the concept of the multiplier and multiplicand (know how many

groups and how many items in a group) Use properties of addition and multiplication to multiply whole numbers

(associative, commutative, distributive property, property of zero) Practice recall of 1-digit multiplication facts with cards and games to

mastery

Multiply 2-digit by 1-digit numbers Understand the meaning of division (partitive and quotitive) Divide 2-digit by 1-digit numbers as related to multiplication facts Understand the meaning of fraction as a unit Understand commonly used fractions as part of a whole, part of a set, and on

the number line Read and write numbers up to 1,000,000 Units and relative size of numbers up to 1,000,000 Understand the relative size of large numbers Understand the meaning and structure of decimal notation system Construct and solve simple open sentences involving addition and subtraction

Geometry and Measurement – Students deepen their understanding of linear measurement. Students describe and analyze properties of 2-d shapes.

Identify and describe many plane and solid figures (rectangle, square, triangle, circle, hexagon, octagon, pentagon, and cube, rectangular prism, sphere, cylinder, cone, pyramid)

Know about triangles and quadrilateral by examining the elements that compose the figures

Draw basic geometric shapes (square, rectangles, triangles, right triangles) Understand properties of right angles Understand elements that compose 3-d solids by observing and making

figures Measure length to the 1/2 inch, inch, foot, yard and cm, mm, meter Determine area by counting square units Describe relationships among units of length

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Describe relationships among units of time Tell time to the minute

Process Standards- Students use mathematical processes to solve problems and communicate mathematical ideas. These processes are embedded in the goals above.

Write and interpret number sentences Write two-step problem in one number sentence Explain strategies and present ideas to class Listen and apply others’ strategies

Grade 4

Number and Operation and Algebra – Students continue to practice quick recall of multiplication facts. Students develop recall of related division facts and fluency with multi-digit multiplication.

Review Grade 3 Goals:

Practice recall of 1-digit multiplication facts with cards and games to mastery Multiply 2-digit or 3-digit numbers by 1-digit numbers

In addition:

Multiply 2-digit or 3-digit numbers by 2-digit Understand the meaning of division with larger numbers (e.g. finding the unit,

seeing different models, repeated subtraction, ratio relationship, opposite of multiplication)

Know where division is used Understand the two types of division and how remainder is involved Understand the meaning of dividend, divisor, and quotient Practice recall of basic division facts with cards and games to mastery Divide 2-digit or 3-digit numbers by 1-digit numbers with remainder Understand the meaning of fraction as a unit Notice commonly used fractions as part of a whole, part of a set, and on the

number line Know how to order/compare fractions by finding equivalent Read and write numbers up to 1,000,000,000 and decimals to thousandths Understand the units and relative size of numbers up to a billion and decimals

to thousandths Do simple calculations with money Describe and extend patterns Understand and apply properties of operations and numbers Use concrete and pictorial models to explore basic concept of functions Understand relationship between two quantities (ex. function machines)

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Geometry and Measurement – Students deepen their understanding of linear measurement. Students describe and analyze properties of 2-d shapes.

Use properties of 3-d and 2-d shapes to identify, classify, and describe them Understand and apply concepts involving lines, angles, and circles Understand parallel and perpendicular lines Experience simple transformations (slide, flip, turn) Locate and name points on a grid Select and use appropriate standard units of measure and measurement tools

to solve problems Use strategies to estimate measurements Draw and construct basic shapes with tools (compass and protractor) Solve problems using elapsed time Distinguish between perimeter and area to solve problems Measure and compare volume of 3-d objects using cubes

Process Standards- Students use mathematical processes to solve problems and communicate mathematical ideas. These processes are embedded in the goals above.

Write and interpret more complex number sentences Explain strategies and present ideas to class Listen and apply others’ strategies Understand multiple strategies

Grade 5Number and Operation and Algebra – Students develop an understanding of and fluency with division of multi-digit whole numbers, and addition and subtraction of fractions and decimals.

Review Grade 4 Goal: Practice recall of basic division facts with cards and games to mastery

Divide 2-digit or 3-digit numbers by 2-digit numbers with remainder Convert improper and proper fractions Add and subtract fractions with like and unlike denominators Add and subtract decimals numbers Estimate with large numbers Develop and apply number theory concepts (prime, factors, and multiples) Describe and apply order of operations Describe, extend, and create patterns using tables, verbal rules, simple

equations, and graphs Describe arithmetic operations as functions Use number sentences to model situations (use variable for unknown) Solve simple linear equations with manipulatives and informally

Geometry and Measurement – Students describe 2-d and 3-d shapes and analyzing their properties, including volume and surface area.

Relate 2-d and 3-d shapes and analyze their properties

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Understand how to find volume Understand how to find surface area Develop and apply strategies and formulas for perimeter and area

(Rectangles) Understand and apply concepts involving lines and angles (Parallel,

perpendicular, intersecting, sum of interior angles in a triangle) Identify, describe, compare, and classify polygons Map one figure to another congruent figure using translation, reflection, or

rotation Create shapes on a coordinate grid in the first quadrant Calculate area of rectangles, parallelograms, triangles, and volume of a prism

using formulas Select and use appropriate units to measure angles and area Convert measurement within a system Measure and construct figures using tools (compass and protractor)

Process Standards- Students use mathematical processes to solve problems and communicate mathematical ideas. These processes are embedded in the goals above.

Write and interpret more complex number sentences Explain strategies and present ideas to class Listen and apply others’ strategies Understand and connect multiple strategies

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GRADE 3

Numbers, Number Systems and Number Relationships

A. Count using whole numbers (to 10,000) and by 2’s, 3’s, 5’s, 10’s, 25’s and 100’s.

B. Use whole numbers and fractions to represent quantities.

C. Represent equivalent forms of the same number through the use of concrete objects, drawings, word names and symbols.

D. Use drawings, diagrams or models to show the concept of fraction as part of a whole.

E. Count, compare and make change using a collection of coins and one-dollar bills.

F. Apply number patterns (even and odd) and compare values of numbers on the hundred board.

Students understand different representations of numbers, the relationship between/among numbers and number systems.

1. Know and understand verbal names, written names and standard numerals for whole numbers through hundred thousands or more, as well as common decimals, fractions, and percents.

2. Understand and work with whole numbers through hundred thousands, commonly used fractions, and decimals.

3. Know that two numbers in different forms areequivalent or non-equivalent using whole numbers, fractions, and decimals in the context of money.

4. Recognize and represent equivalent forms offractions, decimals, and percents.

5. Understand place value in base ten system to include decimals up to hundredths.

6. Know the place value of digits in numbers to 100,000 and more, including writing expanded forms of numbers.

7. Explore numbers less than zero.

By the end of 3rd grade students will be able to:

Read, write, and identify whole numbers through hundredthousands or more.

Write story problems using whole numbers through hundred thousands or more.

Use real-life experiences, physical materials, and technology to construct meanings for numbers

(unless otherwise noted, all indicators for grade 3

pertain to these sets of numbers as well) with 80% accuracy

Whole numbers through hundred thousands

Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10) as part of a whole, as a subset of a set, and as a location on a number line

Demonstrate an understanding of whole number place

value concepts with 85% accuracy

Identify whether any whole number is odd or even with 90% accuracy

 

  Explore the extension of the place value system

G. Use concrete objects to count, order and group.

H. Demonstrate an understanding of one-to-one correspondence.

I. Apply place-value concepts and numeration to counting, ordering and grouping.

J. Estimate, approximate, round or use exact numbers as appropriate.

K. Describe the inverse relationship between addition and subtraction.

L. Demonstrate knowledge of basic facts in four basic operations.

8. Recognize other number systems andcompare these to the Decimal System, including the Roman Numeral System, using numerals I, V, X, L, and C.

NS4: Students make reasonable estimates

1. Understand the various strategies used for estimating numbers

2. Justify verbally and in writing estimation choices for real world problems

3. Estimate quantities of objects to 300 or more

to decimals through hundredths.

Understand the various uses of numbers Counting, measuring, labeling (e.g.,

numbers on baseball uniforms)

Compare and order numbers.

Compare, contrast and order decimals using concretematerials, number lines, drawings, numerals, language, and symbols (>, <, =).

Create a picture story showing that two numbers in different forms are equivalent or non-equivalent, using whole numbers, fractions, and decimals in the context of money.

Create a chart that shows the relationship among fractions, decimals, and percents.

Design a large number line that only extends from zero toone and place a number of common fractions and decimals on the number line. Include equivalent fractions on the number line.

Compare and contrast the Decimal (base 10) numbersystem to the Roman Numeral System. Use a chart to

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contrast the Roman numerals I, V, X, L, and C.

Estimate size of quantities and explain the estimation. Look at three piles of buttons. There are 12 buttons in the middle pile. Guess how many more buttons are in the biggest pile and how many fewer buttons there are in the smallest pile. Explain the guess.

Key Elements

A. Apply addition and subtraction in everyday situations using concrete objects.

B. Solve single- and double-digit addition and subtraction problems with regrouping in vertical form.

C. Demonstrate the concept of multiplication as repeated addition and arrays.

D. Demonstrate the concept of division as repeated subtraction and as sharing.

COMPUTATIONStudents understand operations and learn operation algorithms

1. Know addition and subtraction of wholenumbers with three digit numbers and the relationship between these two operations.

2. Begin to understand addition and subtraction of decimals and common fractions.3. Understand multiplication and division of whole numbers (multiply one digit by two digits; division with one digit divisor).

4. Begin to understand the inverse relationship ofmultiplication and division.

5. Recall from memory basic multiplication facts up to 12 X 12.

By the end of 3rd grade students will be able to:

Develop the meanings of addition and subtraction by concretely modeling and discussing a large variety of problems.

Explore the meanings of multiplication and division by modeling and discussing problems.Develop proficiency with basic addition and subtraction number facts using a variety of fact strategies (such as “counting on” and “near doubles”) and then commit them to memory.

Construct, use, and explain procedures for performing addition and subtraction calculations with:

Pencil-and-paper Mental math  Calculator

Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

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E. Use estimation skills to arrive at conclusions.

F. Determine the reasonableness of calculated answers.

G. Explain addition and subtraction algorithms with regrouping.

6. Understand the concept of division and remainders

7. Choose the correct operation from addition,subtraction, multiplication, and division to solve specific real world problems.

