Preparation and Customization of a Lesson using Eureka in

Preparation and Customization of a Lesson using Eureka Math® in SyncTM:

A Story of Ratios ® and A Story of Functions®

Virtual Engagement Materials

October 2020

Table of Contents

In the live virtual session we will be taking a deep look at Grade 7 Module 3 (G7 M3).

Pre-Engagement Materials: Complete the following pages and come prepared with your notes and thoughts to have meaningful collaborative conversations throughout this virtual session. Directions for each task are listed at the top of the relevant page.

Professional Reading 1, Page 3Note-catcher, Pages 4-5G7 M3 Table of Contents, Page 6G7 M3 Overview Narrative, Pages 7-8G7 M3 Module Level Learn Anywhere Plan, Pages 9-15 G7 M3 Topic B Overview Narrative, Pages 16-17G7 M3 L7-15 Exit Tickets, Pages 18-27G7 M3 Topic B Discourse Facilitation Slides, Pages 28-29 G7 M3 Mid-Module Assessment, Pages 30-36G7 M3 Lesson 10 Student Edition, Pages 37-42

Session 1 Materials

Interim Materials-Complete the following pages and come prepared with your notes and thoughts to have meaningful collaborative conversations throughout this virtual session. Directions Directions for each task are listed at the top of the relevant page.

Watch the Lesson Videos for L10-11 (found on landing page)G7 M3 L10-11 Lesson Level Learn Anywhere Plans, Pages 43-44Talking Teachers, Page 45

Session 2 Materials

Interim Materials, Pages 43-45G7 M3 L10 Teacher Edition, Pages 46-57 G7 M3 L10-11 Topic Discourse Student Debrief Slides, Page 58-59G7 M3 L10 Problem Set, Pages 60-62G7 M3 L11 Problem Set, Pages 63-65Professional Reading 2, Page 66

Post-Engagement Materials

Plan an upcoming lesson using the three-step Preparation & Customization process

Virtual Engagement MaterialsPreparation and Customization of a Lesson using Eureka Math® in SyncTM

2 Copyright © 2020 Great Minds® greatminds.org

G7 M3 L9-11 Exit Tickets, Pages 23-25G7 M3 L10 Student Edition, Pages 37-42

3

of Fundamental Mathematics in China and the United States (Ma 1999, 131).1

The textbook is the material on which Chinese teachers spend most of their time and devote most

of their efforts to “study intensively.” They study it constantly throughout the school year when

they teach it. First of all, they work for an understanding of “what it is.” They study how it

interprets and illustrates the ideas in the [standards], why the authors structured the book in a

certain way, what the connections among the contents are, what the connections are between the

content of a certain textbook compared with an old version and why changes have been made, and

so on. At a more detailed level, they study how each [module] of the textbook is organized, how

the content was presented by the authors, and why. They study what examples are in a [topic],

why these examples were selected, and why the examples were presented in a certain order. They

review the exercises in each [lesson] of a [topic], the purpose for each exercise … and so on.

Indeed, they conduct a very careful and critical investigation of the textbook. Although teachers

usually find the authors’ ideas ingenious and inspiring, they also sometimes find parts of the

textbook that from their perspective are unsatisfactory, or inadequate illustrations of ideas in the

[Standards].

§ “At a more detailed level, they study how each [module] of the textbook is organized, how the content waspresented by the authors, and why. They study what examples are in a [topic].”

§ “They review the exercises in each [lesson] of a [topic], the purpose for each exercise … and so on.”

§ “Although teachers usually find the authors’ ideas ingenious and inspiring, they also sometimes find parts ofthe textbook that from their perspective are unsatisfactory, or inadequate illustrations of ideas in the[Standards].”

1 Ma, Liping. 1999. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum.

Directions: Read and highlight anything that stands out to you.

Professional Reading 1

Read the following excerpt from Knowing and Teaching Elementary Mathematics: Teachers’ Understanding

Which of the following excerpts resonates most with you?

Copyright © 2020 Great Minds® greatminds.org


Discern the Plot:

What makes Discern the Plot an essential step in preparing a lesson?

Find the Ladder:

What makes Find the Ladder an essential step in preparing a lesson?



Hone the Lesson:

How did the intensive study in the Discern the Plot and Find the Ladder steps change the way you customized in the Hone the Lesson step?



6G R A D E Mathematics Curriculum

1Table of Contents1

.............................................................................................................................. .................... 3

Topic A: Representing and Reasoning About Ratios (6.RP.A.1, 6.RP.A.3a) ........................................................ 12

Lessons 1–2: Ratios ......................................................................................................... ........................ 14

Lessons 3–4: Equivalent Ratios............................................................................................................... 28

Lessons 5–6: Solving Problems by Finding Equivalent Ratios................................................................. 41

Lesson 7: Associated Ratios and the Value of a Ratio ............................................................................ 51

Lesson 8: Equivalent Ratios Defined Through the Value of a Ratio ....................................................... 57

Topic B: Collections of Equivalent Ratios (6.RP.A.3a) ......................................................................................... 63

Lesson 9: Tables of Equivalent Ratios ....................................................................................... .............. 65

Lesson 10: The Structure of Ratio Tables—Additive and Multiplicative ................................................ 71

Lesson 11: Comparing Ratios Using Ratio Tables ................................................................................... 80

Lesson 12: From Ratio Tables to Double Number Line Diagrams .......................................................... 88

Lesson 13: From Ratio Tables to Equations Using the Value of a Ratio ................................................. 99

Lesson 14: From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on theCoordinate Plane .............................................................................................................. .. 109

Lesson 15: A Synthesis of Representations of Equivalent Ratio Collections ........................................ 117

Mid-Module Assessment and Rubric ................................................................................................................ 126 Topics A through B (assessment 1 day, return 1 day, remediation or further applications 1 day)

Topic C: Unit Rates (6.RP.A.2, 6.RP.A.3b, 6.RP.A.3d) ....................................................................................... 132

Lesson 16: From Ratios to Rates........................................................................................................... 134

Lesson 17: From Rates to Ratios............................................................................................. .............. 139

Lesson 18: Finding a Rate by Dividing Two Quantities ........................................................................ . 145

Lessons 19–20: Comparison Shopping—Unit Price and Related Measurement Conversions ............. 150

Lessons 21–22: Getting the Job Done—Speed, Work, and Measurement Units................................. 165

Lesson 23: Problem-Solving Using Rates, Unit Rates, and Conversions............................................... 179

1Each lesson is ONE day, and ONE day is considered a 45-minute period.

Module 1: Ratios and Unit Rates

A STORY OF RATIOS

©2015 Great Minds. eureka-math.orgG6-M1-TE-B1-1 .3 .1-01.2016

1Module 3: Expressions and Equations 6

Copyright © Great Minds PBCVirginia Edition greatminds.org/math

Module Overview ��3

Topic A: Use Properties of Operations to Generate Equivalent Expressions (7.11, 7.12, 7.13) ��12

Lessons 1–2: Generating Equivalent Expressions ��13

Lessons 3–4: Writing Products as Sums and Sums as Products��48

Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions ��73

Lesson 6: Collecting Rational Number Like Terms ��85

Topic B: Solve Problems Using Expressions, Equations, and Inequalities (7.3, 7.5, 7.12, 7.13) ��98

Lesson 7: Understanding Equations ��100

Lessons 8–9: Using If-Then Moves in Solving Equations ��113

Lessons 10–11: Angle Problems and Solving Equations ��150

Lesson 12: Properties of Inequalities ��171

Lesson 13: Inequalities ��184

Lesson 14: Solving Inequalities ��193

Lesson 15: Graphing Solutions to Inequalities ��203Mid-Module Assessment and Rubric ��217Topics A through B (assessment 2 days, return 1 day, remediation or further applications 1 day)

Topic C: Use Equations to Solve Geometry Problems (7.4) ��239

Lesson 16: Volume with Fractional Edge Lengths and Unit Cubes ��241

Lesson 17: From Unit Cubes to the Formulas for Volume ��261

Lesson 18: The Formulas for Volume of Rectangular Prisms ��277

Lesson 19: Volume in the Real World ��290

Lesson 20: Representing Three-Dimensional Figures Using Nets ��302

Lesson 21: Constructing Nets ��337

Lesson 22: From Nets to Surface Area of Rectangular Prisms ��353


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Grade7Module3(G7M3)TableofContents2Grade7•Module3:ExpressionsandEquations

Directions: While reading, annotate and consider your responses to the following Guiding Questions:1. What can you discern about the teaching sequence by reviewing the Lesson Titles?2. What other information can we gain by reviewing the Table of Contents?


Lesson 23: Determining Surface Area of Three-Dimensional Figures .................................................368

Lesson 24: Surface Area and Volume in the Real World ......................................................................379End-of-Module Assessment and Rubric ...........................................................................................................391 Topics A through C (assessment 1 day, return 1 day, remediation or further applications 2 day)

In Grade 6, students interpreted expressions and equations as they reasoned about one-variable equations (6.6, 6.13). This module consolidates and expands upon students’ understanding of equivalent expressions as they apply the properties of operations (associative, commutative, and distributive) to write expressions in both standard form (by expanding products into sums) and in factored form (by expanding sums into products). They use linear equations to solve unknown angle problems and other problems presented within context to understand that solving algebraic equations is all about the numbers. It is assumed that a number already exists to satisfy the equation and context; we just need to discover it. A number sentence is an equation that is said to be true if both numerical expressions evaluate to the same number; it is said to be false otherwise. Students use the number line to understand the properties of inequality and recognize when to preserve the inequality and when to reverse the inequality when solving problems leading to inequalities. They interpret solutions within the context of problems. Students extend their sixth-grade study of geometric figures and the relationships between them as they apply their work with expressions and equations to solve problems involving area of a circle and composite area in the plane, as well as volume and surface area of right prisms. In this module, students discover the most famous ratio of all, π, and begin to appreciate why it has been chosen as the symbol to represent the Grades 6–8 mathematics curriculum, A Story of Ratios.

