activities incorporate concept and skill development and guided practice, then move on to the ... Integrated Pathway: Mathematics I Program are design...

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This program was developed and reviewed by experienced math educators who have both academic and professional backgrounds in mathematics. This ensures: freedom from mathematical errors, grade level appropriateness, freedom from bias, and freedom from unnecessary language complexity. Developers and reviewers include: Joyce Hale

Zachary Lien

Cameron Larkins

Mike May

Shelly Northrop Sommer

Valerie Ackley

Jennifer Blair

James Quinlan

Vanessa Sylvester

Laura McPartland

Doug Kühlmann

Peter Tierney-Fife

The classroom teacher may reproduce these materials for classroom use only. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored, or recorded in any form without written permission from the publisher.

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

1 2 3 4 5 6 7 8 9 10 ISBN 978-0-8251-7109-3 Copyright © 2012 J. Weston Walch, Publisher Portland, ME 04103 www.walch.com Printed in the United States of America

WALCH

EDUCATION

Table of Contents Program Overview Introduction to the Program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Unit Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Standards Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Station Activities Guide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Online and Digital Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Unit 1: Relationships Between Quantities Lesson 1: Interpreting Structure in Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-1 Lesson 2: Creating Equations and Inequalities in One Variable. . . . . . . . . . . . . . . . . . . . . . . U1-32 Lesson 3: Creating and Graphing Equations in Two Variables. . . . . . . . . . . . . . . . . . . . . . . . U1-90 Lesson 4: Representing Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-166 Lesson 5: Rearranging Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-186 Unit 1 Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-203 Answer Key Teacher Resource. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-211 Student Resource Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-218 Station Activities Set 1: Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-225 Set 2: Solving Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-238 Set 3: Solving Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-249

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CCSS IP Math I Teacher Resource

PROGRAM OVERVIEW

Introduction to the Program Introduction The Common Core State Standards Integrated Pathway: Mathematics I Program is a complete set of materials developed around the Common Core State Standards (CCSS), the overview of the Integrated Pathway for the Common Core State Mathematics Standards, and the Mathematics I content map found in Appendix A of the Common Core State Standards. Topics are built around accessible core curricula, ensuring that the CCSS Integrated Pathway: Mathematics I Program is useful for striving students and diverse classrooms. This program realizes the benefits of exploratory and investigative learning and employs a variety of instructional models to meet the learning needs of students with a range of abilities. The CCSS Integrated Pathway: Mathematics I Program includes components that support problem-based learning, instruct and coach as needed, provide practice, and assess students’ skills. Instructional tools and strategies are embedded throughout. The set of unit materials or digital version of the program includes: •

More than 150 hours of lessons, addressing the six units of CCSS IP: Mathematics I

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Essential Questions for each instructional topic

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Vocabulary

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Instruction and Guided Practice

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Problem-based Tasks and Coaching questions

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Step-by-step graphing calculator instructions for the TI-Nspire and the TI-83/84

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Station activities to promote collaborative learning and problem-solving skills

Purpose of Materials The CCSS Integrated Pathway: Mathematics I Program has been organized to coordinate with the CCSS Integrated Pathway: Mathematics I content map and specifications from Appendix A of the Common Core State Standards. Each lesson includes activities that offer opportunities for exploration and investigation. These activities incorporate concept and skill development and guided practice, then move on to the application of new skills and concepts in problem-solving situations. Throughout the lessons and activities, problems are contextualized to enhance rigor and relevance.

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PROGRAM OVERVIEW Introduction to the Program This program includes all the topics addressed in the CCSS Integrated Pathway: Mathematics I content map. These include: •

Relationships Between Quantities

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Linear and Exponential Relationships

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Reasoning with Equations

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Descriptive Statistics

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Congruence, Proof, and Constructions

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Connecting Algebra and Geometry Through Coordinates

The eight Mathematical Practices described in the Common Core are infused throughout and are as follows: • CCSS.MP.1: Make sense of problems and persevere in solving them. • CCSS.MP.2: Reason abstractly and quantitatively. • CCSS.MP.3: Construct viable arguments and critique the reasoning of others. • CCSS.MP.4: Model with mathematics. • CCSS.MP.5: Use appropriate tools strategically. • CCSS.MP.6: Attend to precision. • CCSS.MP.7: Look for and make use of structure. • CCSS.MP.8: Look for and express regularity in repeated reasoning. Structure of the Teacher Resource The CCSS Integrated Pathway: Mathematics I Program is provided as a collection of unit books and an overview book, or in binder format. The materials are completely reproducible. You may also have purchased the CCSS Integrated Pathway: Mathematics I Teacher Resource in digital format. In this case, electronic “bookmarks” allow you to access the sections quickly and easily. The digital format also facilitates printing and copying student pages. The Program Overview is the first section. Written for you, this section helps you to navigate the materials, offers several graphic organizers and suggested strategies for their use, and shows how the lessons correlate to the Common Core State Standards and the CCSS Integrated Pathway: Mathematics I content map found in Appendix A of the Common Core State Standards. The remaining materials focus on content, knowledge, and application of the 6 units in the CCSS Integrated Pathway Mathematics I curriculum: Relationships Between Quantities; Linear and Exponential Relationships; Reasoning with Equations; Descriptive Statistics; Congruence, Proof, and 2 CCSS IP Math I Teacher Resource

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PROGRAM OVERVIEW Introduction to the Program Constructions; and Connecting Algebra and Geometry Through Coordinates. The units in the CCSS Integrated Pathway: Mathematics I Program are designed to be flexible so that you can mix and match activities as the needs of your students and your instructional style dictate. The Station Activities correspond to the content in the units and provide students with the opportunity to apply concepts and skills, while you have a chance to circulate, observe, speak to individuals and small groups, and informally assess and plan. Each lesson begins with a pre-assessment and ends with a progress assessment. These allow you to assess students’ progress as you move from lesson to lesson, enabling you to gauge how well students have understood the material and to differentiate as appropriate. Glossary The Glossary contains vocabulary terms and formulas from throughout the program, organized alphabetically. Each listing provides the term and the definition in both English and Spanish to support ELL students. The listings include the page number(s) where the term can be found in the Words to Know.

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Unit Structure All of the instructional units have some common features. Each lesson begins with a pre-assessment, followed by the list of standards addressed in the lesson; Essential Questions; vocabulary (titled “Words to Know”); and a list of recommended websites to be used as additional resources. Each sub-lesson begins with a list of identified prerequisite skills that students need to have mastered in order to be successful with the new material in the upcoming sub-lesson. This is followed by an introduction, key concepts, common errors/misconceptions, guided practice examples, a problem-based task with coaching questions and sample responses, a closure activity, and practice. Each lesson ends with a progress assessment to evaluate students’ learning. All of the components are described below and on the following pages for your reference. Pre-Assessment This can be used to gauge students’ prior knowledge and to inform instructional planning. Common Core State Standards for the Lesson All standards that are addressed in the entire lesson are listed. Essential Questions These are intended to guide students’ thinking as they proceed through the lesson. By the end of each lesson, students should be able to respond to the questions. Words to Know Vocabulary terms and formulas are provided as background information for instruction or to review key concepts that are addressed in the lesson. Recommended Resources This is a list of websites that can be used as additional resources. Some websites are games; others provide additional examples and/or explanations. The links for these resources are live in the PDF version of the Teacher Resource. (Note: These website links will be monitored and repaired or replaced as necessary.) Common Core State Standards for the Sub-Lesson When lessons are broken down into sub-lessons, the specific standard or standards that are addressed are presented at the beginning of the instructional portion of the lesson. 5 © Walch Education

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PROGRAM OVERVIEW Unit Structure Warm-Up Each warm-up takes approximately 5 minutes and addresses either prerequisite and critical-thinking skills or previously taught math concepts. Warm-Up Debrief Each debrief provides the answers to the warm-up questions, and offers suggestions for situations in which students might have difficulties. A section titled Connection to the Lesson is also included in the debrief to help answer students’ questions about the relevance of the particular warm-up activity to the upcoming instruction. Identified Prerequisite Skills Presented at the beginning of each sub-lesson, this list cites the skills necessary to be successful with the new material. Introduction This brief paragraph gives a description of the concepts about to be presented and often contains some Words to Know. Key Concepts Provided in bulleted form, this instruction highlights the important ideas and/or processes for meeting the standard. Common Errors/Misconceptions This is a list of the common errors students make when applying Key Concepts. This list suggests what to watch for when students arrive at an incorrect answer or are struggling with solving the problems. Guided Practice This section provides step-by-step examples of applying the Key Concepts. Problem-Based Task This activity can be used to walk students through the application of the standard, prior to traditional instruction or at the end of instruction. The task makes use of critical-thinking skills.

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PROGRAM OVERVIEW Unit Structure Problem-Based Task Coaching Questions These questions scaffold the task and guide students to solving the problem(s) presented in the task. Problem-Based Task Coaching Questions Sample Responses These are the answers and suggested appropriate responses to the coaching questions. In some cases answers may vary, but a sample answer is given for each question. Recommended Closure Activity Students are given the opportunity to synthesize and reflect on the lesson through a journal entry or discussion of one or more of the Essential Questions. Practice Each sub-lesson includes practice problems to support students’ achievement of the learning objectives. These sheets are written for the student. They can be used in any combination of teacherled instruction, cooperative learning, or independent application of knowledge. Progress Assessment Each lesson ends with 10 multiple-choice questions, as well as one extended-response question that incorporates critical thinking and writing components. This can be used to document the extent to which students grasp the concepts and skills addressed during instruction. Unit Assessment Each unit ends with 12 multiple-choice questions and three extended-response questions that incorporate critical thinking and writing components. This can be used to document the extent to which students grasped the concepts and skills of each unit. Answer Key Answers for all of the Warm-Ups, Assessments, and Practice problems from the Teacher Resource and all of the problems from the Student Resource Book are provided at the end of each unit. (Student editions include odd answers for the exercises in the student book.)

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PROGRAM OVERVIEW Unit Structure Station Activities Each unit provides at least one set of hands-on activities that correspond to instructional topics. These activities can be used to introduce new concepts or to culminate a sequence of instructional experiences. Graphing Calculators Step-by-step instructions for using a TI-Nspire and a TI-83/84 are provided whenever graphing calculators are referenced. Optional Digital/Online Components If you have the enhanced version of the Common Core State Standards Integrated Pathway: Mathematics I Program, and have purchased student subscriptions to online assessments, the following components are included: Online Pre-Assessment, Progress Assessment, Unit Assessment These versions of the assessments are provided as options for students to take online. Upon completion, students will receive immediate feedback and can forward their scoring data to you. Digital Warm-Ups These PowerPoint versions of the warm-ups and debriefs each include a video clip for student engagement. The video clip can be played as students enter the classroom. The answer key slides can be used as you debrief the warm-up. Digital Instruction Delivered via PowerPoint, this instruction adds interactive applets to the sub-lessons and guided practice to illuminate and illustrate key concepts. This can be used in preparation for the class, for teaching, or for helping students catch up after missing class.

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PROGRAM OVERVIEW

Standards Correlations Each lesson in this Integrated Pathway: Mathematics I program was written specifically to address the Common Core State Standards. Each lesson lists the standards covered in all the sub-lessons, and each sub-lesson lists the standards addressed in that particular section. In this section, you’ll find a comprehensive list mapping the sub-lessons to the CCSS.

Guide to Common Core State Standards Annotation As you use this program, you will come across a symbol included with the Common Core standards for some of the lessons and activities. The description of the star symbol is found below, taken verbatim from the Common Core State Standards Initiative website, at www.corestandards.org. Symbol: ★ Denotes: Modeling Standards The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. Specific modeling standards appear throughout the high school standards indicated by a star symbol (★). From http://www.walch.com/CCSS/00006

Symbol: (+) Denotes: College and Career Readiness Standards Advanced mathematics standards that are required in higher-level courses such as advanced statistics may also be included in lower-level courses. These additional standards are denoted by (+). According to the Common Core State Standards Initiative, “the evidence concerning college and career readiness shows clearly that the knowledge, skills, and practices important for readiness include a great deal of mathematics prior to the boundary defined by (+) symbols in these standards. Indeed, some of the highest priority content for college and career readiness comes from Grades 6–8.” From http://www.walch.com/CCSS/00004

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CCSS IP Math I Teacher Resource

Lesson 5

Lesson 4

Lesson 3

Lesson 2

Lesson 1

Lesson

Creating Linear Equations in One Variable

Creating and Graphing Exponential Equations

1.3.2

Representing Constraints 1.4.1 Representing Constraints Rearranging Formulas 1.5.1 Rearranging Formulas

Creating and Graphing Linear Equations in Two Variables

1.3.1

1.2.2 Creating Linear Inequalities in One Variable 1.2.3 Creating Exponential Equations Creating and Graphing Equations in Two Variables

1.2.1

Sub-lesson Title number Interpreting Structure in Expressions 1.1.1 Identifying Terms, Factors, and Coefficients 1.1.2 Interpreting Complicated Expressions Creating Equations and Inequalities in One Variable Standard(s)

A–CED.4★

A–CED.3★

A–CED.2★ N–Q.1★ A–CED.2★ N–Q.1★

A–CED.1★ N–Q.2★ N–Q.3★ A–CED.1★ A–CED.1★

A–SSE.1a★ A–SSE.1b★

Unit 1: Relationships Between Quantities

CCSS INTEGRATED PATHWAY: MATHEMATICS I STANDARDS CORRELATIONS

U1-188–199

U1-168–182

U1-130–150

U1-95–129

U1-56–69 U1-70–86

U1-35–55

U1-5–16 U1-17–28

Pages

PROGRAM OVERVIEW Standards Correlations

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PROGRAM OVERVIEW

Station Activities Guide Introduction Each unit includes a collection of station-based activities to provide students with opportunities to practice and apply the mathematical skills and concepts they are learning. You may use these activities in addition to the instructional lessons, or, especially if the pre-test or other formative assessment results suggest it, instead of direct instruction in areas where students have the basic concepts but need practice. The debriefing discussions after each set of activities provide an important opportunity to help students reflect on their experiences and synthesize their thinking. Debriefing also provides an additional opportunity for ongoing, informal assessment to guide instructional planning.

Implementation Guide The following guidelines will help you prepare for and use the activity sets in this section. Setting Up the Stations Each activity set consists of four or five stations. Set up each station at a desk, or at several desks pushed together, with enough chairs for a small group of students. Place a card with the number of the station on the desk. Each station should also contain the materials specified in the teacher’s notes, and a stack of student activity sheets (one copy per student). Place the required materials (as listed) at each station. When a group of students arrives at a station, each student should take one of the activity sheets to record the group’s work. Although students should work together to develop one set of answers for the entire group, each student should record the answers on his or her own activity sheet. This helps keep students engaged in the activity and gives each student a record of the activity for future reference. Forming Groups of Students All activity sets consist of four or five stations. You might divide the class into four or five groups by having students count off from 1 to 4 or 5. If you have a large class and want to have students working in small groups, you might set up two identical sets of stations, labeled A and B. In this way, the class can be divided into eight groups, with each group of students rotating through the “A” stations or “B” stations.

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PROGRAM OVERVIEW Station Activities Guide Assigning Roles to Students Students often work most productively in groups when each student has an assigned role. You may want to assign roles to students when they are assigned to groups and change the roles occasionally. Some possible roles are as follows: •

Reader—reads the steps of the activity aloud

•

F acilitator—makes sure that each student in the group has a chance to speak and pose questions; also makes sure that each student agrees on each answer before it is written down

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aterials Manager—handles the materials at the station and makes sure the materials are put M back in place at the end of the activity

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imekeeper—tracks the group’s progress to ensure that the activity is completed in the T allotted time

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Spokesperson—speaks for the group during the debriefing session after the activities

Timing the Activities The activities in this section are designed to take approximately 10 minutes per station. Therefore, you might plan on having groups change stations every 10 minutes, with a two-minute interval for moving from one station to the next. It is helpful to give students a “5-minute warning” before it is time to change stations. Since each activity set consists of four or five stations, the above time frame means that it will take about 50 to 60 minutes for groups to work through all stations. Guidelines for Students Before starting the first activity set, you may want to review the following “ground rules” with students. You might also post the rules in the classroom. •

ll students in a group should agree on each answer before it is written down. If there is a A disagreement within the group, discuss it with one another.

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You can ask your teacher a question only if everyone in the group has the same question.

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I f you finish early, work together to write problems of your own that are similar to the ones on the activity sheet.

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L eave the station exactly as you found it. All materials should be in the same place and in the same condition as when you arrived.

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PROGRAM OVERVIEW Station Activities Guide Debriefing the Activities After each group has rotated through every station, bring students together for a brief class discussion. At this time, you might have the groups’ spokespersons pose any questions they had about the activities. Before responding, ask if students in other groups encountered the same difficulty or if they have a response to the question. The class discussion is also a good time to reinforce the essential ideas of the activities. The questions that are provided in the teacher’s notes for each activity set can serve as a guide to initiating this type of discussion. You may want to collect the student activity sheets before beginning the class discussion. However, it can be beneficial to collect the sheets afterward so that students can refer to them during the discussion. This also gives students a chance to revisit and refine their work based on the debriefing session. If you run out of time to hold class discussions, you might want to have students journal about their experiences and follow up with a class discussion the next day.

