Overview

Student's learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean theorem based on the angle-angle criterion for similar triangles.

Topic A: Dilation

Topic B: Similar Figures

Topic C: The Pythagorean Theorem

Advanced Preparation:

- Compass (required)
- Transparency or patty paper
- Wet or dry erase markers for use with transparency
- Geometry software (optional)
- Ruler
- Protractor

Students extend what they already know about unit rates and proportional relationships to linear equations and their graphs. Students understand the connections between proportional relationships, lines, and linear equations in this module.

Topic A: Writing and Solving Linear Equations

Topic B: Linear Equations in Two Variables and Their Graphs

Topic C: Slope and Equations of Lines

Topic D: Systems of Linear Equations and Their Solutions

Advanced Preparation:

- Scientific calculator
- Online graphing calculator
- Graph paper
- Straightedge

Module Tip Sheet for Parents

These documents provide parents an opportunity to see what their children are learning in the current module including:

- Standards
- Key vocabulary
- Example Problems
- Pictorial Models
- Ideas for how to can help children at home
- Coherence of learning with past and future modules

Week 1: November 6

Notes:

Lesson 3 (P)

Exercise 3 can be omitted if time is an issue. The word reciprocal can be emphasized when discussing manipulation of a dilation.

Lesson 4 (S)

To further assist the learning, graph paper rather than notebook paper can be used to have both horizontal and vertical line. FTS can be added to an anchor chart or other tracking device for theorems that are established during course of year.

Lesson 5 (P)

Direct students to use centimeters for exercise 1.

Week 2: November 13

Assessment

Mid Module Assessment: Administer after lesson 7

Biweekly 1: Exit ticket from lesson 6

Standards Addressed in Week's Lessons

Topic A

8.G.A.3

Topic B

8.G.A.4

8.G.A.5

Notes:

Lesson 6 (P)

For exercise 3, allow students to grapple with answer mathematically prior to using coordinate graph. Exercise 6-7 can be omitted if time is an issue. Similar Problem Set problems can be assigned for homework.

Lesson 7 (S)

During the Discussion section, students may use a protractor to confirm similar angles.

Lesson 8 (P)

Students will need to reference an anchor chart that describes the different type of rigid motions.

One day this week is provided for Mid Module assessment.

Standards Addressed in Assessment

Mid Module

8.G.A.3

Week 3: November 20

Notes:

Lesson 9 (E)

Ask each student to verbally or in written form share what they have learned about similarity and dilation in closing of lesson.

Lesson 10 (S)

Include the AA criteria for similarity on anchor chart that included FTS from Topic A of this module. In exercise 1 & 2, if students hesitate to begin, suggest specific side lengths for them to use.

Week 4: November 27

Assessment

End of Module Assessment: Administer after lesson 12

Biweekly 2: Exit ticket from lesson 13

Standards Addressed in Week's Lessons

Topic B

8.G.A.4

8.G.A.5

Topic C

8.G.B.6

8.G.B.7

Notes:

Lesson 11 (P)

Students use what they know about dilation, congruence, the fundamental theorem of similarity (FTS), and the angle-angle (AA) criterion to determine if two triangles are similar

Lesson 12 (M)

Real world application is used for students to demonstrate knowledge of module 3 topics.

Lesson 13 (S)

Pythagorean Theorem lessons come after the end of module, however they are part of the major works of the grade and a prerequisite for later modules in 8th grade as well as high school math classes. Scaffold abstract understanding of the theorem with concrete numbers when needed.

One day this week is provided for End of Module assessment.

Standards Addressed in Assessment

End of Module

8.G.A.3

8.G.A.4

8.G.A.5

Week 5: December 4

Notes:

Lesson 14 (P)

It may be necessary to demonstrate how to use the squared button on a calculator.

Lesson 1 (P)

Encourage students throughout lesson to compare and share with table partners about how to translate mathematical statements to symbols. Encourage multiple accurate solutions.

Lesson 2 (P)

Carefully differentiate from using expressions and equations with students when describing problems. Can tell students that non-linear equations will be discussed later in the year during module 7. Examples 2-6 can be written on the board and different colors used to identify the correct responses to questions posed.