To begin this module, students will generate equivalent expressions using the fact that addition and multiplication can be done in any order with any grouping and will extend this understanding to subtraction (adding the inverse) and division (multiplying by the multiplicative inverse, also known as the reciprocal) (7.12, 7.13). They extend the properties of operations with numbers (learned in earlier grades) and recognize how the same properties hold true for letters that represent numbers. Knowledge of rational number operations from Module 2 is demonstrated as students collect like terms containing both positive and negative integers.

An area model is used as a tool for students to rewrite products as sums and sums as products and to provide a visual representation leading students to recognize the repeated use of the distributive property in factoring and expanding linear expressions (7.12, 7.13). Students examine situations where more than one form of an expression may be used to represent the same context, and they see how looking at each form can bring a new perspective (and thus deeper understanding) to the problem. Students recognize and use the identity properties and the existence of additive inverses to efficiently write equivalent expressions in standard form, for example, 2x + (−2x) + 3 = 0 + 3 = 3 (7.11, 7.12, 7.13). By the end of the topic, students have the opportunity to practice knowledge of operations with rational numbers gained in Module 2 (7.1, 7.2, 7.11, 7.12, 7.13) as they collect like terms with rational number coefficients (7.12, 7.13).

In Topic B, students use linear equations and inequalities to solve problems (7.12, 7.13). They continue to use tape diagrams from earlier grades where they see fit, but will quickly discover that some problems would more reasonably be solved algebraically (as in the case of large numbers). Guiding students to arrive at this

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Grade 7 Module 3 Overview Narrative

Directions: While reading, annotate and consider your responses to the following Guiding Questions:1. What new information did you acquire from reading the Module Overview?2. How has your idea of the sequence of learning improved or changed after studying the Narrative?



realization on their own develops the need for algebra. This algebraic approach builds upon work in Grade 6 with equations (6.13) to now include multi-step equations and inequalities containing rational numbers (7.3, 7.12, 7.13). Students solve problems involving consecutive numbers; total cost; age comparisons; distance, rate, and time; area and perimeter; and missing angle measures. Solving equations with a variable is all about numbers, and students are challenged with the goal of finding the number that makes the equation true. When given in context, students recognize that a value exists, and it is simply their job to discover what that value is. Even the angles in each diagram have a precise value, which can be checked with a protractor to ensure students that the value they find does indeed create a true number sentence.In Grade 5, students recognized volume as an attribute of solid figures. They measured volume by packing right rectangular prisms with unit cubes and found that determining volumes was the same as multiplying the edge lengths of the prisms (5.6). Students extend this knowledge in Topic C, where they continue packing right rectangular prisms with unit cubes; however, this time the right rectangular prism has fractional lengths. (It should be noted that students will explore the volume of cylinders in Module 6.) Students decompose a one cubic unit prism in order to conceptualize finding the volume of a right rectangular prism with fractional edge lengths using unit cubes (7.4). They connect those findings to apply the formula V = lwh and multiply fractional edge lengths. Students extend and apply the volume formula to V = the area of the base × height or simply V = bh, where b represents the area of the base. Students explore the bases of right rectangular prisms and find the area of the base first and then multiply by the height. They determine that two formulas can be used to find the volume of a right rectangular prism. Students apply both formulas to application problems. Real-life application of the volume formula as students extend the notion that volume is additive and find the volume of composite solid figures. They apply volume formulas and use their previous experience with solving equations to find missing volumes and missing dimensions (7.4)

Module 3 concludes with deconstructing the faces of solid figures to determine surface area of right rectangular prisms (7.4). (Students will determine the surface area of cylinders in Module 6.) Students note the difference between finding the volume of right rectangular prisms and finding the surface area of such prisms. Students build solid figures using nets. They note which nets compose specific solid figures and also understand when nets cannot compose a solid figure. From this knowledge, students deconstruct solid figures into nets to identify the measurements of the solids’ face edges. With this knowledge students are prepared to use nets to determine the surface area of solid figures. They find that adding the areas of each face of the solid results in a combined surface area. Students find that each right rectangular prism has a front, a back, a top, and bottom, and two sides. They determine that surface area is obtained by adding the areas of all the faces. They understand that the front and back of the prism have the same surface area, the top and bottom have the same surface area, and the sides have the same surface area. Thus, students develop the formula SA = 2lw + 2lh + 2wh (7.4). To wrap up the module, students apply the surface area formula to real-life contexts and distinguish between the need to find surface area or volume within contextual situations.

This module is comprised of 24 lessons; 8 days are reserved for administering the Mid-Module and End-of-Module Assessments, returning the assessments, and remediating or providing further applications of the concepts. The Mid-Module Assessment follows Topic B, and the End-of-Module Assessment follows Topic C.




Module Level Learn Anywhere Plan

Grade 7 Module 3: Expressions and Equations The Learn Anywhere Plan (LAP) optimizes learning for circumstances that reduce or eliminate in-person instruction. Leveraging asynchronous video lessons and combining them with prioritized content for teacher-facilitated instruction, the plans ensure the most essential elements of student learning are maintained for all students.

The timing and frequency of synchronous facilitation is intended to be flexible to fit a variety of structures. For example, teachers might combine two to five lessons into one facilitated session, as needed, to be delivered shortly before or after the lessons' asynchronous learning is assigned and completed.

Used flexibly, the LAP can reduce the total instructional time required to complete the content of the year while tending to the objectives of every lesson. If circumstances require, the recommendations found in the resource Eureka Math® Grade 7 Consolidation Guide outline additional options for consolidating grade level content while maintaining coherence and providing emphasis on the most integral learning for college and career readiness.

Lesson Types Each lesson in Grades 6 through 12 is designed with one of four structures: Exploration, Modeling Cycle, Problem Set, or Socratic. Lesson types are driven by content and provide variation in the structure of lessons. The icon identifying a lesson type can be found within each lesson of the curriculum beside the lesson title in the Teacher Edition.

Facilitating the Student Debrief The Lesson Debrief is an opportunity to engage students in meaningful discussion. There are four approaches to facilitating discussion in a Lesson Debrief: Sample Solution, Error Analysis, Connection, or Impact.

Sample Solution - Analyze samples of accurate solutions (student or teacher created) to reinforce procedures, strategies, pictorial representations, etc.

Exploration Lesson: Students work independently or in small groups on a challenging problem followed by a debrief to clarify, expand, or develop math knowledge.

Modeling Cycle Lesson: Students practice all or part of the modeling cycle with real-world or mathematical problems that are either well- or ill-defined

Problem Set Lesson: The teacher and students work through examples and complete exercises to develop or reinforce a concept.

Socratic Lesson: The teacher leads students in a conversation to develop a specific concept or proof.




Error Analysis - Identify and correct errors and misconceptions in sample solutions.

Connection - Recognize a connection or relationship across problems or key concepts within a lesson or from previous learning.

Impact - Observe and understand the impact of a key concept within the lesson. The Impact approach may often be paired with another approach to provide a visual support.

Sequence of Grade 7 Modules

Module 1: Ratios and Proportional Relationships Module 2: Rational Numbers Module 3: Expressions and Equations Module 4: Percent and Proportional Relationships Module 5: Statistics and Probability Module 6: Geometry

Topic A: Use Properties of Operations to Generate Equivalent Expressions

Topic A Facilitation Slides

Lesson and

Lesson Type

Student Outcomes Video Lesson Additional Concept

Development

Student Debrief

Lesson 1 Students generate equivalent expressions using the fact that addition and multiplication can be done in any order (commutative property) and any grouping (associative property).

Students recognize how any order, any grouping can be applied in a subtraction problem by using additive inverse relationships (adding the opposite) to form a sum and likewise with division problems by using the multiplicative inverse relationships (multiplying by the reciprocal) to form a product.

Students recognize that any order does not apply to expressions mixing addition and multiplication, leading to the need to follow the order of operations.

Connection

Lesson 2 Students generate equivalent expressions using the fact that addition and multiplication can be done in any order (commutative property) and any grouping (associative property).

Students recognize how any order, any grouping can be applied in a subtraction problem by using additive inverse relationships

Error Analysis




(adding the opposite) to form a sum and likewise with division problems by using the multiplicative inverse relationships (multiplying by the reciprocal) to form a product.

Students recognize that any order does not apply to expressions mixing addition and multiplication, leading to the need to follow the order of operations.

Lesson 3 Students use area and rectangular array models and the distributive property to write products as sums and sums as products.

Students use the fact that the opposite of a number is the same as multiplying by −1 to write the opposite of a sum in standard form.

Students recognize that rewriting an expression in a different form can shed light on the problem and how the quantities in it are related.

Sample Solution

Lesson 4 Students use an area model to write products as sums and sums as products.

Students use the fact that the opposite of a number is the same as multiplying by −1 to write the opposite of a sum in standard form.

Students recognize that rewriting an expression in a different form can shed light on the problem and how the quantities in it are related.

Error Analysis

Lesson 5 Students recognize the identity properties of 0 and 1 and the existence of inverses (opposites and reciprocals) to write equivalent expressions.

Sample Solution

Lesson 6 Students rewrite rational number expressions by collecting like terms and combining them by repeated use of the distributive property. Connection

Affirm™ Topic A Quiz

Topic B: Solve Problems Using Expressions, Equations, and Inequalities Topic B Facilitation Slides

Lesson and

Lesson Type


Development

Student Debrief




Lesson 7 Students understand that an equation is a statement of equality between two expressions.

Students build an algebraic expression using the context of a word problem and use that expression to write an equation that can be used to solve the word problem.