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PROGRAM OVERVIEW

Online and Digital Components Introduction If you have the “enhanced” version of the Common Core State Standards Integrated Pathway: Mathematics I Teacher Resource, you have access to the following digital components. Further, if you purchase annual subscriptions, your students have access to online versions of all the assessments included in the program. Each of these additional components is described here, along with guidelines and suggestions for implementation. Online Assessments These versions of the assessments are provided as an option for students to take online. Students will be prompted to enter an e-mail address and username. Upon completion of the assessment, student scoring data will be sent to this e-mail address. You may wish to have students enter in your e-mail address so that feedback is sent directly to you, or have students enter their own e-mail address and then forward the results to you. As students finish each assessment, a score report will be sent to the e-mail address that was provided. The data from these assessments may serve as feedback to students and can be used to inform instructional decisions. If you wish to aggregate and analyze assessment data, you can cut and paste the information from individual score reports into a spreadsheet. For more informal analysis, simply reviewing the reports and noting common errors or patterns of difficulty can help you to plan for your students’ needs. Digital Warm-Ups Each of the warm-ups in this collection includes a video clip to enhance student engagement. Hard copies of all warm-ups are also included in the Teacher Resource. These optional versions of the warmups and debriefs run on a PowerPoint platform. (Please note: Computers may render PowerPoint images differently. For best viewing and display, use a PowerPoint Viewer and adjust your settings to optimize images and text.) We suggest that the video clip be played as students enter the class. The green “play” button on the first PowerPoint slide will link your computer to a browser and then to the video, so you will need an Internet connection. Make sure that you use the “full screen” option for display and adjust the volume appropriately. The video clips have been selected to catch students’ attention by addressing the context (not the content) of the warm-up problem in an engaging way. After the video, each warm-up takes approximately 5 minutes and addresses critical-thinking skills and/or previously taught math concepts that are prerequisites for the new content in the lesson. The warm-ups include debriefs that reveal the answers to the warm-up questions. The answers slide into view and can be used to facilitate discussion.

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PROGRAM OVERVIEW Online and Digital Components In the notes section, you will find the URL for the video (to copy and paste into a web browser as needed); the applicable CCSS correlation for the warm-up; and “Connections to the Lesson,” which describes concepts students will glean or skills they will need in the upcoming lesson. The “Connections” help transition from the warm-up to instruction. Digital Instruction In addition to the hard copy materials found in the Teacher Resource, “enhanced” instruction can be delivered via PowerPoint. Each PowerPoint includes the Introduction and Key Concepts from the sub-lesson, as well as Guided Practice. Two of the guided practice examples from the hard copy are included, enhanced with interactive applets that illuminate and illustrate key concepts. You can view these PowerPoint sets in preparation for teaching a class, display them during instruction, post them on websites or blogs as a resource for parents and students at home, or assign them to help students catch up after an absence. The visual components used to illustrate the guided practice examples include animated tables or graphs and applets; i.e., interactive models created with GeoGebra. An Internet connection will be needed to access most of these visual components. In these cases, the PowerPoint slide will display a green “play” button; clicking it will connect to your browser and subsequently to the visual model. Please adjust your view to maximize the image. You will note that each applet illustrates the specific problem addressed in the guided practice example. The applets allow you to walk through the solution process by visually demonstrating the steps, such as defining points and drawing lines. Variable components of the applets (usually fill-in boxes or sliders) allow you to substitute different values in order to explore the mathematics. For example, “What happens to the line when we increase the amount of time?” or “What if we cut the number of students in half?” This experimentation and discussion supports development of conceptual understanding. Finally, a slide describing the Common Errors and Misconceptions for the particular lesson is also included. This information will alert you and your students to the difficulties most often associated with the mathematics in the lesson. GeoGebra for PC/MAC GeoGebra is not required for using the applets in the Enhanced Instruction PowerPoints, but can be downloaded for free for further exploration at the following link: http://www.geogebra.org/cms/en/download Java Troubleshooting GeoGebra is a Java-based website. If you are experiencing any difficulty in using the applets in your browser, please visit the following link for our troubleshooting document. http://www.walch.com/javahelp 16 CCSS IP Math I Teacher Resource

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Assessment Pre-Assessment Circle the letter of the best answer. 1. How many terms are in the expression 36x3 + 27x2 – 18x – 9? a. 3

c. 4

b. 7

d. 9

2. What are the factors in the expression 11x2 + 7x – 4? a. 11 and x2, 7 and x b. 11 and 7 c. There aren’t any factors in this expression. d. x

3. W hat are the term(s), coefficient, and constant described by the phrase, “the cost of 4 tickets to the football game, t, and a service charge of $10”? a. term: 4t, coefficient: 4, constant: 10 b. terms: 4t and 10, coefficient: 10, constant: 4 c. term: 14t, coefficient: 14, constant: none d. terms: 4t and 10, coefficient: 4, constant: 10

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CCSS IP Math I Teacher Resource 1.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Assessment 4. N ewer books can be downloaded for $8 each, while older books can be downloaded for $4 each. The total cost, in dollars, of 12 books is represented by the expression 8n + 4(12 – n), where n represents the number of new books downloaded. How does changing the value of n change the value of the term 4(12 – n)? a. F or values of n less than 12, the term 4(12 – n) will be negative; for values of n greater than 12, the term will be positive; for values of n equal to 12, the term will equal 0. b. F or values of n less than 12, the term 4(12 – n) will be positive; for values of n greater than 12, the term will be positive; for values of n equal to 12, the term will equal 0. c. F or values of n less than 12, the term 4(12 – n) will be positive; for values of n greater than 12, the term will be negative; for values of n equal to 12, the term will equal 0. d. F or values of n less than 12, the term 4(12 – n) will be negative; for values of n greater than 12, the term will be negative; for values of n equal to 12, the term will equal 0.

5. A n amount of money deposited in a bank account earns interest according to the following expression: P(1 + r)t, where P represents the initial deposit, r represents the rate, and t represents the period of time. How does increasing the period of time change the value of P? a. Increasing the period of time will increase the value of P. b. Increasing the period of time does not change the value of P. c. Increasing the period of time will decrease the value of P. d. It is not possible to increase the period of time.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES

Lesson 1: Interpreting Structure in Expressions Instruction Common Core State Standards A–SSE.1

Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. I nterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Essential Questions 1. How are algebraic expressions different from algebraic equations? 2. How is the order of operations applied to expressions and simple formulas at specific values? 3. How are verbal phrases translated into algebraic expressions? WORDS TO KNOW algebraic expression a mathematical statement that includes numbers, operations, and variables to represent a number or quantity the factor being multiplied together in an exponential expression; in the base expression ab, a is the base the number multiplied by a variable in an algebraic expression coefficient a quantity that does not change constant the number of times a factor is being multiplied together in an exponential exponent expression; in the expression ab, b is the exponent one of two or more numbers or expressions that when multiplied produce factor a given product terms that contain the same variables raised to the same power like terms order of operations the order in which expressions are evaluated from left to right (grouping symbols, evaluating exponents, completing multiplication and division, completing addition and subtraction) a number, a variable, or the product of a number and variable(s) term a letter used to represent a value or unknown quantity that can change or variable vary

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Recommended Resources •

Math-Play.com. “Algebraic Expressions Millionaire Game.” http://walch.com/rr/CAU1L1Expressions “Algebraic Expressions Millionaire Game” can be played alone or in two teams. For each question, players have to identify the correct mathematical expression that models a given expression.

•

Quia. “Algebraic Symbolism Matching Game.” http://walch.com/rr/CAU1L1AlgSymbolism In this matching game, players pair each statement with its algebraic interpretation. There are 40 matches to the provided game.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Lesson 1.1.1: Identifying Terms, Factors, and Coefficients Warm-Up 1.1.1 Ella purchased 2 DVDs and 3 CDs from Tyler’s Electronics at the prices listed below. After taxes, her total cost increased by $5.60. Item CD DVD

Cost ($) c d

1. How can you write the cost of 2 DVDs as an algebraic expression?

2. How can you write the cost of 2 DVDs and 3 CDs as an algebraic expression?

3. H ow can you write the cost of 2 DVDs and 3 CDs, increased by $5.60 for taxes, as an algebraic expression?

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CCSS IP Math I Teacher Resource 1.1.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Lesson 1.1.1: Identifying Terms, Factors, and Coefficients Common Core State Standard A–SSE.1

Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients.

Warm-Up 1.1.1 Debrief 1. How can you write the cost of 2 DVDs as an algebraic expression? The cost of 2 DVDs can be written as an expression of multiplication, 2d . 2. How can you write the cost of 2 DVDs and 3 CDs as an algebraic expression? The cost of 2 DVDs and 3 CDs can be written as the expression 2d + 3c. 3. H ow can you write the cost of 2 DVDs and 3 CDs, increased by $5.60 for taxes, as an algebraic expression? “Increased by $5.60 for taxes” means “to add 5.60.” 2d + 3c + 5.60 would represent the cost of 2 DVDs and 3 CDs increased by $5.60. Connection to the Lesson •

In this lesson, students will be asked to identify parts of expressions given in context.

•

I t will be necessary for students to be able to first translate a given context into an algebraic expression prior to identifying its parts.

U1-6 CCSS IP Math I Teacher Resource 1.1.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Prerequisite Skills This lesson requires the use of the following skills: •

translating verbal expressions to algebraic expressions

•

evaluating expressions following the order of operations

Introduction Thoughts or feelings in language are often conveyed through expressions; however, mathematical ideas are conveyed through algebraic expressions. Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. Variables are letters used to represent values or unknown quantities that can change or vary. One example of an algebraic expression is 3x – 4. Notice the variable, x. Key Concepts •

E xpressions are made up of terms. A term is a number, a variable, or the product of a number and variable(s). An addition or subtraction sign separates each term of an expression.

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In the expression 4x2 + 3x + 7, there are 3 terms: 4x2, 3x, and 7.

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he factors of each term are the numbers or expressions that when multiplied produce a T given product. In the example above, the factors of 4x2 are 4 and x2. The factors of 3x are 3 and x.

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4 is also known as the coefficient of the term 4x2. A coefficient is the number multiplied by a variable in an algebraic expression. The coefficient of 3x is 3.

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he term 4x2 also has an exponent. Exponents indicate the number of times a factor is being T multiplied by itself. In this term, 2 is the exponent and indicates that x is multiplied by itself 2 times.

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erms that do not contain a variable are called constants because the quantity does not T change. In this example, 7 is a constant. Expression Terms Factors Coefficients Constants

4x2 4 and x2 4

4x2 + 3x + 7 3x 3 and x 3

7

7 U1-7

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CCSS IP Math I Teacher Resource 1.1.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction erms with the same variable raised to the same exponent are called like terms. In the T example 5x + 3x – 9, 5x and 3x are like terms. Like terms can be combined following the order of operations by evaluating grouping symbols, evaluating exponents, completing multiplication and division, and completing addition and subtraction from left to right. In this example, the sum of 5x and 3x is 8x.

•

Common Errors/Misconceptions •

incorrectly following the order of operations

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incorrectly identifying like terms

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incorrectly combining terms involving subtraction

U1-8 CCSS IP Math I Teacher Resource 1.1.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Guided Practice 1.1.1 Example 1 Identify each term, coefficient, constant, and factor of 2(3 + x) + x(1 – 4x) + 5. 1. Simplify the expression. The expression can be simplified by following the order of operations and combining like terms. 2(3 + x) + x(1 – 4x) + 5 6 + 2x + x(1 – 4x) + 5 6 + 2x + x – 4x2 + 5 11 + 3x – 4x2

Distribute 2 over 3 + x. Distribute x over 1 – 4x. Combine like terms: 2x and x; 6 and 5.

It is common to rearrange the expression so the powers are in descending order, or go from largest to smallest power. –4x2 + 3x + 11 2. Identify all terms. There are three terms in the expression: –4x2, 3x, and 11. 3. Identify any factors. The numbers or expressions that, when multiplied, produce the product –4x2 are –4 and x2. The numbers or expressions that, when multiplied, produce the product 3x are 3 and x. 4. Identify all coefficients. The number multiplied by a variable in the term –4x2 is –4; the number multiplied by a variable in the term 3x is 3; therefore, –4 and 3 are coefficients. 5. Identify any constants. The number that does not change in the expression is 11; therefore, 11 is a constant.

U1-9 © Walch Education

CCSS IP Math I Teacher Resource 1.1.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Example 2 A smartphone is on sale for 25% off its list price. The sale price of the smartphone is $149.25. What expression can be used to represent the list price of the smartphone? Identify each term, coefficient, constant, and factor of the expression described. 1. Translate the verbal expression into an algebraic expression. Let x represent the unknown list price. Describe the situation. The list price is found by adding the discounted amount to the sale price: sale price + discount amount The discount amount is found by multiplying the discount percent by the unknown list price. The expression that represents the list price of the smartphone is 149.25 + 0.25x. 2. Identify all terms. There are two terms described in the expression: the sale price of $149.25, and the discount of 25% off the list price, or 149.25 and 0.25x. 3. Identify the factors. 0.25x is the product of the factors 0.25 and x. 4. Identify all coefficients. 0.25 is multiplied by the variable, x; therefore, 0.25 is a coefficient. 5. Identify any constants. The number that does not change in the expression is 149.25; therefore, 149.25 is a constant.

U1-10 CCSS IP Math I Teacher Resource 1.1.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Example 3 Helen purchased 3 books from an online bookstore and received a 20% discount. The shipping cost was $10 and was not discounted. Write an expression that can be used to represent the total amount Helen paid for 3 books plus the shipping cost. Identify each term, coefficient, constant, and factor of the expression described. 1. Translate the verbal expression into an algebraic expression. Let x represent the unknown price. The expression used to represent the total amount Helen paid for the 3 books plus shipping is 3x – 0.20(3x) + 10. 2. Simplify the expression. The expression can be simplified by following the order of operations and combining like terms. 3x – 0.20(3x) + 10 3x – 0.60x + 10 2.4x + 10

Multiply 0.20 and 3x. Combine like terms: 3x and –0.60x.

3. Identify all terms. There are two terms in the described expression: the product of 2.4 and x, and the shipping charge of $10: 2.4x and 10. 4. Identify the factors. 2.4x is the product of the factors 2.4 and x. 5. Identify all coefficients. 2.4 is multiplied by the variable, x; therefore, 2.4 is a coefficient. 6. Identify any constants. The number that does not change in the expression is 10; therefore, 10 is a constant.

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CCSS IP Math I Teacher Resource 1.1.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Problem-Based Task 1.1.1: Identifying Parts of an Expression in Context Tara and two friends had dinner at a Spanish tapas restaurant that charged $6 per tapa, or appetizer. The three of them shared several tapas. The total bill included taxes of $4.32. What are the terms, factors, and coefficients of the algebraic expression that represents the number of tapas ordered, including taxes?

What are the terms, factors, and coefficients of the algebraic expression that represents the number of tapas ordered, including taxes?

U1-12 CCSS IP Math I Teacher Resource 1.1.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Problem-Based Task 1.1.1: Identifying Parts of an Expression in Context Coaching a. What was the cost of each tapa without including taxes?

b. W hat algebraic expression can be used to represent the number of tapas ordered if each one cost $6?

c. What algebraic expression can be used to represent the cost of the tapas ordered including taxes?

d. How many terms does the expression from part c include?

e. What are the factors that make up each of the terms?

f. What are the coefficients of each term?

U1-13 © Walch Education

CCSS IP Math I Teacher Resource 1.1.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Problem-Based Task 1.1.1: Identifying Parts of an Expression in Context Coaching Sample Responses a. W hat was the cost of each tapa without including taxes? Each tapa cost $6. b. W hat algebraic expression can be used to represent the number of tapas ordered if each one cost $6? The number of tapas ordered is unknown. Let n represent the unknown number of tapas. The total cost of n tapas is represented by 6n. c. W hat algebraic expression can be used to represent the cost of the tapas ordered including taxes? The total bill included $4.32 in taxes. The cost of the tapas is 6n. The expression that represents the cost of the tapas ordered and the taxes is 6n + 4.32. d. H ow many terms does the expression from part c include? There are two terms in the expression 6n + 4.32: 6n and 4.32. e. W hat are the factors that make up each of the terms? The factors are the products of the quantities of each term. The second term, 4.32, is a constant and does not have factors. The product of 6n is made up of the factors 6 and n. f. W hat are the coefficients of each term? The coefficient of the term 6n is 6. The second term, 4.32, is a constant and does not have a coefficient. Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

U1-14 CCSS IP Math I Teacher Resource 1.1.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Practice 1.1.1: Identifying Terms, Factors, and Coefficients For problems 1 and 2, identify the terms, coefficients, constants, and factors of the given expressions. 1. 12a3 + 16a + 4

2. 21x2 + 3x – 15x2 + 9

For problems 3 and 4, translate each verbal expression to an algebraic expression. Then, identify the terms, coefficients, and constants of the given expressions. 3. half the sum of x and y decreased by one-third y

4. the product of 5 and the cube of x, increased by the difference of 6 and x3

5. Write an expression with 4 terms, containing the coefficients 3, 6, and 9.

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CCSS IP Math I Teacher Resource 1.1.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

For problems 6–10, write an algebraic expression to describe each situation. Then, identify the terms, coefficients, constants, and factors. 6. G avin agrees to buy a 6-month package deal of monthly gym passes, and in turn receives a 15% discount. Write an algebraic expression to represent the total cost of the monthly passes with the discount, if x represents the cost of each monthly pass.

7. A ndre purchased 10 cans of tennis balls from an online store and received a 25% discount. Shipping cost $5.99. Write an algebraic expression to represent the total cost of the tennis balls with the shipping cost, if x represents the cost of each can.

8. N adia and some friends went to a movie. Their total cost was $30.24, which included taxes of $2.24. Write an algebraic expression to represent the price of each movie ticket, not including taxes. Let x represent the number of Nadia’s friends that went to the movies.

9. T he area of a trapezoid can be found by multiplying the height of the trapezoid by half of the sum of base1 and base2.

10. T he surface area of a cylinder with radius r and height h is twice the product of π and the square of the radius plus twice the product of π, the radius, and the height.

U1-16 CCSS IP Math I Teacher Resource 1.1.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Lesson 1.1.2: Interpreting Complicated Expressions Warm-Up 1.1.2 Read the scenario below. Use words and symbols to explain your answers to the questions that follow. At the beginning of the school year, Javier deposited $750 in an account that pays 3% of his initial deposit each year. He left the money in the bank for 5 years. 1. How much interest did Javier earn in 5 years?

