Curriculum Activity Packets Correlation to Daily Lesson...

Curriculum Activity Packets Correlation to Daily Lesson Plans

The Georgia Department of Juvenile Justice

8th Grade Mathematics

Units of Instruction Resource Manual

Table of Contents

8th Grade Mathematics

Acknowledgments

Superintendents Letter

Mission and Vision Statements

Chapter 1: Introduction

Chapter 2: Teachers Guide

Chapter 3: Instructional Rotation

Chapter 4: Georgia Performance Standards

Chapter 5: Curriculum Map

Chapter 6: Essential Questions and Enduring Understandings

Chapter 7: Units of Instruction

Unit 1: Principles of Algebra & Rational Numbers

Task 1

Task 2

Task 3

Task 4 Focus CAPs

Unit 2: Graphs, Functions & Sequences

Task 1

Task 2

Task 3

Task 4

Task 5

Task 6 Focus CAPs

Unit 3: Exponents& Roots/ Ratios, Proportions & Similarity/ Percents

Task 1

Task 2

Task 3

Task 4

Task 5 Focus CAPs

Unit 4: Foundation of Geometry & Perimeter, Area & Volume

Task 1

Task 2

Task 3

Task 4 Focus CAPs

Unit 5: Data & Statistics

Task 1

Task 2

Task 3 Focus CAPs

Unit 6: Probability

Task 1

Task 2

Task 3

Task 4 Focus CAPs

Unit 7: Multi-Step Equations & Inequalities

Task 1

Task 2

Task 3

Task 4

Task 5 Focus CAPs

Unit 8: Graphing Lines, Sequences & Functions & Polynomials

Task 1

Task 2

Task 3

Task 4 Focus CAPs

Chapter 8: Task websites

Acknowledgements

The Georgia Department of Juvenile Justice Department of Education would like to thank the many educators who have helped to create this 8th Grade Math Units of Instruction Resource Manual. The educators have been particularly helpful in sharing their ideas and resources to ensure the completion and usefulness of this manual.

Students served by the DJJ require a special effort if they are to become contributing and participating members of their communities. Federal and state laws, regulations, and rules will mean nothing in the absence of professional commitment and dedication by every staff member.

The Georgia Department of Juvenile Justice is very proud of its school system. The school system is Georgias 181st and is accredited by the Southern Association of Colleges and Schools (SACS). The DJJ School System has been called exemplary by the US Department of Justice. This didnt just happen by chance; rather it was the hard work of many teachers, clerks, instructors and administrators that earned DJJ these accolades and accreditations. The DJJ education programs operate well because of the dedicated staff. These dedicated professionals are the heart of our system.

These Content Area Units of Instruction were designed to serve as a much needed tool for delivering meaningful whole group instruction. In addition, this resource will serve as a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs).

I would like to thank all the DJJ Teaching Staff, the Content Area Leadership Teams, Kimberly Harrison, DJJ Special Education/Curriculum Consultant and Martha Patton, Curriculum Director for initiating this project and seeing it through. Thank you all for your hard work and dedication to the youth we serve.

Sincerely yours,

James Jack Catrett, Ed.D.

Associate Superintendent

Mission

The mission of Department of Juvenile Justice Math Consortium (DJJMC) is to build a multiparty effort statewide to achieve continuous, systemic and sustainable improvements in the education system serving the Math students of the Department of Juvenile Justice (DJJ).

Vision

To achieve the mission of the DJJMC, members work collaboratively in examining the Georgia Performance Standards. These guidelines speak specifically to teachers being able to: deliver meaning content pertaining to the Characteristics of Math and its content standards across the Math Units of Instruction Resource Manual. The DJJMC will master and develop whole-group unit lessons built around Curriculum Activity Packets (CAPs), critique student work, and work as a team to solve the common challenges of teaching within DJJ. Additionally, the DJJMC jointly analyzes student test data in order to: develop strategies to eradicate common academic deficits among students, align curriculum, and create a coherent learning pathway across grade levels. The DJJMC also reviews research articles, attends workshops or courses, and invites consultants to assist in the acquisition of necessary knowledge and skills. Finally, DJJMC members observe one another in the classroom through focus walks.

Introduction

The 8th Grade Math Units of Instruction Resource Manual is a tool that has been created to serve as a much needed tool for delivering meaningful whole group instruction. This manual is a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs). It is imperative that our students learn to reason mathematically, to evaluate mathematical arguments both formally and informally, to use the language of mathematics to communicate ideas and information precisely, and to work in cooperative learning groups. Best practices in education indicate that teachers should first model new skills for students. Next, teachers should provide opportunities for guided practice. Only then should teachers expect students to successfully complete an activity independently. The 8th Grade Math Units of Instruction meets that challenge.

The Georgia Department of Juvenile Justice

Office of Education

Direct Instruction Lesson Plan

Teacher:

Subject:______________________________

Date:_____________to__________________

Period

1st

2nd

3rd

4th

5th

6th

Students will engage in:

Independent activities pairing

Cooperative learning hands-on

Peer tutoring Visuals

technology integration Simulations

a project centers

lecture Other

Essential Question(s):

Standards:

CAPs Covered:

Grade Level:____ Unit:______

RTI Tier for data collection: 2 or 3

Tier 2 Students:

Tier 3 Students:

Time

Procedures Followed:

Material/Text

_______

Minutes

Review of Previously Learned Material/Lesson Connections:

Recommended Time: 2 Minutes

_______

Minutes

Display the Georgia Performance Standard(s) (project on

blackboard via units of instruction located at

http://thevillage411.weebly.com/units-of-instruction3.html, or print on blackboard) Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.

Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard). Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.


_______

Minutes

Introduce task by stating the purpose of todays lesson.


_______

Minutes

Engage students in conversation by asking open ended questions related to the essential question(s).


