Lesson 2 Mathematics Gymnastics M3-31 Answer Key Carnegie

Unformatted text preview: Middle School Math Solution Course 2 Student Edition Sandy Bartle Finocchi and Amy Jones Lewis with Kelly Edenfield and Josh Fisher C02_SE_FM.indd FM-1 1/17/19 4:33 PM 501 Grant St., Suite 1075 Pittsburgh, PA 15219 Phone 888.851.7094 Customer Service Phone 412.690.2444 Fax 412.690.2444 Cover Design by Anne Milliron Copyright © 2018–2020 by Carnegie Learning, Inc. All rights reserved. Carnegie Learning and MATHia are registered marks of Carnegie Learning, Inc. All other company and product names mentioned are used for identification purposes only and may be trademarks of their respective owners. Permission is granted for photocopying rights within licensed sites only. Any other usage or reproduction in any form is prohibited without the expressed consent of the publisher. ISBN: 978-1-60972-890-8 Student Edition Printed in the United States of America 1 2 3 4 5 6 7 8 9 CC 21 20 19 18 17 C02_SE_FM.indd FM-2 1/18/19 8:57 PM Manifesto Manifesto • FM-3 C02_SE_FM.indd FM-3 1/17/19 4:33 PM Acknowledgments Middle School Math Solution Authors ● Sandy Bartle Finocchi, Senior Academic Officer ● Amy Jones Lewis, Director of Instructional Design ● Kelly Edenfield, Instructional Designer ● Josh Fisher, Instructional Designer Foundation Authors (2010) ● William S. Hadley, Algebra and Proportional Reasoning ● Mary Lou Metz, Data Analysis and Probability ● Mary Lynn Raith, Number and Operations ● Janet Sinopoli, Algebra ● Jaclyn Snyder, Geometry and Measurement Vendors ● Lumina Datamatics, Ltd. ● Cenveo Publisher Services, Inc. Images ● Special Thanks ● Alison Huettner for project management and editorial review. ● Jacyln Snyder and Janet Sinopoli for their contributions to the Teacher's Implementation Guide facilitation notes. ● Victoria Fisher for her review of content and contributions to all the ancillary materials. ● Valerie Muller for her contributions and review of content. ● The members of Carnegie Learning's Cognitive Scientist Team—Brendon Towle, John Connelly, Bob Hausmann, Chas Murray, and Martina Pavelko—for their insight in learning science and review of content. ● Bob Hausmann for his contributions to the Family Guide. ● John Jorgenson, Chief Marketing Officer, for all his insight and messaging. ● Carnegie Learning's Education Services Team for content review and providing customer feedback. ● In Memory of David Dengler, Director of Curriculum Development (Deceased), who made substantial contributions to conceptualizing Carnegie Learning's middle school software. FM-4 • Acknowledgements C02_SE_FM.indd FM-4 1/17/19 4:33 PM Mathematics is so much more than memorizing rules. It is learning to reason, to make connections, and to make sense of the world. We believe in Learning by Doing(TM)—you need to actively engage with the content if you are to benefit from it. The lessons were designed to take you from your intuitive understanding of the world and build on your prior experiences to then learn new concepts. My hope is that these instructional materials help you build a deep understanding of math. Sandy Bartle Finocchi, Senior Academic Officer My hope is that as you work through this course, you feel capable—capable of exploring new ideas that build upon what you already know, capable of struggling through challenging problems, capable of thinking creatively about how to fix mistakes, and capable of thinking like a mathematician. Amy Jones Lewis, Director of Instructional Design At Carnegie Learning we have created an organization whose mission and culture is defined by your success. Our passion is creating products that make sense of the world of mathematics and ignite a passion in you. Our hope is that you will enjoy our resources as much as we enjoyed creating them. Barry Malkin, CEO Acknowledgements • FM-5 C02_SE_FM.indd FM-5 1/17/19 4:33 PM Table of Contents Module 1: Thinking Proportionally Topic 1: Circles and Ratio 1.1 Pi: The Ultimate Ratio Exploring the Ratio of Circle Circumference to Diameter ................................M1-7 1.2 That's a Spicy Pizza! Area of Circles.................................................................................................M1-19 1.3 Circular Reasoning Solving Area and Circumference Problems ....................................................M1-33 Topic 2: Fractional Rates 2.1 Making Punch Unit Rate Representations ..............................................................................M1-51 2.2 Eggzactly! Solving Problems with Ratios of Fractions ......................................................M1-59 2.3 Tagging Sharks Solving Proportions Using Means and Extremes ............................................M1-69 Topic 3: Proportionality 3.1 How Does Your Garden Grow? Proportional Relationships ..............................................................................M1-91 3.2 Complying with Title IX Constant of Proportionality ...........................................................................M1-109 3.3 Fish-Inches