2. After 5 years, what is the total amount of money that Javier has?

U1-17 © Walch Education

CCSS IP Math I Teacher Resource 1.1.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Lesson 1.1.2: Interpreting Complicated Expressions Common Core State Standard A–SSE.1

Interpret expressions that represent a quantity in terms of its context.★ b. I nterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Warm-Up 1.1.2 Debrief At the beginning of the school year, Javier deposited $750 in an account that pays 3% of his initial deposit each year. He left the money in the bank for 5 years. 1. How much interest did Javier earn in 5 years? The amount of interest Javier earned in 5 years is $112.50. 5(0.03 • 750) = 112.50 I t is not necessary at this point for students to create and solve an equation to represent this situation, but it is important that the interest amount is calculated based on the initial deposit of $750. Look out for students who try to calculate compound interest. There are several methods to calculating interest; compound interest will be discussed later in the lesson. Focus the discussion on calculating 3% simple interest over five years. 2. After 5 years, what is the total amount of money that Javier has? The total amount of money that Javier has after 5 years is the sum of the simple interest and the initial deposit, or $862.50. 112.50 + 750.00 = 862.50 Connection to the Lesson •

I n this lesson, students will interpret the parts of given expressions and how changes to each part affect the expression.

•

This warm-up provides an opportunity for students to begin looking at changes in expressions.

U1-18 CCSS IP Math I Teacher Resource 1.1.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Prerequisite Skills This lesson requires the use of the following skills: •

evaluating expressions using order of operations

•

evaluating expressions for a given value

•

identifying parts of an expression

Introduction Algebraic expressions, used to describe various situations, contain variables. It is important to understand how each term of an expression works and how changing the value of variables impacts the resulting quantity. Key Concepts •

I f a situation is described verbally, it is often necessary to first translate each expression into an algebraic expression. This will allow you to see mathematically how each term interacts with the other terms.

•

s variables change, it is important to understand that constants will always remain the same. A The change in the variable will not change the value of a given constant.

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Similarly, changing the value of a constant will not change terms containing variables.

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I t is also important to follow the order of operations, as this will help guide your awareness and understanding of each term.

Common Errors/Misconceptions •

incorrectly translating given verbal expressions

U1-19 © Walch Education

CCSS IP Math I Teacher Resource 1.1.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Guided Practice 1.1.2 Example 1 A new car loses an average value of $1,800 per year for each of the first six years of ownership. When Nia bought her new car, she paid $25,000. The expression 25,000 – 1800y represents the current value of the car, where y represents the number of years since Nia bought it. What effect, if any, does the change in the number of years since Nia bought the car have on the original price of the car? 1. Refer to the expression given: 25,000 – 1800y. The term 1800y represents the amount of value the car loses each year, y. As y increases, the product of 1800 and y also increases. 2. 25,000 represents the price of the new car. As y increases and the product of 1800 and y increases, the original cost is not affected. 25,000 is a constant and remains unchanged. Example 2 To calculate the perimeter of an isosceles triangle, the expression 2s + b is used, where s represents the length of the two congruent sides and b represents the length of the base. What effect, if any, does increasing the length of the congruent sides have on the expression? 1. Refer to the expression given: 2s + b. Changing only the length of the congruent sides, s, will not impact the length of base b since b is a separate term. 2. I f the value of the congruent sides, s, is increased, the product of 2s will also increase. Likewise, if the value of s is decreased, the value of 2s will also decrease. 3. I f the value of s is changed, the result of the change in the terms is a doubling of the change in s while the value of b remains the same.

U1-20 CCSS IP Math I Teacher Resource 1.1.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Example 3 Money deposited in a bank account earns interest on the initial amount deposited as well as any interest earned as time passes. This compound interest can be described by the expression P(1 + r)n, where P represents the initial amount deposited, r represents the interest rate, and n represents the number of months that pass. How does a change in each variable affect the value of the expression? 1. Refer to the given expression: P(1 + r)n. Notice the expression is made up of one term containing the factors P and (1 + r)n. 2. C hanging the value of P does not change the value of the factor (1 + r)n, but it will change the value of the expression by a factor of P. In other words, the change in P will multiply by the result of (1 + r)n. 3. S imilarly, changing r changes the base of the exponent (the number that will be multiplied by itself), but does not change the value of P. This change will affect the value of the overall expression. 4. C hanging n changes the number of times (1 + r) will be multiplied by itself, but does not change the value of P. This change will affect the value of the overall expression.

U1-21 © Walch Education

CCSS IP Math I Teacher Resource 1.1.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Problem-Based Task 1.1.2: Searching for a Greater Savings Austin plans to open a savings account. The amount of money in a savings account can be found by using the equation s = p • (1 + r)t, where p is the principal, or the original amount deposited into the account; r is the rate of interest; and t is the amount of time. Austin is considering two savings accounts. He will deposit $1,000.00 as the principal into either account. In Account A, the interest rate will be 0.015 per year for 5 years. In Account B, the interest rate will be 0.02 per year for 3 years. If he could, would it be wise for Austin to leave his money in the account that has less savings for an additional year? Explain your reasoning.

If he could, would it be wise for Austin to leave his money in the account that has less savings for an additional year?

U1-22 CCSS IP Math I Teacher Resource 1.1.2

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Problem-Based Task 1.1.2: Searching for a Greater Savings Coaching a. What is the total amount in Austin’s savings if he chooses Account A?

b. What is the total amount in Austin’s savings if he chooses Account B?

c. Which account has more money at the end of the term?

d. I f Austin left his money in the account that has less savings for an additional year, would this change which account he might select? Explain your answer.

U1-23 © Walch Education

CCSS IP Math I Teacher Resource 1.1.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction Problem-Based Task 1.1.2: Searching for a Greater Savings Coaching Sample Responses a. What is the total amount in Austin’s savings if he chooses Account A? Use the equation s = p • (1 + r)t to determine the amount in Account A. Identify p, r, and t from the problem statement. The principal, p, is $1,000; the interest rate, r, is 0.015; and the time, t, is 5 years. Replace the variables in the equation with the given quantities. s = p • (1 + r)t = (1000) • (1 + 0.015)5 Begin to evaluate the expression using the order of operations. Start by performing any operations in parentheses. (1 + 0.015) is in parentheses. Find the sum. (1000) • (1 + 0.015)5 = (1000) • (1.015)5 Next, evaluate any exponential expressions. The expression (1.015)5 is an exponential expression. (1000) • (1.015)5 ≈ (1000) • (1.08) Next, find any products or quotients, evaluating all multiplication and division from left to right. The final expression, (1000) • (1.08), is a product. (1000) • (1.08) = 1080 s = (1000) • (1 + 0.015)5 ≈ 1080 If he selects Account A, after five years Austin will have approximately $1,080.00 in savings.

U1-24 CCSS IP Math I Teacher Resource 1.1.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction b. What is the total amount in Austin’s savings if he chooses Account B? Use the equation s = p • (1 + r)t to determine the amount in Account B. Identify p, r, and t from the problem statement. The principal, p, is $1,000; the interest rate, r, is 0.02; and the time, t, is 3 years. Replace the variables in the equation with the given quantities. s = p • (1 + r)t = (1000) • (1 + 0.02)3 Begin to evaluate the expression using the order of operations. Start by performing any operations in parentheses. (1 + 0.02) is in parentheses. Find the sum. (1000) • (1 + 0.02)3 = (1000) • (1.02)3 Next, evaluate any exponential expressions. The expression (1.02)3 is an exponential expression. (1000) • (1.02)3 ≈ (1000) • (1.06) Next, find any products or quotients, evaluating all multiplication and division from left to right. The final expression, (1000) • (1.06), is a product. (1000) • (1.06) = 1060 s = (1000) • (1 + 0.02)3 ≈ 1060 If he selects Account B, after three years Austin will have approximately $1,060.00 in savings. c. Which account has more money at the end of the term? Compare the results from parts a and b. If Austin leaves the money in Account A for 5 years, he will have $1,080.00. If Austin leaves the money in Account B for 3 years, he will have $1,060.00. Account A has more money at the end of the term.

U1-25 © Walch Education

CCSS IP Math I Teacher Resource 1.1.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions Instruction d. I f Austin left his money in the account that has less savings for an additional year, would this change which account he might select? Explain your answer. Re-evaluate the total amount in savings if Austin left the money in Account B, the account with the lower total savings, for one extra year. The new value of t in the equation s = p • (1 + r)t would be 4. The values of p and r will be the same as the ones used in part b: p = 1000 and r = 0.02. Replace the values of p, r, and t in the equation s = p • (1 + r)t to find the total amount in Account B after 4 years. s = (1000) • (1 + 0.02)4 Evaluate the equation using the order of operations. Begin by performing any operations in parentheses. (1 + 0.02) is in parentheses. Find the sum. (1000) • (1 + 0.02)4 = (1000) • (1.02)4 Next, evaluate any exponential expressions. The expression (1.02)4 is an exponential expression. (1000) • (1.02)4 ≈ (1000) • (1.08) Next, find any products or quotients, evaluating all multiplication and division from left to right. The final expression, (1000) • (1.08), is a product. (1000) • (1.08) = 1080 s = (1000) • (1 + 0.02)4 ≈ 1080 After four years, Austin will have approximately $1,080.00 in savings in Account B. This is the same amount that would be in Account A after five years, but the money would be available one year sooner. Austin should put his money in Account B. Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

U1-26 CCSS IP Math I Teacher Resource 1.1.2

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Practice 1.1.2: Interpreting Complicated Expressions Use your understanding of terms, coefficients, factors, exponents, and the order of operations to answer each of the following questions. 1. Is the expression

5 + 3x equal to the expression 4x? Explain your answer. 2

2. Is the expression 2 • 4x equal to the expression 8x? Explain your answer.

3. Is the expression (5 • 2) x equal to the expression 10 x ? Explain your answer.

4. A transfer station charges $15 for a waste disposal permit and an additional $5 for each cubic yard of garbage it disposes of. This relationship can be described using the expression 15 + 5x. What effect, if any, does changing the value of x have on the cost of the permit?

5. A bsolute Cable company bills on a monthly basis. Each bill includes a $30.00 service fee plus $4.75 in taxes and $2.99 for each movie purchased. The following expression describes the cost of the cable service per month: 34.75 + 2.99m. If Absolute Cable lowers the service fee, how will the expression change?

continued U1-27 © Walch Education

CCSS IP Math I Teacher Resource 1.1.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

6. I n order to lose weight in a healthy manner, a veterinarian suggested an overweight large-breed dog lose no more than 2 pounds per week. If the expression x – 2y represents this situation, what must be true about the value of y?

7. T he product of 7, x, and y is represented by the expression 7xy. If the value of x is negative, what can be said about the value of y in order for the product to remain positive?

8. A bank account balance for an account with an initial deposit of P dollars earns interest at an annual rate of r. The amount of money in the account after n years is described using the following expression: P(1 + r)n. What effect, if any, does decreasing the value of r have on the amount of money after n years?

9. For what values of x will the result of 5 x be greater than 1?

10. A tire can hold C cubic feet of air. It loses air at a rate of r for a period of time, t. This situation can be described using the following formula: C(1 – r)t. What effect, if any, does increasing the value of r have on the value C ?

U1-28 CCSS IP Math I Teacher Resource 1.1.2

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Assessment Progress Assessment Circle the letter of the best answer. 1. How many terms are in the expression 14 x 2 − 12 x + 5 ? a. 3

c. 2

b. 7

d. 4

2. What are the factors in the expression 8 x 2 + 10 x − 3 ? a. 8 and x 2

c. 8 and x 2 , 10 and x

b. x

d. There are no factors in this expression.

3. What are the coefficients in the expression 5 x 3 + 14 x 2 − 2 x + 6 ? a. 5, 14, 2, and 6

c. 5, 14, and –2

b. 5, 14, and 2

d. 5, 14, –2, and 6

4. What is the constant in the expression 6 x 3 + 4 x − 3 ? a. 6

c. 3

b. 4

d. –3

5. T imely Taxi Service charges $3.50 for pickup, plus an additional $0.20 per mile. What is the constant in this situation? a. $0.20

c. $3.70

b. $3.50

d. $3.30

continued U1-29 © Walch Education

CCSS IP Math I Teacher Resource 1.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Assessment 6. A shirt that costs C dollars with 7% sales tax can be described using the expression C + 0.07C. Which expression below is NOT the same as this expression? a. 1.07C

c. 1.7C

b. C(1 + 0.07)

d. 0.07C + C

7. T he product of 5, a, and b is represented by the expression 5ab. If the value of a is positive, what must be said about the value of b in order for the product to remain positive? a. b must be positive.

c. b must be 0.

b. b must be negative.

d. The value of b does not matter.

5 8. The expression ( F - 32) is used to convert a Fahrenheit temperature to Celsius. What values of 9 F will NOT result in a Celsius temperature below 0? a. F must be greater than 0.

c. F must be greater than 32.

b. F must be less than 0.

d. F must be less than 32.

9. The population of bacteria in a Petri dish is 550 and increases according to the expression 550(3.4 0.005 t ) , where t is the number of hours. What effect would increasing the initial population of the bacteria have on the rate at which the number of bacteria increases? a. An increase in the initial population will lower the rate at which the population increases. b. A n increase in the initial population does not affect the rate at which the population increases. c. A n increase in the initial population will increase the rate at which the population increases. d. An increase in the initial population increases the time the bacteria are increasing.

continued U1-30 CCSS IP Math I Teacher Resource 1.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 1: Interpreting Structure in Expressions

Date:

Assessment 10. For what values of x will the result of 3x be greater than 1? a. x can be any number greater than 1.

c. x can be any number less than 0.

b. x can be any number greater than 3.

d. x can be any number greater than 0.

Use what you know about the structure of expressions to complete the following problem. 11. Explain the differences among each of these expressions: a x b , ab x , and ( ab) x .

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CCSS IP Math I Teacher Resource 1.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Assessment Pre-Assessment Circle the letter of the best answer. 1. M ina bought a plane ticket to New York City and used a coupon for 15% off the ticket price. The total cost of her ticket, with the discount, was $253.30. What equation could she use to find the price of the ticket without the discount? a. 0.15x = 253.30

c. x = 253.30 + 0.15

b. x – 0.15(x) = 253.30

d. x + 0.15(x) = 253.30

2. L ucas bought a refrigerator. His total cost of $1,331 included sales tax at the rate of 8% and an additional, untaxed delivery charge of $35. How much sales tax did he pay? a. $72

c. $120

b. $106

d. $96

3. Y our cell phone plan allows you 400 minutes to talk per month. So far this month, you have used 265 minutes and you have 7 days left on this month’s plan. Which inequality could you use to determine how many minutes at most you can use per day so that you don’t go over your monthly plan minutes? a. 7x + 265 < 400

c. 7x + 265 ≤ 400

b. 7x + 265 > 400

d. 7x + 265 ≥ 400

4. S ydney has a $75 mall gift card. She wants to buy a sundress and a movie ticket. The movie ticket with tax costs $11.50. The sales tax on the sundress will be 4%. How much can the ticketed price of the sundress be? a. less than or equal to $61.06

c. less than or equal to $45.36

b. less than $61.06

d. less than $45.36

5. A population of rabbits doubles every 4 months. If the population starts out with 8 rabbits, how many rabbits will there be in 1 year? a. 128 rabbits

c. 32 rabbits

b. 64 rabbits

d. 16 rabbits

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES

Lesson 2: Creating Equations and Inequalities in One Variable Instruction Common Core State Standards N–Q.2

Define appropriate quantities for the purpose of descriptive modeling.★

N–Q.3

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.★

A–CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

Essential Questions 1. How are quantities modeled with equations and inequalities? 2. How are equations and inequalities alike and different? 3. What makes creating an exponential equation different from creating a linear equation? WORDS TO KNOW equation

a mathematical sentence that uses an equal sign (=) to show that two quantities are equal

exponential decay

an exponential equation with a base, b, that is between 0 and 1 (0 < b < 1); can be represented by the formula y = a(1 – r) t, where a is the initial value, (1 – r) is the decay rate, t is time, and y is the final value

exponential equation

an equation that has a variable in the exponent; the general form is y = a • b x, where a is the initial value, b is the base, x is the time, and y x

is the final output value. Another form is y = ab t , where t is the time it takes for the base to repeat. exponential growth

an exponential equation with a base, b, greater than 1 (b > 1); can be represented by the formula y = a(1 + r) t, where a is the initial value, (1 + r) is the growth rate, t is time, and y is the final value

inequality

a mathematical sentence that shows the relationship between quantities that are not equivalent

linear equation

an equation that can be written in the form ax + by = c, where a, b, and c are rational numbers U1-33

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CCSS IP Math I Teacher Resource 1.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction quantity

something that can be compared by assigning a numerical value

rate

a ratio that compares different kinds of units

solution

a value that makes the equation true

solution set

the value or values that make a sentence or statement true

unit rate

a rate per one given unit

variable

a letter used to represent a value or unknown quantity that can change or vary

Recommended Resources •

APlusMath. “Algebra Planet Blaster.” http://walch.com/rr/CAU1L2LinEquations Players solve each multi-step linear equation to find the correct planet to “blast.” Incorrect answers cause players to destroy their own ship.

•

Figure This! Math Challenges for Families. “Challenge 24: Gasoline Tanks.” http://walch.com/rr/CAU1L2Rates This website features a description of the math involved, related occupations, a hint to get started, complete solutions, and a “Try This” section, as well as additional related problems with answers, questions to think about, and resources for further exploration. These resources could be used to lead a group discussion or as an investigation. Students could work in small groups to create a presentation related to each challenge.

•

Purplemath.com. “Exponential Functions: Introduction.” http://walch.com/rr/CAU1L2ExpEquations This website gives an introduction of exponential equations and provides a few examples of tables of input and output values, with integers as inputs. The introduction goes into more depth about the shapes of the graphs of exponential functions and continues on to develop the concept of compound interest. It also introduces the number e.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Lesson 1.2.1: Creating Linear Equations in One Variable Warm-Up 1.2.1 Read the scenario and answer the questions that follow. Andrew is practicing for a tennis tournament and needs more tennis balls. He bought 10 cans of tennis balls online and received a 25% discount. The shipping cost was $5.99. Let x represent the cost of each can. 1. Write an algebraic expression to represent the cost of the tennis balls.

2. Write an algebraic expression to represent the cost of the tennis balls with the discount.

3. W rite an algebraic expression to represent the total cost of the tennis balls with the shipping cost and the discount. Simplify the expression.