_______

Minutes

Begin whole group instruction with corrective feedback:


_______

Minutes

Lesson Review/Reteach:


_______

Minutes

Independent Work CAPs:


Teacher Reflections:

The Instructional Rotation Matrix has been designed to assist language arts teachers in providing a balanced approach to utilizing the Math Units of Instruction across all grade levels on a rotating schedule.

Monday

Tuesday

Wednesday

Thursday

6th Grade Content

Middle School

9th Grade Content

High School

7th Grade Content

Middle School

10th Grade Content

High School

8th Grade Content

Middle School

11th Grade Content

High School

6th Grade Content

Middle School

12th Grade Content

High School

7th Grade Content

Middle School

9th Grade Content

High School

8th Grade Content

Middle School

10th Grade Content

High School

6th Grade Content

Middle School

11th Grade Content

High School

7th Grade Content

Middle School

12th Grade Content

High School

Georgia Performance Standards

M8A1 Students will use algebra to represent, analyze, and solve problems.

a. Represent a given situation using algebraic expressions or equations in one variable.

b. Simplify and evaluate algebraic expressions.

c. Solve algebraic equations in one variable, including equations involving absolute values.

d.Solve equations involving several variables for one variable in terms of the others.

e.Interpret solutions in problem contexts.

M8A2 Students will understand and graph inequalities in one variable.

a. Represent a given situation using an inequality in one variable.

b. Use the properties of inequality to solve inequalities.

c. Graph the solution of an inequality on a number line.

d. Interpret solutions in problem contexts.

M8A3 Students will understand relations and linear functions.

a. Recognize a relation as a correspondence between varying quantities.

b. Recognize a function as a correspondence between inputs and outputs where the output for each input must be unique.

c. Distinguish between relations that are functions and those that are not functions.

d. Recognize functions in a variety of representations and a variety of contexts.

e. Use tables to describe sequences recursively and with a formula in closed form.

f. Understand and recognize arithmetic sequences as linear functions with whole-number input values.

h. Interpret the constant difference in an arithmetic sequence as the slope of the associated linear function.

i. Identify relations and functions as linear or nonlinear.

j. Translate among verbal, tabular, graphic, and algebraic representations of functions.

M8A4 Students will graph and analyze graphs of linear equations and inequalities.

a. Interpret slope as a rate of change.

b. Determine the meaning of the slope and y-intercept in a given situation.

c. Graph equations of the form y = mx + b.

d. Graph equations of the form ax + by = c.

e.Graph the solution set of a linear inequality, identifying whether the solution set is an open or a closed half-plane.

f.Determine the equation of a line given a graph, numerical information that defines the line, or a context involving a linear relationship.

g.Solve problems involving linear relationships.

M8A5 Students will understand systems of linear equations and inequalities and use them to solve problems.

a.Given a problem context, write an appropriate system of linear equations or inequalities.

b. Solve systems of equations graphically and algebraically, using technology as appropriate.

c.Graph the solution set of a system of linear inequalities in two variables.

d.Interpret solutions in problem contexts.

M8D1 Students will apply basic concepts of set theory.

a. Demonstrate relationships among sets through use of Venn diagrams.

b. Determine subsets, complements, intersection, and union of sets.

c. Use set notation to denote elements of a set.

M8D2 Students will determine the number of outcomes related to a given event.

a. Use tree diagrams to find the number of outcomes.

b. Apply the addition and multiplication principles of counting.

M8D3 Students will use the basic laws of probability.

a. Find the probability of simple independent events.

b. Find the probability of compound independent events.

M8D4 Students will organize, interpret, and make inferences from statistical data.

a. Gather data that can be modeled with a linear function.

b. Estimate and determine a line of best fit from a scatter plot.

M8G1 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.

a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.

b. Apply properties of angle pairs formed by parallel lines cut by a transversal.

c. Understand the properties of the ratio of segments of parallel lines cut by one or more transversals.

d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.

M8G2 Students will understand and use the Pythagorean theorem.

a. Apply properties of right triangles, including the Pythagorean theorem.

b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right triangle.

M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.

a. Find square roots of perfect squares.

b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.

c. Recognize square roots as points and as lengths on a number line.

d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in sign.

e. Recognize and use the radical symbol to denote the positive square root of a positive number.

f. Estimate square roots of positive numbers.

g. Simplify, add, subtract, multiply, and divide expressions containing square roots.

h. Distinguish between rational and irrational numbers.

i. Simplify expressions containing integer exponents.

j. Express and use numbers in scientific notation.

k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.

M8P1 Students will solve problems (using appropriate technology).

a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

d. Monitor and reflect on the process of mathematical problem solving.

M8P2 Students will reason and evaluate mathematical arguments.

a. Recognize reasoning and proof as fundamental aspects of mathematics.

b. Make and investigate mathematical conjectures.

c. Develop and evaluate mathematical arguments and proofs.

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

M8P3 Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

M8P4 Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

M8P5 Students will represent mathematics in multiple ways.

a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.

c. Use representations to model and interpret physical, social, and mathematical phenomena.

M8RC1 Students will enhance reading in all curriculum areas by:

a. Reading in All Curriculum Areas

Read a minimum of 25 grade-level appropriate books per year from a variety of subject disciplines and participate in discussions related to curricular learning in all areas.

Read both informational and fictional texts in a variety of genres and modes of discourse.

Read technical texts related to various subject areas.

b. Discussing books

Discuss messages and themes from books in all subject areas.

Respond to a variety of texts in multiple modes of discourse.

Relate messages and themes from one subject area to messages and themes in another area.

Evaluate the merit of texts in every subject discipline.

Examine authors purpose in writing.

Recognize the features of disciplinary texts.

c. Building vocabulary knowledge

Demonstrate an understanding of contextual vocabulary in various subjects.

Use content vocabulary in writing and speaking.

Explore understanding of new words found in subject area texts.

d. Establishing context

Explore life experiences related to subject area content.

Discuss in both writing and speaking how certain words are subject area related.