Error Analysis

Lesson 8 Students understand and use the addition, subtraction, multiplication, division, and substitution properties of equality to solve word problems leading to equations of the form 𝑝𝑥 + 𝑞 = 𝑟 and 𝑝(𝑥 + 𝑞) = 𝑟 where 𝑝, 𝑞, and 𝑟 are specific rational numbers.

Students understand that any equation with rational coefficients can be written as an equation with expressions that involve only integer coefficients by multiplying both sides by the least common multiple of all the rational number terms.

Connection

Lesson 9 Students understand and use the addition, subtraction, multiplication, division, and substitution properties of equality to solve word problems leading to equations of the form 𝑝𝑥 + 𝑞 = 𝑟 and 𝑝(𝑥 + 𝑞) = 𝑟 where 𝑝, 𝑞, and 𝑟 are specific rational numbers.

Students understand that any equation can be written as an equivalent equation with expressions that involve only integer coefficients by multiplying both sides by the correct number.

Sample Solution

Lesson 10 Students use vertical angles, adjacent angles, angles on a line, and angles at a point in a multistep problem to write and solve simple equations for an unknown angle in a figure.

Sample Solution

Lesson 11 Students use vertical angles, adjacent angles, angles on a line, and angles at a point in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Error Analysis

Lesson 12 Students justify the properties of inequalities that are denoted by < (less than), ≤ (less than or equal to), > (greater than), and ≥ (greater than or equal to).

Connection

Lesson 13 Students understand that an inequality is a statement that one expression is less than (or equal to) or greater than (or equal to) another expression, such as 2𝑥 + 3 < 5 or 3𝑥 + 50 ≥100.

Connection




Students interpret a solution to an inequality as a number that makes the inequality true when substituted for the variable.

Students convert arithmetic inequalities into a new inequality with variables (e.g., 2 × 6 + 3 >12 to 2𝑚 + 3 > 12) and give a solution, such as 𝑚 = 6, to the new inequality. They check to see if different values of the variable make an inequality true or false.

Lesson 14 Students solve word problems leading to inequalities that compare 𝑝𝑥 + 𝑞 and 𝑟, where 𝑝, 𝑞, and 𝑟 are specific rational numbers.

Students interpret the solutions in the context of the problem.

Error Analysis

Lesson 15 Students graph solutions to inequalities taking care to interpret the solutions in the context of the problem.

Sample Solution Impact

Affirm™ Topic B Quiz

Mid Module Assessment Task

Discussion-Based Assessment: Problem 7

Topic C: Use Equations and Inequalities to Solve Geometry Problems

Topic C Facilitation Slides

Lesson and

Lesson Type


Development

Student Debrief

Lesson 16 Students develop the definition of a circle using diameter and radius.

Students know that the distance around a circle Is called the circumference and discover that the ratio of the circumference to the diameter of a circle is a special number called pi, written π.

Students know the formula for the circumference 𝐶 of a circle, of diameter 𝑑, and radius 𝑟. They use scale models to derive these formulas.

Connection




Students use !!"

and 3.14 as estimates for π and informally show that π is slightly greater than 3.

Lesson 17 Students give an informal derivation of the relationship between the circumference and are of a circle. Students know the formula for the area of a circle and use it to solve problems.

Connection

Lesson 18 Students examine the meaning of quarter circle and semicircle.

Students solve area and perimeter problems for regions made out of rectangles, quarter circles, semicircles, and circles, including solving for unknown lengths when the area or perimeter is given.

Connection

Lesson 19 Students find the area of triangles and simple polygonal regions in the coordinate plane with vertices at grid points by composing into rectangles and decomposing into triangles and quadrilaterals.

Impact

Lesson 20 Students find the area of regions in the coordinate plane with polygonal boundaries by decomposing the plane into triangles and quadrilaterals including regions with polygonal holes.

Students find composite areas of regions in the coordinate plane by decomposing the plane into familiar figures (triangles, quadrilaterals, circles, semicircles, and quarter circles).

Sample Solution

Lesson 21 Students find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals. They use polyhedron nets to understand that surface area is simply the sum of the area of the lateral faces and the area of the base(s).

Connection

Lesson 22 Students find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals, specifically focusing on pyramids. They use polyhedron nets to understand that surface area is simply the sum of the area of the lateral faces and the area of the base(s).

Connection

Lesson 23 Students use the known formula for the volume of a right rectangular prism (length ×width × height).Students understand the volume of a right prism to be the area of the base times the height.

Sample Solution

Impact




Students compute volumes of right prisms involving fractional values for length.

Lesson 24 Students use the formula for the volume of a right rectangular prism to answer questions about the capacity of tanks.Students compute volumes of right prisms involving fractional values for length.

Impact

Lesson 25 Students solve real-world and mathematical problems involving volume and surface areas of three-dimensional objects composed of cubes and right prisms.

Connection

Lesson 26 Students solve real-world and mathematical problems involving volume and surface areas of three-dimensional objects composed of cubes and right prisms.

Impact

Affirm™ Topic C Quiz

End of Module Assessment Task

Discussion-Based Assessment: Problem 6(a), 6(d)



6G R A D E Mathematics Curriculum

1Table of Contents1

.............................................................................................................................. .................... 3

Topic A: Representing and Reasoning About Ratios (6.RP.A.1, 6.RP.A.3a) ........................................................ 12

Lessons 1–2: Ratios ......................................................................................................... ........................ 14

Lessons 3–4: Equivalent Ratios............................................................................................................... 28

Lessons 5–6: Solving Problems by Finding Equivalent Ratios................................................................. 41

Lesson 7: Associated Ratios and the Value of a Ratio ............................................................................ 51

Lesson 8: Equivalent Ratios Defined Through the Value of a Ratio ....................................................... 57

Topic B: Collections of Equivalent Ratios (6.RP.A.3a) ......................................................................................... 63

Lesson 9: Tables of Equivalent Ratios ....................................................................................... .............. 65

Lesson 10: The Structure of Ratio Tables—Additive and Multiplicative ................................................ 71

Lesson 11: Comparing Ratios Using Ratio Tables ................................................................................... 80

Lesson 12: From Ratio Tables to Double Number Line Diagrams .......................................................... 88

Lesson 13: From Ratio Tables to Equations Using the Value of a Ratio ................................................. 99

Lesson 14: From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on theCoordinate Plane .............................................................................................................. .. 109

Lesson 15: A Synthesis of Representations of Equivalent Ratio Collections ........................................ 117

Mid-Module Assessment and Rubric ................................................................................................................ 126 Topics A through B (assessment 1 day, return 1 day, remediation or further applications 1 day)

Topic C: Unit Rates (6.RP.A.2, 6.RP.A.3b, 6.RP.A.3d) ....................................................................................... 132

Lesson 16: From Ratios to Rates........................................................................................................... 134

Lesson 17: From Rates to Ratios............................................................................................. .............. 139

Lesson 18: Finding a Rate by Dividing Two Quantities ........................................................................ . 145

Lessons 19–20: Comparison Shopping—Unit Price and Related Measurement Conversions ............. 150

Lessons 21–22: Getting the Job Done—Speed, Work, and Measurement Units................................. 165

Lesson 23: Problem-Solving Using Rates, Unit Rates, and Conversions............................................... 179


Module 1: Ratios and Unit Rates

A STORY OF RATIOS

©2015 Great Minds. eureka-math.orgG6-M1-TE-B1-1 .3 .1-01.2016

1

7.3, 7.5, 7.12, 7.13

Focus Standards: 7 .3 The student will solve single-step and multistep practical problems, using proportional reasoning.

7.5 The student will solve problems, including practical problems, involving the relationship between corresponding sides and corresponding angles of similar quadrilaterals and triangles.

7.12 The student will solve two-step linear equations in one variable, including practical problems that require the solution of a two-step linear equation in one variable.

7.13 The student will solve one- and two-step linear inequalities in one variable, including practical problems, involving addition, subtraction, multiplication, and division, and graph the solution on a number line.

Instructional Days: 9

Lesson 7: Understanding Equations (P)3

Lessons 8–9: Using If-Then Moves in Solving Equations (P)

Lessons 10–11: Angle Problems and Solving Equations (P)

Lesson 12: Properties of Inequalities (E)

Lesson 13: Inequalities (P)

Lesson 14: Solving Inequalities (P)

Lesson 15: Graphing Solutions to Inequalities (P)

3Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson


Directions: While reading, annotate and consider your response to the following Guiding Question:1. What new information did you acquire from reading the Topic B Overview?


G7 M3 Topic B:

SolveProblemsUsingExpressions,Equations,andInequalities

16Copyright © 2020 Great Minds® greatminds.org

Topic B begins in Lesson 7 with students evaluating equations and problems modeled with equations for given rational number values to determine whether the value makes a true or false number sentence. In Lessons 8 and 9, students are given problems of perimeter; total cost; age comparisons; and distance, rate, and time to solve. Students will discover that modeling these types of problems with an equation becomes an efficient approach to solving the problem, especially when the problem contains rational numbers (7.3, 7.12, 7.13). Students apply the properties of equality to isolate the variable in these equations as well as those created to model missing angle problems in Lessons 10 and 11. All problems provide a real-world or mathematical context so that students can connect the (abstract) variable, or letter, to the number that it actually represents in the problem. The number already exists; students just need to find it.

Lesson 12 introduces students to situations that are modeled in the form px + q > r and px + q < r. Initially, students start by translating from verbal to algebraic, choosing the inequality symbol that best represents the given situation. Students then find the number(s) that make each inequality true. To better understand how to solve an inequality containing a variable, students look at statements comparing numbers in Lesson 13. They discover when (and why) multiplying by a negative number reverses the inequality symbol when this symbol is preserved. In Lesson 14, students extend the idea of isolating the variable in an equation to solve problems modeled with inequalities using the properties of inequality. This topic concludes with students modeling inequality solutions on a number line and interpreting what each solution means within the context of the problem (7.13).