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CCSS IP Math I Teacher Resource 1.2.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Lesson 1.2.1: Creating Linear Equations in One Variable Common Core State Standards N–Q.2

Define appropriate quantities for the purpose of descriptive modeling.★

N–Q.3

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.★

A–CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

Warm-Up 1.2.1 Debrief Andrew is practicing for a tennis tournament and needs more tennis balls. He bought 10 cans of tennis balls online and received a 25% discount. The shipping cost was $5.99. Let x represent the cost of each can. 1. Write an algebraic expression to represent the cost of the tennis balls. Andrew purchased 10 cans of tennis balls at an unknown price, x. Therefore, the expression to represent the cost of the tennis balls is 10x. 2. Write an algebraic expression to represent the cost of the tennis balls with the discount. First, Andrew will be charged the cost of the tennis balls (10x). Then, 25% will be discounted or taken off the cost of the tennis balls, so –0.25(10x). Add these amounts to arrive at the price of the tennis balls. 10x – 0.25(10x) 10x – 2.5x 7.5x Students might have trouble remembering to convert the percentage to a decimal. Students might not have simplified.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 3. W rite an algebraic expression to represent the total cost of the tennis balls with the shipping cost and the discount. Simplify the expression. The shipping cost was $5.99. Add this to the expression found above. 10x – 0.25(10x) + 5.99 10x – 2.5x + 5.99 7.5x + 5.99 Students might try to apply the discount to the shipping cost, but the discount does not apply to shipping. Encourage students to read the problem scenario carefully before and while writing their expressions.

Connection to the Lesson •

Students will extend writing expressions to writing equations in the upcoming lesson.

•

The process of translating an expression is the same for translating an equation.

•

he warm-up is scaffolded to help students think about breaking down the context T into smaller pieces, providing an example for how to think about doing this with an equation context.

•

S tudents will be asked to take a scenario like this a step further and solve for the unknown given a total cost.

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CCSS IP Math I Teacher Resource 1.2.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Prerequisite Skills This lesson requires the use of the following skills: •

applying the order of operations

•

creating ratios

•

translating verbal sentences into expressions

Introduction Creating equations from context is important since most real-world scenarios do not involve the equations being given. An equation is a mathematical sentence that uses an equal sign (=) to show that two quantities are equal. A quantity is something that can be compared by assigning a numerical value. In this lesson, contexts will be given and equations must be created from them and then used to solve the problems. Since these problems are all in context, units are essential because without them, the numbers have no meaning. Key Concepts •

linear equation is an equation that can be written in the form ax + b = c, where a, b, and A c are rational numbers. Often, the most difficult task in turning a context into an equation is determining what the variable is and how to represent that variable.

•

he variables are letters used to represent a value or unknown quantity that can change or T vary. Once the equation is determined, solving for the variable is straightforward.

•

The solution will be the value that makes the equation true.

•

I n some cases, the solution will need to be converted into different units. Multiplying by a unit rate or a ratio can do this.

•

unit rate is a rate per one given unit, and a rate is a ratio that compares different kinds A of units.

•

se units that make sense, such as when reporting time; for example, if the time is less than U 1 hour, report the time in minutes.

•

hink about rounding and precision. The more numbers you list to the right of the decimal T place, the more precise the number is.

•

hen using measurement in calculations, only report to the nearest decimal place of the least W accurate measurement. See Guided Practice Example 5.

U1-38 CCSS IP Math I Teacher Resource 1.2.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Creating Equations from Context 1. Read the problem statement first. 2. Reread the scenario and make a list or a table of the known quantities. 3. Read the statement again, identifying the unknown quantity or variable. 4. Create expressions and inequalities from the known quantities and variable(s). 5. Solve the problem. 6. I nterpret the solution of the equation in terms of the context of the problem and convert units when appropriate, multiplying by a unit rate.

Common Errors/Misconceptions •

a ttempting to solve the problem without first reading/understanding the problem statement

•

incorrectly setting up the equation

•

misidentifying the variable

•

forgetting to convert to the correct units

•

setting up the ratio inversely when converting units

•

r eporting too many decimals—greater precision comes with more precise measuring and not with calculations

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CCSS IP Math I Teacher Resource 1.2.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Guided Practice 1.2.1 Example 1 James earns $15 per hour as a teller at a bank. In one week he pays 17% of his earnings in state and federal taxes. His take-home pay for the week is $460.65. How many hours did James work? 1. Read the statement carefully. 2. Reread the scenario and make a list of the known quantities. James earns $15 per hour. James pays 17% of his earning in taxes. His pay for the week is $460.65. 3. Read the statement again and look for the unknown or the variable. The scenario asks for James’s hours for the week. The variable to solve for is hours. 4. C reate expressions and inequalities from the known quantities and variable(s). James’s pay for the week was $460.65. ______ = 460.65 He earned $15 an hour. Let h represent hours. 15h He paid 17% in taxes. –0.17(15h) Put this information all together. 15h – 0.17(15h) = 460.65

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 5. Solve the equation. 15h – 0.17(15h) = 460.65

Multiply –0.17 and 15h.

15h – 2.55h = 460.65

Combine like terms 15h and –2.55h.

12.45h = 460.65

Divide both sides by 12.45.

12.45 h

=

460.65

12.45 12.45 37 hours hours hh == 37 James worked 37 hours. 6. Convert to the appropriate units if necessary. The scenario asked for hours and the quantity given was in terms of hours. No unit conversions are necessary. Example 2 Brianna has saved $600 to buy a new TV. If the TV she wants costs $1,800 and she saves $20 a week, how many years will it take her to buy the TV? 1. Read the statement carefully. 2. Reread the scenario and make a list of the known quantities. The TV costs $1,800. Brianna saved $600. Brianna saves $20 per week. 3. Read the statement again and look for the unknown or the variable. The scenario asks for the number of years. This is tricky because the quantity is given in terms of weeks. The variable to solve for first, then, is weeks.

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CCSS IP Math I Teacher Resource 1.2.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 4. C reate expressions and inequalities from the known quantities and variable(s). Brianna needs to reach $1,800. _____ = 1800 Brianna has saved $600 so far and has to save more to reach her goal. 600 + ______ = 1800 Brianna is saving $20 a week for some unknown number of weeks to reach her goal. Let x represent the number of weeks. 600 + 20x = 1800 5. S olve the problem for the number of weeks it will take Brianna to reach her goal. 600 + 20 x = 1800 −600 − 600 20 x = 1200 20 x 1200 = 20 20 x = 60 weeks Brianna will need 60 weeks to save for her TV. 6. Convert to the appropriate units. The problem statement asks for the number of years it will take Brianna to save for the TV. There are 52 weeks in a year. 1 year 52 weeks 60 weeks •

1 year

52 weeks 1 year ≈ 1.15 years 60 weeks • 52 weeks Brianna will need approximately 1.15 years, or a little over a year, to save for her TV. U1-42 CCSS IP Math I Teacher Resource 1.2.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Example 3 Suppose two brothers who live 55 miles apart decide to have lunch together. To prevent either brother from driving the entire distance, they agree to leave their homes at the same time, drive toward each other, and meet somewhere along the route. The older brother drives cautiously at an average speed of 60 miles per hour. The younger brother drives faster, at an average speed of 70 mph. How long will it take the brothers to meet each other? 1. Read the statement carefully. 2. Reread the scenario and make a table of the known quantities. Problems involving “how fast,” “how far,” or “how long” require the distance equation, d = rt, where d is distance, r is rate of speed, and t is time. Complete a table of the known quantities. Rate (r)

Distance (d)

Older brother

60 mph

55 miles

Younger brother

70 mph

55 miles

3. Read the statement again and look for the unknown or the variable. The scenario asks for how long, so the variable is time, t.

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CCSS IP Math I Teacher Resource 1.2.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 4. C reate expressions and inequalities from the known quantities and variable(s). Step 2 showed that the distance equation is d = rt or rt = d. Together the brothers will travel a distance, d, of 55 miles. (older brother’s rate)(t) + (younger brother’s rate)(t) = 55 The rate r of the older brother = 60 mph and the rate of the younger brother = 70 mph. 60t + 70t = 55 Expand the table from step 2 to see this another way. Older brother Younger brother

Rate (r) 60 mph 70 mph

Time (t) t t

Distance (d) d = 60t d = 70t

Together, they traveled 55 miles, so add the distance equations based on each brother’s rate. 60t + 70t = 55 5. S olve the problem for the time it will take for the brothers to meet each other. 60t + 70t = 55

Equation

130t = 55

Combine like terms 60t and 70t.

130t

=

55

Divide both sides by 130. 130 130 t ≈ 0.42 hours It will take the brothers 0.42 hours to meet each other. Note: The answer was rounded to the nearest hundredth of an hour because any rounding beyond the hundredths place would not make sense. Most people wouldn’t be able to or need to process that much precision. When talking about meeting someone, it is highly unlikely that anyone would report a time that is broken down into decimals, which is why the next step will convert the units.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 6. Convert to the appropriate units if necessary. Automobile speeds in the United States are typically given in miles per hour (mph). Therefore, this unit of measurement is appropriate. However, typically portions of an hour are reported in minutes unless 1 the time given is of an hour. 2 Convert 0.42 hours to minutes using 60 minutes = 1 hour. 60 min = 1 hr 60 min 0.42 hr • 1 hr 60 min 0.42 hr • = 25.2 minutes 1 hr Here again, rarely would a person report that they are meeting someone in 25.2 minutes. In this case, there is a choice of rounding to either 25 or 26 minutes. Either answer makes sense. The two brothers will meet each other in 25 or 26 minutes. Example 4 Think about the following scenarios. In what units should they be reported? Explain the reasoning. 1. Water filling up a swimming pool A swimming pool, depending on the size, has between several gallons and hundreds of thousands of gallons of water. Think about water flowing out of a faucet and picture filling up a milk jug. How long does it take? Less than a minute? The point is that gallons of water can be filled in minutes. Report the filling of a swimming pool in terms of gallons per minute.

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CCSS IP Math I Teacher Resource 1.2.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 2. The cost of tiling a kitchen floor Think about how big rooms are. They can be small or rather large, but typically they are measured in feet. When calculating the area, the measurement units are square feet. Tiles cost in the dollar range. Report the cost of tiling a kitchen floor in dollars per square foot. 3. The effect of gravity on a falling object Think about how fast an object falls when you drop it from shoulderheight. How far is it traveling from your shoulder to the ground? It travels several feet (or meters). How long does it take before the object hits the ground? It only takes a few seconds. Report gravity in terms of feet or meters per second. 4. A snail traveling across the sidewalk The context of the problem will determine the correct units. Think about how slowly a snail moves. Would a snail be able to travel at least one mile in an hour? Perhaps it makes more sense to report the distance in a smaller unit. Report the snail traveling across the sidewalk in feet per minute. If comparing speeds of other animals to the snail’s rate, and the animals’ rates are being reported in miles per hour, then it makes sense to report the snail’s rate in miles per hour, too. 5. Painting a room Think about how long it takes to paint a room. It takes longer than several minutes. It would probably take hours. How is the surface area of a wall typically measured? It’s usually measured in square feet. Report the painting of a room in square feet per hour.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Example 5 Ernesto built a wooden car for a soap box derby. He is painting the top of the car blue and the sides black. He already has enough black paint, but needs to buy blue paint. He needs to know the approximate area of the top of the car to determine the size of the container of blue paint he should 1 1 buy. He measured the length to be 9 feet 11 inches, and the width to be inch less than 3 feet. 4 2 What is the surface area of the top of the car? What is the most accurate area Ernesto can use to buy his paint? 1. Read the statement carefully. 2. Reread the scenario and make a list of the known quantities. Length = 9 feet 11.25 inches Width = 35.5 inches (3 feet = 36 inches; 36 –

1 2

inch = 35.5 inches)

3. Read the statement again and look for the unknown or the variable. The scenario asks for the surface area of the car’s top. Work with the accuracy component after calculating the surface area.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 4. C reate expressions and inequalities from the known quantities and variable(s). The surface area will require some assumptions. A soap box derby car is tapered, meaning it is wider at one end than it is at another. To be sure Ernesto has enough paint, he assumes the car is rectangular with the width being measured at the widest location. A = length × width = lw For step 2, we listed length and width, but they are not in units that can be multiplied. Convert the length to inches. Length = 9 feet 11.25 inches; 9(12) + 11.25 = 119.25 inches Width = 35.5 inches 5. Solve the problem. Substitute length and width into the formula A = lw. A = lw A = 119.25 • 35.5 = 4233.375 This gives a numerical result for the surface area, but the problem asks for the most accurate surface area measurement that can be calculated based on Ernesto’s initial measurements. Since Ernesto only measured to the hundredths place, the answer can only be reported to the hundredths place. The surface area of the top of Ernesto’s car is 4,233.38 square inches.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 6. Convert to the appropriate units if necessary. When buying paint, the hardware store associate will ask how many square feet need to be covered. Ernesto has his answer in terms of square inches. Convert to square feet. There are 144 square inches in a square foot. 1 ft 2 = 144 in 2 1 ft 2

2

4233.38 in • 2

144 in 2

4233.38 in •

1 ft 2 144 in

2

= 29.398472 ft 2

7. R ounding must take place here again because Ernesto can only report to the hundredths place. Ernesto’s surface area = 29.40 ft2

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Problem-Based Task 1.2.1: Rafting and Hiking Trip To celebrate graduation, you and 4 of your closest friends have decided to take a 5-day white-water rafting and hiking trip. During your 5-day trip, 2 days are spent rafting. If the rafting trip covers a distance of 60 miles and you are expected to raft 8 hours each day, how many miles must you raft each hour? For the hiking portion of your trip, you and your friends carry the same amount of equipment, which works out to 35 pounds of equipment each. For extra money, you can hire an assistant, who will carry 50 pounds of equipment. Each assistant charges a flat fee of $50 and an additional $22 for each mile. The total amount you would have to pay the assistant is $512. How many miles will your group be hiking? Is it worth hiring two assistants to help you and your friends carry the equipment? Justify your answers.

How many miles will your group be hiking? Is it worth hiring two assistants to help you and your friends carry the equipment?

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Problem-Based Task 1.2.1: Rafting and Hiking Trip Coaching a. I f the rafting trip covers a distance of 60 miles and you are expected to raft 8 hours each day, how many miles must you raft each hour? What is the ratio of miles to days? What is the ratio you are looking for? What is the ratio of days to hours? How do you convert the original ratio of miles to days into miles per hour? b. How many miles will your group be hiking? What is the equation of the cost of hiring an assistant? What is the solution to this equation? c. Is it worth hiring two assistants to help you and your friends carry the equipment? How much weight will each of you carry without assistants? How much weight will each of you carry with two assistants? What is the difference in the cost per day? Are you willing to pay more money to have someone carry your equipment?

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Problem-Based Task 1.2.1: Rafting and Hiking Trip Coaching Sample Responses a. I f the rafting trip covers a distance of 60 miles and you are expected to raft 8 hours each day, how many miles must you raft each hour? What is the ratio of miles to days? 60 miles 2 days What is the ratio you are looking for? miles hour What is the ratio of days to hours? 1 day 8 hours How do you convert the original ratio of miles to days into miles per hour? Multiply the two numeric ratios together. 60 miles 2 days

•

1 day 8 hours

=

60 miles 16 hours

= 3.75 miles/hour

b. How many miles will your group be hiking? What is the equation of the cost of hiring an assistant? 22x + 50 = 512, where x = miles What is the solution to this equation? x = 21 miles c. Is it worth hiring two assistants to help you and your friends carry the equipment? How much weight will each of you carry without assistants? Each person will carry 35 pounds. There are 5 people. 35 × 5 = 175 pounds How much weight will each of you carry with two assistants?

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction First, determine how much less there would be to carry among the 5 of you with two assistants who each carry 50 pounds. 175 – 2(50) = 175 – 100 = 75 pounds Now, divide 75 pounds by 5 people. 75

= 15 pounds 5 35 – 15 = 20 If you hire assistants, each of you will carry 15 pounds, which is 20 pounds less than the original amount. What is the difference in the cost per day? First, determine the cost for 2 assistants if each will be paid $512. 2(512) = 1024 The cost is $1,024 for 2 assistants. Next, determine the cost per person. $1024 5 people

= $204.80 / person

The difference is $204.80 per person or almost $205 per person. Are you willing to pay more money to have someone carry your equipment? Answers will vary, but should reflect and be justified by students’ calculations. Sample answer: I am willing to pay about an extra $205 to be free of 20 pounds of equipment during my 21-mile hike, so that I can enjoy my experience without being weighed down. Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Practice 1.2.1: Creating Linear Equations in One Variable For the problem below, read each scenario and give the units you would use to work with each situation. 1. What units would you use for each scenario that follows? a. riding a bicycle

b. rainfall during a storm

c. water coming from a fire hydrant

d. watching caloric intake

For problems 2–8, read each scenario, write an equation, and then solve the problem. Remember to include the appropriate units. 2. Y ou need to buy new tile for your kitchen. It measures 13.25 feet by 7.5 feet. What is the area of the kitchen that you calculated? What is the most accurate area you can report to your hardware store in order to purchase enough tile?

3. Z ach watches TV 3 times as much as Joel. Joel watches TV 2 hours a day. How many hours a day does Zach watch TV?

4. I t costs Raquel $5 in tolls to drive to work and back each day, plus she uses 3 gallons of gas. It costs her a total of $15.50 to drive to work and back each day. How much per gallon is Raquel paying for her gas? How do you know?

continued U1-54 CCSS IP Math I Teacher Resource 1.2.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

5. H ayden bought 4 tickets to a football game. He paid a 5% service charge for buying them from a broker. His total cost was $105.00. What was the price of each ticket, not including the service charge?

6. I t cost Justin $100 to have cable TV installed in his house. Each month he pays an access fee plus tax of 7% of his monthly bill. After 6 months, Justin had paid a total of $350.38 for his access fee, taxes, and his initial installation. What is Justin’s monthly access fee not including taxes?

7. Y ou and 3 friends divide the proceeds of a garage sale equally. The garage sale earned $412. How much money did you receive?

8. The area of Sofia’s herb garden is

1

the area of her vegetable garden. The area of her herb 8 garden is 6 square feet. What is the area of her vegetable garden?