Determine strategies for finding content and contextual meaning for unknown words.

DJJ 8th Grade Mathematics

Georgia Performance Standards: Curriculum Map

1st Semester

2nd Semester

Principles of Algebra & Rational Numbers

Graphs, Function & Sequences

Exponents & Roots Ratios, Proportions

& Similarity

& Percents

Foundations of Geometry & Perimeter, Area & Volume

Data & Statistics

Probability

Multi-Step Equations & Inequalities

Graphing Lines, Sequences & Functions & Polynomials

Chapter

1

CAPs

1-6

Chapter

3

CAPs

12-15

Chapter

4

CAPs

16-20

Chapter

7

CAPs

31-36

Chapter

9

CAPs

43-47

Chapter

10

CAPs

48-52

Chapter

11

CAPs

54-57

Chapter

12

CAPs

58-62

2

7-11

5

21-25

8

37-42

13

63-67

6

26-30

14

68-71

GPS:

M8A1a,b,c,d

M8P1a,b,c,d

M8P4a,b,c

M8P3a,c

M8P5a,b,c

M8P2c

M8A2b,c

M8A1a,b

M8P3c

GPS:

M8A1b,d

M8P1a,b

M8P5a,b,c

M8P2c,d

M8P3a,c

M8A4c,e

M8P4a,b,c

M8A3b,c,d,e,i

GPS:

M8N1a,b,c,d,e,f,g,h,i,j,k

M8A1a,b,c,d

M8P1a,b,c,d

M8P2a,b,c,d

M8P3a,c,d

M8P4a,b,c

M8P5a,b,c

M8G2a,b

M8G1a,b,d

M8A4.b

M8N1i,k

GPS:

M8A1a,b,c,d

M8P3a,b,c,d

M8P1a,b,c

M8P2a,c,d

M8G1a,b,d

M8P4a,b,c

M8A1a,b,c,d

M8P5a,b,c

M8G2a

M8N1.k

GPS:

M8P2.c

M8P3a,c,d

M8P4a,b,c

M8P1a,b,c,d

M8P5a,b,c

M8D4.a

GPS:

M8D3a,b

M8P1a,b,c,d

M8P3a,c,d

M8P4a,b ,c

M8P2.c

M8P5a,b,c

M8A1a,c,d

M8D2a,b

GPS:

M8A1a,b,c,d

M8P1b,c,d

M8P5a,b,c

M8P3a,c

M8P4c

M8A2a,b,c,d

M8A5b,c

GPS:

M8A3h,d,e,f,

M8A4a,b,c,d,e,f

M8P1a,b,c,d

M8P3a,c

M8P4a,b,c

M8P5a,b,c

M8A3d,h,i

M8D4b

M8A1a,b,c,d

M8P2a,b,c,d

M8N1i,k

Focus CAPs:

Chapter 1

2 & 6

Chapter 2

7 & 11

Focus CAPs:

12 & 15

Focus CAPs:

Chapter 4

16 & 20

Chapter 5

21

Chapter 6

26

Focus CAPs:

Chapter 7

31 & 36

Chapter 8

37 & 42

Focus CAPs:

43

Focus CAPs:

48 & 52

Focus CAPs:

54 & 57

Focus CAPs:

Chapter 12

58

Chapter 13

63

Chapter 14

68

Enduring Understandings & Essential Question

Principles of Algebra & Rational Numbers

Enduring Understandings:

Algebraic expressions, equations and inequalities are used to represent relationships

between numbers.

Absolute value is used to represent distances between numbers.

Graphs can be used to represent all of the possible solutions to a given situation.

Many problems encountered in everyday life can be solved using equations or

inequalities.

Essential Questions:

How can I simplify and evaluate an algebraic expression?

How can I solve an equation or inequality?

How can I tell the difference between an expression, equation and an inequality?

How can I determine the absolute value of an expression?

How can I represent absolute value on a number line?

Graphs, Functions & Sequences


Collecting and examining data can sometimes help one discover patterns in the way in which two quantities vary.

Changes in varying quantities are often related by patterns which, once discovered, can be used to predict outcomes and solve problems.

Written descriptions, tables, graphs and equations are useful in representing and investigating relationships between varying quantities.

Different representations (written descriptions, tables, graphs and equations) of the relationships between varying quantities may have different strengths and weaknesses.

Linear functions may be used to represent and generalize real situations.

Slope and y-intercept are keys to solving real problems involving linear relationships.


What does the data tell me?

How does a change in one variable affect the other variable in a given situation?

Which tells me more about the relationship I am investigating a table, a graph or an equation? Why?

What strategies can I use to help me understand and represent real situations involving linear relationships?

How can the properties of lines help me to understand graphing linear functions?

How is a linear inequality like a linear equation? How are they different?

Exponents& Roots/ Ratios, Proportions & Similarity/ Percents


An irrational number is a real number that cannot be written as a ratio of two integers.

All real numbers can be plotted on a number line.

Exponents are useful for representing very large or very small numbers.

Square roots can be rational or irrational.

Some properties of real numbers hold for all irrational numbers.

There are many relationships between the lengths of the sides of a right triangle.


When are exponents used and why are they important?

Why is it useful for me to know the square root of a number?

How do I simplify and evaluate algebraic expressions involving integer exponents and square roots?

What is the Pythagorean Theorem and when does it hold?

Foundation of Geometry & Perimeter, Area & Volume


Parallel lines have the same slope and perpendicular lines have opposite, reciprocal

slopes.

When two lines intersect, vertical angles are congruent and adjacent angles are

supplementary.

When parallel lines are cut by a transversal, corresponding, alternate interior and alternate

exterior angles are congruent.

The length of segments formed by two non-parallel transversals cutting parallel lines is

proportional to the distances of the parallel lines from the intersection of the transversals.

Parallel lines can be constructed using the properties of parallel lines cut by a transversal.

In Euclidean Geometry, there is exactly one line through a given point parallel to a

second given line.