G7M3Lesson7:UnderstandingEquations

ExitTicket

Checkwhetherthegivenvalueof𝑥isasolutiontotheequation.Justifyyouranswer.

a. 2F𝑥 + 4 = 20 𝑥 = 48

b. 3𝑥 − 1 = 5𝑥 + 10 𝑥 = −5 12

Thetotalcostof4pensand7mechanicalpencilsis$13.25.Thecostofeachpencilis75cents.a. Usinganarithmeticapproach,findthecostofapen.



b. Letthecostofapenbe𝑝dollars.Writeanexpressionforthetotalcostof4pensand7mechanicalpencilsintermsof𝑝.

c. Writeanequationthatcouldbeusedtofindthecostofapen.

d. Determineavaluefor𝑝forwhichtheequationyouwroteinpart(b)istrue.

e. Determineavaluefor𝑝forwhichtheequationyouwroteinpart(b)isfalse.



G7M3L8:UsingIf–ThenMovesinSolvingEquations

ExitTicket

Mrs.Canale’sclassissellingfrozenpizzastoearnmoneyforafieldtrip.Foreverypizzasold,theclassmakes$5.35.Theyhavealreadyearned$182.90towardtheir$750goal.Howmanymorepizzasmusttheyselltoearn$750?Solvethisproblemfirstbyusinganarithmeticapproach,thenbyusinganalgebraicapproach.Comparethecalculationsyoumadeusingeachapproach.



G7M3L9:UsingIf–ThenMovesinSolvingEquations

ExitTicket

BrandAscooterhasatopspeedthatgoes2milesperhourfasterthanBrandB.Ifafter3hours,BrandAscootertraveled24milesatitstopspeed,atwhatratedidBrandBscootertravelatitstopspeedifittraveledthesamedistance?Writeanequationtodeterminethesolution.Identifytheif–thenmovesusedinyoursolution.

2. Ateachscooter’stopspeed,BrandAscootergoes2milesperhourfasterthanBrandB.Ifaftertravelingatitstopspeedfor3hours,BrandAscootertraveled40.2miles,atwhatratedidBrandBscootertravelifittraveledthesamedistanceasBrandA?Writeanequationtodeterminethesolutionandthenwriteanequivalentequationusingonlyintegers.


Directions: Please complete the Lesson 9 Exit Ticket


Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Thatis,describetheanglerelationshipsyoucouldusetodeterminethevalueof𝑥.

Usetheanglerelationshipsdescribedabovetowriteanequationtosolvefor𝑥.Then,determinethemeasurementsof∠𝐽𝐴𝐻and∠𝐻𝐴𝐺.

45o

30o

E

30°

3x° 5x°

G

FJ

H

K

A


G7M3L10:AngleProblemsandSolvingEquations

ExitTicket



G7M3L11:AngleProblemsandSolvingEquations

ExitTicket

Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝑥.Findthemeasuresof∠𝑅𝑄𝑆and∠𝑇𝑄𝑈.

A

G

F

H

225°3x°

J2x°

E18o

27o

90o

x=9

3x°

4x°

139o 21o

221°R

S

T

U

Q




G7M3L12:PropertiesofInequalities

ExitTicket

1. Giventheinitialinequality−4 < 7,statepossiblevaluesfor𝑐thatwouldsatisfythefollowinginequalities.a. 𝑐(−4) < 𝑐(7)

b. 𝑐(−4) > 𝑐(7)

c. 𝑐(−4) = 𝑐(7)

Giventheinitialinequality2 > −4,identifywhichoperationpreservestheinequalitysymbolandwhichoperationreversestheinequalitysymbol.Writethenewinequalityaftertheoperationisperformed.

a. Multiplybothsidesby−2.

b. Add−2tobothsides.

c. Dividebothsidesby2.

d. Multiplybothsidesby−12.

e. Subtract−3frombothsides.



G7M3L13:Inequalities

ExitTicket

Shaggyearned$7.55perhourplusanadditional$100intipswaitingtablesonSaturday.Heearnedatleast$160inall.Writeaninequalityandfindtheminimumnumberofhours,tothenearesthour,thatShaggyworkedonSaturday.



G7M3L14:SolvingInequalities

ExitTicket

Gamesatthecarnivalcost$3each.Theprizesawardedtowinnerscost$145.65.Howmanygamesmustbeplayedtomakeatleast$50?



G7M3L15:GraphingSolutionstoInequalities

ExitTicket

Thejuniorhighartclubsellscandlesforafundraiser.Thefirstweekofthefundraiser,theclubsells7casesofcandles.Eachcasecontains40candles.Thegoalistosellatleast13cases.Duringthesecondweekofthefundraiser,theclubmeetsitsgoal.Write,solve,andgraphaninequalitythatcanbeusedtofindthepossiblenumberofcandlessoldthesecondweek.



Grade 7 Module 3 Topic B: Topic Discourse Facilitation Slides

Lesson 15 Problem Set 2(b)Stronger and Clearer Each Time

Do you agree with this claim? Justify your answer. • Answers will vary• The claim is incorrect. Gary must work more than 22.58 hours in order to earn more than

$400.• Students may answer based on a general sense of reasonableness or estimation.

Have students exchange their justifications with a peer.

Have students ask clarifying questions and critique one another’s responses.

Improve your justification based on the feedback from your peer. What other ways could you justify your response?• Students should incorporate inequalities and graphs as justification• Students may also substitute values to show true and false statements.

Repeat the process until satisfied with students’ justifications.



Grade 7 Module 3 Topic B: Topic Discourse Facilitation Slides

Lesson 11 Problem Set 5

Math Chat

Ask the question: Without solving, brainstorm the approach you would take to solve for the unknown angle measures. What approach would you take? Why?

• Answers will vary• ∠EAD and ∠HAC are vertical angles 90+20=2x+3x• ∠EAF, ∠FAD, and ∠DAH are angles on a line m∠DAH=70° ∠DAH, ∠HAB, and∠CAB are angles on a line 70+3x+2x=180

• Once the value of x is known, it must be substituted to find the measures of ∠HAB and ∠CAB.

How could you check for the reasonableness of your answers?• Answers may vary• Compare the measures to their approximate angle measure.• Compare the found measurements proportionally to other angles in the diagram.

How could you mathematically check your solutions?• Answers may vary• Sum angles on a line, or at a point, with the found measurements to confirm they sum to the

correct value.• Compare the value of vertical angles with found measurements to confirm equivalence.



G7M3Mid-ModuleAssessmentTask

Name___________________________________________________ Date____________________

1. Usetheexpressionbelowtoanswerparts(a)and(b).

4𝑥 − 3 𝑥 − 2𝑦 +12(6𝑥 − 8𝑦)

a. Writeanequivalentexpressioninstandardform,andcollectliketerms.

b. Expresstheanswerfrompart(a)asanequivalentexpressioninfactoredform.

2. Usetheinformationtosolvetheproblemsbelow.

a. Thelongestsideofatriangleis6moreunitsthantheshortestside.Thethirdsideistwicethelengthoftheshortestside.Iftheperimeterofthetriangleis25units,writeandsolveanequationtofindthelengthsofallthreesidesofthetriangle.



b. Thelengthofarectangleis(𝑥 + 3)incheslong,andthewidthis3 )Tinches.Iftheareais15 F

2Usquareinches,writeandsolveanequationtofindthelengthoftherectangle.

3. Apicture10 2Vfeetlongistobecenteredonawallthatis14 2

)feetlong.Howmuchspaceistherefrom

theedgeofthewalltothepicture?

a. Solvetheproblemarithmetically.



b. Solvetheproblemalgebraically.

c. Comparetheapproachesusedinparts(a)and(b).Explainhowtheyaresimilar.

4. InAugust,Corybeginsschoolshoppingforhistripletdaughters.

a. Oneday,hebought10pairsofsocksfor$2.50eachand3pairsofshoesfor𝑑dollarseach.Hespentatotalof$135.97.Writeandsolveanequationtofindthecostofonepairofshoes.



b. ThefollowingdayCoryreturnedtothestoretopurchasesomemoresocks.Hehad$40tospend.Whenhearrivedatthestore,theshoeswereonsalefor2

Foff.Whatisthegreatestnumberofpairs

ofsocksCorycanpurchaseifhepurchasesanotherpairofshoesinadditiontothesocks?

5. Benwantstohavehisbirthdayatthebowlingalleywithafewofhisfriends,buthecanspendnomorethan$80.Thebowlingalleychargesaflatfeeof$45foraprivatepartyand$5.50perpersonforshoerentalsandunlimitedbowling.

a. WriteaninequalitythatrepresentsthetotalcostofBen’sbirthdayfor𝑝peoplegivenhisbudget.



b. HowmanypeoplecanBenpayfor(includinghimself)whilestayingwithinthelimitationsofhisbudget?

c. Graphthesolutionoftheinequalityfrompart(a).

6. JennyinvitedGiannatogowatchamoviewithherfamily.Themovietheaterchargesoneratefor3Dadmissionandadifferentrateforregularadmission.JennyandGiannadecidedtowatchthenewestmoviein3D.Jenny’smother,father,andgrandfatheraccompaniedJenny’slittlebrothertotheregularadmissionmovie.

a. Writeanexpressionforthetotalcostofthetickets.Definethevariables.



b. Thecostofthe3Dticketwasdoublethecostoftheregularadmissionticket.Writeanequationtorepresenttherelationshipbetweenthetwotypesoftickets.

c. Thefamilypurchasedrefreshmentsandspentatotalof$18.50.Ifthetotalamountofmoneyspentonticketsandrefreshmentswas$94.50,useanequationtofindthecostofoneregularadmissionticket.