9. D riving to your friend’s house, you travel at an average rate of 35 miles per hour. On your way home, you travel at an average rate of 40 miles per hour. If the round trip took you 45 minutes, how far is it from your house to your friend’s house?

10. T wo trains heading toward each other are 400 miles apart. One train travels 15 miles per hour faster than the other train. If they arrive at the same station in 5 hours, how fast is each train traveling?

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CCSS IP Math I Teacher Resource 1.2.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Lesson 1.2.2: Creating Linear Inequalities in One Variable Warm-Up 1.2.2 Read the scenario below, write an equation that models the situation, and then use it to answer the questions that follow. Two people can balance on a seesaw even if they are different weights. The balance will occur when the following equation, w1d1 = w2d2, is satisfied or true. In this equation, w1 is the weight of the first person, d1 is the distance the first person is from the center of the seesaw, w2 is the weight of the second person, and d2 is the distance the second person is from the center of the seesaw. 1. Eric and his little sister Amber enjoy playing on the seesaw at the playground. Amber weighs 65 pounds. Eric and Amber balance perfectly when Amber sits about 4 feet from the center and 1 Eric sits about 2 feet from the center. About how much does Eric weigh? 2

2. T heir little cousin Aleah joins them and sits right next to Amber. Can Eric balance the seesaw with both Amber and Aleah on one side, if Aleah weighs about the same as Amber? If so, where should he sit? If not, why not?

U1-56 CCSS IP Math I Teacher Resource 1.2.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Lesson 1.2.2: Creating Linear Inequalities in One Variable Common Core State Standard A–CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

Warm-Up 1.2.2 Debrief Two people can balance on a seesaw even if they are different weights. The balance will occur when the following equation, w1d1 = w2d2, is satisfied or true. In this equation, w1 is the weight of the first person, d1 is the distance the first person is from the center of the seesaw, w2 is the weight of the second person, and d2 is the distance the second person is from the center of the seesaw. 1. Eric and his little sister Amber enjoy playing on the seesaw at the playground. Amber weighs 65 pounds. Eric and Amber balance perfectly when Amber sits about 4 feet from the center and 1 Eric sits about 2 feet from the center. About how much does Eric weigh? 2 First set up the equation. Students might struggle with which weight is represented by w1 and which is represented by w2. If time allows, show students that as long as the associated weight and distance stay together on the same side of the equation, it doesn’t matter which weight is w1 and which is w2. Given: w1d1 = w2d2 w1 = Amber’s weight = 65 pounds d1 = Amber’s distance = 4 feet w2 = Eric’s weight = x pounds 1 d2 = Eric’s distance = 2 feet 2 The unknown is Eric’s weight. Now, make the substitutions. 1 (65)(4) = (x) 2 2 Simplify. 260 = 2.5x

U1-57 © Walch Education

CCSS IP Math I Teacher Resource 1.2.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Solve. x = 104 Interpret the solution. In this equation, x represented Eric’s weight. Eric weighs about 104 pounds. 2. T heir little cousin Aleah joins them and sits right next to Amber. Can Eric balance the seesaw with both Amber and Aleah on one side, if Aleah weighs about the same as Amber? If so, where should he sit? If not, why not? Set up the equation using w1d1 = w2d2. w1 = Amber and Aleah’s combined weight = 2(65) pounds d1 = Amber and Aleah’s distance = 4 feet w2 = Eric’s weight = 104 pounds d2 = Eric’s distance = x feet This time, the unknown is Eric’s distance. Now, make the substitutions. 2(65)(4) = 104(x) Simplify. 520 = 104x Solve. x=5 Interpret the solution. In this equation, x represented Eric’s distance from the center of the seesaw. The question asks if it’s possible for Eric to balance with Amber and Aleah. It’s possible if the seesaw is at least 5 feet long from the center, because Eric needs to sit at least 5 feet away from the center. If the seesaw is shorter than 5 feet, then he cannot balance with his sister and his cousin. If the seesaw is 5 feet or longer, then Eric can balance the seesaw. Connection to the Lesson •

Students are asked to create equations much like they will be asked to create inequalities.

•

Solving equations is similar to solving inequalities with a few exceptions.

•

The second question’s response deals with the inequality concept “at least.”

U1-58 CCSS IP Math I Teacher Resource 1.2.2

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Prerequisite Skills This lesson requires the use of the following skills: •

solving simple linear equations

•

comparing rational numbers

Introduction Inequalities are similar to equations in that they are mathematical sentences. They are different in that they are not equal all the time. An inequality has infinite solutions, instead of only having one solution like a linear equation. Setting up the inequalities will follow the same process as setting up the equations did. Solving them will be similar, with two exceptions, which will be described later. Key Concepts •

he prefix in- in the word inequality means “not.” Inequalities are sentences stating that two T things are not equal. Remember earlier inequalities such as 12 > 2 and 1 < 7.

•

Remember that the symbols >, <, ≥, ≤, and ≠ are used with inequalities.

•

se the table below to review the meanings of the inequality symbols and the provided examples U with their solution sets, or the value or values that make a sentence or statement true. Symbol Description > greater than, more than ≥ < ≤ ≠

•

Example Solution set x>3 all numbers greater than 3; does not include 3 greater than or equal to, at least x≥3 all numbers greater than or equal to 3; includes 3 less than x<3 all numbers less than 3; does not include 3 less than or equal to, no more than x ≤ 3 all numbers less than or equal to 3; includes 3 not equal to x≠3 includes all numbers except 3

S olving a linear inequality is similar to solving a linear equation. The processes used to solve inequalities are the same processes that are used to solve equations.

U1-59 © Walch Education

CCSS IP Math I Teacher Resource 1.2.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction •

ultiplying or dividing both sides of an inequality by a negative number requires reversing M the inequality symbol. Here is a number line to show the process. •

First, look at the example of the inequality 2 < 4. 2<4 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0

•

1

2

3

4

5

6

7

8

9 10

x

Multiply both sides by –2 and the inequality becomes 2(–2) < 4(–2) or –4 < –8. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0

1

2

3

4

5

6

7

8

9 10

x

•

Is –4 really less than –8?

•

To make the statement true, you must reverse the inequality symbol: –4 > –8 –4 > –8 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0

1

2

3

4

5

6

7

8

9 10

x

Creating Inequalities from Context 1. 2. 3. 4. 5. 6.

Read the problem statement first. Reread the scenario and make a list or a table of the known quantities. Read the statement again, identifying the unknown quantity or variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

Common Errors/Misconceptions •

n ot translating the words into the correct symbols, especially with the phrases no fewer than, no more than, at least as many, and so on

•

forgetting to switch the inequality symbol when multiplying or dividing by a negative

•

n ot interpreting the solution in terms of the context of the problem—students can often mechanically solve the problem but don’t know what the solution means

U1-60 CCSS IP Math I Teacher Resource 1.2.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Guided Practice 1.2.2 Example 1 Juan has no more than $50 to spend at the mall. He wants to buy a pair of jeans and some juice. If the sales tax on the jeans is 4% and the juice with tax costs $2, what is the maximum price of jeans Juan can afford? 1. Read the problem statement first. 2. Reread the scenario and make a list or a table of the known quantities. Sales tax is 4%. Juice costs $2. Juan has no more than $50. 3. R ead the statement again, identifying the unknown quantity or variable. The unknown quantity is the cost of the jeans. 4. C reate expressions and inequalities from the known quantities and variable(s). The price of the jeans + the tax on the jeans + the price of the juice must be less than or equal to $50. x + 0.04x + 2 ≤ 50

U1-61 © Walch Education

CCSS IP Math I Teacher Resource 1.2.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 5. Solve the problem. x + 0.04x + 2 ≤ 50

Add like terms.

1.04x + 2 ≤ 50

Subtract 2 from both sides.

1.04x ≤ 48

Divide both sides by 1.04.

x ≤ 46.153846 Normally, the answer would be rounded down to 46.15. However, when dealing with money, round up to the nearest whole cent as a retailer would. x ≤ 46.16 6. I nterpret the solution of the inequality in terms of the context of the problem. Juan should look for jeans that are priced at or below $46.16. Example 2 Alexis is saving to buy a laptop that costs $1,100. So far she has saved $400. She makes $12 an hour babysitting. What’s the least number of hours she needs to work in order to reach her goal? 1. Read the problem statement first. 2. Reread the scenario and make a list or a table of the known quantities. Alexis has saved $400. She makes $12 an hour. She needs at least $1,100. 3. R ead the statement again, identifying the unknown quantity or variable. You need to know the least number of hours Alexis must work to make enough money. Solve for hours.

U1-62 CCSS IP Math I Teacher Resource 1.2.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 4. C reate expressions and inequalities from the known quantities and variable(s). Alexis’s saved money + her earned money must be greater than or equal to the cost of the laptop. 400 + 12h ≥ 1100 5. Solve the problem. 400 + 12h ≥ 1100

Subtract 400 from both sides.

12h ≥ 700

Divide both sides by 12.

h ≥ 58.33 6. I nterpret the solution of the inequality in terms of the context of the problem. In this situation, it makes sense to round up to the nearest half hour since babysitters usually get paid by the hour or half hour. Therefore, Alexis needs to work at least 58.5 hours to make enough money to save for her laptop. Example 3 A radio station is giving away concert tickets. There are 40 tickets to start. They give away 1 pair of tickets every hour for a number of hours until they have at most 4 tickets left for a grand prize. If the contest runs from 11:00 a.m. to 1:00 p.m. each day, for how many days will the contest last? 1. Read the problem statement first. 2. Reread the scenario and make a list or a table of the known quantities. The contest starts with 40 tickets. The station gives away 2 tickets every hour. The contest ends with at most 4 tickets left.

U1-63 © Walch Education

CCSS IP Math I Teacher Resource 1.2.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 3. Read the statement again, identifying the unknown quantity or variable(s). For how many days will the contest last? This is tricky because the tickets are given away in terms of hours. First, solve for hours. 4. C reate expressions and inequalities from the known quantities and variable(s). 40 tickets – 2 tickets given away each hour must be less than or equal to 4 tickets. 40 – 2h ≤ 4 5. Solve the problem. 40 – 2h ≤ 4

Subtract 40 from both sides.

–2h ≤ –36

Divide both sides by –2 and switch the inequality symbol.

h ≥ 18 6. I nterpret the solution of the inequality in terms of the context of the problem. The inequality is solved for the number of hours the contest will last. The contest will last at least 18 hours, or 18 hours or more. The problem asks for the number of days the contest will last. If the contest lasts from 11:00 a.m. to 1:00 p.m. each day, that is 3 hours per day. Convert the units. 1 day = 3 hours 18 hours •

1 day

3 hours 1 day 18 hours • = 6 days 3 hours

The contest will run for 6 days or more. U1-64 CCSS IP Math I Teacher Resource 1.2.2

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Problem-Based Task 1.2.2: Free Checking Accounts The time has come for you to open a checking account. A local bank is offering you a free checking account if you maintain a minimum balance of $200. You already have a savings account with this bank and you have $60 saved. You decide to keep saving money until you have enough to open a checking account, plus keep some money in savings. If you deposit $15 a week into the savings account, what is the minimum number of weeks it will take for you to be able to open a checking account with at least $200 and still have $25 in your savings account?

If you deposit $15 a week into the savings account, what is the minimum number of weeks it will take for you to be able to open a checking account with at least $200 and still have $25 in your savings account?

U1-65 © Walch Education

CCSS IP Math I Teacher Resource 1.2.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Problem-Based Task 1.2.2: Free Checking Accounts Coaching a. What is the minimum amount of money you want to have to open your checking account?

b. How much are you depositing each week?

c. What inequality can be written to model the scenario?

d. What is the solution to the inequality?

e. What does the solution of the inequality represent in terms of the context of the problem?

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© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Problem-Based Task 1.2.2: Free Checking Accounts Coaching Sample Responses a. What is the minimum amount of money you want to have to open your checking account? $225 Students at first might be inclined to write $200 because this is the minimum amount needed for a free checking account, but the problem also states that there needs to be $25 left in the savings account. b. How much are you depositing each week? $15/week This is given in the problem. c. What inequality can be written to model the scenario? 60 + 15x ≥ 225 You start with $60 and deposit $15 each week and need at least $225. d. What is the solution to the inequality? 60 + 15x ≥ 225

Subtract 60 from both sides.

15x ≥ 165

Divide each side by 15.

x ≥ 11 e. What does the solution of the inequality represent in terms of the context of the problem? It will take at least 11 weeks (11 weeks or more) to reach the savings goal of having $200 to open the checking account while still having $25 in the savings account. Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

U1-67 © Walch Education

CCSS IP Math I Teacher Resource 1.2.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Practice 1.2.2: Creating Linear Inequalities in One Variable Translate each phrase into an algebraic inequality. 1. A tour bus can seat 55 passengers.

2. An energy-efficient lamp can only be used with light bulbs that are 60 watts or less.

Read each scenario, write an inequality to model the scenario, and then use the inequality to solve the problem. 3. C amilla is saving to purchase a new pair of bowling shoes that will cost at least $39. She has already saved $19. What is the least amount of money she needs to save for the shoes?

4. S uppose you earn $20 per hour working part time at a tax office. You want to earn at least $1,800 this month, before taxes. How many hours must you work?

5. H iram earned a score of 83 on his semester algebra test. He needs to have a total of at least 180 points from his semester and final tests to receive an A for his grade. What score must Hiram earn on his final test to ensure his A?

6. C laire purchases DVDs from an online entertainment store. Each DVD that she orders costs $15 and shipping for her order is $10. Claire can spend no more than $100. How many DVDs can Claire purchase?

continued U1-68 CCSS IP Math I Teacher Resource 1.2.2

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

7. T he temperature outside is dropping at a rate of 1.5 degrees each hour. The weather station predicts a changeover from rain to freezing rain when the temperature reaches 41°F or below. Currently the temperature is 53°F and the time is 8:30 a.m. When do you predict the changeover to freezing rain will occur?

8. A recreation center holds a soccer game every Saturday morning for older teens. The group agreed that there should be at least 5 players on each team. One team started out with 17 players. After an hour of playing, 3 players started leaving every 5 minutes. For at least how long can the team keep playing?

9. A website has no more than $15,000 to give away for a contest. They have decided to give away $1,000 twice a day every day until they have at least $3,000 left to award as a grand prize. How many days will the contest run?

For problem 10, create your own context for the given inequality, and then solve the inequality. Be sure to express your solution in terms of the context of the problem. 10. 2x – 5 ≤ 9

U1-69 © Walch Education

CCSS IP Math I Teacher Resource 1.2.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Lesson 1.2.3: Creating Exponential Equations Warm-Up 1.2.3 Read the scenario and follow the directions. This year, Zachary has been babysitting his young cousins after school for $70 a month. His uncle also gave him an extra bonus of $100 for his excellent work. Since school started, Zachary has earned more than $500. How many months ago did school start? Write an inequality that represents this situation. Solve it and show all your work.

U1-70 CCSS IP Math I Teacher Resource 1.2.3

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Lesson 1.2.3: Creating Exponential Equations Common Core State Standard A–CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

Warm-Up 1.2.3 Debrief This year, Zachary has been babysitting his young cousins after school for $70 a month. His uncle also gave him an extra bonus of $100 for his excellent work. Since school started, Zachary has earned more than $500. How many months ago did school start? Write an inequality that represents this situation. Solve it and show all your work. •

S tudents will hopefully recognize this scenario as modeling a linear inequality: 70x + 100 > 500, where x is the number of months Zachary has been babysitting.

•

Solving this equation for x gives x > 5 months.

•

E ncourage students to make meaning of the numerical answer, “School started more than 5 months ago.”

Connection to the Lesson •

S tudents will be required to create equations from context just as in the warm-up, except the equations will be exponential instead of linear.

•

S tudents will also be required to make meaning of the solution in the upcoming lesson rather than just stopping at solving the equation for the numerical answer.

U1-71 © Walch Education

CCSS IP Math I Teacher Resource 1.2.3

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Prerequisite Skills This lesson requires the use of the following skills: •

working with exponents (raising a base to a power)

•

applying the order of operations

Introduction Exponential equations are equations that have the variable in the exponent. Exponential equations are found in science, finance, sports, and many other areas of daily living. Some equations are complicated, but some are not. Key Concepts •

he general form of an exponential equation is y = a • b x, where a is the initial value, b is the T base, and x is the time. The final output value will be y.

•

Since the equation has an exponent, the value increases or decreases rapidly.

•

The base, b, must always be greater than 0 (b > 0).

•

I f the base is greater than 1 (b > 1), then the exponential equation represents exponential growth.

•

I f the base is between 0 and 1 (0 < b < 1), then the exponential equation represents exponential decay.

•

If the time is given in units other than 1 (e.g., 1 month, 1 hour, 1 minute, 1 second), use the x

equation y = ab t , where t is the time it takes for the base to repeat. •

nother form of the exponential equation is y = a(1 ± r) t, where a is the initial value, r is the A rate of growth or decay, and t is the time.

•

se y = a(1 + r)t for exponential growth (notice the plus sign). For example, if a population U grows by 2% then r is 0.02, but this is less than 1 and by itself does not indicate growth.

•

S ubstituting 0.02 for b into the formula y = a • b x requires the expression (1 + r) to arrive at the full growth rate of 102%, or 1.02.

•

se y = a(1 – r) t for exponential decay (notice the minus sign). For example, if a population U decreases by 3%, then 97% is the factor being multiplied over and over again. The population from year to year is always 97% of the population from the year before (a 3% decrease). Think of this as 100% minus the rate, or in decimal form (1 – r).

U1-72 CCSS IP Math I Teacher Resource 1.2.3

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction •

L ook for words such as double, triple, half, quarter—such words give the number of the base. For example, if an experiment begins with 1 bacterium that doubles (splits itself in two) every hour, determining how many bacteria will be present after x hours is solved with the following equation: y = (1)2 x, where 1 is the starting value, 2 is the rate, x is the number of hours, and y is the final value.

•

Look for the words initial or starting to substitute in for a.

•

Look for the words ended with and after—these words will be near the final value given.