How can I be certain whether lines are parallel, perpendicular, or skew lines?

Why do I always get a special angle relationship when any two lines intersect?

When I draw a transversal through parallel lines, what are the special angle and segment

relationships that occur?

What information is necessary before I can conclude two figures are congruent?

How can my knowledge of constructing congruent triangles be used to construct

perpendicular and parallel lines?

Can I find parallel lines that intersect? Why or why not?

Data & Statistics


Relations show any correspondence between sets, while functions are special relations in which each input from a fixed set is associated with a single output.

Linear functions are defined by constant slope.

Arithmetic sequences are numerical representations of linear functions.

Venn diagrams are visual tools for organizing members of related sets.

Like functions and relations, visualizing sets in multiple representations often reveals unexpected patterns.


How can I identify a function?

How can I tell the difference between a relation and a function?

How can I relate arithmetic sequences to linear functions?

When working with sets, when do I use a union, and when do I use an intersection?

Probability


Tree diagrams are useful for describing relatively small sample spaces and computing

probabilities, as well as for visualizing why the number of outcomes can be extremely large.


How do I determine a sample space?

How can a tree diagram help me to find the number of possible outcomes related to a given event?

When and why do I use addition to determine sample space size?

When and why do I use multiplication to determine sample space size?

When and why do I use addition to determine probabilities?

When and why do I use multiplication to determine probabilities?

How can I use probability to determine if a game is fair or to figure my chances of winning the lottery?

Multi-Step Equations & Inequalities


There are situations that require two or more equations to be satisfied simultaneously.

There are several methods for solving systems of equations.

Solutions to systems can be interpreted algebraically, geometrically, and in terms of

problem contexts.

In some problem contexts, the constraints that must be satisfied are modeled by

inequalities rather than equations.

The number of solutions to a system of equations or inequalities can vary from no

solution to an infinite number of solutions.


How can I interpret the meaning of a system of equations algebraically and

geometrically?

How does mathematical notation indicate that equations are to be treated as a system?

What does it mean to solve a system of linear equations?

How can the solution to a system be interpreted geometrically?

How can I recognize how many solutions a system of equations has prior to solving?

How do I decide which method would be easier to use to solve a particular system of

equations?

Why is graphing a system of inequalities a good way to show the solution set?

How can I translate a problem situation into a system of equations or inequalities?

What does the solution to a system tell me about the answer to a problem situation?

Graphing Lines, Sequences & Functions & Polynomials


Relations show any correspondence between sets, while functions are special relations in which each input from a fixed set is associated with a single output.

Linear functions are defined by constant slope.

Arithmetic sequences are numerical representations of linear functions.

Venn diagrams are visual tools for organizing members of related sets.

Like functions and relations, visualizing sets in multiple representations often reveals unexpected patterns.


How can I identify a function?

How can I tell the difference between a relation and a function?

How can I relate arithmetic sequences to linear functions?

When working with sets, when do I use a union, and when do I use an intersection?

Unit: Principles of Algebra & Rational Numbers

Georgia Performance Standards:
























M8A2 Students will understand and graph inequalities in one variable.

b. Use the properties of inequality to solve inequalities.

c. Graph the solution of an inequality on a number line.






Selected Terms and Symbols:

Independent events: Events for which the occurrence of one has no impact on the occurrence of the other.

Relative frequency: The number of times an outcome occurs divided by the total number of trials.

Sample space: All possible outcomes of a given experiment.

Event: A subset of a sample space.

Simple Event: An event consisting of just one outcome. A simple event can be represented by a single branch of a tree diagram.

Compound Event: A sequence of simple events.

Complement: The complement of event E, sometimes denoted E (E prime), occurs when E doesnt. The probability of E equals 1 minus the probability of E: P(E) = 1 P(E).

Counting Principle: If an event A can occur in m ways and for each of these m ways, an event B can occur in n ways, then events A and B can occur in ways. This counting principle can be generalized to more than two events that happen in succession. So, if for each of the m and n ways A and B can occur respectively, there is also an event C that can occur in s ways, then events A, B, and C can occur in ways.

Tree diagram: A tree-shaped diagram that illustrates sequentially the possible outcomes of a given event.

Teachers Place:

Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.

1. Explain the activity (activity requirements)

2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at http://thevillage411.weebly.com/units-of-instruction3.html.

3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.

4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)

5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.

6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard.

7. Discuss answers with the students using the following questioning techniques as applicable:

Questioning Techniques:

Memory Questions

Signal words: who, what, when, where?

Cognitive operations: naming, defining, identifying, designating

Convergent Thinking Questions


Cognitive operations: explaining, stating relationships, comparing and

contrasting

Divergent Thinking Questions

Signal words: imagine, suppose, predict, if/then

Cognitive operations: predicting, hypothesizing, inferring, reconstructing

Evaluative Thinking Questions

Signal words: defend, judge, justify (what do you think)?

Cognitive operations: valuing, judging, defending, justifying

8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.

9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter)

10. At the end of the *whole group learning session, students will transition into independent CAP assignments.

*The phrase, whole group learning session is utilized rather than, the end of the activity because all of the activities may not be completed in one day.

Task: 1

Resources:

http://www.purplemath.com/modules/nextnumb2.htm

Activity

Part 1:

a) Do the following sequence of operations in order:

b) What did you get for your final number?

c) Check with your partner, what did that person get for their final number?

d) Everyone should have the same number. What number is that?

e) Why did everyone end with the same number?

f) How does this trick work?

Write down any number.

(This is your start number.)

Add to it the number that comes before it.

Add 11. Divide by 2.

Subtract your start number.

Part 2:

Now try this one:

Take the number of your birth month. Add 32.

Add the difference between your birth month number and 12.

Divide by 4.

Add 2.

This is your Lucky Number!

Do you feel lucky? Why or why not? Explain what made this trick work.