7. Thethreelinesshowninthediagrambelowintersectatthesamepoint.Themeasuresofsomeoftheanglesindegreesaregivenas3 𝑥 − 2 °, F

T𝑦 °,12°,42°.

a. Writeandsolveanequationthatcanbeusedtofindthevalueof𝑥.

b. Writeandsolveanequationthatcanbeusedtofindthevalueof𝑦.


Directions: While reading, annotate and consider your response to the following Guiding Question:How does this assessment item demonstrate the learning and application of concepts from Topic B?


G7M3StudentEdition

Lesson10:AngleProblemsandSolvingEquationsClasswork

AngleFactsandDefinitionsNameofAngleRelationship AngleFact Diagram

AdjacentAngles

VerticalAngles(vert.∠s)

AnglesonaLine(∠sonaline)

AnglesataPoint(∠satapoint)

C

A B Da° b°

a°b°

B

A

C

D A

D

C

B Ea°

b°c°

a°b°

B

A

C

Dc°

b°

A A

C

D

b°a°

E

D

F

G

C

a°b°

B

A

C

D A

D

C

B Ea°

b°c°

a°b°

B

A

C

Dc°

Directions: Please complete the following tasks with G7 M3 L101. Review the 'Classwork' (page 37)2. Complete the Opening Exercise (page 38)3. Complete all Examples and Exercises (page 39-42)



RelevantVocabulary(Note:RelevantVocabularyappearsattheendofthelessonintheStudentMaterials)

ADJACENTANGLES:Twoangles∠𝐵𝐴𝐶and∠𝐶𝐴𝐷withacommonside𝐴𝐶areadjacentanglesif𝐶belongstotheinteriorof∠𝐵𝐴𝐷.

VERTICALANGLES:Twoanglesareverticalangles(orverticallyoppositeangles)iftheirsidesformtwopairsofoppositerays.

ANGLESONALINE:Thesumofthemeasuresofadjacentanglesonalineis180°.

ANGLESATAPOINT:Thesumofthemeasuresofadjacentanglesatapointis360°.

OpeningExercise

Usethediagramtocompletethechart.

Nametheanglesthatare…

Vertical

Adjacent

Anglesonaline

Anglesatapoint



Example1

Estimatethemeasurementof𝑥. ________

Inacompletesentence,describetheanglerelationshipinthediagram.

Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝑥.Then,findthemeasuresof∠𝐵𝐴𝐶andconfirmyouranswersbymeasuringtheanglewithaprotractor.

Exercise1


Findthemeasurementsof∠𝐵𝐴𝐶and∠𝐷𝐴𝐸.

A

DC

B E3x° 2x°

b°

36o

90o



Example2


Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝑥and𝑦.Findthemeasurementsof∠𝐿𝐸𝐵and∠𝐾𝐸𝐵.

Exercise2

Inacompletesentence,describetheanglerelationshipsinthediagram.

Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝑥. 85o

16o

E

L

J

K

M

3x°

N16°

85°



Example3


Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝑥.Findthemeasurementof∠𝐸𝐾𝐹andconfirmyouranswersbymeasuringtheanglewithaprotractor.

Exercise3


Findthemeasurementof∠𝐺𝐴𝐻.

103o

E

(x+1)°

59°

F

G

H

A

103°

167°

59o

31 o

E

x°

F

G K135°

135o



Example4

Thefollowingtwolinesintersect.Theratioofthemeasurementsoftheobtuseangletotheacuteangleinanyadjacentanglepairinthisfigureis2 ∶ 1.Inacompletesentence,describetheanglerelationshipsinthediagram.

Labelthediagramwithexpressionsthatdescribethisrelationship.Writeanequationthatmodelstheanglerelationshipandsolvefor𝑥.Findthemeasurementsoftheacuteandobtuseangles.

Exercise4

Theratioof𝑚∠𝐺𝐹𝐻to𝑚∠𝐸𝐹𝐻is2 ∶ 3.Inacompletesentence,describetheanglerelationshipsinthediagram.

Findthemeasuresof∠𝐺𝐹𝐻and∠𝐸𝐹𝐻.



Directions: Read and annotate the Lessons 10 and 11 Learn Anywhere Plans with the Guiding Question in mind: How does the work in Lesson 10 lay the foundation for the work in Lesson 11?



Eureka Math in Sync™ Learn Anywhere Plan Grade 7 Module 3: Expressions and Equations Lesson 11: Angle Problems and Solving Equations

Student Outcomes • Students use vertical angles, adjacent

angles, angles on a line, and angles at apoint in a multi-step problem to write andsolve simple equations for an unknownangle in a figure.

Terminology adjacent angles angles at a point angles on a line vertical angles

Materials Protractor (optional)

Video Class Facilitation Description

• Solves for unknown angle measurementsand uses a protractor to confirm in theOpening Exercise

• Uses the relationship of vertically oppositeangles to solve for unknownmeasurement in Example 1

• Finds the measurement of multipleunknown angles in Exercise 1

• Writes equations to represent anglerelationships as determined from a diagram in Exercise 2

• Utilizes knowledge of angles at a point tosolve for the variable and then calculatethe measure of unknown angles inExample 3

• Solidifies understanding of angles at apoint in Exercise 3

Assignment • Problem Set 2, 5, 10

Additional Concept Development • n/a

Student Debrief • Which angle relationship was incorrectly

identified? Which angle relationships could beused to correctly solve for the unknown angles?(Error Analysis)

Closing n/a

Looking Ahead Lesson 12 introduces students to situations modeled mathematically using inequality symbols.



Anticipated Difficulty Must Do Remedial Problem Suggestion

The first problem of the lesson is too challenging.

Write a short sequence of problems on the board that provides a ladder to Problem 1. Direct students to complete those first problems to empower them to begin the lesson.

The jump in complexity between two problems is too big.

Provide a problem or set of problems that bridges student understanding between the two problems.

Students lack fluency or foundational skills necessary for the lesson.

Before beginning the lesson, do a quick, engaging fluency exercise, such as a Rapid Whiteboard Exchange or Sprint. Before beginning any fluency activity, assess whether students have conceptual understanding of the problems in the set and whether they will have success with the easiest problem in the set.

Students need more work at the concrete or pictorial level.

Provide manipulatives or the opportunity for students to draw solution strategies.

Students need more work at the abstract level.

Add a Rapid Whiteboard Exchange of abstract problems for students to complete toward the end of the lesson.

I will have my students warm up with some problems that are

foundational to the work of this lesson.

I think I’ll develop an RWBE for this skill so

students can get immediate feedback.

My students need more practice. I’ll have to

insert some extra problems in the

exercises.

Directions: While reading, annotate and consider your response to the following Guiding Question: 1. How can you help students access Must Do problems?



Lesson10:AngleProblemsandSolvingEquations

StudentOutcomes

§ Studentsuseverticalangles,adjacentangles,anglesonaline,andanglesatapointinamultistepproblemtowriteandsolvesimpleequationsforanunknownangleinafigure.

LessonNotes

InLessons10and11,studentsapplytheirunderstandingofequationstounknownangleproblems.Thegeometrytopicisanaturalcontextwithinwhichalgebraicskillsareapplied.Studentsunderstandthattheunknownangleisanactual,measureableangle;theysimplyneedtofindthevaluethatmakeseachequationtrue.TheysetuptheequationsbasedontheanglefactstheylearnedinGrade4.TheproblemspresentedarenotassimpleasinGrade4becausediagramsincorporateanglefactsincombination,ratherthaninisolation.Encouragestudentstoverifytheiranswersbymeasuringrelevantanglesineachdiagram―alldiagramsaredrawntoscale.

Classwork

Opening(5minutes)

Discussthewaysinwhichanglesarenamedandnotated.

§ Whatdoyounoticeaboutthethreefiguresonthenextpage?Whatisthesameaboutallthreefigures;whatisdifferent?

ú Therearethreeanglesthatappeartobethesamemeasurementbutarenotateddifferently.

§ Whatisalikelyimplicationofthethreedifferentkindsofnotation?

ú Theyindicatethedifferentwaysoflabelingoridentifyingtheangle.

StudentsarefamiliarwithaddressingFigure1as𝑏andhavingameasurementof𝑏°andaddressingFigure2asangle𝐴.ElicitthisfromstudentsandsaythatinacaselikeFigure1,theangleisnamedbythemeasureofthearc,andinacaselikeFigure2,theangleisnamedbythesingleletter.

§ InacaselikeFigure3,weusethreeletterswhenwenametheangle.Whyusethreepointstonameanangle?

ú Inafigurewhereseveralanglessharethesamevertex,namingaparticularanglebythevertexpointisnotsufficientinformationtodistinguishthatangle.Twoadditionalpoints,onebelongingtoeachsideoftheintendedangle,arenecessarytoidentifyit.

Encouragestudentstousebothmultipleformsofanglenotationinthetabletodemonstrateeachanglerelationship.



b°

A A

C

D

b°a°

E

D

F

G

C

C

A B Da° b°

Figure1 Figure2 Figure3

Namingbythearc. Namingbythevertex. Namingbythreepoints.

∠𝑏 ∠𝐴 ∠𝐶𝐴𝐷or∠𝐷𝐴𝐶

Recallthedefinitionsofadjacentandverticalandthefactsregardinganglesonalineandanglesatapoint.Ifanabbreviationexists,studentsshouldincludetheabbreviationoftheanglefactunderthenameofeachrelationship.IntheAngleFactcolumn,studentswritethedefinitionsandpracticethedifferentanglenotationswhendescribingtherelationshipintheanglefact.

Note:Theanglesonalinefactappliestotwoormoreangles.

AngleFactsandDefinitions

NameofAngleRelationship AngleFact Diagram

AdjacentAngles

Twoangles,∠𝑩𝑨𝑪and∠𝑪𝑨𝑫withacommonside𝑨𝑪,areadjacentanglesif𝑪belongstotheinteriorof∠𝑩𝑨𝑫.