8. Select appropriate method of computing, such as mental math, pencil and paper, calculator.

Students understand and apply properties of numbers and operations

1. Understand, describe, and apply real number theory concepts.

2. Understand, describe, and apply properties of operations such as commutative, associative, and distributive.

3. Understand the strategies for choosing a particular computing method in any real world problem.

4. Understand the various strategies used formultiplication and division.

Students make reasonable estimates

1. Understand the various strategies used for

Addition of 2-digit numbers Subtraction of 2-digit numbers

Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.

Check the reasonableness of results of computations.

 Understand and use the inverse relationship between addition and subtraction.

Explain and demonstrate the addition and subtraction of whole numbers (up to three digits or more), decimals, and fractions using concrete materials, drawings, symbols, and algorithms.

Solve one step word problems involving money.

Explain the inverse relationship of multiplication anddivision, and create a chart with the related fact families.

Compete in Mental Math Mania with classmates, demonstrating the knowledge of the multiplication facts.Explain and demonstrate the meaning of division and ofremainders using manipulatives, drawings, number

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estimating numbers

2. Justify verbally and in writing estimation choices for real world problems

3. Estimate quantities of objects to 300 or moreMeasurement and Estimation

A. Compare measurable characteristics of different objects on the same dimensions (e.g., time, temperature, area, length, weight, capacity, and perimeter).

B. Determine the measurement of objects with non-standard and standard units (e.g., US customary and metric).

C. Determine and compare elapsed times.

D. Tell time (analog and digital) to the minute.

E. Determine the appropriate unit of measure.

F. Use concrete objects to determine area and perimeter.

MEASUREMENTStudents make comparisons and conversions within measurement systems1. Compare and convert within a system of measurement

2. Begin to use customary, metric, and nonstandard units of measurement to calculate and compare objects.

3. Select and use appropriate standard units and tools for measurement

Students choose appropriate units and tools for measuring

1. Select appropriate unit of measurement tosolve real world problems.

2. Select appropriate instruments and technology to measure in real life situations.

3. Know how to tell time to the minute (both AM and PM) using analog and digital clocks.

By the end of 3rd grade students will be able to:

Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set with 90% accuracy.

Determine the reasonableness of an answer by estimating the result of computations (e.g., 15 + 16 is not 211 with 90% accuracy).

Explore a variety of strategies for estimating both quantities (e.g., the number of marbles in a jar) and results of computation.

Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.

Know how to tell time to the minute (both AM and PM) using analog and digital clocks with 100% accuracy.

Determine capacity and effective instruments for measuring capacity in a given context: The class has decided to participate in the collect-a-million-pennies project. Before collecting pennies, the student must decide how much space

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G. Estimate and verify measurements.

H. Demonstrate that a single object has different attributes that can be measured in different ways (e.g., length, mass, weight, time, area, temperature, capacity, perimeter).

4. Use ruler, graduated cylinder, and thermometer accurately.

Students solve problems utilizing geometric modeling and spatial reasoning

1. Know how to build and draw geometric objects2. Recognize shapes from different perspectives3. Begin to understand concept of area and

perimeter

the pennies will fill if they are stored in gallon milk jugs and how much they will all weigh to determine where they can be stored. Write a report of the findings, including recommendations for different sized storage containers.

Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.

Identify and describe spatial relationships of two or more objects in space.

Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)        Relative shapes and sizes

Mathematical Reasoning and Connections

A. Make, check and verify predictions about the quantity, size and shape of objects and groups of objects.

B. Use measurements in everyday situations (e.g., determine the geography of the school building).

MATHEMATICAL REASONINGStudents estimate measurements in real world contexts

1. Explore and derive strategies for approximating perimeter and area of irregular shapes.

2. Identify and use benchmarks to estimate measurements.

3. Compare and evaluate measurement estimation strategies.

By the end of 3rd grade students will be able to:

Work in groups and use an assortment of various sized boxes to estimate how many “objects” (cubes, large beans, and paper clips) will Fit inside the boxes. Record the estimates and then fill the boxes and record the actual. Report findings to the class.

Use reasoning to support their mathematical conclusions and problem solutions.

Select and use various types of reasoning and methods of proof.

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Problem Solving

A. Use appropriate problem-solving strategies (e.g., guess and check, working backwards).

B. Determine when sufficient information is present to solve a problem and explain how to solve a problem.

C. Select and use an appropriate method, materials and strategy to solve problems, including mental mathematics, paper and pencil and concrete objects.

PROBLEM SOLVINGStudents understand and apply properties of numbers and operations

1. Understand, describe, and apply real number theory concepts.

2. Understand, describe, and apply properties of operations such as commutative, associative, and distributive.

3. Understand the strategies for choosing a particular computing method in any real world problem.

4. Understand the various strategies used formultiplication and division.

By the end of 3rd grade students will be able to:

Solve problems that arise in mathematics and in other contexts .

     Open-ended problems     Non-routine problems       Problems with multiple solutions      Problems that can be solved in several

ways

Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or   “make a diagram”) to solve problems.

Pose problems of various types and levels of difficulty.

Monitor their progress and reflect on the process of their problem solving activity.  

A. Gather, organize and display data using pictures, tallies, charts, bar graphs and pictographs.

B. Formulate and answer questions based on data shown on graphs.

C. Predict the likely number of times a condition will occur based on analyzed data.

DATA ANALYSISStudents generate questions and collect data.

1. Identify different types of graphs.

2. Know the different parts of a graph.

3. Explore information from different types of graphs including graphs from various subject areas and from periodicals.

4. Collect, organize and represent data to communicate results.

By the end of 3rd grade students will:

Use the textbook to explore bar graphs, circle graphs, pictographs; discuss the difference between these graphs and the similarities.

Survey the class for their favorite pizza toppings. Use a data table to organize the information. Create a bar graph and discuss the findings.

Create a graph with height vs. shoe size. Measure students’ height and shoes with inches.

Create a class set of ordered pairs, graph these pairs in the first quadrant of a graph (be sure to use title, labels,

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D. Form and justify an opinion on whether a given statement is reasonable based on a comparison to data.

5. Differentiate between categorical and numerical data.

6. Design statistical experiments and evaluate how the data collection methods affect the nature of the data set.

numbers on the scale).

Collect, generate, organize, and display data in response to questions, claims, or curiosity

Predictions

A. Predict and measure the likelihood of events and recognize that the results of an experiment may not match predicted outcomes.

B. Design a fair and an unfair spinner.

C. List or graph the possible results of an experiment.

D. Analyze data using the concepts of largest, smallest, most often, least often and middle.

Students make predictions and draw reasonable conclusions from collected data.

1. Justify conclusions and make predictions based on given or collected data, including line graphs and circle graphs.

2. Use statistical data to recognize trends and make and explain generalizations.

Students understand and utilize basic concepts of probability.

1. Know counting strategies to determine the possible outcomes of a particular event.

2. Conduct probability experiments by determining the possible outcomes and by making predictions based on those probabilities.

3. Determine the probability through experimentation or simulations, and compare the results with mathematicalexpectation.

By the end of 3rd grade students will be able to:

Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data             Pictograph, bar graph, table

Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color)

What students think will happen (intuitive) Collect data and use that data to predict

the probability (experimental) Design projects to answer questions, collect data and evaluate results.

Use spinners, with different areas of color to generate data. Discuss the most likely color to show up.

Flip a coin 100 times and see if the results approach 50-50.

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4. Consider the degree of likelihood of an event occurring, using terms such as certain, equally likely, and impossible.

Functions

A. Recognize, describe, extend, create and replicate a variety of patterns including attribute, activity, number and geometric patterns.

B. Use concrete objects and trial and error to solve number sentences and check if solutions are sensible and accurate.

C. Substitute a missing addend in a number sentence.

D. Create a story to match a given combination of symbols and numbers.

E. Use concrete objects and symbols to model the concepts of variables, expressions, equations and inequalities.

ALGEBRAIC THINKINGStudents recognize, understand and extend patterns.

1. Describe, extend and create numerical and geometric patterns using models.

2. Generalize a pattern, relation or function in words, tables, and/or graphs.

3. Analyze patterns and identify mathematical relationships to pose and solve problems.

Students utilize symbols and mathematical expressions to represent mathematical situations

1. Select appropriate properties and apply them in computation.

2. Introduce the term ‘variable’ for an unknown value, represented by a letter.

By the end of grade 3 students will be able to:

Use color tiles or cubes to design a step pattern and determine the number of tiles or cubes that must come next.

Investigate function machines. Create a function machine from an old box – put in a 3, out comes a 5. Put in a 5, out comes a 7. Name that function.

Practice the patterns found in the multiples of 2, 3, 5, and 10.

Discuss and explain rules that apply to patterns in completing tables or charts.

Write number sentences from a story problem.

Create a story problem from a number sentence.

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F. Explain the meaning of solutions and symbols.

G. Use a table or a chart to display information.

H. Describe and interpret the data shown in tables and charts.

I. Demonstrate simple function rules.

J. Analyze simple functions and relationships and locate points on a simple grid.

3. Represent problem situations using variables represented by a letter.

Students use mathematical models to represent math relationships

1. Express real world mathematic relationships using physical models and equations.

2. Use information from physical models, charts, tables and graphs to solve problems.

Write and solve number sentences that represent classroom situations.

Look at age vs. height on a classroom chart. Discuss concept of change in relation to chart.

A. Name and label geometric shapes in two and three dimensions (e.g., circle/sphere, square/cube, triangle/pyramid, rectangle/prism).

B. Build geometric shapes using concrete objects (e.g., manipulatives).

C. Draw two- and three-dimensional geometric shapes and construct rectangles, squares and triangles on graph paper satisfying specific criteria.

D. Find and describe geometric figures in real life.

GEOMETRYStudents build geometric understanding by analyzing 2- and 3- dimensional shapes.

1. Compare and analyze 2- and 3-dimensional shapes based on characteristics and properties.

2. Know appropriate vocabulary to describe characteristics of geometric shapes, including parallel and perpendicular lines, quadrilateral, and right angle.

3. Investigate and develop strategies for subdividing, combining, and changing shapes.

Students apply transformations and symmetry to other math courses and to the visual arts.

By the end of 3rd grade students will:

Use real-life experiences and physical materials to describe, classify, compare, and sort geometric figures, including squares, rectangles, triangles, circles, cubes, rectangular solids, spheres, pyramids, cylinders, and prisms, according to the number of faces, edges, bases and corners.