•

Follow the same procedure as with setting up linear equations and inequalities in one variable: Creating Exponential Equations from Context 1. Read the problem statement first. 2. Reread the scenario and make a list or a table of the known quantities. 3. Read the statement again, identifying the unknown quantity or variable. 4. Create expressions and inequalities from the known quantities and variable(s). 5. Solve the problem. 6. I nterpret the solution of the exponential equation in terms of the context of the problem.

Common Errors/Misconceptions •

multiplying a and b together before raising to the power

•

misidentifying the base

•

creating a linear equation instead of an exponential equation

U1-73 © Walch Education

CCSS IP Math I Teacher Resource 1.2.3

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Guided Practice 1.2.3 Example 1 A population of mice quadruples every 6 months. If a mouse nest started out with 2 mice, how many mice would there be after 2 years? Write an equation and then use it to solve the problem. 1. R ead the scenario and then reread it again, this time identifying the known quantities. The initial number of mice = 2. The base = quadruples, so that means 4. The amount of time = every 6 months for 2 years. 2. R ead the statement again, identifying the unknown quantity or variable. The unknown quantity is the number of mice after 2 years. Solve for the final amount of mice after 2 years. 3. C reate expressions and equations from the known quantities and variable(s). The general form of the exponential equation is y = a • b x, where y is the final value, a is the initial value, b is the base, and x is the time. a=2 b=4 x = every 6 months for 2 years Since the problem is given in months, you need to convert 2 years into 6-month time periods. How many 6-month time periods are there in 2 years? To determine this, think about how many 6-month time periods there are in 1 year. There are 2. Multiply that by 2 for each year. Therefore, there are four 6-month time periods in 2 years. (continued)

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Another way to determine the period is to set up ratios. 2 years •

12 months 1 year

24 months •

= 24 months

1 time period 6 months

= 4 time periods

Therefore, x = 4. 4. Substitute the values into the general form of the equation y = a • b x. x

y = a • b

OR

x

y = ab t 24

y = (2) • (4) OR 4

y =(2)(4) 6

5. Follow the order of operations to solve the problem. y = (2) • (4)4

Raise 4 to the 4th power.

y = (2) • 256

Multiply 2 and 256.

y = 512 6. Interpret the solution in terms of the context of the problem. There will be 512 mice after 2 years if the population quadruples every 6 months.

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CCSS IP Math I Teacher Resource 1.2.3

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Example 2 In sporting tournaments, teams are eliminated after they lose. The number of teams in the tournament then decreases by half with each round. If there are 16 teams left after 3 rounds, how many teams started out in the tournament? 1. R ead the scenario and then reread it again, this time identifying the known quantities. The final number of teams = 16. 1 The reduction = . 2 The amount of time = 3 rounds. 2. Read the statement again, identifying the unknown quantity or variable. The unknown quantity is the number of teams with which the tournament began. Solve for the initial or starting value, a. 3. C reate expressions and equations from the known quantities and variable(s). The general form of the exponential equation is y = a • b x, where y is the final value, a is the initial value, b is the rate of decay or growth, and x is the time. y = 16 1 b= 2 x = 3 rounds 4. Substitute the values into the general form of the equation y = a • b x. y = a • b x 1 16 = a • 2

3

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 5. Follow the order of operations to solve the problem. 1 16 = a • 2 16 = a •

3

1 8

Raise the base to the power of 3. Multiply by the reciprocal.

a = 128 6. Interpret the solution in terms of the context of the problem. The tournament started with 128 teams. Example 3 The population of a small town is increasing at a rate of 4% per year. If there are currently about 6,000 residents, about how many residents will there be in 5 years at this growth rate? 1. R ead the scenario and then reread it again, this time identifying the known quantities. The initial number of residents = 6,000. The growth = 4%. The amount of time = 5 years. 2. Read the statement again, identifying the unknown quantity or variable. The unknown quantity is the number of residents after 5 years. Solve for the final value after 5 years.

U1-77 © Walch Education

CCSS IP Math I Teacher Resource 1.2.3

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 3. C reate expressions and equations from the known quantities and variable(s). The general form of the exponential growth equation with a percent increase is y = a (1 + r) t, where y is the final value, a is the initial value, r is the rate of growth, and t is the amount of time. a = 6000 r = 4% = 0.04 t = 5 years 4. Substitute the values into the general form of the equation y = a(1 + r)t. y = a(1 + r)t y = 6000(1 + 0.04)5 5. Follow the order of operations to solve the problem. y = 6000(1 + 0.04)5

Add inside the parentheses first.

y = 6000(1.04)5

Raise the base to the power of 5.

y = 6000(1.21665)

Multiply.

y ≈ 7300 6. Interpret the solution in terms of the context of the problem. If this growth rate continues for 5 years, the population will increase by more than 1,000 residents to about 7,300 people.

U1-78 CCSS IP Math I Teacher Resource 1.2.3

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Example 4 You want to reduce the size of a picture to place in a small frame. You aren’t sure what size to choose on the photocopier, so you decide to reduce the picture by 15% each time you scan it until you get it to the size you want. If the picture was 10 inches long at the start, how long is it after 3 scans? 1. R ead the scenario and then reread it again, this time identifying the known quantities. The initial length = 10 inches. The reduction = 15% = 0.15. The amount of time = 3 scans. 2. Read the statement again, identifying the unknown quantity or variable. The unknown quantity is the length of the picture after 3 scans. Solve for the final value after 3 scans. 3. C reate expressions and equations from the known quantities and variable(s). The general form of the exponential growth equation with a percent decrease is y = a (1 – r) t , where y is the final value, a is the initial value, r is the rate of decay, and t is the amount of time. a = 10 r = 15% = 0.15 t = 3 scans 4. Substitute the values into the general form of the equation y = a (1 – r) t. y = a (1 – r) t y = 10(1 – 0.15)3

U1-79 © Walch Education

CCSS IP Math I Teacher Resource 1.2.3

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction 5. Follow the order of operations to solve the problem. y = 10(1 – 0.15)3

Subtract inside the parentheses first.

y = 10(0.85)3

Raise the base to the power of 3.

y = 10(0.614125)

Multiply.

y ≈ 6.14 6. Interpret the solution in terms of the context of the problem. After 3 scans, the length of the picture is about 6 inches.

U1-80 CCSS IP Math I Teacher Resource 1.2.3

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Problem-Based Task 1.2.3: Population Change On opposite sides of a major city two suburban towns are experiencing population changes. One town, Town A, is growing rapidly at 5% per year and has a current population of 39,000. Town B has a declining population at a rate of 2% per year. Its current population is 55,000. Economists predict that in 5 years the populations of these two towns will be about the same, but the residents of both towns are in disbelief. The economists also claim that ten years after that, Town A will double the size of Town B. Can you verify the predictions based on the data given? Do you think these predictions will come true?

Can you verify the predictions based on the data given? Do you think these predictions will come true?

U1-81 © Walch Education

CCSS IP Math I Teacher Resource 1.2.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Problem-Based Task 1.2.3: Population Change Coaching a. I s Town A experiencing population growth or decay? What is the equation for Town A’s population change after 5 years?

b. What is the solution to the equation in part a?

c. I s Town B experiencing population growth or decay? What is the equation for Town B’s population change after 5 years?

d. What is the solution to the equation in part c?

e. A re your solutions to parts b and d similar? What can you conclude about the economists’ prediction about the populations of the two towns being about the same after 5 years?

f. W hat will the variable, t, equal if the towns experience the same rates of growth or decline for 10 more years after that?

g. What are the equations for each town’s population change?

h. B ased on your calculations, was the economists’ prediction for the town populations after 10 more years correct?

i. What factors might influence whether or not the economists’ predictions come true?

U1-82 CCSS IP Math I Teacher Resource 1.2.3

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction Problem-Based Task 1.2.3: Population Change Coaching Sample Responses a. I s Town A experiencing population growth or decay? What is the equation for Town A’s population change after 5 years? Town A is experiencing growth. Fill in the information we know to find the equation: y = a(1 + r)t •

a = 39,000 people

•

r = 5% = 0.05

•

t = 5 years

y = 39,000(1 + 0.05)5 is the equation. b. What is the solution to the equation in part a? y = a(1 + r)t y = 39,000(1 + 0.05)5 y ≈ 49,775 people Round up to the nearest whole person. c. I s Town B experiencing population growth or decay? What is the equation for Town B’s population change after 5 years? Town B is experiencing decay. Fill in the information we know to find the equation: y = a(1 – r)t •

a = 55,000 people

•

r = 2% = 0.02

•

t = 5 years

y = 55,000(1 – 0.02)5 is the equation. d. What is the solution to the equation in part c? y = a(1 – r)t y = 55,000(1 – 0.02)5 y ≈ 49,716 people

U1-83 © Walch Education

CCSS IP Math I Teacher Resource 1.2.3

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable Instruction e. A re your solutions to parts b and d similar? What can you conclude about the economists’ prediction about the populations of the two towns being about the same after 5 years? The numbers are similar, especially when considering the magnitude. If rounded to the nearest 100 people, Town A would have 49,800 and Town B would have 49,700 people. A difference in 100 people compared to tens of thousands is a small number. Based on these calculations, it would seem that the economists are correct that the populations will be about equal in 5 years. f. W hat will the variable, t, equal if the towns experience the same rates of growth or decline for 10 more years after that? The amount of time passed since the initial population count will be 5 years plus 10, which equals 15 years. t = 5 + 10 t = 15 g. What are the equations and solutions for each town’s population change? The equations below use the town’s current populations, with t = 15. Alternatively, students could use the value t = 10 and the population values they calculated at the 5-year mark for a. Town A

Town B

y = a(1 + r)t

y = a(1 – r)t

y = 39,000(1 + 0.05)15

y = 55,000(1 – 0.02)15

y ≈ 81,079 people

y ≈ 40,622 people

h. B ased on your calculations, was the economists’ prediction for the town populations after 10 more years correct? Yes, 81,079 is about twice as much as 40,622. i. What factors might influence whether or not the economists’ predictions come true? Many factors might contribute to the economists’ predictions coming true or not, such as the economy itself. If the economy starts to decline, the population of Town A might not continue to grow as fast. Or, a new industry might relocate to Town B, bringing jobs, and perhaps people will start moving back into that town. Or, Town A might reach its capacity before it hits 81,000 people. Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt. U1-84 CCSS IP Math I Teacher Resource 1.2.3

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Practice 1.2.3: Creating Exponential Equations Use what you know about linear and exponential equations to complete problems 1 and 2. 1. Determine whether each scenario can be modeled by a linear or an exponential equation. a. The price of a loaf of bread increases by $0.25 each week.

b. Each week, a loaf of bread costs twice as much as it did the week before.

2. Determine whether each scenario can be modeled by a linear or an exponential equation. a. 10 people leave a football game every minute after the third quarter.

b.

1 4

of the people leave a football game every minute after the third quarter.

For problems 3–10, write an equation to model each scenario. Then use the equation to solve the problem. 3. A population of insects doubles every month. If there are 100 insects to start with, how many will there be after 7 months?

4. A type of bacteria in a Petri dish doubles every hour. If there were 1,073,741,824 bacteria after 24 hours, how many were there to start with?

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CCSS IP Math I Teacher Resource 1.2.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

5. A stock loses half its value every week. If the stock was worth $125 starting out, what is it worth after 4 weeks at this rate of decline?

6. A computer depreciates (loses its value) at a rate of

1

the original value every 2 years. If the 2 computer now costs $75 after 6 years, how much did it cost when you bought it new?

7. A new car depreciates as soon as you drive it out of the parking lot. A certain car depreciates to half its original value in 4 years. If you bought a car 8 years ago and it is now worth $5,000, how much did you pay for the car originally?

8. T he number of dandelions growing on your lawn triples every 3 days. If your lawn started out with 15 dandelions, how many dandelions would you have after 3 weeks?

9. T he population of a town is increasing by 2% each year. The current population is 12,000. How many people will there be in 4 years?

10. T he population of New York City doubles during the workday. At the end of the workday, the population decreases by half. If the population in New York City is roughly 4,000,000 people before rush hour, and the morning rush hour lasts 3 hours, what would be the population if the rush-hour growth rate continued for 12 hours instead of 3? People leave the city for home at the same rate as they come into the city. What would be the population if the decay rate continued for 12 hours at the end of the day when everyone goes home?

U1-86 CCSS IP Math I Teacher Resource 1.2.3

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Assessment Progress Assessment Circle the letter of the best answer. 1. Y ou are redecorating your room and the only thing left to paint is your door. You’re only going to paint the side that faces the inside of your room. The door is 6 feet 8 inches tall and 30 inches wide. You need to know the surface area of the side of the door to determine how much paint to buy. The hardware store sells paint by how much covers a square foot. What is the surface area you should report to the hardware store? (Hint: 1 square foot = 144 square inches) a. about 16 square feet

c. 16.7 square feet

b. 16.67 square feet

d. about 17 square feet

2. Y ou are participating in a fund-raiser in which you read for donations. People can donate money based on a flat fee or based on the number of minutes you read. So far, you have two donors. Your grandma has agreed to donate $10 and your mom has agreed to donate $0.05 per minute. If together they donated $22.35, what equation represents this situation? a. (10 + 0.05)x = 22.35

c. 0.05x + 10 = 22.35

b. 10x + 0.05 = 22.35

d. 22.35x = 10 + 0.05

3. T wo friends live 253 miles apart. They drive toward each other to meet for a weekend. The first friend drives the speed limit and it takes her 2 hours to meet her friend. The second friend leaves 15 minutes late, so she drives 5 miles per hour over the speed limit. What was the second friend’s speed? a. about 55 mph

c. about 65 mph

b. about 60 mph

d. about 70 mph

4. A carafe at a banquet holds about 12 cups of coffee. When the carafe has 1 cup or less of coffee left, the waiter dumps the coffee out and pours in a fresh pot. Each coffee mug at the banquet holds 6 ounces of coffee. Assuming each person fills his or her mug, what inequality represents the number of people who can fill their mugs before the carafe needs to be refilled? (Hint: 1 cup = 8 ounces) a. 12 – 0.75x ≤ 1

c. 12 – 0.75x ≥ 1

b. 12 – x ≤ 1

d. 12 – x ≥ 1

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CCSS IP Math I Teacher Resource 1.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Assessment 5. G eraldine brings a large printing job to her local print shop. She doesn’t want to sit and wait for the job to completed, so she asks when she should come back to pick up her order. The machines print 30 pages per minute and 225 pages printed while she paid for her order. She has less than 1,000 pages left to go. How much longer will it take her order to print? a. more than about 33 minutes

c. less than about 33 minutes

b. more than about 41 minutes

d. less than about 41 minutes

6. J uan is saving for a new computer and has saved $213. He is saving $3 a day and needs at least $654 for the computer he wants. How many weeks will it take him to save for the computer he wants? a. at most 147 weeks

c. at least 147 weeks

b. at most 21 weeks

d. at least 21 weeks

7. A new grocery store is giving away four $20 gift certificates to the store every hour to celebrate its grand opening. The store manager wants to save at least 24 of the certificates to give to a local food bank. If there are 120 gift certificates to start with, for how many hours should the store continue giving them away? a. greater than or equal to 24 hours

c. less than 24 hours

b. less than or equal to 24 hours

d. greater than 24 hours

8. A certain type of bacteria doubles every 7 hours. If you started with 36 bacteria, how many would you have after 28 hours? a. 9,663,676,416 bacteria

c. 26,873,856 bacteria

b. 576 bacteria

d. 4,608 bacteria

9. A soccer tournament starts out with 128 teams. After each round, half of the teams are eliminated. After 6 rounds, how many teams are left? a. 2 teams

c. 8 teams

b. 4 teams

d. 16 teams

continued U1-88 CCSS IP Math I Teacher Resource 1.2

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 2: Creating Equations and Inequalities in One Variable

Date:

Assessment 10. A rural town is declining in population at a rate of 2% per year. If the current population is 7,000 people, what will the population be in 6 years? a. about 6,201 people

c. about 1,835 people

b. about 7,884 people

d. about 6,160 people

Read the scenario, write an equation to model the scenario, and then use the equation to solve the problem. 11. Y our car broke down, and the final bill was $261.50. The part that was replaced cost $99, and the charge for the mechanic’s labor is $65 per hour. Write an equation to model this situation, then solve the equation for the number of hours the mechanic worked on your car.

U1-89 © Walch Education

CCSS IP Math I Teacher Resource 1.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment Pre-Assessment Circle the letter of the best answer. 1. I t costs $80 to buy an air conditioner and about $0.40 per minute to run it. Which equation models the total cost of using an air conditioner? a. x + y = 80.40

c. y = 80x + 0.40

b. y = 80.40x

d. y = 0.40x + 80

2. A ringtone company charges $10 a month for the service plus $1.50 for each ringtone downloaded. What is the graph of the equation that models the total fees? a.

30

y

c.

28

100

24

90

22

80

Fee in dollars ($)

Fee in dollars ($)

26

20 18 16 14 12 10 8 6

70 60 50 40 30 20

4

10

2 0

y

110

0

1

2

3

4

5

6

7

8

9

x 10

0

0

1

Ringtones downloaded

b.

90

d.

15

4

5

6

7

9

x 10

9

x 10

8

y

14 13 12

Fee in dollars ($)

70

Fee in dollars ($)

3

Ringtones downloaded

y

80

60 50 40 30 20

11 10 9 8 7 6 5 4 3 2

10 0

2

0

1

2

3

4

5

6

7

8

Ringtones downloaded

9

x 10

1 0

0

1

2

3

4

5

6

7

8

Ringtones downloaded

continued U1-90 CCSS IP Math I Teacher Resource 1.3

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 3. A 12-inch candle burns at a rate of 2 inches per hour. What is the graph of the equation that models the height of the candle over time? a.

15

y

c.

13

12

12

11

11

10 9 8 7 6 5 4

10 9 8 7 6 5 4

3

3

2

2

1 0

1

x 0

1

2

3

4

5

6

y

14

13

Height in inches

Height in inches

14

15

0

7

x 0

1

2

3

Minutes

b.

28

y

d.