Lesson 3 (P)

Consider developing a word bank or word wall to be used throughout the module.

Week 6: December 11

Notes:

Lesson 4 (P)

Be sure to use the names distributive, associative, and commutative when discussing how to manipulate an equation. Allow students to use multiple roads to a solution emphasizing only that properties are used correctly.

Lesson 5 (P)

Encourage students to draw diagrams to represent the situations presented in the word problems.

Lesson 6 (P)

Checking solutions should become a regular part of practice for whole class demonstrating and independent student work. Example 5 can be omitted for pacing considerations.

Lesson 7 (S)

Closing questions should include a way for each student to justify to an elbow partner the difference between unique, infinite, and no solution.

Standards Addressed in Week's Lessons

Topic A

8.EE.C.7

Week 7: December 18

Notes:

Lesson 8 (P)

This lesson on solving rational equations is included because of the types of equations students see in later topics of this module related to slope.

Lesson 9 (S)

If the Discussion is too challenging for students, use Exercises 3–11, which is a series of more accessible applications of linear equations. If using that alternative, use only question 2 from the Exit Ticket.

Lesson 10 (S)

Make clear to students that constant rate must be assumed in order to write linear equations in two variables.

Lesson 11 (P)

To scaffold for students, facilitate discussion on prior understanding from 7th grade modules of constant rate and proportions.

Week 8: January 8

Module

Topic

Notes:

Lesson 12 (E)

Model for students and ask them to consistently display independently, that answers to linear equations in two variables should be written in an ordered pair.

Lesson 13 (S)

Students need graph paper to complete Exercises 1–2

Lesson 14 (S)

The term standard form has been introduced in preceding lessons. Continue to refer to this term with students even when one variable is equal to zero.

One day this week is provided for Mid Module assessment.

Assessment

Mid Module Assessment: Administer after lesson 14

Standards Addressed in Week's Lessons

Topic B

8.EE.B.5

Standards Addressed in Assessment

Mid Module

8.EE.C.7

8.EE.B.5

Week 9: January 15

Module

Topic

Notes:

Lesson 15 (P)

Two days are provided for lesson 15. Teach lesson whole group and provide differentiation through centers work with slope using problem set problems on day 2.

Lesson 16 (S)

Slope formula can be represented in various ways for students and at this point teachers can consider using the words "slope" and "rate of change" as interchangeable. Consider using problem set questions during small group time this week.

Standards Addressed in Week's Lessons

Topic C

8.EE.B.5

8.EE.B.6

Week 10: January 22

Module

Topic

Notes:

Lesson 17 (S)

The slope intercept formula is introduced and that vocabulary, including "m" being equal to the slope, can be used from here on when discussing linear equations.

Lesson 18 (P)

Two days are provided for lesson 18 to provide additional practice time using information about slope and intercepts to draw a line. The Opening Exercise requires students to examine part (f) from the Problem Set of Lesson 17. Coordinate planes are provided for students in the exercises of this lesson, but they need graph paper to complete the problem set.

Assessment

ANet

Standards Addressed in Week's Lessons

Topic C

8.EE.B.5

8.EE.B.6

Standards Addressed in Assessment

ANET

Major

8.EE.B.5

8.EE.B.6

8.EE.B.7a,b

8.G.A.4

8.G.A.5

Review

8.EE.A.3

8.EE.A.4

8.G.A.3

Week 11: January 29

Notes:

Lesson 19 (S)

The theorems in this lesson are presented to students and, based on the readiness of each class, the teacher can choose to discuss the proof of the theorem, or students can explore the theorem experimentally by completing the exercises of the alternate activity.

Lesson 20 (P)

Two days are provided for this lesson. All problem set questions should be included during those two days.

The proof that every line is the graph of a linear equation in the Discussion is optional. If using the Discussion, skip the Opening Exercise, and resume the lesson with Example 1. Complete all other examples and exercises that follow. As an alternative to the Discussion, complete the Opening Exercise by showing a graph of a line on the coordinate plane and having students attempt to name the equation of the line. Graphs can be found on p. 317.