Given real objects in geometric shapes, such as cereal boxes or rectangular solids, classify them according to specified criteria and explain the groupings.

Make and test predictions and draw conclusions regarding geometric properties and relationships.

Find the distance between two points (horizontal or vertical lines only).

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E. Identify and draw lines of symmetry in geometric figures.

F. Identify symmetry in nature.

G. Fold paper to demonstrate the reflections about a line.

H. Show relationships between and among figures using reflections.

I. Predict how shapes can be changed by combining or dividing them.

1. Understand concepts of symmetry, congruency, and similarity.

2. Understand concept of flips, slides and turns.

Students solve problems utilizing geometric modeling and spatial reasoning.

1. Know how to build and draw geometric objects.

2. Recognize shapes from different perspectives.

3. Begin to understand concept of area and perimeter.

Identify figures that are symmetrical, congruent or similar.

Predict, draw, and confirm figure as a result of one of the rigid transformations (reflection, rotation, translation).

Draw and classify two-dimensional figures having up to six or more sides.

Create a 3-dimensional object from a 2-dimensional representation.

Trigonometry

C. Identify right angles in the environment.

D. Model right angles and right triangles using concrete objects.

E. Draw two- and three-dimensional geometric shapes and construct rectangles, squares and triangles on the geoboard and on graph paper satisfying specific criteria.

F. Find and describe geometric figures

Students solve problems utilizing geometric modeling and spatial reasoning

1. Know how to build and draw geometric objects2. Recognize shapes from different perspectives3. Begin to understand concept of area and

perimeter

By the end of grade 3 students will:

Identify figures that are symmetrical, congruent or similar.

Predict, draw, and confirm figure as a result of one of the rigid transformations (reflection, rotation, translation).

Draw and classify two-dimensional figures having up to six or more sides.

Create a 3-dimensional object from a 2-dimensional representation.

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in real life.

G. Identify and draw lines of symmetry in geometric figures.

H. Identify symmetry in nature.

I. Fold paper to demonstrate the reflections about a line.

J. Show relationships between and among figures using reflections.

K. Predict how shapes can be changed by combining or dividing them.

GRADE 4

A. Multiple representations of numbers deepen the understanding of place value (i.e. Place value blocks, number lines, standard, expanded, and word form)

B. Our numerical system is organized around a base of ten. The system is arranged in groups of three place values called periods. It allows for the creation of an infinite number of numbers using only the digits 0-9. Place value charts arrange numbers in a way that allows one to better understand the value of each digit.

NUMBERS AND OPERATIONS Students will be able to:Thousands – represent numbers with place value blocks and

number lines. They write numbers in standard form, expanded form, and written form.

Millions – represent numbers in the millions using a place value chart. They write numbers in expanded form, using periods to help write numbers in word form. (

Comparing and Ordering Whole Numbers –apply their knowledge of place value to compare and order numbers.

Rounding Whole Numbers – show how to use place value to round whole numbers.

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C. Counting, estimating, and rounding allow for mental math computation and lead to a better understanding of when the operations can be beneficial

D. Computational fluency includes understanding not only the meaning, but also the appropriate use of numerical operations.

E. The magnitude of numbers affects the outcome of operations on them.

F. The use of arrays develops meaning for multiplication leading to a better understanding of arithmetic operations by modeling ad discussing a large variety of problems.

G. Proficiency with basic multiplication is aided in using a variety of fact strategies such as skip counting, repeated addition, and finding patterns.

H. For a given set of numbers, there are relationships that are always true called properties, and these are the rules that govern arithmetic and algebra.

.

Using Money to Understand Decimals – use place value charts to read, write, and compare decimals in tenths and hundredths using money.

Counting Money and Making Change – convert a collection of coins and bills into a total amount and are able to make change.

Problem Solving: Make an Organized List – systematically find and record all possible outcomes for a situation.

Using Mental Math to Add and Subtract – apply a variety of methods to add and subtract whole numbers mentally.

Estimating Sums and Differences of Whole Numbers – round whole numbers to estimate sums and differences.

Problem Solving: Missing or Extra Information – identify what information in a problem is not needed or not present.

Adding Whole Numbers – add numbers to hundred thousands with or without regrouping.

Subtracting Whole Numbers – subtract numbers to thousands with and without regrouping

Subtracting Across Zeros – subtract numbers with zeros to thousands.

Problem Solving: Draw a Picture and Write an Equation –

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use a picture or diagram to translate an everyday situation into a number sentence or equation.

Recognize multiplication as repeated addition of equal groups used in arrays and comparisons.

Use patterns to find products with factors of 2, 5, and 9.

Use multiplication properties to simplify computations. (

Use the Distributive Property to simplify multiplication problems involving 3s and 4s by rewriting one of the factors as a sum of two numbers.

Use the distributive Property and other regrouping properties to simplify multiplication involving 6s, 7s, and 8s by rewriting one of the factors.

Use patterns to mastery of facts and multiples of 10, 11, 12.

Draw pictures to problem solve multiplication situations and use their pictures to write number sentences.

A. Sharing and repeated subtraction involves separating equal groups and are two ways to think about division.

B. Multiplication and division have an inverse relationship.

C. The inverse relationship between multiplication and division can be used to find division facts; every division fact has a related

Division Use and draw models to solve division problems. Use arrays to write and complete multiplication and division

fact families.

Use multiplication facts with 0 and 1 to learn about special division rules with 0 and 1.

Identify multiplication facts related to division facts in order to solve division problems.

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multiplication factDraw pictures and write related number sentences to solve problems.

A. Breaking apart calculations into simpler ones is the idea used in all algorithms for rational numbers. This directly correlates with the Associative Property which states that the way in which factors are grouped does not affect the product.

B. Answers to problems should always be checked for reasonableness and this can be done through estimation and to check the answer against the questions ad conditions in the problem.

Multiplying by 1-Digit Numbers The student will be able to:Construct, use, and explain procedures for performing whole

number compuations in multiplication.

Use basic multiplication facts and number patterns to multiply by multiples of 10 and 100.

Use compatible numbers with adjustment, breaking apart,

and other strategies to multiply numbers mentally.

Use compatible numbers and rounding to estimate solutions to multiplication problems.

Check for reasonableness by making sure their calculations answer the questions asked and by using estimation to make sure the calculation was performed correctly.

A. The symbolic language of algebra is used to communicate and generalize the patterns in mathematics.

B. Algebraic representation using variables can be used to generalize patterns and relationships.

C. Algebra provides language through which we communicate the patterns in mathematics.

ALGEBRAPatterns and Expressions

Students will be able to:Understand how to work with variables in a table.

Study completed tables to determine a rule and write it as an addition or subtraction expression.

Study completed tables to determine a rule and write it as a multiplication or division expression.

Solve problems by using objects to show the action.

A. Basic facts and place-value patterns can be used to mentally multiply a

Multiplying 2-digit Numbers By the end of 4th grade students will be able to:

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two-digit number by a multiple of 10, 100, or 1,000.

B. Context is critical when using estimation.

C. There is more than one algorithm to solve a problem.

Discover, understand, and use patterns to multiply by 10 and 100.

Use rounding and compatible numbers to estimate solutions to multiplication problems.

Use arrays and expanded algorithms to multiply two-digit numbers by two-digit numbers to find the product.

Use grids and patterns to multiply 2-digit numbers and multiples of 10.

Use partial products to multiply two-digit numbers by two-digit numbers and find the products.

Multiply greater numbers. Solve two-question problems.

A. Multiples of 10, 100, and 1,000 provide place-value patterns that can be used to divide using mental math.

B. Models can be used to explore what happens when a group is separated into smaller groups, and there are remainders.

C. Using estimation, rounding, and mental math before dividing will help in the placement of the first digit when dividing. Regrouping with models will further show how to know where to begin dividing.

Dividing by 1-digit divisors By the end of 4th grade students will be able to:

Construct, use, and explain procedures for performing whole number compuations in division.

Use basic facts and patterns of zeros to solve division problems with 3-digit dividends and 1-digit divisors.

Use compatible numbers and rounding to estimate quotients.

Divide whole numbers by 1-digit divisors resulting in quotients with remainders.

Use place value to understand the algorithm for long division.

Use the standard algorithm to divide two-digit numbers by one-digit numbers.

Use the standard algorithm to divide 3-digit numbers by 1-digit numbers.

Use the standard algorithm to divide 3-digit numbers by 1-digit numbers and properly decide where to start dividing.

Learn how to factor whole numbers.Learn to identify prime and composite numbers. Identify the hidden question in a multi-step problem.

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A. Point, line, and plane are the core attributes of space objects, and real-world situations can be used to think about these attributes.

B. Line segments and rays are sets of points that describe parts of lines, shapes, and solids. Angles are formed by two intersecting lines or by rays with a common endpoint and are classified by size.

C. Plane shapes have many properties that make them different from one another. Polygons can be described and classified by their sides and angles.

Geometry

1. Communicate mathematical ideas in clear, concise, organized language that varies in content, format and form for different audiences and purposes.

2. Comprehend, understand, analyze, evaluate,

critique, solve, and respond to a variety of real-life, meaningful problems.

3. Investigate, research, and synthesize various information from a variety of media sources.

By the end of 4th grade students will be able to:

Describe, classify, and analyze lines, angles, and shapes according to their attributes.

Identify and describe points, lines, and planes. Learn Geometric terms to describe parts of lines and types

of angles. Learn to measure and draw angles. Identify polygonsLearn to identify and classify triangles. Learn to identify quadrilaterals. Solve problems by making and testing generalizations Learn to Identify terms used in circles (ie. Diameter, radius, center)Identify and draw fractional parts of a region and a set, and

divide sets to show fractional parts. Describe and compare fractional parts of whole objects and

setsEstimate fractional parts of regions and sets, and estimate

fractions for points on the number line. Use models and objects to show equivalent fractions.

simplest form. Identify and write mixed numbers as improper fractions and

improper fractions as mixed numbers. Use benchmark fractions to compare fractions with unlike

denominators.

A. A fraction describes the division of a whole region into equal parts and can be interpreted in more than one way.

B. Benchmark fractions can be used to

Numeration

Communicate mathematical ideas in clear, concise, organized language that varies in content, format

By the end of 4th grade students will be able to:Identify and draw fractional parts of a region and a set, and

divide sets to show fractional parts. Describe and compare fractional parts of whole objects and

sets.