Angles𝒂and𝒃areadjacentangles;∠𝑩𝑨𝑪and∠𝑪𝑨𝑫areadjacentangles.

VerticalAngles(vert.∠s)

Twoanglesareverticalangles(orverticallyoppositeangles)iftheirsidesformtwopairsofoppositerays.

𝒂 = 𝒃

𝒎∠𝑫𝑪𝑭 = 𝒎∠𝑮𝑪𝑬

AnglesonaLine(∠sonaline)

Thesumofthemeasuresoftwoanglesthatsharearayandformalineis𝟏𝟖𝟎°.

𝒂 + 𝒃 = 𝟏𝟖𝟎

𝒎∠𝑨𝑩𝑪 +𝒎∠𝑪𝑩𝑫 = 𝟏𝟖𝟎°

AnglesataPoint(∠satapoint)

Themeasureofallanglesformedbythreeormorerayswiththesamevertexis𝟑𝟔𝟎°.

𝒂 + 𝒃 + 𝒄 = 𝟑𝟔𝟎

𝒎∠𝑩𝑨𝑪 +𝒎∠𝑪𝑨𝑫 +𝒎∠𝑫𝑨𝑩 = 𝟑𝟔𝟎°

b°

A A

C

D

b°

A A

C

Db°

A A

C

D

a°b°

B

A

C

D A

D

C

B Ea°

b°c°

a°b°

B

A

C

Dc°

a°b°

B

A

C

D A

D

C

B Ea°

b°c°

a°b°

B

A

C

Dc°



OpeningExercise(4minutes)

OpeningExercise

Usethediagramtocompletethechart.

Example1(4minutes)

Studentsdescribetheanglerelationshipinthediagramandsetupandsolveanequationthatmodelsit.Havestudentsverifytheiranswersbymeasuringtheunknownanglewithaprotractor.

Example1

Estimatethemeasurementof𝒙.


∠𝑩𝑨𝑪and∠𝑪𝑨𝑫areanglesonalineandtheirmeasureshaveasumof𝟏𝟖𝟎°.

Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝒙.Then,findthemeasuresof∠𝑩𝑨𝑪andconfirmyouranswersbymeasuringtheanglewithaprotractor.

𝒙 + 𝟏𝟑𝟐 = 𝟏𝟖𝟎𝒙 + 𝟏𝟑𝟐 − 𝟏𝟑𝟐 = 𝟏𝟖𝟎 − 𝟏𝟑𝟐

𝒙 = 𝟒𝟖

𝒎∠𝑩𝑨𝑪 = 𝟒𝟖°

Nametheanglesthatare…

Vertical ∠𝑨𝑬𝑪and∠𝑩𝑬𝑫,∠𝑪𝑬𝑩and∠𝑫𝑬𝑨

AdjacentAnswersinclude:∠𝑨𝑬𝑪and∠𝑪𝑬𝑭∠𝑪𝑬𝑭and∠𝑭𝑬𝑩

AnglesonalineAnswersinclude:∠𝑩𝑬𝑫,∠𝑫𝑬𝑮,and∠𝑮𝑬𝑨∠𝑨𝑬𝑪,∠𝑪𝑬𝑭,and∠𝑭𝑬𝑩



Exercise1(4minutes)


Exercise1


∠𝑩𝑨𝑪,∠𝑪𝑨𝑫,and∠𝑫𝑨𝑬areanglesonalineandtheirmeasureshaveasumof𝟏𝟖𝟎°.

Findthemeasurementsof∠𝑩𝑨𝑪and∠𝑫𝑨𝑬.

𝟑𝒙 + 𝟗𝟎 + 𝟐𝒙 = 𝟏𝟖𝟎𝟓𝒙 + 𝟗𝟎 = 𝟏𝟖𝟎

𝟓𝒙 + 𝟗𝟎 − 𝟗𝟎 = 𝟏𝟖𝟎 − 𝟗𝟎𝟏𝟓(𝟓𝒙) =

𝟏𝟓(𝟗𝟎)

𝒙 = 𝟏𝟖

𝒎∠𝑩𝑨𝑪 = 𝟑 𝟏𝟖° = 𝟓𝟒°

𝒎∠𝑫𝑨𝑬 = 𝟐 𝟏𝟖° = 𝟑𝟔°

Example2(4minutes)


Example2


∠𝑨𝑬𝑳and∠𝑳𝑬𝑩areanglesonalineandtheirmeasureshaveasumof𝟏𝟖𝟎°.∠𝑨𝑬𝑳and∠𝑲𝑬𝑩areverticalanglesandareofequalmeasurement.

Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝒙and𝒚.Findthemeasurementsof∠𝑳𝑬𝑩and∠𝑲𝑬𝑩.

𝒚 = 𝟏𝟒𝟒°;𝒎∠𝑲𝑬𝑩 = 𝟏𝟒𝟒°(orvert.∠sare=)

𝒙 + 𝟏𝟒𝟒 = 𝟏𝟖𝟎𝒙 + 𝟏𝟒𝟒 − 𝟏𝟒𝟒 = 𝟏𝟖𝟎 − 𝟏𝟒𝟒

𝒙 = 𝟑𝟔

𝒎∠𝑳𝑬𝑩 = 𝟑𝟔°



Exercise2(4minutes)


Exercise2


∠𝑱𝑬𝑵and∠𝑵𝑬𝑴areadjacentanglesand,whenaddedtogether,arethemeasureof∠𝑱𝑬𝑴;∠𝑱𝑬𝑴and∠𝑲𝑬𝑳areverticalanglesandareofequalmeasurement.

Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝒙.

𝟑𝒙 + 𝟏𝟔 = 𝟖𝟓𝟑𝒙 + 𝟏𝟔 − 𝟏𝟔 = 𝟖𝟓 − 𝟏𝟔

𝟑𝒙 = 𝟔𝟗𝟏𝟑𝟑𝒙 = 𝟔𝟗

𝟏𝟑

𝒙 = 𝟐𝟑

Example3(4minutes)


Example3


∠𝑮𝑲𝑬,∠𝑬𝑲𝑭,and∠𝑮𝑲𝑭areanglesatapointandtheirmeasureshaveasumof𝟑𝟔𝟎°.

Writeanequationfortheanglerelationshipshowninthefigureandsolvefor𝒙.Findthemeasurementof∠𝑬𝑲𝑭andconfirmyouranswersbymeasuringtheanglewithaprotractor.

𝒙 + 𝟗𝟎 + 𝟏𝟑𝟓 = 𝟑𝟔𝟎𝒙 + 𝟐𝟐𝟓 = 𝟑𝟔𝟎

𝒙 + 𝟐𝟐𝟓 − 𝟐𝟐𝟓 = 𝟑𝟔𝟎 − 𝟐𝟐𝟓𝒙 = 𝟏𝟑𝟓

𝒎∠𝑬𝑲𝑭 = 𝟏𝟑𝟓°

85o

16o

E

L

J

K

M

3x°

N16°

85°

103o

E

(x+1)°

59°

F

G

H

A

103°

167°

59o

31 o

E

x°

F

G K135°

135o



Exercise3(4minutes)


Exercise3


∠𝑬𝑨𝑯,∠𝑮𝑨𝑯,∠𝑮𝑨𝑭,and∠𝑭𝑨𝑬areanglesatapointandtheirmeasuressumto𝟑𝟔𝟎°.

Findthemeasurementof∠𝑮𝑨𝑯.

𝒙 + 𝟏 + 𝟓𝟗 + 𝟏𝟎𝟑 + 𝟏𝟔𝟕 = 𝟑𝟔𝟎𝒙 + 𝟏 + 𝟓𝟗 + 𝟏𝟎𝟑 + 𝟏𝟔𝟕 = 𝟑𝟔𝟎

𝒙 = 𝟑𝟎

𝒎∠𝑮𝑨𝑯 = 𝟑𝟎 + 𝟏 ° = 𝟑𝟏°

Example4(5minutes)

§ Listpairsofangleswhosemeasurementsareinaratioof2 ∶ 1.ú Examplesinclude:90°and45°,60°and30°,150°and75°.

§ Whatdoesitmeanfortheratioofthemeasurementsoftwoanglestobe2 ∶ 1?ú Themeasurementofoneangleistwotimesthemeasureoftheotherangle.

Ifthesmallerangleisdefinedas𝑥°,thenthelargerangleis2𝑥°.Ifthelarger

angleisdefinedas𝑥°,thenthesmallerangleis2)𝑥°.

§ Basedonthefollowingfigure,whichanglerelationship(s)canbeutilizedtofindthemeasureofanobtuseandacuteangle?

ú Anyadjacentanglepairareonalineandhaveasumof180°.

Studentsdescribetheanglerelationshipinthequestionandsetupandsolveanequationthatmodelsit.Havestudentsverifytheiranswersbymeasuringtheunknownanglewithaprotractor.

Example4

Thefollowingtwolinesintersect.Theratioofthemeasurementsoftheobtuseangletotheacuteangleinanyadjacentanglepairinthisfigureis𝟐 ∶ 𝟏.Inacompletesentence,describetheanglerelationshipsinthediagram.

Themeasurementofanobtuseangleistwicethemeasurementofanacuteangleinthediagram.

Scaffolding:

Studentsmayfindithelpfultohighlightthepairsofequalverticalangles.

MP.8

15o 15o

60o

x°

E

G F

H

xx

2x°3x°

2x°



Labelthediagramwithexpressionsthatdescribethisrelationship.Writeanequationthatmodelstheanglerelationshipandsolvefor𝒙.Findthemeasurementsoftheacuteandobtuseangles.