12

20

Height in inches

Height in inches

13

18 16 14 12 10 8

11 10 9 8 7 6 5 4

6

3

4

2

2

x 2

3

4

7

14

22

1

6

5

6

7

y

15

24

0

5

Minutes

26

0

4

1 0

x 0

1

2

3

4

5

6

Minutes

Minutes

4. A town’s population increases at a rate of 2.3% every year. The current population is 7,500 people. Which equation models this scenario? a. y = 7500(1.23) x

c. y = 7500(0.023) x

b. y = 7500(1.023) x

d. y = 7500(0.23) x

continued U1-91 © Walch Education

CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 5. A n investment of $900 earns 3% interest and is compounded semi-annually. Which graph models the worth of the investment over time? y

930 928 926 924 922 920 918 916 914 912 910 908 906 904 902 900

c. Investment worth in dollars ($)

Investment worth in dollars ($)

a.

0

1

2

3

4

5

6

7

8

9

x 10

950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 50 0

y

0

1

2

y

d.

2850

1200

Investment worth in dollars ($)

Investment worth in dollars ($)

3000 2700 2550 2400 2250 2100 1950 1800 1650 1500 1350 1200 1050 900 0

1

2

4

5

Years

Years

b.

3

3

Years

4

x 5

y

1175 1150 1125 1100 1075 1050 1025 1000 975 950 925 900 0

1

2

3

4

5

6

7

8

9

x 10

Years

U1-92 CCSS IP Math I Teacher Resource 1.3

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES

Lesson 3: Creating and Graphing Equations in Two Variables Instruction Common Core State Standards A–CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

N–Q.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.★

Essential Questions 1. 2. 3. 4.

What do the graphs of equations in two variables represent? How do you determine the scales to use for the x- and y-axes on any given graph? How do the graphs of linear equations and exponential equations differ? How are they similar? How can graphing equations help you to make decisions?

WORDS TO KNOW coordinate plane dependent variable exponential decay

exponential equation

a set of two number lines, called the axes, that intersect at right angles labeled on the y-axis; the quantity that is based on the input values of the independent variable an exponential equation with a base, b, that is between 0 and 1 (0 < b < 1); can be represented by the formula y = a(1 – r) t, where a is the initial value, (1 – r) is the decay rate, t is time, and y is the final value an equation that has a variable in the exponent; the general form is y = a • bx, where a is the initial value, b is the base, x is the time, and y x

exponential growth

independent variable linear equation

is the final output value. Another form is y = ab t , where t is the time it takes for the base to repeat. an exponential equation with a base, b, greater than 1 (b > 1); can be represented by the formula y = a(1 + r)t, where a is the initial value, (1 + r) is the growth rate, t is time, and y is the final value labeled on the x-axis; the quantity that changes based on values chosen an equation that can be written in the form ax + by = c, where a, b, and c are rational numbers; can also be written as y = mx + b, in which m is the slope, b is the y-intercept, and the graph is a straight line

U1-93 © Walch Education

CCSS IP Math I Teacher Resource 1.3

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction slope

x-intercept y-intercept

the measure of the rate of change of one variable with respect to another y2 − y1 ∆ y rise variable; slope = = = x2 − x1 ∆ x run the point at which the line intersects the x-axis at (x, 0) the point at which the line intersects the y-axis at (0, y)

Recommended Resources •

Math-Play.com. “Hoop Shoot.” http://walch.com/rr/CAU1L3SlopeandIntercept This one- or two-player game includes 10 multiple-choice questions about slope and y-intercept. Correct answers result in a chance to make a 3-point shot in a game of basketball.

•

Oswego City School District Regents Exam Prep Center. “Equations and Graphing.” http://walch.com/rr/CAU1L3GraphLinear This site contains a thorough summary of the methods used to graph linear equations.

•

Ron Blond Mathematics Applets. “The Exponential Function y = ab x.” http://walch.com/rr/CAU1L3ExponentialFunction This applet provides sliders for the variables a and b, and shows how changing the values of these variables results in changes in the graph.

U1-94 CCSS IP Math I Teacher Resource 1.3

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Lesson 1.3.1: Creating and Graphing Linear Equations in Two Variables Warm-Up 1.3.1 Read the information that follows and use it to complete the problems. A cell phone company charges a $20 flat fee plus $0.05 for every minute used for calls. 1. M ake a table of values from 0 to 60 minutes in 10-minute intervals that represent the total amount charged.

2. Write an algebraic equation that could be used to represent the situation.

3. What do the unknown values in your equation represent?

U1-95 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Lesson 1.3.1: Creating and Graphing Linear Equations in Two Variables Common Core State Standards A–CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

N–Q.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.★

Warm-Up 1.3.1 Debrief A cell phone company charges a $20 flat fee plus $0.05 for every minute used for calls. 1. M ake a table of values from 0 to 60 minutes in 10-minute intervals that represent the total amount charged. Minutes used 0 10 20 30 40 50 60

Total amount charged ($) 20 + 0(0.05) = 20.00 20 + 10(0.05) = 20.50 20 + 20(0.05) = 21.00 20 + 30(0.05) = 21.50 20 + 40(0.05) = 22.00 20 + 50(0.05) = 22.50 20 + 60(0.05) = 23.00

2. Write an algebraic equation that could be used to represent the situation. y = 0.05x + 20 3. What do the unknown values in your equation represent? x represents the number of minutes used, and y represents the total amount charged. Connection to the Lesson •

S tudents will be creating equations just like these in the upcoming lesson but will be given the option of skipping the step of creating the table of values.

•

Students gain exposure to working with input and output pairs in the warm-up.

•

Students will take this type of problem a step further and graph the equation.

U1-96 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Prerequisite Skills This lesson requires the use of the following skills: •

plotting points in all four quadrants

•

understanding slope as a rate of change

Introduction Many relationships can be represented by linear equations. Linear equations in two variables can be written in the form y = mx + b, where m is the slope and b is the y-intercept. The slope of a linear graph is a measure of the rate of change of one variable with respect to another variable. The y-intercept of the equation is the point at which the graph crosses the y-axis and the value of x is zero. Creating a linear equation in two variables from context follows the same procedure at first for creating an equation in one variable. Start by reading the problem carefully. Once you have created the equation, the equation can be graphed on the coordinate plane. The coordinate plane is a set of two number lines, called the axes, that intersect at right angles. Key Concepts Reviewing Linear Equations: •

he slope of a linear equation is also defined by the ratio of the rise of the graph compared to T the run. Given two points on a line, (x1, y1) and (x2, y2), the slope is the ratio of the change in the y-values of the points (rise) to the change in the corresponding x-values of the points (run). slope =

•

y − y1 rise = 2 run x2 − x1

he slope-intercept form of an equation of a line is often used to easily identify the slope T and y-intercept, which then can be used to graph the line. The slope-intercept form of an equation is shown below, where m represents the slope of the line and b represents the y-value of the point where the line intersects the y-axis at point (0, y). y = mx + b

•

Horizontal lines have a slope of 0. They have a run but no rise. Vertical lines have no slope.

•

The x-intercept of a line is the point where the line intersects the x-axis at (x, 0).

•

If a point lies on a line, its coordinates make the equation true.

U1-97 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction •

he graph of a line is the collection of all points that satisfy the equation. The graph of the T linear equation y = –2x + 2 is shown, with its x- and y-intercepts plotted. 5

y

4 3 2 1 -5

-4

-3

-2

-1

0

1

2

3

4

x 5

-1 -2 -3 -4 -5

Creating Equations 1. Read the problem statement carefully before doing anything. 2. Look for the information given and make a list of the known quantities. 3. D etermine which information tells you the rate of change, or the slope, m. Look for words such as each, every, per, or rate. 4. D etermine which information tells you the y-intercept, or b. This could be an initial value or a starting value, a flat fee, and so forth. 5. Substitute the slope and y-intercept into the linear equation formula, y = mx + b. Determining the Scale and Labels When Graphing: •

I f the slope has a rise and run between –10 and 10 and the y-intercept is 10 or less, use a grid that has squares equal to 1 unit.

•

djust the units according to what you need. For example, if the y-intercept is 10,000, each A square might represent 2,000 units on the y-axis. Be careful when plotting the slope to take into account the value each grid square represents.

U1-98 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction •

S ometimes you need to skip values on the y-axis. It makes sense to do this if the y-intercept is very large (positive) or very small (negative). For example, if your y-intercept is 10,000, you could start your y-axis numbering at 0 and “skip” to 10,000 at the next y-axis number. Use a short, zigzag line starting at 0 to about the first grid line to show that you’ve skipped values. Then continue with the correct numbering for the rest of the axis. For an illustration, see Guided Practice Example 3, step 4.

•

nly use x- and y-values that make sense for the context of the problem. Ask yourself if O negative values make sense for the x-axis and y-axis labels in terms of the context. If negative values don’t make sense (for example, time and distance can’t have negative values), only use positive values.

•

Determine the independent and dependent variables.

•

he independent variable will be labeled on the x-axis. The independent variable is the T quantity that changes based on values you choose.

•

he dependent variable will be labeled on the y-axis. The dependent variable is the quantity T that is based on the input values of the independent variable.

Graphing Equations Using a Table of Values Using a table of values works for any equation when graphing. For an example, see Guided Practice Example 1, step 7. 1. Choose inputs or values of x. 2. Substitute those values in for x and solve for y. 3. The result is an ordered pair (x, y) that can be plotted on the coordinate plane. 4. Plot at least 3 ordered pairs on the line. 5. Connect the points, making sure that they lie in a straight line. 6. A dd arrows to the end(s) of the line to show when the line continues infinitely (if continuing infinitely makes sense in terms of the context of the problem). 7. Label the line with the equation.

U1-99 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Graphing Equations Using the Slope and y-intercept For an example, see Guided Practice Example 2, step 6. 1. Plot the y-intercept first. The y-intercept will be on the y-axis. rise 2. Recall that slope is . Change the slope into a fraction if you need to. run 3. To find the rise when the slope is positive, count up the number of units on your coordinate 3 plane the same number of units in your rise. (So, if your slope is , you count up 3 on 5 the y-axis.) 4. For the run, count over to the right the same number of units on your coordinate plane in your 3 run, and plot the second point. (For the slope , count 5 to the right and plot your point.) 5 5. To find the rise when the slope is negative, count down the number of units on your coordinate plane the same number of units in your rise. For the run, you still count over to the right the same number of units on your coordinate plane in your run and plot the second point. (For a 4 slope of - , count down 4, right 7, and plot your point.) 7 6. Connect the points and place arrows at one or both ends of the line when it makes sense to have arrows within the context of the problem. 7. Label the line with the equation. Graphing Equations Using a TI-83/84: Step 1: Press [Y=] and key in the equation using [X, T, θ, n] for x. Step 2: Press [WINDOW] to change the viewing window, if necessary. Step 3: Enter in appropriate values for Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl, using the arrow keys to navigate. Step 4: Press [GRAPH].

U1-100 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Graphing Equations Using a TI-Nspire: Step 1: Press the home key. Step 2: Arrow over to the graphing icon (the picture of the parabola or the U-shaped curve) and press [enter]. Step 3: At the blinking cursor at the bottom of the screen, enter in the equation and press [enter]. Step 4: To change the viewing window: press [menu], arrow down to number 4: Window/ Zoom, and click the center button of the navigation pad. Step 5: Choose 1: Window settings by pressing the center button. Step 6: Enter in the appropriate XMin, XMax, YMin, and YMax fields. Step 7: Leave the XScale and YScale set to auto. Step 8: Use [tab] to navigate among the fields. Step 9: Press [tab] to “OK” when done and press [enter].

Common Errors/Misconceptions •

switching the slope and y-intercept when creating the equation from context

•

switching the x- and y-axis labels

•

incorrectly graphing the line with the wrong y-intercept or the wrong slope

U1-101 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Guided Practice 1.3.1 Example 1 A local convenience store owner spent $10 on pencils to resell at the store. What is the equation of the store’s revenue if each pencil sells for $0.50? Graph the equation. 1. R ead the problem and then reread the problem, determining the known quantities. Initial cost of pencils: $10 Charge per pencil: $0.50 2. Identify the slope and the y-intercept. The slope is a rate. Notice the word “each.” Slope = 0.50 The y-intercept is a starting value. The store paid $10. The starting revenue then is –$10. y-intercept = –10 3. S ubstitute the slope and y-intercept into the equation y = mx + b, where m is the slope and b is the y-intercept. m = 0.50 b = –10 y = 0.50x – 10 4. Change the slope into a fraction in preparation for graphing. 0.50 =

50

=

1

100 2

U1-102 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 5. Rewrite the equation using the fraction. 1 y = x − 10 2 6. S et up the coordinate plane and identify the independent and dependent variables. In this scenario, x represents the number of pencils sold and is the independent variable. The x-axis label is “Number of pencils sold.” The dependent variable, y, represents the revenue the store will make based on the number of pencils sold. The y-axis label is “Revenue in dollars ($).” Determine the scales to be used. Since the slope’s rise and run are within 10 units and the y-intercept is –10 units, a scale of 1 on each axis is appropriate. Label the x-axis from 0 to 10 since you will not sell a negative amount of pencils. Label the y-axis from –15 to 15, to allow space to plot the $10 the store owner paid for the pencils (–10). 15

y

14 13 12 11 10 9 8 7

Revenue in dollars ($)

6 5 4 3 2 1 0 -1

x 1

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10

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15

Number of pencils sold

U1-103 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 7. Plot points using a table of values.

1 Substitute x values into the equation y = x − 10 and solve for y. 2 Choose any values of x to substitute. Here, it’s easiest to use values of 1 x that are even since after substituting you will be multiplying by . 2 Using even-numbered x values will keep the numbers whole after you multiply. x

y 1

0

2

(0) − 10 = −10

2

–9

4

–8

6

–7 15

y

14 13 12 11 10 9 8 7

Revenue in dollars ($)

6 5 4 3 2 1 0 -1

x 1

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-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15

Number of pencils sold

U1-104 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 8. C onnect the points with a line and add an arrow at the right end of the line to show that the line of the equation goes on infinitely in that direction. Be sure to write the equation of the line next to the line on the graph. 15

y

14 13 12 11 10 9 8 7

Revenue in dollars ($)

6 5 4 3 2 1 0 -1

x 1

2

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10

-2 -3 -4 -5 -6 -7 -8 -9 -10

y = 1 x – 10 2

-11 -12 -13 -14 -15

Number of pencils sold

U1-105 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Example 2 A taxi company in Kansas City charges $2.50 per ride plus $2 for every mile driven. Write and graph the equation that models this scenario. 1. R ead the problem statement and then reread the problem, determining the known quantities. Initial cost of taking a taxi: $2.50 Charge per mile: $2 2. Identify the slope and the y-intercept. The slope is a rate. Notice the word “every.” Slope = 2 The y-intercept is a starting value. It costs $2.50 initially to hire a cab driver. y-intercept = 2.50 3. S ubstitute the slope and y-intercept into the equation y = mx + b, where m is the slope and b is the y-intercept. m=2 b = 2.50 y = 2x + 2.50

U1-106 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 4. Set up the coordinate plane. In this scenario, x represents the number of miles traveled in the cab and is the independent variable. The x-axis label is “Miles traveled.” The dependent variable, y, represents the cost of taking a cab based on the number of miles traveled. The y-axis label is “Cost in dollars ($).” Determine the scales to be used. Since the slope’s rise and run are within 10 units and the y-intercept is within 10 units of 0, a scale of 1 on each axis is appropriate. Label the x-axis from 0 to 10, since miles traveled will only be positive. Label the y-axis from 0 to 10, since cost will only be positive. 10

y

9

Cost in dollars ($)

8 7 6 5 4 3 2 1

x 0

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Miles traveled

U1-107 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 5. G raph the equation using the slope and y-intercept. Plot the y-intercept first. The y-intercept is 2.5. Remember that the y-intercept is where the graph crosses the y-axis and the value of x is 0. Therefore, the coordinate of the y-intercept will always have 0 for x. In this case, the coordinate of the y-intercept is (0, 2.5). To plot points that lie in between grid lines, use estimation. Since 2.5 is halfway between 2 and 3, plot the point halfway between 2 and 3 on the y-axis. Estimate the halfway point. 10

y

9

Cost in dollars ($)

8 7 6 5 4 3 2 1

x 0

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Miles traveled

U1-108 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 6. G raph the equation using the slope and y-intercept. Use the slope to find the second point. rise . In this case, the slope is 2. Write 2 as Remember that the slope is run a fraction. 2 rise 2= = 1 run The rise is 2 and the run is 1. Point your pencil at the y-intercept. Move the pencil up 2 units, since the slope is positive. Remember that the y-intercept was halfway between grid lines. Be sure that you move your pencil up 2 complete units by first going to halfway between 3 and 4 (3.5) and then halfway between 4 and 5 (4.5) on the y-axis. Now, move your pencil to the right 1 unit for the run and plot a point. This is your second point. 10

y

9

Cost in dollars ($)

8 7 6 5 4

rise

3 2

run

1

x 0

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Miles traveled

U1-109 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 7. Connect the points and extend the line. Then, label your line. Draw a line through the two points and add an arrow to the right end of the line to show that the line of the equation continues infinitely in that direction. Label the line with the equation, y = 2x + 2.5. 10

y

9

Cost in dollars ($)

8

y = 2 x + 2.5

7 6 5 4 3 2 1

x 0

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Miles traveled

U1-110 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Example 3 Miranda gets paid $300 a week to deliver groceries. She also earns 5% commission on any orders she collects while out on her delivery run. Write an equation that represents her weekly pay and then graph the equation. 1. R ead the problem statement and then reread the problem, determining the known quantities. Weekly payment: $300 Commission: 5% = 0.05 2. Identify the slope and the y-intercept. The slope is a rate. Notice the symbol “%,” which means percent, or per 100. Slope = 0.05 The y-intercept is a starting value. She gets paid $300 a week to start with before taking any orders. y-intercept = 300 3. S ubstitute the slope and y-intercept into the equation y = mx + b, where m is the slope and b is the y-intercept. m = 0.05 b = 300 y = 0.05x + 300