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estimate fractional amounts.C. The same fractional amount can be

represented by an infinite set of different but equivalent fractions.

D. Fractional amounts greater than 1 can be represented using a whole number and a fraction

and form for various audiences and purposes.

Comprehend, understand, analyze, evaluate, critique, solve, and respond to a variety of meaningful real life problems.

Investigate, research, and synthesize various information from a variety of sources.

Estimate fractional parts of regions and sets, and estimate fractions for points on the number line.

Use models and objects to show equivalent fractions. Express equivalent fractions in simplest form.Identify and write mixed numbers as improper fractions and

improper fractions as mixed numbers. Use benchmark fractions to compare fractions with unlike

denominators.Use common denominators and equivalent fractions to order

fractions with unlike denominators. Write to explain whether an answer is correct or not.

A. To add or subtract or subtract fractions with like denominators, add or subtract the numerators and write the sum over the common denominator.

B. To add with unlike denominators, change to an equivalent fraction with like denominators.

Numbers and Operations 31Adding and Subtracting Fractions

By the end of 4th grade students will be able to:The student will be able to use concrete models to explore

addition and subtraction of fractions and relate them to real life situations.

Add and subtract fractions with like denominators using models and paper and pencil.

Add fractions with unlike denominators using models and paper and pencil.

Subtract fractions with unlike denominators using models and paper and pencil.

Draw a picture and write an equation to solve a problem.

A. Decimal numeration is just an extension of whole number numeration.

B. Place value can be used to compare and order numbers.

C. A decimal is another name for a fraction.

D. Each fraction, mixed number, and decimal can be associated with a unique point on the number line.

NumerationsUnderstanding decimals

By the end of 4th grade students will be able to:

Use models and place value charts to represent decimals to hundredths. Read and write decimals in expanded, standard, and word form.

Use models and place value charts to compare decimals to hundredths. They use greater-than and less-than symbols to order decimal numbers.

Write fractions as decimals and decimals as fractions. Locate and name fractions and decimals on a number line. Students will understand how to graph decimals and mixed

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E. Information in a problem can often be shown using a picture or diagram to understand and solve the problem.

numbers on the number line. Solve problems using the strategy Draw a Picture

A. Rounding decimals is a process for finding the multiple of .1, .01, etc., closest to a given number.

B. Techniques for estimating calculations with whole numbers can also be used to estimate calculations with decimals

C. Models and algorithms for adding or subtracting multi-digit decimals are just an extension of using models and algorithms for adding and subtracting whole numbers

D. Multiplying and dividing decimals is similar to multiplying and dividing whole numbers. Estimation can help determine where to place the decimal point.

Number and OperationsOperations with decimals

By the end of 4th grade students will be able to:Round two-place decimal numbers to one place or the

nearest whole number.Round decimal numbers to estimate sums and differences. Add and subtract decimals in tenths and hundredths using

models. Estimate and compute the sum or difference of whole

numbers and positive decimals to two places. Multiply a decimal by a whole number. Divide a decimal by a whole number.Try and check solutions and if it is not correct, revise the

solution, following the same method until the correct solution is determined via checking.

A. The amount of space inside a shape is its area and area can be estimated or found using square units. Formulas exist for finding the area of some polygons.

B. Some measurements can be approximated using known references as the unit in the measurement process.

C. Measurements provide a real world context for revisiting mathematics from other strands of the curriculum since the process of measurement

MeasurementArea

By the end of 4th grade students will be able to:Measure the area of a figure by counting the number of

square units that cover a region. Find the area of rectangles by counting square units or by using a formulaFind the area of irregular shapes. Use the formula for the area of a rectangle in order to find a formula for the area of a parallelogram. Use the relationship between triangles and parallelograms to find the area of triangles. Compare different rectangles with the same perimeter to discover the change in area.Compare different rectangles with the same area to discover

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is the same for every attribute that is measurable.

D. Break a problem into smaller, more manageable pieces and find a pattern to fit.

the change in perimeter.

A. Three dimensional or solid figures have length, width, and height. Many can be described, classified, and analyzed by their faces, edges, and vertices. Many everyday object closely approximate standard geometric solids.

B. Volume is the amount of space inside a solid figure. Volume can be measured by counting the number of cubic units needed to fill the three-dimensional object.

GeometrySolids

The goal is to identify, classify, construct, and describe two and three-dimensional shapes.

By the end of 4th grade students will be able to:Describe and classify solids. Use two dimensional shape to represent a three-dimensional

object. Interpret views of solids as seen from different perspectives. Measure the volume of a solid either by counting cubic units

or by using a formula.Recognize patterns and are able to continue the pattern.

GRADE 5

GRADE 5 Math

Numbers, Number Systems and Number Relationships

A. Use expanded notation to represent whole numbers or decimals.

B. Apply number theory concepts to rename a number quantity (e.g., six, 6,

, 3 2, 10 4).

C. Demonstrate that mathematical operations can represent a variety of

NUMBER SENSEStudents understand different representations of numbers, the relationship between/among numbers and number systems.

1. Identify and use verbal names, written names and standard numerals for whole numbers through millions or more.

2. Know and identify fractions and mixed numbers with denominators including 2,3,4,5,6,8,10, 12, 20, 25, 100 and1000.

By the end of 5th grade students will be able to:

Read and write whole numbers through millions.

Represent, order and compare whole numbers through millions or more.

Read and write fractions and mixed numbers, through thousandths.

Represent order and compare commonly used fractions and decimals to thousandths using concrete materials or drawings.

Read and write decimals through thousandths.

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problem situations.

D. Use models to represent fractions and decimals.

E. Explain the concepts of prime and composite numbers.

F. Use simple concepts of negative numbers (e.g., on a number line, in counting, in temperature).

G. Develop and apply number theory concepts (e.g., primes, factors, multiples, composites) to represent numbers in various ways.

H. Explain budgeting

3. Know and identify whole numbers and decimals through thousandths.

4. Understand the meaning of and identify percents including 10, 20, 25, 30, 40, 50, 60, 70, 75, 80, 90, and 100 percent.

5. Translate problem situations into using whole numbers, fractions, mixed numbers, decimals and percents.

6. Know that two numbers in different forms are equivalent or non-equivalent, using whole numbers, decimals, fractions,mixed numbers, and percents.

7. Understand place value in base ten system to include decimals, and know that place value relates to powers of

8. Express numbers to millions or more in expanded form using powers of ten, with or without exponential notation.

9. Explore numbers less than 0.

10. Compare and contrast the decimal number system and other number systems that do or do not use place value.

Students understand and apply properties of

Read and write common percents including 10%, 20%, 25%, 30%, 40%, 50%, 60%, 70%, 75% , 80%, 90%, and 100%.

Compare and order whole numbers, commonly used fractions, percents, and decimals (to thousandths) using concrete materials, number lines, drawings, numerals, and symbols (>, <, =).

Explain and justify the solution to problem situations with the use of diagrams, models, and numerals.

Compare and contrast numbers decimals, fractions, mixed numbers, and percents in different forms, both equivalent and nonequivalent, using tables charts and graph paper.

Explain how place value relates to the powers of 10, using charts, diagrams, or manipulatives.

Write numbers to millions or more in expanded form using powers of ten, in chart or table or display models with graph paper.Explain and demonstrate the multiplication of numbers between zero and one, (for example, when two numbers less than one are multiplied, the result is a number less than either factor) using concrete materials, drawings, story problems, symbols, and algorithms. (Group project with oral report).

Explain the similarities and differences between the decimal (base 10) number system and other number systems (Research this on the Internet and report, individually or by groups).

Use real life experiences, physical material, and technology to construct meanings for numbers. All fractions as part of a whole, as subset of a

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numbers and operations.

1. Understand, describe, and apply properties of operations including the identity, commutative, and associativeproperties of addition; the zero and identity properties of multiplication; the commutative, associative, and distributiveproperties of multiplication.

2. Differentiate between prime numbers and composite numbers.

3. Determine the prime factors of numbers through 100 and write the numbers as the product of their prime factors.

4. Determine the greatest common factor or the least common multiple of two numbers up to 100 or more.

5. Know rules of divisibility.

6. Use models to identify perfect squares to 144.

7. Explain the reason for choosing a particular computing method for a particular problem.

8. Use a table or chart to differentiate between the greatest common factor and the least common multiple of two numbers up to 100 or more.

set, as a location on a number line, and as divisions of whole numbers

Develop and apply number theory concepts in problem solving situations Primes, factors, multiplesGiven specific numbers, students identify them as prime or composite.

Recognize the appropriate use of each arithmetic operation in problem situations

Make posters and display a number story to explain and demonstrate the commutative, associative, and distributive properties of multiplication.

Discover patterns in the various products when a whole number is displayed as a product of its prime factors.

Display the perfect squares to 144 with the use of square arrays on graph paper.

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GRADE 4

Numbers, Number Systems and Number Relationships

A. Use whole numbers and fractions to represent quantities.

B. Represent equivalent forms of the same number through the use of concrete objects, drawings, word names and symbols.

M. Use drawings, diagrams or models to show the concept of fraction as part of a whole.

N. Count, compare and make change using a collection of coins and one-dollar bills.

O. Apply number patterns (even and odd) and compare values of numbers on the hundred board.

P. Use concrete objects to count, order and group.

Q. Demonstrate an understanding of one-to-one correspondence.

Students understand different representations of numbers, the relationship between/among numbers and number systems.

1. Know and understand verbal names, written names and standard numerals for whole numbers through hundred thousands or more, as well as common decimals, fractions, and percents.

2. Understand and work with whole numbers through hundred thousands, commonly used fractions, and decimals.

3. Know that two numbers in different forms areequivalent or non-equivalent using whole numbers, fractions, and decimals in the context of money.

4. Recognize and represent equivalent forms offractions, decimals, and percents.

5. Understand place value in base ten system to include decimals up to hundredths.

6. Know the place value of digits in numbers to 100,000 and more, including writing expanded forms of numbers.

7. Explore numbers less than zero.

By the end of 4th grade students will be able to:

Read, write, and identify whole numbers through hundredthousands or more.

Write story problems using whole numbers through hundred thousands or more.

Use real-life experiences, physical materials, and technology to construct meanings for numbers

(unless otherwise noted, all indicators for grade 3

pertain to these sets of numbers as well) with 80% accuracy

Whole numbers through hundred thousands

Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10) as part of a whole, as a subset of a set, and as a location on a number line

Demonstrate an understanding of whole number place

value concepts with 85% accuracy

Identify whether any whole number is odd or even with 90% accuracy

 

  Explore the extension of the place value system

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R. Apply place-value concepts and numeration to counting, ordering and grouping.