Identifying the Constant of Proportionality in Graphs ..................................M1-127 3.4 Minding Your Ps and Qs Constant of Proportionality in Multiple Representations ..............................M1-139 Topic 4: Proportional Relationships 4.1 Markups and Markdowns Introducing Proportions to Solve Percent Problems .....................................M1-161 4.2 Perks of Work Calculating Tips, Commissions, and Simple Interest ....................................M1-177 4.3 No Taxation Without Calculation Sales Tax, Income Tax, and Fees...................................................................M1-197 FM-6 • Table of Contents C02_SE_FM.indd FM-6 1/17/19 4:33 PM 4.4 More Ups and Downs Percent Increase and Percent Decrease .......................................................M1-209 4.5 Pound for Pound, Inch for Inch Scale and Scale Drawings .............................................................................M1-223 Module 2: Operating with Signed Numbers Topic 1: Adding and Subtracting Rational Numbers 1.1 Math Football Using Models to Understand Integer Addition ................................................M2-7 1.2 Walk the Line Adding Integers, Part I ....................................................................................M2-17 1.3 Two-Color Counters Adding Integers, Part II ...................................................................................M2-31 1.4 What's the Difference? Subtracting Integers .......................................................................................M2-49 1.5 All Mixed Up Adding and Subtracting Rational Numbers ....................................................M2-69 Topic 2: Multiplying and Dividing Rational Numbers 2.1 Equal Groups Multiplying and Dividing Integers ..................................................................M2-89 2.2 Be Rational! Quotients of Integers ....................................................................................M2-103 2.3 Building a Wright Brothers' Flyer Simplifying Expressions to Solve Problems ..................................................M2-113 2.4 Properties Schmoperties Using Number Properties to Interpret Expressions with Signed Numbers...M2-125 Module 3: Reasoning Algebraically Topic 1: Algebraic Expressions 1.1 No Substitute for Hard Work Evaluating Algebraic Expressions .....................................................................M3-7 1.2 Mathematics Gymnastics Rewriting Expressions Using the Distributive Property ...................................M3-19 1.3 All My Xs Combining Like Terms ....................................................................................M3-33 Table of Contents • FM-7 C02_SE_FM.indd FM-7 1/17/19 4:33 PM Topic 2: Two-Step Equations and Inequalities 2.1 Picture Algebra Modeling Equations as Equal Expressions .....................................................M3-53 2.2 Expressions That Play Together… Solving Equations on a Double Number Line ................................................M3-65 2.3 Formally Yours Using Inverse Operations to Solve Equations ................................................M3-77 2.4 Be Greater Than Solving Inequalities with Inverse Operations ..................................................M3-95 Topic 3: Multiple Representations of Equations 3.1 Put It on the Plane Representing Equations with Tables and Graphs .........................................M3-125 3.2 Stretches, Stacks, and Structure Structure of Linear Equations........................................................................M3-139 3.3 Deep Flight I Building Inequalities and Equations to Solve Problems ...............................M3-155 3.4 Texas Tea and Temperature Using Multiple Representations to Solve Problems ......................................M3-169 Module 4: Analyzing Populations and Probabilities Topic 1: Introduction to Probability 1.1 Rolling, Rolling, Rolling… Defining and Representing Probability .............................................................M4-7 1.2 Give the Models a Chance Probability Models ..........................................................................................M4-23 1.3 Toss the Cup Determining Experimental Probability of Simple Events ................................M4-33 1.4 A Simulating Conversation Simulating Simple Experiments ......................................................................M4-47 Topic 2: Compound Probability 2.1 Evens or Odds? Using Arrays to Organize Outcomes ..............................................................M4-73 2.2 Three Girls and No Boys? Using Tree Diagrams .......................................................................................M4-89 FM-8 • Table of Contents C02_SE_FM.indd FM-8 1/17/19 4:33 PM 2.3 Pet Shop Probability Determining Compound Probability ............................................................M4-101 2.4 On a Hot Streak Simulating Probability of Compound Events ................................................M4-113 Topic 3: Drawing Inferences 3.1 We Want