𝟐𝒙 + 𝟏𝒙 = 𝟏𝟖𝟎𝟑𝒙 = 𝟏𝟖𝟎

𝟏𝟑

𝟑𝒙 =𝟏𝟑

𝟏𝟖𝟎

𝒙 = 𝟔𝟎

Acuteangle= 𝟔𝟎°

Obtuseangle= 𝟐𝒙° = 𝟐 𝟔𝟎° = 𝟏𝟐𝟎°

Exercise4(4minutes)


Theratioof𝒎∠𝑮𝑭𝑯to𝒎∠𝑬𝑭𝑯is𝟐 ∶ 𝟑.Inacompletesentence,describetheanglerelationshipsinthediagram.

Themeasurementof∠𝑮𝑭𝑯is𝟐𝟑themeasurementof∠𝑬𝑭𝑯;Themeasurementsof∠𝑮𝑭𝑯and∠𝑬𝑭𝑯haveasumof𝟗𝟎°.

Findthemeasuresof∠𝑮𝑭𝑯and∠𝑬𝑭𝑯.

𝟐𝒙 + 𝟑𝒙 = 𝟗𝟎𝟓𝒙 = 𝟗𝟎

𝟏𝟓

𝟓𝒙 =𝟏𝟓

𝟗𝟎

𝒙 = 𝟏𝟖

𝒎∠𝑮𝑭𝑯 = 𝟐 𝟏𝟖° = 𝟑𝟔°

𝒎∠𝑬𝑭𝑯 = 𝟑 𝟏𝟖° = 𝟓𝟒°

RelevantVocabulary

ADJACENTANGLES:Twoangles∠𝑩𝑨𝑪and∠𝑪𝑨𝑫withacommonside𝑨𝑪areadjacentanglesif𝑪belongstotheinteriorof∠𝑩𝑨𝑫.

VERTICALANGLES:Twoanglesareverticalangles(orverticallyoppositeangles)iftheirsidesformtwopairsofoppositerays.

ANGLESONALINE:Thesumofthemeasuresofadjacentanglesonalineis𝟏𝟖𝟎°.

ANGLESATAPOINT:Thesumofthemeasuresofadjacentanglesatapointis𝟑𝟔𝟎°.

ExitTicket(3minutes)



Name___________________________________________________ Date____________________

Lesson10:AngleProblemsandSolvingEquations

ExitTicket

Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Thatis,describetheanglerelationshipsyoucouldusetodeterminethevalueof𝑥.

Usetheanglerelationshipsdescribedabovetowriteanequationtosolvefor𝑥.Then,determinethemeasurementsof∠𝐽𝐴𝐻and∠𝐻𝐴𝐺.

45o

30o

E

30°

3x° 5x°

G

FJ

H

K

A



ExitTicketSampleSolutions

Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Thatis,describetheanglerelationshipsyoucouldusetodeterminethevalueof𝒙.

∠𝑲𝑨𝑬and∠𝑬𝑨𝑭areadjacentangleswhosemeasurementsareequalto∠𝑲𝑨𝑭;∠𝑲𝑨𝑭and∠𝑱𝑨𝑮areverticalanglesandareofequalmeasurement.

Usetheanglerelationshipsdescribedabovetowriteanequationtosolvefor𝒙.Then,determinethemeasurementsof∠𝑱𝑨𝑯and∠𝑯𝑨𝑮.

𝟓𝒙 + 𝟑𝒙 = 𝟗𝟎 + 𝟑𝟎𝟖𝒙 = 𝟏𝟐𝟎

𝟏𝟖

𝟖𝒙 =𝟏𝟖(𝟏𝟐𝟎)

𝒙 = 𝟏𝟓

𝒎∠𝑱𝑨𝑯 = 𝟑 𝟏𝟓° = 𝟒𝟓°

𝒎∠𝑯𝑨𝑮 = 𝟓 𝟏𝟓° = 𝟕𝟓°

ProblemSetSampleSolutions

Foreachquestion,useanglerelationshipstowriteanequationinordertosolveforeachvariable.Determinetheindicatedangles.Youcancheckyouranswersbymeasuringeachanglewithaprotractor.

1. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasurementof∠𝑫𝑨𝑬.

Onepossibleresponse:∠𝑪𝑨𝑫,∠𝑫𝑨𝑬,and∠𝑭𝑨𝑬areanglesonalineandtheirmeasuressumto𝟏𝟖𝟎°.

𝟎 + 𝒙 + 𝟔𝟓 = 𝟏𝟖𝟎𝒙 + 𝟏𝟓𝟓 = 𝟏𝟖𝟎

𝒙 + 𝟏𝟓𝟓 − 𝟏𝟓𝟓 = 𝟏𝟖𝟎 − 𝟏𝟓𝟓𝒙 = 𝟐𝟓

𝒎∠𝑫𝑨𝑬 = 𝟐𝟓°

45o

30o

E

30°

3x° 5x°

G

FJ

H

K

A



2. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasurementof∠𝑸𝑷𝑹.

∠𝑸𝑷𝑹,∠𝑹𝑷𝑺,and∠𝑺𝑷𝑻areanglesonalineandtheirmeasuressumto𝟏𝟖𝟎°.

𝒇 + 𝟏𝟓𝟒 + 𝒇 = 𝟏𝟖𝟎𝟐𝒇 + 𝟏𝟓𝟒 = 𝟏𝟖𝟎

𝟐𝒇 + 𝟏𝟓𝟒 − 𝟏𝟓𝟒 = 𝟏𝟖𝟎 − 𝟏𝟓𝟒𝟐𝒇 = 𝟐𝟔

𝟏𝟐𝟐𝒇 =

𝟏𝟐𝟐𝟔

𝒇 = 𝟏𝟑

𝒎∠𝑸𝑷𝑹 = 𝟏𝟑°

3. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasurementsof∠𝑪𝑸𝑫and∠𝑬𝑸𝑭.

∠𝑩𝑸𝑪,∠𝑪𝑸𝑫,∠𝑫𝑸𝑬,∠𝑬𝑸𝑭,and∠𝑭𝑸𝑮areanglesonalineandtheirmeasuressumto𝟏𝟖𝟎°.

𝟏𝟎 + 𝟐𝒙 + 𝟏𝟎𝟑 + 𝟑𝒙 + 𝟏𝟐 = 𝟏𝟖𝟎𝟓𝒙 + 𝟏𝟐𝟓 = 𝟏𝟖𝟎

𝟓𝒙 + 𝟏𝟐𝟓 − 𝟏𝟐𝟓 = 𝟏𝟖𝟎 − 𝟏𝟐𝟓𝟓𝒙 = 𝟓𝟓

𝟏𝟓𝟓𝒙 =

𝟏𝟓𝟓𝟓

𝒙 = 𝟏𝟏

𝒎∠𝑪𝑸𝑫 = 𝟐 𝟏𝟏° = 𝟐𝟐°

𝒎∠𝑬𝑸𝑭 = 𝟑 𝟏𝟏° = 𝟑𝟑°

4. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasureof𝒙.

Alloftheanglesinthediagramareanglesatapointandtheirmeasuressumto𝟑𝟔𝟎°.

𝟒 𝒙 + 𝟕𝟏 = 𝟑𝟔𝟎𝟒𝒙 + 𝟐𝟖𝟒 = 𝟑𝟔𝟎

𝟒𝒙 + 𝟐𝟖𝟒 − 𝟐𝟖𝟒 = 𝟑𝟔𝟎 − 𝟐𝟖𝟒𝟒𝒙 = 𝟕𝟔

𝟏𝟒𝟒𝒙 =

𝟏𝟒𝟕𝟔

𝒙 = 𝟏𝟗



5. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasuresof𝒙and𝒚.

∠𝑪𝑲𝑬,∠𝑬𝑲𝑫,and∠𝑫𝑲𝑩areanglesonalineandtheirmeasuressumto𝟏𝟖𝟎°.Since∠𝑭𝑲𝑨and∠𝑨𝑲𝑬formastraightangleandthemeasurementof∠𝑭𝑲𝑨is𝟗𝟎°,∠𝑨𝑲𝑬is𝟗𝟎°,making∠𝑪𝑲𝑬and∠𝑨𝑲𝑪formarightangleandtheirmeasureshaveasumof𝟗𝟎°.

𝒙 + 𝟐𝟓 + 𝟗𝟎 = 𝟏𝟖𝟎𝒙 + 𝟏𝟏𝟓 = 𝟏𝟖𝟎

𝒙 + 𝟏𝟏𝟓 − 𝟏𝟏𝟓 = 𝟏𝟖𝟎 − 𝟏𝟏𝟓𝒙 = 𝟔𝟓

(𝟔𝟓) + 𝒚 = 𝟗𝟎𝟔𝟓 − 𝟔𝟓 + 𝒚 = 𝟗𝟎 − 𝟔𝟓

𝒚 = 𝟐𝟓

6. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasuresof𝒙and𝒚.

∠𝑬𝑨𝑮and∠𝑭𝑨𝑲areverticalanglesandareofequalmeasurement.∠𝑬𝑨𝑮and∠𝑮𝑨𝑫formarightangleandtheirmeasureshaveasumof𝟗𝟎°.

𝟐𝒙 + 𝟐𝟒 = 𝟗𝟎𝟐𝒙 + 𝟐𝟒 − 𝟐𝟒 = 𝟗𝟎 − 𝟐𝟒

𝟐𝒙 = 𝟔𝟔𝟏𝟐𝟐𝒙 =

𝟏𝟐𝟔𝟔

𝒙 = 𝟑𝟑

𝟑𝒚 = 𝟔𝟔𝟏𝟑𝟑𝒚 =

𝟏𝟑𝟔𝟔

𝒚 = 𝟐𝟐

7. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasuresof∠𝑪𝑨𝑫and∠𝑫𝑨𝑬.

∠𝑪𝑨𝑫and∠𝑫𝑨𝑬formarightangleandtheirmeasureshaveasumof𝟗𝟎°.