Part 3:

Work with a partner to write your own Number Trick. Be sure to list your instructions in the appropriate order. Swap Number Tricks with another group or share your Tricks one at a time with the entire class. See if your friends can describe how your Number Trick works.

Discussion, Suggestions, Possible Solutions

The challenge is to see if students can discover how the trick works. If students need a hint, suggest that instead of using an actual number, they use a box or a letter to begin with.

Part 1:

Write down any number. (This is your start number.)

Start with N. N

Add to it the number that comes before it.

The previous number is N-1 N + (N 1) = 2N 1

Add 11.

Add the 11 to the 2N 1 2N 1 + 11 = 2N + 10

Divide by 2.

Dividing 2N + 10 by 2 2N + 10 = N + 5

2

Subtract your start number.

The start number was N. N + 5 N = 5

This means that everyone should have ended with a 5 regardless of the start number.

Part 2:

Take the number of your birth month.

The birth month will be a number between 1 and 12.

If x is the number of a students birth, then 1 x 12.

Add 32.

Adding 32 gives x + 32.

Add the difference between your birth month number and 12.

When you add the difference between your birth month and 12 you get x + 32 + (12 x) which is equal to 44.

Divide by 4.

Dividing by 4 leaves 11.

Add 2.

Adding 2 yields 13; this is usually considered an unlucky number.

Part 3:

Work with a partner to write your own Number Trick. Be sure to list your instructions in the appropriate order. Swap Number Tricks with another group or share your Tricks one at a time with the entire class. See if your friends can describe how your Number Trick works.

Task: 2

Resources:

http://www.math.com/school/subject2/lessons/S2U4L1GL.html

Activity

The students at Eastman RYDC and Eastman YDC are participating in a game room survey. Eastman RYDC is located 5 miles from the game room and Eastman YDC 3 miles from the game room. The owner, of the game room wonders how far apart the centers are.

On grid paper, pick a point to represent the location of the game room.

Illustrate all of the possible points where Eastman RYDC could be located on the grid paper.

Using a different color, illustrate all of the possible points where Eastman YDC could be located.

What is the smallest distance, d, that could separate the centers?

How did you know?

What is the largest distance, d, that could separate the centers?

How did you know?

Write and graph an inequality in terms of d to show the owner of the game room all of the possible

distances that could separate the two centers.


Students should understand that the centers could be located anywhere on the circle with the game room as the center and the radius as the distance that they are from the game room.

Eastman RYDC

Eastman YDC

Game Room

Therefore, the closest the centers could be would be 5 3 = 2 miles and the farthest apart that they could be would be 5 + 3 = 8 miles. This may be written as 5 3 = d where d represents the distance from Eastman

Task: 3

Resources:

http://www.math.com/school/subject2/lessons/S2U4L1GL.html

Activity

Part 1:

Your science teacher says the grades, g, in your class can be represented by the inequality |g 85| < 10. What is the lowest grade and what is the highest grade in the class? Explain your thinking.

Part 2:

Suppose the grades in your Language Arts class range from 68 to 94. Represent this information on a number line. Then write a compound inequality to represent the information. Finally, write an absolute value inequality to represent the same information.


Part 1:

This absolute value inequality means that g 85 10 and g 85 -10.

Another way to write this is known as a compound inequality -10 g 85 10 and may be graphed on a number line.

Using the same thinking process as used when solving equations we have

-10 g 85 10

+85+85 +85

75 g 95

74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Part 2:

Suppose the grades in your Language Arts class range from 68 to 94. Represent this information on a number line. Then write a compound inequality to represent the information. Finally, write an absolute value inequality to represent the same information.

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

This means that the lowest score included 68 and the highest score included 94. That is why the points are shaded along with all points in between.

One possible answer could be determined by 68 x 94

-81-81-81

-13 x 81 13

Course Title: Course 3 8th grade Ga DJJ

State Code: 27.0230000 CAP: 2

Georgia Performance Standard(s):

M8P1.b Solve problems that arise in mathematics and in other contexts.

M8P4.c Recognize and apply mathematics in contexts outside of mathematics.

M8P5.b Select, apply, and translate among mathematical representations to solve

problems.

M8P5.c Use representations to model and interpret physical, social, and mathematical

phenomena.

M8P2.c Develop and evaluate mathematical arguments and proofs.

M8P3.a Organize and consolidate their mathematical thinking through communication.

M8A1.b Simplify and evaluate algebraic expressions.

M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.

Objective(s):

The student compares and orders integers and evaluates expressions containing absolute values.

The student will be able to add integers.

Instructional Resources:

Holt Mathematics 8th grade Course 3 Pgs. 14-21

Chapter 1 Resource Book (CRB)

One-Stop Planner

Activities:

Read textbook pgs. 14-17.

Complete Think and Discuss, pg. 15.in textbook

Complete Practice and Problem Solving, Problems 1-5, 15-19, and 29-39 on pg. 16.in textbook

Complete Practice A 1-3 CRB, pg. 19.

Complete Reading Strategies 1-3 CRB, pg. 25.



Complete Practice and Problem Solving, Problems 1-8, 13-20, 31-38, and 49-56 on pgs. 20-21.in textbook



Evaluation:

Complete Power Presentations Lesson Quiz 1-3, pg. 17 and 1-4, pg. 21.

Modifications:

Performance Tasks: IDEA works CD




M8A2.b Use the properties of inequality to solve inequalities.

M8A2.c Graph the solution of an inequality on a number line.


problems

M8P5.c Use representations to model and interpret physical, social, and

mathematical

phenomena.

M8P3.a Organize and consolidate their mathematical thinking through

communication

Objective(s):

The student organizes and reviews key concepts and skills presented in Chapter One.

The student assesses mastery of concepts and skills in Chapter One


Holt Mathematics Course Two Textbook


One Stop Planner

Activities:

Complete the odd number problems on the Study Guide, pgs. 53-54 in textbook.