to Hear From You! Collecting Random Samples.........................................................................M4-133 3.2 Tiles, Gumballs, and Pumpkins Using Random Samples to Draw Inferences .................................................M4-151 3.3 Spicy or Dark? Comparing Two Populations.........................................................................M4-169 3.4 Finding Your Spot to Live Using Random Samples from Two Populations to Draw Conclusions ..........M4-181 Module 5: Constructing and Measuring Topic 1: Angles and Triangles 1.1 Here's Lookin' at Euclid Geometric Constructions ..................................................................................M5-7 1.2 Special Delivery Special Angle Relationships ............................................................................M5-19 1.3 Consider Every Side Constructing Triangles Given Sides ................................................................M5-39 1.4 Unique or Not? Constructing Triangles Given Angles..............................................................M5-53 Topic 2: Three-Dimensional Figures 2.1 Slicing and Dicing Cross-Sections of Rectangular Prisms .............................................................M5-75 2.2 Dissecting a Pyramid Cross-Sections of Rectangular Pyramids.........................................................M5-97 2.3 Hey, Mister, Got Some Bird Seed? Volume of Pyramids ......................................................................................M5-107 2.4 The Sound of Surface Area Surface Area of Pyramids ..............................................................................M5-129 2.5 More Than Four Sides of the Story Volume and Surface Area of Prisms and Pyramids .......................................M5-143 Glossary .................................................................................................................. G-1 Index ........................................................................................................................... I-1 Table of Contents • FM-9 C02_SE_FM.indd FM-9 1/17/19 4:33 PM Lesson Structure 1. Learning Goals Learning goals are stated for each lesson to help you take ownership of the learning objectives. 2 Stretches, Stacks, and Structure 2. Connection Each lesson begins with a statement connecting what you have learned with a question to ponder. Return to this question at the end of this lesson to gauge your understanding. Structure of Linear Equations 1 WARM UP LEARNING GOALS Use properties to rewrite. • Write and solve two-step equations. • Compare two linear problem situations. • Rewrite expressions in different forms in problem contexts in order to interpret how quantities are related. • Compare graphs of linear problem situations. • Compare and interpret forms of linear equations. 1. 3(x 2 1) 2. 29(22 1 x) 1 3. __ (x 2 6) 2 4. 6 1 3(x 1 4) 2 All of the linear equations you have written for problem situations have been in the form y 5 ax 1 b. Are there other common forms of equations used to express linear problem situations? LESSON 2: Stretches, Stacks, and Structure • M3-139 AL7_SE_M03_T03_L02.indd 139 1/8/19 10:08 PM FM-10 • Lesson Structure C02_SE_FM.indd FM-10 1/17/19 4:33 PM 3. Getting Started 3 Each lesson begins with a Getting Started. When working on the Getting Started, use what you know about the world, what you have learned previously, or your intuition. The goal is just to get you thinking and ready for what's to come. Getting Started Learning the Limo Business Katie is starting her own limousine rental company. She wisely decides to check her competitors' pricing plans before setting her own plan. The table shows the fees from two rival limousine rental companies. Examine the fee schedule for the two limousine companies provided in the table. Number of Hours Rented Limousines by Lilly Fees (in dollars) Transportation with Class Fees (in dollars) 1 99.99 89.99 2 123.74 126.54 3 147.49 163.09 4 171.24 199.64 5 194.99 236.19 1. Which company would you choose if you were renting a limousine? Support your answer with information from the table. M3-140 • TOPIC 3: Multiple Representations of Equations AL7_SE_M03_T03_L02.indd 140 1/8/19 10:08 PM Lesson Structure • FM-11 C02_SE_FM.indd FM-11 1/17/19 4:33 PM 4. Activities You are going to build a deep understanding of mathematics through a variety of activities in an environment where collaboration and conversations are important and expected. You will learn how to solve new problems, but you will also learn why those strategies work and how they are connected to other strategies you already know. Remember: • • • It's not just about answer-getting. The process is important. Making mistakes is a critical part of learning, so take risks. There is often more than one way to solve a problem. Activities may include real-world problems, sorting activities, worked examples, or analyzing sample student work. 4 AC T I V I T Y 2.1 Different Forms, Same Equation Katie starts by analyzing the cost structure of Limousines by Lilly. 