𝟑𝟐𝒙 + 𝟐𝟎 + 𝟐𝒙 = 𝟗𝟎

𝟕𝟐𝒙 + 𝟐𝟎 = 𝟗𝟎

𝟕𝟐𝒙 + 𝟐𝟎 − 𝟐𝟎 = 𝟗𝟎 − 𝟐𝟎

𝟕𝟐𝒙 = 𝟕𝟎

𝟐𝟕𝟕𝟐𝒙 = 𝟕𝟎

𝟐𝟕

𝒙 = 𝟐𝟎

𝒎∠𝑪𝑨𝑫 =𝟑𝟐𝟐𝟎° + 𝟐𝟎° = 𝟓𝟎°

𝒎∠𝑫𝑨𝑬 = 𝟐 𝟐𝟎° = 𝟒𝟎°

90o

KE F

D

25°

25ox°

y°

B

C A

24o

10o

DAC

E

F

24°x°

x°15o

y°y°y°

33o

22o

K

H

G

F

J



8. Inacompletesentence,describetherelevantanglerelationshipsinthefollowingdiagram.Findthemeasureof∠𝑪𝑸𝑮.

∠𝑫𝑸𝑬and∠𝑪𝑸𝑭areverticalanglesandareofequalmeasurement.∠𝑪𝑸𝑮and∠𝑮𝑸𝑭areadjacentanglesandtheirmeasuressumtothemeasureof∠𝑪𝑸𝑭.

𝟑𝒙 + 𝟓𝟔 = 𝟏𝟓𝟓𝟑𝒙 + 𝟓𝟔 − 𝟓𝟔 = 𝟏𝟓𝟓 − 𝟓𝟔

𝟑𝒙 = 𝟗𝟗𝟏𝟑𝟑𝒙 =

𝟏𝟑𝟗𝟗

𝒙 = 𝟑𝟑

𝒎∠𝑪𝑸𝑮 = 𝟑 𝟑𝟑° = 𝟗𝟗°

9. Theratioofthemeasuresofapairofadjacentanglesonalineis𝟒 ∶ 𝟓.a. Findthemeasuresofthetwoangles.

∠𝟏 = 𝟒𝒙,∠𝟐 = 𝟓𝒙

𝟒𝒙 + 𝟓𝒙 = 𝟏𝟖𝟎𝟗𝒙 = 𝟏𝟖𝟎

𝟏𝟗𝟗𝒙 =

𝟏𝟗𝟏𝟖𝟎

𝒙 = 𝟐𝟎

∠𝟏 = 𝟒 𝟐𝟎° = 𝟖𝟎°

∠𝟐 = 𝟓(𝟐𝟎°) = 𝟏𝟎𝟎°

b. Drawadiagramtoscaleoftheseadjacentangles.Indicatethemeasurementsofeachangle.

10. Theratioofthemeasuresofthreeadjacentanglesonalineis𝟑 ∶ 𝟒 ∶ 𝟓.a. Findthemeasuresofthethreeangles.

∠𝟏 = 𝟑𝒙,∠𝟐 = 𝟒𝒙,∠𝟑 = 𝟓𝒙

𝟑𝒙 + 𝟒𝒙 + 𝟓𝒙 = 𝟏𝟖𝟎𝟏𝟐𝒙 = 𝟏𝟖𝟎

𝟏𝟏𝟐

𝟏𝟐𝒙 =𝟏𝟏𝟐

𝟏𝟖𝟎

𝒙 = 𝟏𝟓

∠𝟏 = 𝟑 𝟏𝟓° = 𝟒𝟓°

∠𝟐 =𝟒 𝟏𝟓° = 𝟔𝟎°

∠𝟑 = 𝟓 𝟏𝟓° = 𝟕𝟓°

b. Drawadiagramtoscaleoftheseadjacentangles.Indicatethemeasurementsofeachangle.

56o

96°

25°40°

3x°

y°

50o

56°

155°C D

E

F

G

Q



Grade 7 Module 3 L10: Topic Discourse Student Debrief Slide

Lesson 10 Problem Set 8

Sample Solution

How would you describe to a friend or family member the approach that this student took to solve for the missing angles?• Answers will vary• Used adjacent angles to establish m∠CQG+m∠GQF=3x+56.• Used vertical angles to establish m∠CQG+m∠GQF=m∠DQE. Therefore, 3x+56=155.

Were there any steps which this student took that are correct, but not entirely necessary to find the solution?• Answers will vary• Some students may note combining algebraic steps (not showing all work).• Once it is established that 3x=99 the measure of ∠CQG is found. The remaining steps are

unnecessary (but not incorrect).



Grade 7 Module 3 L11: Topic Discourse Student Debrief Slide

Lesson 11

Error Analysis

What angle relationship was incorrectly identified here?

What angle relationships could be used to correctly solve for the unknown angles?

• Answers may vary• ∠EAD is vertically opposite, and equal in measure, to ∠HAC• ∠EAF, ∠FAD and ∠DAH are angles on a line which sum to 180 degrees. ∠DAH,∠HAB and ∠BAC are angles on a line which sum to 180 degrees.

• ∠C AB and ∠FAD are not vertical angles, so 2x and 20 are not equal



G7 M3 L10 Problem Set

For each question, use angle relationships to write an equation in order to solve for each variable. Determine the indicated angles. You can check your answers by measuring each angle with a protractor.

1. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementof ∠𝐷𝐷𝐵𝐵𝐷𝐷.

2. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementof ∠𝑄𝑄𝑄𝑄𝑄𝑄.

3. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementsof ∠𝐵𝐵𝑄𝑄𝐷𝐷 and ∠𝐷𝐷𝑄𝑄𝐸𝐸.

4. In a complete sentence, describe the relevant angle relationships in thefollowing diagram. Find the measure of 𝑥𝑥.



http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US


5. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of 𝑥𝑥and 𝑦𝑦.

6. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of 𝑥𝑥and 𝑦𝑦.

7. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measuresof ∠𝐵𝐵𝐵𝐵𝐷𝐷 and ∠𝐷𝐷𝐵𝐵𝐷𝐷.





8. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of∠𝐵𝐵𝑄𝑄𝐺𝐺.

9. The ratio of the measures of a pair of adjacent angles on a line is 4 ∶ 5.

a. Find the measures of the two angles.

b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.

10. The ratio of the measures of three adjacent angles on a line is 3 ∶ 4 ∶ 5.

a. Find the measures of the three angles.

b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.





G7 M3 L11 Problem Set

In a complete sentence, describe the angle relationships in each diagram. Write an equation for the angle relationship(s) shown in the figure, and solve for the indicated unknown angle. You can check your answers by measuring each angle with a protractor.

1. Find the measures of ∠𝐸𝐸𝐶𝐶𝐸𝐸, ∠𝐶𝐶𝐶𝐶𝐸𝐸, and ∠𝐶𝐶𝐶𝐶𝐶𝐶.

2. Find the measure of 𝑎𝑎.

3. Find the measures of 𝑥𝑥 and 𝑦𝑦.





4. Find the measure of ∠𝐽𝐽𝐶𝐶𝐽𝐽.

5. Find the measures of ∠𝐽𝐽𝐶𝐶𝐸𝐸 and ∠𝐶𝐶𝐶𝐶𝐸𝐸.

6. The measure of ∠𝑆𝑆𝐸𝐸𝑆𝑆 is 𝑏𝑏°. The measure of ∠𝑆𝑆𝐸𝐸𝑇𝑇 is five more than two times ∠𝑆𝑆𝐸𝐸𝑆𝑆. The measure of ∠𝑄𝑄𝐸𝐸𝑆𝑆 is twelve less than eight times the measure of ∠𝑆𝑆𝐸𝐸𝑆𝑆. Find the measures of ∠𝑆𝑆𝐸𝐸𝑆𝑆, ∠𝑆𝑆𝐸𝐸𝑇𝑇, and ∠𝑄𝑄𝐸𝐸𝑆𝑆.





7. Find the measures of ∠𝐽𝐽𝑄𝑄𝐸𝐸 and ∠𝐶𝐶𝑄𝑄𝐺𝐺.

8. The measures of three angles at a point are in the ratio of 2 ∶ 3 ∶ 5. Find the measures of the angles.

9. The sum of the measures of two adjacent angles is 72°. The ratio of the smaller angle to the larger angle is 1 ∶ 3.Find the measures of each angle.

10. Find the measures of ∠𝐶𝐶𝑄𝑄𝐶𝐶 and ∠𝐸𝐸𝑄𝑄𝐸𝐸.





Professional Reading 2

Zuanyan Jiaocai — “Intensive Textbook Study” (Ma 1999)

To study teaching materials is extremely important. To study teaching materials is to study what we

are to teach and how to teach it to our students; in other words, to find links between the knowledge

and the students. The student teachers from normal schools doing their student teaching with me

usually can’t understand why we spend so much time studying teaching materials and what we can

learn from studying. For them, it seems to be too simple and too plain to study: there are just

several example problems, one of which you can solve in a minute and explain to students in two

minutes. But I told them that even after teaching for more than thirty years, every time I study a

textbook I see something new. How to inspire students’ minds, how to explain in a clear way, how to

spend less time and let students benefit more, how to motivate students to learn these topics. …

Your answers for all these questions are supported by a deep and broad understanding of what the

teaching material is about. And every time you study it, you get a better idea of what it is and how to

teach it. You never will feel you have nothing more to learn from studying teaching materials

(Teacher Mao as quoted in Ma 1999, 135).

Probably anyone who knows Chinese and English would translate the Chinese term jiaocai as

“teaching materials” because jiao literally means “teaching” and cai means “materials.” But I would

say that in fact jiaocai is more like what “curriculum” means in the United States. Generally, when

Chinese teachers refer to zuanyan jiaocai, the term consists of three main components—the

[standards] (jiaoxue dagang), textbooks (keben), and teacher’s manuals (beike fudao cailiao) (130).



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Preparation and Customization of a Lesson using Eureka in