Have teacher check your work on review. If score is at least 80%, then go on to

Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in

CRB to cover concepts not understood.

Complete Chapter Test pg.55 in textbook.

Have teacher check your work on test. If score is at least 80%, then go on to next CAP. If score is less than 80%, then teacher will give worksheets in CRB covering concepts still not understood.

Evaluation:

Complete Chapter Test pg.55 in textbook with 80% accuracy

Modifications: IDEA Works CD







phenomena.




M8P4.a Recognize and use connections among mathematical ideas.

M8P4.b Understand how mathematical ideas interconnect and build on one another to

produce a coherent whole.


problems.

Objective(s):

The student writes rational numbers in equivalent forms. The student will be able to

compare and order positive and negative rational numbers written as fractions, decimals,

and integers.


Holt Mathematics 8th grade Course 3, Pgs. 60-71


One-Stop Planner

Activities:

Complete Are You Ready in textbook pg., pg. 61.

Read textbook pgs. 60-67


Complete Practice and Problem Solving, Problems 1-10, 29-38, and 67-79 on pgs. 66-

67.in textbook




Complete Think and Discuss on pg 69.in textbook

Complete Practice and Problem Solving, Problems 1-4, 10-17, 27, and 42-53 on pgs. 70-

71.in textbook



Evaluation:


Modifications:





M8A1.a Represent a given situation using algebraic expressions or equations in one

variable.

M8A1.c Solve algebraic equations in one variable, including equations involving absolute values.

M8A1.d Interpret solutions in problem contexts.



problems.




phenomena.

M8P1.a Build new mathematical knowledge through problem solving.

M8P1.b Solve problems that arise in mathematics and in other contexts

Objective(s):

The student organizes and reviews key concepts and skills presented in Chapter Two.

The student assesses mastery of concepts and skills in Chapter Two




One Stop Planner

Activities:






Have teacher check your work on test. If score is at least 80%, then go on to next CAP.

If score is less than 80%, then teacher will give worksheets in CRB covering concepts

still not understood.

Evaluation:

Complete Chapter Test pg.55 in textbook with 80% accuracy.

Modifications: IDEA Works CD

Unit: Graphs, Functions & Sequences




d. Solve equations involving several variables for one variable in terms of the others.












c. Graph equations of the form y = mx + b.

e.Graph the solution set of a linear inequality, identifying whether the solution set is an open or a closed half-plane.





M8A3 Students will understand relations and linear functions.

b. Recognize a function as a correspondence between inputs and outputs where the output for each input must be unique.

c. Distinguish between relations that are functions and those that are not functions.

d. Recognize functions in a variety of representations and a variety of contexts.

e. Use tables to describe sequences recursively and with a formula in closed form.

i. Identify relations and functions as linear or nonlinear.


Additive Inverse: The sum of a number and its additive inverse is zero. Also called the opposite of a number. Example: 5 and -5 are additive inverses of each other.

Exponent: The number of times a base is used as a factor of repeated multiplication.

Exponential Notation: See Scientific Notation below.

Hypotenuse: The side opposite to the right angle in a right triangle.

Irrational: A real number whose decimal form is non-terminating and non-repeating that cannot be written as the ratio of two integers.

Leg: Either of the two shorter sides of a right triangle. The two legs form the right angle of the triangle.

Pythagorean Theorem: A theorem that relates the lengths of the sides of a right triangle: The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse.

Radical: A symbol that is used to indicate square roots.

Rational: A number that can be written as the ratio of two integers with a nonzero denominator.

Scientific Notation: A representation of real numbers as the product of a number between 1 and 10 and a power of 10, used primarily for very large or very small numbers.

Significant Digits: A way of describing how precisely a number is written.

Square root: One of two equal factors of a nonnegative number. For example, 5 is a square root of 25 because 55 = 25. Another square root of 25 is -5 because (-5)(-5) = 25. The +5 is called the principle square root of 25 and is always assumed when the radical symbol is used.

Teachers Place:

Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.

1. Explain the activity (activity requirements)

2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at http://thevillage411.weebly.com/units-of-instruction3.html.

3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.

4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)

5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.

6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard.

7. Discuss answers with the students using the following questioning techniques as applicable:

Questioning Techniques:

Memory Questions


Cognitive operations: naming, defining, identifying, designating

Convergent Thinking Questions


Cognitive operations: explaining, stating relationships, comparing and

contrasting

Divergent Thinking Questions

Signal words: imagine, suppose, predict, if/then

Cognitive operations: predicting, hypothesizing, inferring, reconstructing

Evaluative Thinking Questions

Signal words: defend, judge, justify (what do you think)?

Cognitive operations: valuing, judging, defending, justifying

8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.

9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter)

10. At the end of the *whole group learning session, students will transition into independent CAP assignments.

*The phrase, whole group learning session is utilized rather than, the end of the activity because all of the activities may not be completed in one day.

Task: 1

Resources:

Stopwatches or clocks with minute hands

Large grid paper or large grid transparencies

Activity

Part 1:

In the sixth and seventh grade, you studied proportional relationships. Do you think that the number of heartbeats you can count is proportional to the number of seconds that you check your pulse? Explain why or why not.

Part 2:

Work with your partner to take measurements and test your conjecture. One of you will be the timer and the other will count their own number of heart beats per period of time.

First, count and record the number of beats in 10 seconds, and repeat the experiment counting the number of beats in 20 seconds and 40 seconds.

Heartbeat Data for ___________________________

Number of Seconds

Number of Heartbeats

10

20

40

Predict how many times your heart would beat in 25 seconds, in 60 seconds, and in 120 seconds. Explain how you made your predictions.

Part 3:

After gathering this data, change jobs. The person who kept time now checks his/her pulse rate for 10 seconds, 20 seconds, and 40 seconds.


Number of Seconds


10

20

40

Predict how many times your heart would beat in 25 seconds, in 60 seconds, and in 120 seconds. Explain how you made your predictions.