1. Consider the cost of renting a limousine from Limousines by Lilly. T I V I TLimousines Y a. What does the first hour of a rentalAC from Comparing by Lilly cost? 2.2 Graphs of Linear Equations b. What does each additional rental Your hourjob cost atfrom Storage Pros is to create new boxes to ship the Limousines by Lilly after the first hour? company's plastic containers. Storage Pros makes all different shapes and sizes of plastic containers. To ship the containers, the lids are removed, allowing the containers to be stacked. Storage Pros wants to design its shipping boxes so that they will hold two AC T I V I T Y c. What would it cost to rent a limo dozen from Limousines byplastic Lilly containers Interpreting stacks of the without lids in stacksForms of two for 10 hours? Explain your reasoning. Do you an of Equations dozen, regardless of the size or shape ofhave the container. 2.3 estimate for thefrom costmeasuring to The table shows the data gathered the heights of rent from containers. Lilly different-sized stacks of the various plastic In the limousine and container scenarios, you represented the for 10 hours? situations with two different equations. Stack Height Number of (centimeters) 1. Complete the table to summarize the different forms of the Containers Round Square x and y for the independent and equations. Use the variables dependent variables. 9 15 d. What would it cost to rent a limo from Limousines by1Lilly for 13 hours? Explain your reasoning. 2 9.8 15.4 Limousines by Lilly y 5 ax 1 b 10.6 15.8 3 4 Transportation with Class e. Explain how you calculated each cost. Containers Round 5 Square Containers 11.4 y 5 c 1 d(x 2 1) y 5 99. 99 1 23.75(x 2 1) 16.2 y 5 0.8x 1 8.2 12.2 16.6 6 7 13 2. Use your equations to explain the meaning of the c and d terms in y 5 c 1 d(x 2 1). LESSON 2: Stretches, Stacks, and Structure • M3-141 1. What are the variable quantities in this problem situation? AL7_SE_M03_T03_L02.indd 141 1/8/19 10:08 PM 2. What quantity depends on the other? 3. Use your equations to explain the meaning of the a and b terms in y 5 ax 1 b. Any letter can be used as a variable. It is common to use of Equations M3-144 • TOPIC 3: Multiple Representations a and b in forms of equations, but the different variables AL7_SE_M03_T03_L02.indd 144 were used to reduce 1/8/19 10:08 PM the possibility of confusing the equations. M3-148 • TOPIC 3: Multiple Representations of Equations AL7_SE_M03_T03_L02.indd 148 1/8/19 10:08 PM Be prepared to share your solutions and methods with your classmates. FM-12 • Lesson Structure C02_SE_FM.indd FM-12 1/17/19 4:33 PM 5. Talk the Talk NOTES 5 Talk the Talk gives you an opportunity to reflect on the main ideas of the lesson. TALK the TALK Back to the Limos! • At the beginning of the lesson, you wrote equations for the fee schedule of Limousines by Lilly and Transportation with Class. Be honest with yourself. • Ask questions to clarify anything you don't understand. • Show what you know! Total Cost (in dollars) 1. Determine which graph represents each equation. Use your equations to explain your reasoning. Don't forget to revisit the question posed on the lesson opening page to gauge your understanding. m n Number of Hours 2. Suppose Katie decides to charge $124.99 for the first three hours and then $49.99 for each additional hour. Write an equation to represent Katie's fee schedule. M3-150 • TOPIC 3: Multiple Representations of Equations AL7_SE_M03_T03_L02.indd 150 1/8/19 10:08 PM Lesson Structure • FM-13 C02_SE_FM.indd FM-13 1/17/19 4:33 PM Assignment 6. Write Reflect on your work and clarify your thinking. Assignment 6 7 Write Remember Write a problem situation that Different forms of an equation reveal different information about a 7. Remember could be modeled by a linear problem situation and about other representations of the problem equation in x and y that includes situation. Take note of the key concepts from the lesson. the expression x 2 c, where c is a positive integer. 8 Practice Write an equation to represent each situation. Define your variables and solve the equation. 1. At the Namaste Yoga Studio, the first two yoga classes are free with a registration fee of $15. Each class 8. Practice Use the concepts learned in the lesson to solve problems. after that is $45. How many classes can you take for $1185? 2. Clara has a coupon for $10 off at her favorite clothing store. The coupon is applied before any discounts are taken. The store is having a sale, and offering 15% off everything. If Clara has $50 to spend, how much can her purchases total before applying the discount and her coupon? Round to the nearest cent. 5. Geoffrey owns the Super Company. Heeach makes custom 3. A dog kennel charges $40 to board a dog for Backyard one night Shed and $35 per night night after built that. sheds Henryfor residential homeowners, and For he buys the majority ...
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Lesson 2 Mathematics Gymnastics M3-31 Answer Key Carnegie

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