S. Estimate, approximate, round or use exact numbers as appropriate.

T. Describe the inverse relationship between addition and subtraction.

U. Demonstrate knowledge of basic facts in four basic operations.

8. Recognize other number systems andcompare these to the Decimal System, including the Roman Numeral System, using numerals I, V, X, L, and C.

NS4: Students make reasonable estimates

4. Understand the various strategies used for estimating numbers

5. Justify verbally and in writing estimation choices for real world problems

6. Estimate quantities of objects to 300 or more

to decimals through hundredths.

Understand the various uses of numbers Counting, measuring, labeling (e.g.,

numbers on baseball uniforms)

Compare and order numbers.

Compare, contrast and order decimals using concretematerials, number lines, drawings, numerals, language, and symbols (>, <, =).

Create a picture story showing that two numbers in different forms are equivalent or non-equivalent, using whole numbers, fractions, and decimals in the context of money.

Create a chart that shows the relationship among fractions, decimals, and percents.

Design a large number line that only extends from zero toone and place a number of common fractions and decimals on the number line. Include equivalent fractions on the number line.

Compare and contrast the Decimal (base 10) numbersystem to the Roman Numeral System. Use a chart to

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contrast the Roman numerals I, V, X, L, and C.

Estimate size of quantities and explain the estimation. Look at three piles of buttons. There are 12 buttons in the middle pile. Guess how many more buttons are in the biggest pile and how many fewer buttons there are in the smallest pile. Explain the guess.

Computation and Estimation

H. Apply addition and subtraction in everyday situations using concrete objects.

I. Solve single- and double-digit addition and subtraction problems with regrouping in vertical form.

J. Demonstrate the concept of multiplication as repeated addition and arrays.

K. Demonstrate the concept of division as repeated subtraction and as sharing.

Students understand operations and learn operation algorithms

1. Know addition and subtraction of wholenumbers with three digit numbers and the relationship between these two operations.

2. Begin to understand addition and subtraction of decimals and common fractions.3. Understand multiplication and division of whole numbers (multiply one digit by two digits; division with one digit divisor).

4. Begin to understand the inverse relationship ofmultiplication and division.

5. Recall from memory basic multiplication facts up to 12 X 12.

6. Understand the concept of division and

By the end of 3rd grade students will be able to:

Develop the meanings of addition and subtraction by concretely modeling and discussing a large variety of problems.

Explore the meanings of multiplication and division by modeling and discussing problems.Develop proficiency with basic addition and subtraction number facts using a variety of fact strategies (such as “counting on” and “near doubles”) and then commit them to memory.

Construct, use, and explain procedures for performing addition and subtraction calculations with:

Pencil-and-paper Mental math  Calculator

Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

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L. Use estimation skills to arrive at conclusions.

M. Determine the reasonableness of calculated answers.

N. Explain addition and subtraction algorithms with regrouping.

remainders

7. Choose the correct operation from addition,subtraction, multiplication, and division to solve specific real world problems.

8. Select appropriate method of computing, such as mental math, pencil and paper, calculator.

Students understand and apply properties of numbers and operations

1. Understand, describe, and apply real number theory concepts.

2. Understand, describe, and apply properties of operations such as commutative, associative, and distributive.

3. Understand the strategies for choosing a particular computing method in any real world problem.

4. Understand the various strategies used formultiplication and division.

Students make reasonable estimates

4. Understand the various strategies used for estimating numbers

Addition of 2-digit numbers Subtraction of 2-digit numbers

Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.

Check the reasonableness of results of computations.

 Understand and use the inverse relationship between addition and subtraction.

Explain and demonstrate the addition and subtraction of whole numbers (up to three digits or more), decimals, and fractions using concrete materials, drawings, symbols, and algorithms.

Solve one step word problems involving money.

Explain the inverse relationship of multiplication anddivision, and create a chart with the related fact families.

Compete in Mental Math Mania with classmates, demonstrating the knowledge of the multiplication facts.Explain and demonstrate the meaning of division and ofremainders using manipulatives, drawings, number

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5. Justify verbally and in writing estimation choices for real world problems

6. Estimate quantities of objects to 300 or moreMeasurement and Estimation

I. Compare measurable characteristics of different objects on the same dimensions (e.g., time, temperature, area, length, weight, capacity, and perimeter).

J. Determine the measurement of objects with non-standard and standard units (e.g., US customary and metric).

K. Determine and compare elapsed times.

L. Tell time (analog and digital) to the minute.

M. Determine the appropriate unit of measure.

N. Use concrete objects to determine area and perimeter.

MEASUREMENTStudents make comparisons and conversions within measurement systems1. Compare and convert within a system of measurement

2. Begin to use customary, metric, and nonstandard units of measurement to calculate and compare objects.

3. Select and use appropriate standard units and tools for measurement

Students choose appropriate units and tools for measuring

1. Select appropriate unit of measurement tosolve real world problems.

2. Select appropriate instruments and technology to measure in real life situations.

3. Know how to tell time to the minute (both AM and PM) using analog and digital clocks.

4. Use ruler, graduated cylinder, and thermometer

By the end of 3rd grade students will be able to:

Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set with 90% accuracy.

Determine the reasonableness of an answer by estimating the result of computations (e.g., 15 + 16 is not 211 with 90% accuracy).

Explore a variety of strategies for estimating both quantities (e.g., the number of marbles in a jar) and results of computation.

Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.

Know how to tell time to the minute (both AM and PM) using analog and digital clocks with 100% accuracy.

Determine capacity and effective instruments for measuring capacity in a given context: The class has decided to participate in the collect-a-million-pennies project. Before collecting pennies, the student must decide how much space the pennies will fill if they are stored in gallon

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O. Estimate and verify measurements.

P. Demonstrate that a single object has different attributes that can be measured in different ways (e.g., length, mass, weight, time, area, temperature, capacity, perimeter).

accurately.

Students solve problems utilizing geometric modeling and spatial reasoning

4. Know how to build and draw geometric objects5. Recognize shapes from different perspectives6. Begin to understand concept of area and

perimeter

milk jugs and how much they will all weigh to determine where they can be stored. Write a report of the findings, including recommendations for different sized storage containers.

Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.

Identify and describe spatial relationships of two or more objects in space.

Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)        Relative shapes and sizes

Mathematical Reasoning and Connections

C. Make, check and verify predictions about the quantity, size and shape of objects and groups of objects.

D. Use measurements in everyday situations (e.g., determine the geography of the school building).

MATHEMATICAL REASONINGStudents estimate measurements in real world contexts

1. Explore and derive strategies for approximating perimeter and area of irregular shapes.

2. Identify and use benchmarks to estimate measurements.

3. Compare and evaluate measurement estimation strategies.

By the end of 3rd grade students will be able to:

Work in groups and use an assortment of various sized boxes to estimate how many “objects” (cubes, large beans, and paper clips) will Fit inside the boxes. Record the estimates and then fill the boxes and record the actual. Report findings to the class.

Use reasoning to support their mathematical conclusions and problem solutions.

Select and use various types of reasoning and methods of proof.

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Problem Solving

D. Use appropriate problem-solving strategies (e.g., guess and check, working backwards).

E. Determine when sufficient information is present to solve a problem and explain how to solve a problem.

F. Select and use an appropriate method, materials and strategy to solve problems, including mental mathematics, paper and pencil and concrete objects.

PROBLEM SOLVINGStudents understand and apply properties of numbers and operations

1. Understand, describe, and apply real number theory concepts.

2. Understand, describe, and apply properties of operations such as commutative, associative, and distributive.

3. Understand the strategies for choosing a particular computing method in any real world problem.

4. Understand the various strategies used formultiplication and division.

By the end of 3rd grade students will be able to:

Solve problems that arise in mathematics and in other contexts .

     Open-ended problems     Non-routine problems       Problems with multiple solutions      Problems that can be solved in several

ways

Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or   “make a diagram”) to solve problems.

Pose problems of various types and levels of difficulty.

Monitor their progress and reflect on the process of their problem solving activity.  

E. Gather, organize and display data using pictures, tallies, charts, bar graphs and pictographs.

F. Formulate and answer questions based on data shown on graphs.

G. Predict the likely number of times a condition will occur based on analyzed data.

H. Form and justify an opinion on

DATA ANALYSISStudents generate questions and collect data.

1. Identify different types of graphs.

2. Know the different parts of a graph.

3. Explore information from different types of graphs including graphs from various subject areas and from periodicals.

4. Collect, organize and represent data to communicate results.

5. Differentiate between categorical and numerical

By the end of 3rd grade students will:

Use the textbook to explore bar graphs, circle graphs, pictographs; discuss the difference between these graphs and the similarities.

Survey the class for their favorite pizza toppings. Use a data table to organize the information. Create a bar graph and discuss the findings.

Create a graph with height vs. shoe size. Measure students’ height and shoes with inches.

Create a class set of ordered pairs, graph these pairs in the first quadrant of a graph (be sure to use title, labels, numbers on the scale).

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whether a given statement is reasonable based on a comparison to data.

data.

6. Design statistical experiments and evaluate how the data collection methods affect the nature of the data set.

Collect, generate, organize, and display data in response to questions, claims, or curiosity

Predictions

E. Predict and measure the likelihood of events and recognize that the results of an experiment may not match predicted outcomes.

F. Design a fair and an unfair spinner.

G. List or graph the possible results of an experiment.

H. Analyze data using the concepts of largest, smallest, most often, least often and middle.

Students make predictions and draw reasonable conclusions from collected data.

1. Justify conclusions and make predictions based on given or collected data, including line graphs and circle graphs.

2. Use statistical data to recognize trends and make and explain generalizations.

Students understand and utilize basic concepts of probability.

1. Know counting strategies to determine the possible outcomes of a particular event.

2. Conduct probability experiments by determining the possible outcomes and by making predictions based on those probabilities.

3. Determine the probability through experimentation or simulations, and compare the results with mathematicalexpectation.

4. Consider the degree of likelihood of an event

By the end of 3rd grade students will be able to:

Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data             Pictograph, bar graph, table

Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color)

What students think will happen (intuitive) Collect data and use that data to predict

the probability (experimental) Design projects to answer questions, collect data and evaluate results.