Part 4:

Develop a function rule (equation) to represent your pulse rate. How does your function rule compare with your partners function rule? Explain to your partner why your function rule is valid.

Draw a graph (scatter plot) to represent your pulse rate. How does your graph compare with your partners graph? Explain to your partner why your graph is valid.

Task: 2

Resources:

http://www.math.com/students/calculators/calculators.html

Graphing calculator

Graphing calculator presentation software

Activity

Part I:

Two students at Sumter YDC Matt and Tyler have been studying graphing linear equations. Matt challenged Tyler to a race to see who could graph y = 3x + 5 in the least amount of time. Matt is going to graph the equation by hand, and Tyler is going to use the graphing calculator. Matt says he can graph the equation in less than ten seconds, in less time than Tyler can enter the equation in the calculator and press the Graph key. Explain how you think Matt intends to graph the equation. Illustrate his method on two more linear equations that you make up to give to Matt and Tyler to use for their race.

Part II:

Who do you think will win the race if Matt and Tyler are given the equation 3x + 4y = 12 to graph? Why? How do you predict Matt will graph the equation?


Regardless of whether students have used graphing calculators to graph linear equations, they should be able to explain how an equation in slope-intercept form may be graphed quickly by first locating the y-intercept and then using the rise and run to locate another point on the line. In the example given, the student should explain that Matt will locate the y-intercept (0, 5), and then go to the right one unit and up three units or up three units and over to the right one unit to locate another point on the line. These two points are then used to sketch the line.

Students might also respond that Matt could quickly get a table of values. Using x = 0, Matt has the y-intercept of 5. Using x = 1, Matt obtains the point (1, 8).

Task: 3

Resources:

Graph paper

Stopwatch or classroom timer

Uncooked spaghetti

Activity

Part : Data Collection

Work with your partner to take measurements and test your conjecture. One of you will be the timer and the other will count their own number of heart beats per period of time.

First, count and record the number of beats in 10 seconds, and repeat the experiment counting the number of beats in 20 seconds, 30 seconds, 40 seconds, 50 seconds, and 60 seconds.


Time (sec)


10

20

30

40

50

60

70

80

90

100

110

120

Part : Data Representation

Graph your data.

Part : Fitting a Line to Data

Method 1: Estimation

Use a piece of raw spaghetti to visualize and estimate a line of best fit.

Choose two points on the line.

Use your two points to write an equation for a line of best fit.

Method 2: Lower and Upper Quartiles

Find the five-number summary for your x-values (time).

Find the five-number summary for your y-values (number of heartbeats).

Record your lower and upper quartiles for your x-values and your y-values.

Time (sec)


Lower Quartile

Upper Quartile

Draw a horizontal box-and-whisker plot using the five-number summary for your x-values (time.) Plot your box-and whisker plot under the x-axis on your graph.

Draw a vertical box-and-whisker plot using the five-number summary for your y-values (number of heartbeats). Plot your box-and whisker plot next to the y-axis on your graph.

Draw vertical lines from the lower and upper quartile values on the x-axis box-and whisker plot.

Draw horizontal lines from the lower and upper quartile values on the y-axis box-and whisker plot.

Find the coordinates of the vertices of the rectangle formed by the intersection of the vertical and horizontal lines. We will refer to these vertices as quartile points. Note: Use only the quartile points (vertices) that follow the direction of the data.

Do the quartile points have to be actual data points? Why or why not?

Draw a line connecting the two quartile points.

Write an equation for this line. This is a line of best fit for your data.

Use one of your equations to predict how many times your heart would beat in 25 seconds, in 240 seconds, and in 3 minutes. Explain how you made your predictions.

Example

(No. of heartbeats) (Time (sec))

Part : Data Collection

After gathering this data, change jobs. The person who kept time now checks his/her pulse rate and repeat part : Fitting a Line to Data.

Part : Analysis How does your line of best fit compare with your partners line of best fit? Explain to your partner why your line of best fit is valid.


Students could come up with possible explanations for variations in heart-rate, such as the effect of exercise, health conditions (thyroid problems, e.g.), etc.

Students should recognize that the variable representing time represents seconds and students will need to convert three minutes to seconds before using substitution.

Task: 4

Resources:

http://www.mathsisfun.com/sets/function.html

Activity

Last summer one of your classmate Brian went to the mountains and panned for gold. Although they didnt find any gold, they did find some pyrite (fools gold) and many other kinds of minerals. Brians friend, who happens to be a geologist, took several of the samples and grouped them together. The he told Brian that all of those minerals were the same. Brian had a hard time believing him, because they are many different colors. He suggested Brian analyze some data about the specimens. Brian carefully weighed each specimen in grams (g) and found the volume of each specimen in milliliters (ml).

Brian has asked his math class at Savannah RYDC to help him analyze the data. Write your analysis of his data given below:

Specimen Number

Mass or weight (g)

Volume (ml)

1

17

7

2

10

4

3

13

5

4

16

6

5

7

3

6

24

10

7

5

2


Students could graph the volume as the independent variable (x) and mass as the dependent variable (y). Their graphs could be produced either by hand on graph paper or as a STAT PLOT on a graphing calculator. Students could choose two points that are on their line of best fit that they determined by eye-balling the data, using the lower and upper quartiles, or they could use the linear regression feature of the calculator to obtain the regression equation y = 2.41x + .40.

Students might divide the mass by the volume by hand for each specimen and then find the average of this value. They could also have the calculator divide the values and find the average. The average value (2.49) is close to the coefficient of the x term in the regression equation. As students explore the meaning of this slope in this problem context, they should come to understand that it means every time the volume goes up by one ml the mass goes up by approximately 2.4 or 2.5 grams. This rate of mass in grams per milliliter of volume is the density of the mineral.

Research could be done to find lists of densities for particular minerals. While earth science references will list the density (also called specific gravity) of quartz as 2.6, the samples used for the data above were quartz. This could lead to a discussion of the precision and accuracy of measurements, as well as to a discussion of impurities and other factors that could influence the results.