Use spinners, with different areas of color to generate data. Discuss the most likely color to show up.

Flip a coin 100 times and see if the results approach 50-50.

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occurring, using terms such as certain, equally likely, and impossible.

Functions

K. Recognize, describe, extend, create and replicate a variety of patterns including attribute, activity, number and geometric patterns.

L. Use concrete objects and trial and error to solve number sentences and check if solutions are sensible and accurate.

M. Substitute a missing addend in a number sentence.

N. Create a story to match a given combination of symbols and numbers.

O. Use concrete objects and symbols to model the concepts of variables, expressions, equations and inequalities.

P. Explain the meaning of solutions and

ALGEBRAIC THINKINGStudents recognize, understand and extend patterns.

1. Describe, extend and create numerical and geometric patterns using models.

2. Generalize a pattern, relation or function in words, tables, and/or graphs.

3. Analyze patterns and identify mathematical relationships to pose and solve problems.

Students utilize symbols and mathematical expressions to represent mathematical situations

1. Select appropriate properties and apply them in computation.

2. Introduce the term ‘variable’ for an unknown value, represented by a letter.

3. Represent problem situations using variables

By the end of grade 3 students will be able to:

Use color tiles or cubes to design a step pattern and determine the number of tiles or cubes that must come next.

Investigate function machines. Create a function machine from an old box – put in a 3, out comes a 5. Put in a 5, out comes a 7. Name that function.

Practice the patterns found in the multiples of 2, 3, 5, and 10.

Discuss and explain rules that apply to patterns in completing tables or charts.

Write number sentences from a story problem.

Create a story problem from a number sentence.

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symbols.

Q. Use a table or a chart to display information.

R. Describe and interpret the data shown in tables and charts.

S. Demonstrate simple function rules.

T. Analyze simple functions and relationships and locate points on a simple grid.

represented by a letter.

Students use mathematical models to represent math relationships

1. Express real world mathematic relationships using physical models and equations.

2. Use information from physical models, charts, tables and graphs to solve problems.

Write and solve number sentences that represent classroom situations.

Look at age vs. height on a classroom chart. Discuss concept of change in relation to chart.

J. Name and label geometric shapes in two and three dimensions (e.g., circle/sphere, square/cube, triangle/pyramid, rectangle/prism).

K. Build geometric shapes using concrete objects (e.g., manipulatives).

L. Draw two- and three-dimensional geometric shapes and construct rectangles, squares and triangles on graph paper satisfying specific criteria.

M. Find and describe geometric figures in real life.

N. Identify and draw lines of symmetry

GEOMETRYStudents build geometric understanding by analyzing 2- and 3- dimensional shapes.

1. Compare and analyze 2- and 3-dimensional shapes based on characteristics and properties.

2. Know appropriate vocabulary to describe characteristics of geometric shapes, including parallel and perpendicular lines, quadrilateral, and right angle.

3. Investigate and develop strategies for subdividing, combining, and changing shapes.

Students apply transformations and symmetry to other math courses and to the visual arts.

By the end of 3rd grade students will:

Use real-life experiences and physical materials to describe, classify, compare, and sort geometric figures, including squares, rectangles, triangles, circles, cubes, rectangular solids, spheres, pyramids, cylinders, and prisms, according to the number of faces, edges, bases and corners.

Given real objects in geometric shapes, such as cereal boxes or rectangular solids, classify them according to specified criteria and explain the groupings.

Make and test predictions and draw conclusions regarding geometric properties and relationships.

Find the distance between two points (horizontal or vertical lines only).

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in geometric figures.

O. Identify symmetry in nature.

P. Fold paper to demonstrate the reflections about a line.

Q. Show relationships between and among figures using reflections.

R. Predict how shapes can be changed by combining or dividing them.

1. Understand concepts of symmetry, congruency, and similarity.

2. Understand concept of flips, slides and turns.

Students solve problems utilizing geometric modeling and spatial reasoning.

1. Know how to build and draw geometric objects.

2. Recognize shapes from different perspectives.

3. Begin to understand concept of area and perimeter.

Identify figures that are symmetrical, congruent or similar.

Predict, draw, and confirm figure as a result of one of the rigid transformations (reflection, rotation, translation).

Draw and classify two-dimensional figures having up to six or more sides.

Create a 3-dimensional object from a 2-dimensional representation.

Trigonometry

A. Identify right angles in the environment.

B. Model right angles and right triangles using concrete objects.

C. Draw two- and three-dimensional geometric shapes and construct rectangles, squares and triangles on the geoboard and on graph paper satisfying specific criteria.

D. Find and describe geometric

Students solve problems utilizing geometric modeling and spatial reasoning

4. Know how to build and draw geometric objects5. Recognize shapes from different perspectives6. Begin to understand concept of area and

perimeter

By the end of grade 3 students will:

Identify figures that are symmetrical, congruent or similar.

Predict, draw, and confirm figure as a result of one of the rigid transformations (reflection, rotation, translation).

Draw and classify two-dimensional figures having up to six or more sides.

Create a 3-dimensional object from a 2-dimensional representation.

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figures in real life.

E. Identify and draw lines of symmetry in geometric figures.

F. Identify symmetry in nature.

G. Fold paper to demonstrate the reflections about a line.

H. Show relationships between and among figures using reflections.

I. Predict how shapes can be changed by combining or dividing them.

GRADE 5 Math

Numbers, Number Systems and Number Relationships

D. Use expanded notation to represent whole numbers or decimals.

E. Apply number theory concepts to rename a number quantity (e.g., six, 6,

, 3 2, 10 4).

F. Demonstrate that mathematical operations can represent a variety of problem situations.

NUMBER SENSEStudents understand different representations of numbers, the relationship between/among numbers and number systems.

1. Identify and use verbal names, written names and standard numerals for whole numbers through millions or more.

2. Know and identify fractions and mixed numbers with denominators including 2,3,4,5,6,8,10, 12, 20, 25, 100 and1000.

3. Know and identify whole numbers and decimals

By the end of 5th grade students will be able to:

Read and write whole numbers through millions.

Represent, order and compare whole numbers through millions or more.

Read and write fractions and mixed numbers, through thousandths.

Represent order and compare commonly used fractions and decimals to thousandths using concrete materials or drawings.

Read and write decimals through thousandths.

Read and write common percents including 10%, 20%, 25%, 30%, 40%, 50%, 60%, 70%, 75% , 80%, 90%, and

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I. Use models to represent fractions and decimals.

J. Explain the concepts of prime and composite numbers.

K. Use simple concepts of negative numbers (e.g., on a number line, in counting, in temperature).

L. Develop and apply number theory concepts (e.g., primes, factors, multiples, composites) to represent numbers in various ways.

M. Explain budgeting

through thousandths.

4. Understand the meaning of and identify percents including 10, 20, 25, 30, 40, 50, 60, 70, 75, 80, 90, and 100 percent.

5. Translate problem situations into using whole numbers, fractions, mixed numbers, decimals and percents.

6. Know that two numbers in different forms are equivalent or non-equivalent, using whole numbers, decimals, fractions,mixed numbers, and percents.

7. Understand place value in base ten system to include decimals, and know that place value relates to powers of

8. Express numbers to millions or more in expanded form using powers of ten, with or without exponential notation.

9. Explore numbers less than 0.

10. Compare and contrast the decimal number system and other number systems that do or do not use place value.

Students understand and apply properties of numbers and operations.

100%.

Compare and order whole numbers, commonly used fractions, percents, and decimals (to thousandths) using concrete materials, number lines, drawings, numerals, and symbols (>, <, =).

Explain and justify the solution to problem situations with the use of diagrams, models, and numerals.

Compare and contrast numbers decimals, fractions, mixed numbers, and percents in different forms, both equivalent and nonequivalent, using tables charts and graph paper.

Explain how place value relates to the powers of 10, using charts, diagrams, or manipulatives.

Write numbers to millions or more in expanded form using powers of ten, in chart or table or display models with graph paper.Explain and demonstrate the multiplication of numbers between zero and one, (for example, when two numbers less than one are multiplied, the result is a number less than either factor) using concrete materials, drawings, story problems, symbols, and algorithms. (Group project with oral report).

Explain the similarities and differences between the decimal (base 10) number system and other number systems (Research this on the Internet and report, individually or by groups).

Use real life experiences, physical material, and technology to construct meanings for numbers. All fractions as part of a whole, as subset of a

set, as a location on a number line, and as

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1. Understand, describe, and apply properties of operations including the identity, commutative, and associativeproperties of addition; the zero and identity properties of multiplication; the commutative, associative, and distributiveproperties of multiplication.

2. Differentiate between prime numbers and composite numbers.

3. Determine the prime factors of numbers through 100 and write the numbers as the product of their prime factors.

4. Determine the greatest common factor or the least common multiple of two numbers up to 100 or more.

5. Know rules of divisibility.

6. Use models to identify perfect squares to 144.

7. Explain the reason for choosing a particular computing method for a particular problem.

8. Use a table or chart to differentiate between the greatest common factor and the least common multiple of two numbers up to 100 or more.

divisions of whole numbers

Develop and apply number theory concepts in problem solving situations Primes, factors, multiplesGiven specific numbers, students identify them as prime or composite.

Recognize the appropriate use of each arithmetic operation in problem situations

Make posters and display a number story to explain and demonstrate the commutative, associative, and distributive properties of multiplication.

Discover patterns in the various products when a whole number is displayed as a product of its prime factors.

Display the perfect squares to 144 with the use of square arrays on graph paper.

2.2. 8 Computation and Estimation NS2: Students understand the effects of By the end of 8th grade students will be able to:

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A. Complete calculations by applying the order of operations.

B. Add, subtract, multiply and divide different kinds and forms of rational numbers including integers, decimal fractions, percents and proper and improper fractions.

C. Estimate the value of irrational numbers.

D. Estimate amount of tips and discounts using ratios, proportions and percents.

E. Determine the appropriateness of overestimating or underestimating in computation.

F. Identify the difference between exact value and approximation and determine which is appropriate for a given situation.

13.3.8 Career Education Work Standards

• Analyze budgets and pay statements, such as, but not limited to:

operations on numbers and the relationships among these operations; select appropriate operations, and are able to compute for various problem-solving situations.

1. Know the effects of the four basic operations on whole numbers, fractions, mixed numbers, decimals, and integers.

2. Apply the properties of real numbers to solve problems (commutative, associative, distributive, identity, equality, inverse, and closure).

3. Solve real-world problems involving percents (for example, discounts, simple interest, taxes, tips).

1. Know proportional relationships.NS3: Students understand and apply properties of numbers and operations.

1. Know the inverse relationship of positive and negative numbers.

2. Know the appropriate operation to solve real-world problems involving integers, ratios, rates, proportions, numbers expressed as percents, decimals, fractions, and square roots.

3. Solve multi-step, real-world problems involving

Express base ten numbers as equivalent numbers in different bases, such as base two, base five, and base eight. Express non-base ten numbers as equivalent numbers in base ten. Discuss the application of the binary (base two) number system in computer technology.

Investigate the structure of number systems other than the decimal number system.

Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

Pencil-and-paper Mental math Calculator

Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.

Estimate square roots and cube roots of numbers

Use equivalent representation of numbers such as fractions, decimals, and percents to facilitate estimation

Recognize the limitations of estimation and assess the amount of error resulting from estimation

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Charitable contributions Expenses Gross pay Net pay Other income Savings Taxes

integers, ratios, proportions, numbers expressed as percents, decimals, and fractions.

4. Solve real-world problems involving percents including percents greater than 100% (for example percent of change,commission).

Use models or pictures to show the effects of addition, subtraction, multiplication, and division on whole numbers, decimals, fractions, mixed numbers, and integers.

Solve real-world problems involving decimals and fractions using two- or three-step problems.

Use tables, graphs, or “constant ratio” relationships to solve and explain problems.

2.3. 8 Measurement and Estimation

A. Develop formulas and procedures for determining measurements (e.g., area, volume, distance).

B. Solve rate problems (e.g., rate time = distance, principal interest rate = interest).

C. Measure angles in degrees and determine relations of angles.

D. Estimate, use and describe measures of distance, rate, perimeter, area, volume, weight, mass and angles.

E. Describe how a change in linear dimension of an object affects its perimeter, area and volume.

ME1: Students measure quantities in the real world and use these measures to solve problems.

1. Understand strategies used to solve real-world problems involving surface area and volume of three-dimensional shapes.

2. Explore and derive formulas for surface area and volume of three-dimensional regular shapes, including pyramids, prisms, and cones.

3. Know and apply formulas for finding rates, distance, time and angle measures.

4. Develop an understanding of rate of change as it applies to real-world problems.

5. Know that new figures can be created by increasing or decreasing the original dimensions.

6. Know how changes in the volume, surface area, area, or perimeter of a figure affect the dimensions

By the end of 8th grade students will be able to:

Use equivalent representation of numbers such as fractions, decimals, and percents to facilitate estimation

Recognize the limitations of estimation and assess the amount of error resulting from estimation

Build three-dimensional solids using the two-dimensional models as faces. Predict the surface areas, and then test predictions.

Using centimeter cubes as a guide, estimate the volume of each model. Working with a group, tests estimations and contribute to group consensus on a working formula for finding the volume of various three-dimensional models.

Describe and use rates of change (for example, temperature as it changes throughout the day, or speed as the rate of change in distance over time) and other derived measures.

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F. Use scale measurements to interpret maps or drawings.

G. Create and use scale models.

of the figure.

7. Solve real world or mathematical problems involving the effects of changes either to the dimensions of a figure orto the volume, surface area, area, perimeter, or circumference of figures.

ME2: Students compare, contrast, and convert within systems of measurement; both standard with nonstandard and metric with customary.

1. Know relationships between metric units of mass and capacity (for example, one cubic centimeter of water weighsone gram).

2. Find measures of length, weight or mass, and capacity or volume using proportional relationships and properties of similar geometric figures.

ME4: Students select and use appropriate units and instruments for measurement to achieve the degree of precision and accuracy required in real-world situations.

1. Solve problems using mixed units within each system, such as feet and inches, hours and minutes.

2. Solve problems using the conversion of

Describe how a change in a figure’s dimensions affects its perimeter, area, circumference, surface area, or volume.

Investigate congruent figures with respect to volume and surface area, and describe the differences in their dimensions.

Describe how the change of a figure in dimensions such as length, width, height, or radius affects its other measurements such as perimeter, area, surface area, and volume.

Measure length, weight or mass, and capacity or volume using customary or metric units.

Apply properties of similarity with shadow measurement and properties of similar triangles to find the height of a flagpole.

Select appropriate units of measurement in a real-world context.

Apply the conversion of measurements within the customary system to real world problems.

Select and use appropriate instruments, technology, and techniques to measure quantities and dimensions to a specified degree of accuracy.

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measurements within the customary system.

3. Determine the appropriate precision unit for a given situation.

4. Measure accurately with measurement tools to the specified degree of accuracy for the task and in keeping with the precision of the measurement tool.

2.4.8 Mathematical Reasoning and Connections

A. Make conjectures based on logical reasoning and test conjectures by using counter-examples.

B. Combine numeric relationships to arrive at a conclusion.

C. Use if...then statements to construct simple, valid arguments.

D. Construct, use and explain algorithmic procedures for computing and estimating with whole numbers, fractions, decimals and integers.

E. Distinguish between inductive and deductive reasoning.

DA2: Students identify patterns and makes predictions from an orderly display of data using concepts ofprobability and statistics.

1. Compare and explain the results of an experiment with the mathematically expected outcomes.

2. Explain observed difference between mathematical and experimental results.

3. Predict the mathematical odds for and against a specified outcome in a given real-world situation.

By the end of 8th grade students will be able to:

Calculate simple mathematical probabilities for independent and dependent events.

Compare the results of an experiment with the expected theoretical outcomes.

Design experiments of chance and predict outcomes of odds, for and against.

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F. Use measurements and statistics to quantify issues (e.g., in family, consumer science situations).

2.5. 8 Mathematical Problem Solving and Communication

A. Invent, select, use and justify the appropriate methods, materials and strategies to solve problems.

B. Verify and interpret results using precise mathematical language, notation and representations, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs and diagrams.

C. Justify strategies and defend approaches used and conclusions reached.

D. Determine pertinent information in problem situations and whether any further information is needed for solution.

NS2: Students understand the effects of operations on numbers and the relationships among these operations; select appropriate operations, and are able to compute for various problem-solving situations.

1. Know the effects of the four basic operations on whole numbers, fractions, mixed numbers, decimals, and integers.

2. Apply the properties of real numbers to solve problems (commutative, associative, distributive, identity, equality, inverse, and closure).

3. Solve real-world problems involving percents (for example, discounts, simple interest, taxes, tips).

NS4: Students make reasonable estimates.(Content(What students should know)

1. Know an appropriate estimation technique for a given situation using whole numbers, fractions and

By the end of 8th grade students will be able to:

Express base ten numbers as equivalent numbers in different bases, such as base two, base five, and base eight. Express non-base ten numbers as equivalent numbers in base ten. Discuss the application of the binary (base two) number system in computer technology.

Investigate the structure of number systems other than the decimal number system.

Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

Pencil-and-paper Mental math Calculator

Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.

Write and simplify expressions from real-world situations using the order of operations.

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Career Education Work Standards13.3.8• Analyze budgets and pay statements,

such as, but not limited to: Charitable contributions Expenses Gross pay Net pay Other income Savings Taxes

13.1.8Analyze the relationship of school subjects, extracurricular activities, and community experiences to career preparation

decimals.

2. Estimate to predict results and check reasonableness of results.

3. Determine whether an exact answer is needed or whether an estimate would be sufficient.

DA2: Students identify patterns and makes predictions from an orderly display of data using concepts of probability and statistics.

1. Compare and explain the results of an experiment with the mathematically expected outcomes.

2. Explain observed difference between mathematical and experimental results.

3. Predict the mathematical odds for and against a specified outcome in a given real-world situation.

Use appropriate methods of computation, such as mental computation, paper and pencil, and calculator.

Justify the choice of method for calculations, such as mental computation, concrete materials, algorithms, or calculators.

Explain a variety of estimation techniques including clustering, compatible number, and front-end.

Execute the 4 steps of problem solving.

Give examples in real world situations where estimation is sufficient for the situation.

Recognize the limitations of estimation and assess the amount of error resulting from estimation

Calculate simple mathematical probabilities for independent and dependent events.

Compare the results of an experiment with the expected theoretical outcomes.

Design experiments of chance and predict outcomes of odds, for and against.

2.6.8 Statistics and Data Analysis

A. Compare and contrast different plots of data using values of mean, median, mode, quartiles and range.

DA1: Students understand the art of managing information for the purpose of data analysis.

1. Read and interpret data displayed in a variety of forms including histograms.

By the end of 8th grade students will be able to:

Interpret and analyze data displayed in a variety of forms including histograms.

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B. Explain effects of sampling procedures and missing or incorrect information on reliability.

C. Fit a line to the scatter plot of two quantities and describe any correlation of the variables.

D. Design and carry out a random sampling procedure.

E. Analyze and display data in stem-and-leaf and box-and-whisker plots.

F. Use scientific and graphing calculators and computer spreadsheets to organize and analyze data.

G. Determine the validity of the sampling method described in studies published in local or national newspapers.

2. Interpret measures of dispersion (range) and of central tendency.

3. Find the mean, median, and mode of a set of data using raw data, tables, charts, or graphs.

4. Describe a set of data by using the measures of central tendency.

5. Construct various graphs, including scatterplots and box-and- whisker graphs, to display a data set.

DA3: Students use statistical methods to make inferences and valid arguments about real-world situations.

1. Understand the application of statistics in the formation, testing and evaluation of a hypothesis.

Determine appropriate measures of central tendency for a given situation or set of data.

Determine the mean, median, mode, and range of a set of real-world data using appropriate technology.

Organize graphs and analyze a set of real-world data using appropriate technology.

Design an experiment, perform the experiment and collect, organize, and display the data. Evaluate the hypothesis by making inferences and drawing conclusions based on statistical results.

Perform an experiment and collect, organize, and display the data conclusions based on statistical results.

2.7.8 Probability and Predictions

A. Determine the number of combinations and permutations for an event.

DA4: Students identify the common uses and misuses of probability or statistical analysis in the everyday world.CONTENT(What students should know)

1. Know appropriate uses of statistics and

By the end of 8th grade students will be able to:

Identify instances in which statistics and probability are used in advertising to mislead the public.

Design several different surveys and use the

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B. Present the results of an experiment using visual representations (e.g., tables, charts, graphs).