The relationship between the mass and volume of the specimens might be described by some students as a proportional relationship. This could lead to the conclusion that a theoretical model for this relationship might be written as y = 2.6x. In other words, a hypothetical sample with a volume of zero would have a mass of zero, so the y-intercept should be zero.

As an extension, ask students where data points would be for samples of minerals that have a density greater than 2.6. These would be in the half-plane above the regression line for the quartz samples. This can provide a transition to graphing inequalities.

Task: 5

Resources:

http://www.coolmath.com/algebra/23-graphing-rational-functions/index.html

Graph paper

Graphing calculator

Colored pencils

Activity

In math class one day Mrs. Smith conducted an experiment. In the first part of the experiment, her students wrote Ts on a sheet of paper and counted how many they were able to make in one minute. Then Mrs. Smith told them to change their pencils to their other hands, and they repeated the experiment. Mrs. Smith then gathered from each student the information about how many Ts were made with his/her right hand and how many were made with his/her left hand in one minute. Using the data for the right hand as the x values and the data for the left hand as the y values, explain what you expect a graph of this data would look like.

In this context, a point on the line y = x would represent data for an ambidextrous person, a person that works equally as well with their left hand as they do their right hand. Left-handed persons would be able to make more Ts with their left hands than with their right hands in one minute, so their data would lie above the line y = x. Since most students are right-handed, most of the points would be graphed below the line y = x.

Questions to ask students:

What is an ambidextrous person?

Can you think of an equation that represents data for an ambidextrous person?

Why does y = x represent data for an ambidextrous person?

Ambidextrous People

(y)

(Left Hand)

(x)

(Right Hand)

Right Handed People Left Handed People

(y) (y)

(x) (x) (Left Hand)

(Right Hand) (Right Hand)


Using a graphing calculator is an efficient way to capture the picture of the actual data. Enter the x values in List 1 (L1), the y values in List 2 (L2), and enter y1 = x. Turn the statplot on to view the data in Lists 1 and 2, select Zoom Stat to set the viewing window appropriately, and select y1 to see the boundary line y = x along with the individual data points.






M8P1.a Build new mathematical knowledge through problem solving.


M8P5.a Create and use representations to organize, record, and communicate

mathematical ideas.


M8P2.d Select and use various types of reasoning and methods of proof.


M8A4.c Graph equations of the form y = mx + b.

M8P4.a Recognize and use connections among mathematical ideas.

M8P4.b Understand how mathematical ideas interconnect and build on one another to

produce a coherent whole.


problems.


Objective(s):

The student writes solutions of equations in two variables as ordered pairs. The student will be able to graph points and lines on the coordinate plane.


Holt Mathematics 8th grade Course 3, Pgs. 115-126


One-Stop Planner

Activities:

Complete Are You Ready in textbook pg., pg. 115.


Complete Think and Discuss, pg. 119 and 123.

Complete Practice and Problem Solving, Problems 1-4, 8-11, 23, 24, and 36-44 on pgs. 120-121.in textbook





Complete Practice and Problem Solving, Problems 1-6, 17-22, and 38-47 on pgs. 124-

125.in textbook



Evaluation:


Modifications:







M8A3.i Translate among verbal, tabular, graphic, and algebraic representations of

functions.

M8A4.c Graph equations of the form y = mx + b.

M8A4.e Determine the equation of a line given a graph, numerical information that defines

the line, or a context involving a linear relationship.

M8P2.d Select and use various types of reasoning and methods of proof.

M8P5.a Create and use representations to organize, record, and communicate

mathematical ideas.


problems.

M8A3.e Use tables to describe sequences recursively and with a formula in closed form.


M8P4.c Recognize and apply mathematics in contexts outside of mathematics

Objective(s):

The student organizes and reviews key concepts and skills presented in Chapter Three

The student assesses mastery of concepts and skills in Chapter Three.




One Stop Planner

Activities:






Have teacher check your work on test. If score is at least 80%, then go on to next CAP.

If score is less than 80%, then teacher will give worksheets in CRB covering concepts

still not understood.

Evaluation:

Complete Chapter Test pg.153 in textbook with 80% accuracy.

Modifications:

IDEA Works CD

Unit: Exponents& Roots/ Ratios, Proportions & Similarity/ Percents



a. Find square roots of perfect squares.

b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.

c. Recognize square roots as points and as lengths on a number line.

d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in sign.

e. Recognize and use the radical symbol to denote the positive square root of a positive number.

f. Estimate square roots of positive numbers.

g. Simplify, add, subtract, multiply, and divide expressions containing square roots.

h. Distinguish between rational and irrational numbers.


j. Express and use numbers in scientific notation.













a. Recognize reasoning and proof as fundamental aspects of mathematics.

b. Make and investigate mathematical conjectures.






d. Use the language of mathematics to express mathematical ideas precisely.









M8G2 Students will understand and use the Pythagorean theorem.

a. Apply properties of right triangles, including the Pythagorean theorem.

b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right triangle.

M8G1 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.

a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.

b. Apply properties of angle pairs formed by parallel lines cut by a transversal.

d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.


b. Determine the meaning of the slope and y-intercept in a given situation.





Absolute Value: The distance a number is from zero on the number line. Examples: |-4| = 4 and |3| = 3

Addition Property of Equality: For real numbers a, b, and c, if a = b, then a + c = b + c. In other words, adding the same number to each side of an equation produces an equivalent equation.

Additive Inverse: Two numbers that when added together equal 0.

Example, 3.2 and -3.2

AlgebraicExpression: A mathematical phrase involving at least one variable. Expressions can contain numbers and operation symbols.

Equation: A mathematical sentence that contains an equals sign.

Evaluate an Algebraic Expression: To perform operations to obtain a single number or value.

Inequality: A mathematical sentence that contains the symbols >,

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