C. Analyze predictions (e.g., election polls).

D. Compare and contrast results from observations and mathematical models.

E. Make valid inferences, predictions and arguments based on probability.

probability in real-world situations.

2. Know when statistics and probability are used in misleading ways.

3. Identify and use different types of sampling techniques (for example, random, systematic, stratified).

4. Know whether a sample is biased.

various sampling techniques for obtaining survey results.

Interpret probabilities as ratios, percents, and decimals.

Determine probabilities of compound events.

Explore the probabilities of conditional events (e.g., if there

are seven marbles in a bag, three red and four green, what

is the probability that two marbles picked from the bag

without replacement, are both red).

Model situations involving probability with simulations

(using spinners, dice, calculators and computers) and

rhetorical models Frequency, relative frequency

Estimate probabilities and make predictions based on experimental and rhetorical probabilities

Play and analyze probability-based games, and discuss the

concepts of fairness and expected value.

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2.8.8 Algebra and Functions

A. Apply simple algebraic patterns to basic number theory and to spatial relations

B. Discover, describe and generalize patterns, including linear, exponential and simple quadratic relationships.

C. Create and interpret expressions, equations or inequalities that model problem situations.

D. Use concrete objects to model algebraic concepts.

E. Select and use a strategy to solve an equation or inequality, explain the solution and check the solution for accuracy.

F. Solve and graph equations and inequalities using scientific and graphing calculators and computer spreadsheets.

AL1: Students recognize, describe, analyze and extend patterns, relations and functions.CONTENT(What students should know)

1. Know the graphical representation of a linear relationship.

2. Determine if a function is linear by making use of the information provided in a table, graph, or rule.

3. Recognize the independent variable and the dependent variable in a real world problem.

4. Know function rules to describe tables of related input-output.

AL2: Students use expressions, equations, inequalities, graphs, and formulas to represent and interpret situations.

1. Interpret and create tables, function tables, and graphs (function tables).

2. Graph solutions to linear equations on the coordinate plane.

3. Write equations and inequalities to express relationships.

By the end of 8th grade students will be able to:

Read, analyze, and describe graphs of linear relationships.

Justify the reason for determining if a function is linear.

Use variables to represent unknown quantities in real-world problems.

Predict outcomes from given tables of related input-output, based upon function rules.

Perform experiments in order to generate data tables that graph functions.

Graph equations and inequalities in order to explain cause-and-effect relationships.

Interpret the meaning of the slope of a line from a graph depicting a real-world situation.

Translate algebraic expressions, equations, or inequalities representing real-world relationships into verbal expressions or sentences.

Graph linear equations on the coordinate plane using tables of values.

Graphically display real-world situations represented by algebraic equations or inequalities.

Evaluate algebraic expressions, equations, and inequalities by substituting integral values for variables and simplifying the results.

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G. Represent relationships with tables or graphs in the coordinate plane and verbal or symbolic rules.

H. Graph a linear function from a rule or table.

I. Generate a table or graph from a function and use graphing calculators and computer spreadsheets to graph and analyze functions.

J. Show that an equality relationship between two quantities remains the same as long as the same change is made to both quantities; explain how a change in one quantity determines another quantity in a functional relationship.

4. Interpret the meaning of the slope of a line from a graph depicting a real-world situation.

5. Translate verbal expressions and sentences into algebraic expressions, equations, and inequalities.

6. Solve single- and multiple-step linear equations and inequalities in concrete or abstract form.

7. Know the relationships represented by algebraic equations or inequalities and their graphic representations.

8. Know how to evaluate algebraic expressions, equations, and inequalities.

9. Know the relationships between the properties of algebraic expressions and real numbers.

10. Evaluate algebraic expressions, equations, and inequalities by substituting integral values for variables and simplifying the results.

Simplify algebraic expressions that represent real-world situations by combining like terms and applying the properties of real numbers.

Translate algebraic expressions, equations, or inequalities representing real-world relationships into verbal expressions or sentences.

Graph linear equations on the coordinate plane using tables of values.

Simplify algebraic expressions that represent real-world situations by combining like terms and applying the properties of real numbers.

Graph functions, and understand and describe their general behavior

Equations involving two variables

Use graphing techniques on a number line Absolute value Arithmetic operations represented by

vectors (arrows)

Solve simple linear inequalities

Create, evaluate, and simplify algebraic expressions involving variables

Order of operations, including appropriate use of parentheses

Distributive property Substitution of a number for a variable Translation of a verbal phrase or

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sentence into algebraic expression, equation, or inequality, and vice versa

2.9.8 Geometry

A. Construct figures incorporating perpendicular and parallel lines, the perpendicular bisector of a line segment and an angle bisector using computer software.

B. Draw, label, measure and list the properties of complementary, supplementary and vertical angles.

C. Classify familiar polygons as regular or irregular up to a decagon.

D. Identify, name, draw and list all properties of squares, cubes, pyramids, parallelograms, quadrilaterals, trapezoids, polygons, rectangles, rhombi, circles, spheres, triangles, prisms and cylinders.

E. Construct parallel lines, draw a transversal and measure and compare angles formed (e.g., alternate interior and exterior angles).

GE1: Students describe, draw, identify, and analyze two and three-dimensional shapes.

1. Compare regular and irregular polygons and two- and three dimensional shapes.

2. Determine the measures of various types of angles based upon geometric relationships in two- and three-dimensional shapes.

3. Represent the properties of two- and three- dimensional figures by drawing them with appropriate tools including astraight edge and a compass.

4. Recognize and draw two-dimensional representations of three-dimensional objects (perspective drawings

GE2: Students use coordinate geometry to locate objects in both two- and three-dimensions and to describe objects algebraically.

1. Know how to find a minimum of three ordered-pair solutions for a given equation.

2. Know the formula for the graph of a line, including the slope of the line and the intercept of

By the end of 8th grade students will be able to:

Draw and build three-dimensional figures from various perspectives (e.g., flat patterns, isometric drawings, and nets).

Draw angles (including acute, obtuse, right, straight, complementary, supplementary, and vertical angles).

Draw three-dimensional figures (including pyramid, cone, sphere, hemisphere, rectangular solids and cylinders).

Given an equation or its graph, finds ordered-pair solutions (for example, y = 2x).

Given the graph of a line, identifies the slope of the line (including the slope of vertical and horizontal lines).

Apply and explain the simple properties of lines on a graph, including parallelism, perpendicularity, and identifying the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

Observe, explain, make and test conjectures regarding geometric properties and relationships (among regular and irregular shapes of two and three dimensions).

Apply the Pythagorean Theorem in real-world problems (for example, finds the relationship among sides in 45 – 45 – 90 and 30 – 60 – 90 right triangles). Use models or diagrams

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F. Distinguish between similar and congruent polygons.

G. Approximate the value of (pi) through experimentation.

H. Use simple geometric figures (e.g., triangles, squares) to create, through rotation, transformational figures in three dimensions.

I. Generate transformations using computer software.

J. Analyze geometric patterns (e.g., tessellations, sequences of shapes) and develop descriptions of the patterns.

K. Analyze objects to determine whether they illustrate tessellations, symmetry, congruence, similarity and scale.

the line, including vertical andhorizontal lines.

3. Know the relationships of linear equations as they apply to: properties of lines on a graph, including parallelism,perpendicularity, the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

4. Know the geometric properties and relationships (among regular and irregular shapes of two and three dimensions).

5. Understand the Pythagorean relationship in special right triangles (45 – 45 – 90 and 30 – 60 – 90).

GE3: Students visualize and illustrate ways in which shapes can be combined, subdivided, and changed.

1. Know and apply the properties of parallelism, perpendicularity and symmetry in real-world contexts.

2. Identify congruent and similar figures in real-world situations.

3. Continue a tessellation pattern using the needed

(manipulatives, dot, graph, or isometric paper).Understand and apply concepts involving lines, angles, and planes

Complementary and supplementary angles

Vertical angles Parallel, perpendicular, and intersecting

planes

Understand and apply properties of polygons Quadrilaterals, including squares,

rectangles, parallelograms, trapezoids Regular polygons Sum of measures of interior angles of a

polygon

Understand and apply transformations Finding the image, given the pre-image

and vice versa Sequence of transformations needed to

map one figure unto another Reflections, rotations, and translations

result in images congruent to the pre-image

Dilations (stretching/shrinking) result in images similar to the pre image

Use the properties of parallelism, perpendicularity, and symmetry in solving real-world problems.

Justify the identification of congruent and similar figures.

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transformations. Create an original tessellating tile and tessellation pattern using a combination of transformations.

2.10. 8 Trigonometry

A. Compute measures of sides and angles using proportions, the Pythagorean Theorem and right triangle relationships.

B. Solve problems requiring indirect measurement for lengths of sides of triangles.

GE2: Students use coordinate geometry to locate objects in both two- and three-dimensions and to describe objects algebraically.

1. Know the formula for the graph of a line, including the slope of the line and the intercept of the line, including vertical and horizontal lines.

3. Know the relationships of linear equations as they apply to: properties of lines on a graph, including parallelism,perpendicularity, the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

By the end of 8th grade students will be able to:

Given an equation or its graph, finds ordered-pair solutions (for example, y = 2x).

Apply and explain the simple properties of lines on a graph, including parallelism, perpendicularity, and identifying the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

Observe, explain, make and test conjectures regarding geometric properties and relationships (among regular and irregular shapes of two and three dimensions).

Develop and apply strategies for finding perimeter and area

Geometric figures made by combining triangles, rectangles and circles or parts of a circle

2.11. 8 Concepts of Calculus

A. Analyze graphs of related quantities for minimum and maximum values and justify the findings.

B. Describe the concept of unit rate, ratio and slope in the context of rate of change.

C. Continue a pattern of numbers or

NS5: Students understand and apply theories related to numbers.CONTENT(What students should know)

1. Determine the appropriate use of number theory concepts, including divisibility rules, to solve real- world ormathematical problems. Describe the concept of unit rate, ratio and slope in the context of rate of change.

By the end of 8th grade students will be able to:

Find the greatest common factor and least common multiple of two or more numbers.

Apply number theory concepts to determine the terms in a real number sequence.

Apply number theory concepts, including divisibility rules, to solve real-world or mathematical problems.

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objects that could be extended infinitely. 2. Continue a pattern of numbers or objects that

could be extended infinitely.

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