U1-111 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 4. Set up the coordinate plane. In this scenario, x represents the amount of money in orders Miranda gets. The x-axis label is “Orders in dollars ($).” The dependent variable, y, represents her total earnings in a week. The y-axis label is “Weekly earnings in dollars ($).” Determine the scales to be used. The y-intercept is in the hundreds and the slope is in decimals. Work with the slope first. The slope is 0.05 or 5 . The rise is a small number, but the run is big. The run is shown 100 on the x-axis, so that will need to be in increments of 100. Start at –100 or 0 since the order amounts will be positive and continue to 1,000. The rise is shown on the y-axis and is small, but remember that the y-intercept is $300. Since there’s such a large gap before the y-intercept, the y-axis will need to skip values so the graph doesn’t become too large. Start the y-axis at 0, then skip to 250 and label the rest of the axis in increments of 5 until you reach 450. Use the zigzag line to show you skipped values between 0 and 250. y

450 445 440 435 430 425 420 415

Weekly earnings in dollars ($)

410 405 400 395 390 385 380 375 370 365 360 355 350 345 340 335 330 325 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250

x -100

0

100

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300

400

500

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700

800

900

1000

Orders in dollars ($) U1-112 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 5. G raph the equation using the slope and y-intercept. Plot the y-intercept first. The y-intercept is 300. Remember that the y-intercept is where the graph crosses the y-axis and the value of x is 0. Therefore, the coordinate of the y-intercept will always have 0 for x. In this case, the coordinate of the y-intercept is (0, 300). y

450 445 440 435 430 425 420 415

Weekly earnings in dollars ($)

410 405 400 395 390 385 380 375 370 365 360 355 350 345 340 335 330 325 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250

x -100

0

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Orders in dollars ($)

U1-113 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 6. G raph the equation using the slope and y-intercept. Use the slope to find the second point. rise . In this case, the slope is 0.05. Rewrite Remember that the slope is run 0.05 as a fraction. 0.05 =

5

=

rise

100 run

The rise is 5 and the run is 100. Place your pencil on the y-intercept. Move the pencil up 5 units, since the slope is positive. On this grid, 5 units is one tick mark. Now, move your pencil to the right 100 units for the run and plot a point. On this grid, 100 units to the right is one tick mark. This is your second point. y

450 445 440 435 430 425 420 415

Weekly earnings in dollars ($)

410 405 400 395 390 385 380 375 370 365 360 355 350 345 340 335 330 325 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250

x -100

0

100

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Orders in dollars ($)

U1-114 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 7. Connect the points and extend the line. Then, label your line. Draw a line through the two points and add an arrow to the right end of the line to show that the line continues infinitely in that direction. Label your line with the equation, y = 0.05x + 300. y

450 445 440 435 430 425 420 415

Weekly earnings in dollars ($)

410 405 400 395 390 385 380 375 370 365 360 355 350 345 340 335 330 325 320

y = 0.05x + 300

315 310 305 300 295 290 285 280 275 270 265 260 255 250

x -100

0

100

200

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Orders in dollars ($)

U1-115 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Example 4 The velocity (or speed) of a ball thrown directly upward can be modeled with the following equation: v = –gt + v0, where v is the speed, g is the acceleration due to gravity, t is the elapsed time, and v0 is the initial velocity at time 0. If the acceleration due to gravity is equal to 32 feet per second per second, and the initial velocity of the ball is 96 feet per second, what is the equation that represents the velocity of the ball? Graph the equation. 1. R ead the problem statement and then reread the problem, determining the known quantities. Initial velocity: 96 ft/s Acceleration due to gravity: 32 ft/s2 Notice that in the given equation, the acceleration due to gravity is negative. This is due to gravity acting on the ball, pulling it back to Earth and slowing the ball down from its initial velocity. 2. Identify the slope and the y-intercept. Notice the form of the given equation for velocity is the same form as y = mx + b, where y = v, m = –g, x = t, and b = v0. Therefore, the slope = –32 and the y-intercept = 96. 3. S ubstitute the slope and y-intercept into the equation y = mx + b, where m is the slope and b is the y-intercept. m = –g = –32 b = v0 = 96 y = –32x + 96

U1-116 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 4. Set up the coordinate plane. In this scenario, x represents the time passing after the ball was dropped. The x-axis label is “Time in seconds.” The dependent variable, y, represents the velocity, or speed, of the ball. The y-axis label is “Velocity in ft/s.” Determine the scales to be used. The y-intercept is close to 100 and the slope is 32. Notice that 96 (the y-intercept) is a multiple of 32. The y-axis can be labeled in units of 32. Since the x-axis is in seconds, it makes sense that these units are in increments of 1. Since time cannot be negative, use only a positive scale for the x-axis. 256

y

224 192 160 128 96

Velocity in ft/s

64 32

0

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x

10

-32 -64 -96 -128 -160 -192 -224 -256

Time in seconds

U1-117 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 5. G raph the equation using the slope and y-intercept. Plot the y-intercept first. The y-intercept is 96. Remember that the y-intercept is where the graph crosses the y-axis and the value of x is 0. Therefore, the coordinate of the y-intercept will always have 0 for x. In this case, the coordinate of the y-intercept is (0, 96). 256

y

224 192 160 128 96

Velocity in ft/s

64 32

0

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x

10

-32 -64 -96 -128 -160 -192 -224 -256

Time in seconds

U1-118 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 6. G raph the equation using the slope and y-intercept. Use the slope to find the second point. rise . In this case, the slope is –32. Remember that the slope is run Rewrite –32 as a fraction. −32 =

−32 rise = 1 run

The rise is –32 and the run is 1. Place your pencil on the y-intercept. Move the pencil down 32 units, since the slope is negative. On this grid, 32 units is one tick mark. Now, move your pencil to the right 1 unit for the run and plot a point. This is your second point. 256

y

224 192 160 128 96

Velocity in ft/s

64 32

0

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-32 -64 -96 -128 -160 -192 -224 -256

Time in seconds

U1-119 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 7. C onnect the points and extend the line toward the right. Then, label your line. Draw a line through the two points and add an arrow to the right end of the line to show that the line of the equation continues infinitely in that direction. Label your line with the equation y = –32x + 96. 256

y

224 192 160 128 96

Velocity in ft/s

64 y = –32x + 96

32

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-32 -64 -96 -128 -160 -192 -224 -256

Time in seconds

U1-120 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Example 5 A Boeing 747 starts out a long flight with about 57,260 gallons of fuel in its tank. The airplane uses an average of 5 gallons of fuel per mile. Write an equation that models the amount of fuel in the tank and then graph the equation using a graphing calculator. 1. R ead the problem statement and then reread the problem, determining the known quantities. Starting fuel tank amount: 57,260 gallons Rate of fuel consumption: 5 gallons per mile 2. Identify the slope and the y-intercept. The slope is a rate. Notice the word “per” in the phrase “5 gallons of fuel per mile.” Since the total number of gallons left in the fuel tank is decreasing at this rate, the slope is negative. Slope = –5 The y-intercept is a starting value. The airplane starts out with 57,260 gallons of fuel. y-intercept = 57,260 Substitute the slope and y-intercept into the equation y = mx + b, where m is the slope and b is the y-intercept. m=5 b = 57,260 y = –5x + 57,260

U1-121 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 3. Graph the equation on your calculator. On a TI-83/84: Step 1: Press [Y=]. Step 2: At Y1, type in [(–)][5][X, T, θ, n][+][57260]. Step 3: Press [WINDOW] to change the viewing window. Step 4: At Xmin, enter [0] and arrow down 1 level to Xmax. Step 5: At Xmax, enter [3000] and arrow down 1 level to Xscl. Step 6: At Xscl, enter [100] and arrow down 1 level to Ymin. Step 7: At Ymin, enter [40000] and arrow down 1 level to Ymax. Step 8: At Ymax, enter [58000] and arrow down 1 level to Yscl. Step 9: At Yscl, enter [1000]. Step 10: Press [GRAPH]. On a TI-Nspire: Step 1: Press the [home] key. Step 2: Arrow over to the graphing icon and press [enter]. Step 3: At the blinking cursor at the bottom of the screen, enter in the equation [(–)][5][x][+][57260] and press [enter]. Step 4: Change the viewing window by pressing [menu], arrowing down to number 4: Window/Zoom, and clicking the center button of the navigation pad. Step 5: Choose 1: Window settings by pressing the center button. Step 6: Enter in the appropriate XMin value, [0], then press [tab]. Step 7: Enter in the appropriate XMax value, [3000], then press [tab]. Step 8: Leave the XScale set to “Auto.” Press [tab] twice to navigate to YMin and enter [40000]. Step 9: Press [tab] to navigate to YMax. Enter [58000]. Press [tab] twice to leave YScale set to “auto” and to navigate to “OK.” Step 10: Press [enter]. Step 11: Press [menu] and select 2: View and 5: Show Grid.

U1-122 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 4. Redraw the graph on graph paper. On the TI-83/84, the scale was entered in [WINDOW] settings. The X scale was 100 and the Y scale was 1,000. Set up the graph paper using these scales. Label the y-axis “Fuel used in gallons.” Show a break in the graph from 0 to 40,000 using a zigzag line. Label the x-axis “Distance in miles.” To show the table on the calculator so you can plot points, press [2nd][GRAPH]. The table shows two columns with values; the first column holds the x-values, and the second column holds the y-values. Pick a pair to plot, and then connect the line. To return to the graph, press [GRAPH]. Remember to label the line with the equation. (Note: It may take you a few tries to get the window settings the way you want. The graph that follows shows an X scale of 200 so that you can easily see the full extent of the graphed line.) y

58,000 57,000 56,000 55,000 54,000

Fuel used in gallons

53,000 52,000 51,000 50,000

y = –5x + 57,260

49,000 48,000 47,000 46,000 45,000 44,000 43,000 42,000 41,000 40,000 0

200

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Distance in miles

(continued)

U1-123 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction If you used a TI-Nspire, determine the scale that was used by counting the dots on the grid from your minimum y-value to your maximum y-value. In this case, there are 18 dots vertically between 40,000 and 58,000. The difference between the YMax and YMin values is 18,000. Divide that by the number of dots (18). The result (1,000) is the scale. 58,000 − 40,000 18,000 = = 1000 Number of dots 18 18 Y Max – Y Min

=

This means each dot is worth 1,000 units vertically. Label the y-axis “Fuel used in gallons.” Use a zigzag line to show a break in the graph from 0 to 40,000. Repeat the same process for determining the x-axis scale. The XMin = 0 and XMax = 3000. The number of dots = 30. 3000 − 0 3000 = = 100 Number of dots 30 30 X Max – X Min

=

This means each dot is worth 100 units horizontally. Set up your graph paper accordingly. Label the x-axis “Distance in miles.” On your calculator, you need to show the table in order to plot points. To show the table, press [tab][T]. To navigate within the table, use the navigation pad. The table shows two columns with values; the first column holds the x-values, and the second column holds the y-values. Pick a pair to plot and then connect the line. Remember to label the line with the equation. To hide the table, navigate back to the graph by pressing [ctrl][tab]. Then press [ctrl][T].

U1-124 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Problem-Based Task 1.3.1: Phone Card Fine Print Write and graph the equation that models the following scenario. You can buy a 6-hour phone card for $5, but the fine print says that each minute you talk actually costs you 1.5 minutes of time. What is the equation that models the number of minutes left on the card compared with the number of minutes you actually talked? What is the graph of this equation?

What is the equation that models the number of minutes left on the card compared with the number of minutes you actually talked? What is the graph of this equation?

U1-125 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Problem-Based Task 1.3.1: Phone Card Fine Print Coaching a. What are the slope and the y-intercept?

b. What is the equation of the line?

c. What are the labels of the x- and y-axes?

d. What are the scales of the x- and y-axes?

e. Which point do you plot first?

f. How can you use the equation to plot the second point?

U1-126 CCSS IP Math I Teacher Resource 1.3.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Problem-Based Task 1.3.1: Phone Card Fine Print Coaching Sample Responses a. What are the slope and the y-intercept? The slope is the rate. Notice the word “each” in the phrase “each minute you talk actually costs you 1.5 minutes of time.” Therefore, the rate at which the time on the card is decreasing is 1.5 minutes. The slope = –1.5 minutes. m = –1.5 The y-intercept is 6 hours. That’s the amount of time you started with, but the rate at which the card is decreasing is given in minutes. You need to convert hours into minutes. 1 hour = 60 minutes 6 hours •

60 minutes 1 hour

= 360 minutes

b = 360 b. What is the equation of the line? y = –1.5x + 360 c. What are the labels of the x- and y-axes? The x-axis label is “Minutes used” and the y-axis label is “Minutes left.” d. What are the scales of the x- and y-axes? Since the minutes on the card are in the hundreds and the slope’s rise and run are in the single digits, the best way to choose the units for both axes is to keep the division of units the same so that you can use the slope to plot the points. Choose the scale on the y-axis first. The y-intercept occurs at 360. Choose a scale that starts at 0 and continues to 400 in increments of 20. This way, the y-intercept will be easy to plot. For the x-axis, since the rate of decreasing minutes is faster than 1, the scale doesn’t need to be as long. Start at 0 and continue to 300, again in increments of 20. This will let you count the rise over the run using the grid marks for the slope to plot the second point. e. Which point do you plot first? Plot the y-intercept first. (0, 360) U1-127 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction f. How can you use the equation to plot the second point? Rewrite the slope as a fraction. −1.5 =

−3 rise = 2 run

Since the units are the same for the x- and y-axes, you can count the number of tick marks for the slope. From the y-intercept, count down by 3 units and to the right by 2 units, then plot the second point. Then connect the points. Extend the line to the edges of the coordinate plane. Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

U1-128 CCSS IP Math I Teacher Resource 1.3.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Practice 1.3.1: Creating and Graphing Linear Equations in Two Variables Graph each equation on graph paper. 1. y = x + 2 1 2. y = x + 2 3 3. A gear on a machine turns at a rate of 2 revolutions per second. Let x = time in seconds and y = number of revolutions. What is the equation that models the number of revolutions over time? Graph this equation. 4. T he relationship between degrees Celsius and degrees Fahrenheit is linear. To convert a temperature in degrees Celsius to degrees Fahrenheit, multiply the temperature by a rate of nine fifths and add 32. What is the equation that models the conversion from degrees Celsius to degrees Fahrenheit? Graph this equation. 5. A cab company charges an initial rate of $2.50 for a ride, plus $0.40 for each mile driven. What is the equation that models the total fee for using this cab company? Graph this equation. 6. M atthew receives a base weekly salary of $300 plus a commission of $50 for each vacuum he sells. What is the equation that models his weekly earnings? Graph this equation. 7. A water company charges a monthly fee of $6.70 plus a usage fee of $2.60 per 1,000 gallons used. What is the equation that models the water company’s total fees? Graph this equation. 8. M addie borrowed $1,250 from a friend to buy a new TV. Her friend doesn’t charge any interest, and Maddie makes $40 payments each month. What is the equation that models the money Maddie owes? Graph this equation. 9. A company started with 3 employees and after 8 months grew to 19. The growth was steady. What is the equation that models the growth of the company’s employees? Graph this equation. 10. Y ou and some friends are hiking the Appalachian Trail. You started out with 70 pounds of food for the group, and eat about 8 pounds each day. What is the equation that models the food you have left? Graph this equation.

U1-129 © Walch Education

CCSS IP Math I Teacher Resource 1.3.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Lesson 1.3.2: Creating and Graphing Exponential Equations Warm-Up 1.3.2 Read the scenario and answer the questions that follow. One form of the element beryllium, beryllium-11, has a half-life of about 14 seconds and decays to the element boron. A chemist starts out with 128 grams of beryllium-11. She monitors the element for 70 seconds. 1. What is the equation that models the amount of beryllium-11 over time?

2. How many grams of beryllium-11 does the chemist have left after 70 seconds?

U1-130 CCSS IP Math I Teacher Resource 1.3.2

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Lesson 1.3.2: Creating and Graphing Exponential Equations Common Core State Standards A–CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

N–Q.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.★

Warm-Up 1.3.2 Debrief One form of the element beryllium, beryllium-11, has a half-life of about 14 seconds and decays to the element boron. A chemist starts out with 128 grams of beryllium-11. She monitors the element for 70 seconds. 1. What is the equation that models the amount of beryllium-11 over time? y = ab x, where y is the final value, a is the initial value, b is the rate of growth or decay, and x is the time. y = unknown a = 128 grams b = 0.5 Time = 70 seconds, but this needs to be converted to time periods before substituting the value for x. Convert 70 seconds into 14-second time periods. 1 time period = 14 seconds. 70 seconds •

1 time period 14 seconds

= 5 time periods

x=5 Substitute all the variables into the equation. y = ab x y = 128(0.5)5

U1-131 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 2. How many grams of beryllium-11 does the chemist have left after 70 seconds? Apply the order of operations to the equation from the end of problem 1. y = 128(0.5)5 y = 4 grams Connection to the Lesson •

As in the warm-up, students will create exponential equations.

•

S tudents will take the equation a step further and graph the set of solutions on the coordinate plane as a curve.

U1-132 CCSS IP Math I Teacher Resource 1.3.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Prerequisite Skills This lesson requires the use of the following skills: •

plotting points in four quadrants

•

applying the order of operations

Introduction Exponential equations in two variables are similar to linear equations in two variables in that there is an infinite number of solutions. The two variables and the equations that they are in describe a relationship between those two variables. Exponential equations are equations that have the variable in the exponent. This means the final values of the equation are going to grow or decay very quickly. Key Concepts Reviewing Exponential Equations: •

he general form of an exponential equation is y = a • b x, where a is the initial value, b is the T rate of decay or growth, and x is the time. The final output value will be y.

•

Since the equation has an exponent, the value increases or decreases rapidly.

•

The base, b, must always be greater than 0 (b > 0).

•

I f the base is greater than 1 (b > 1), then the exponential equation represents exponential growth.

•

If the base is between 0 and 1 (0 < b < 1), then the exponential equation represents exponential decay.

•

If the base repeats after anything other than 1 unit (e.g., 1 month, 1 week, 1 day, 1 hour, x

1 minute, 1 second), use the equation y = ab t , where t is the time when the base repeats. For x

example, if a quantity doubles every 3 months, the equation would be y = 2 . 3

•

nother formula for exponential growth is y = a(1 + r) t, where a is the initial value, (1 + r) is A the growth rate, t is time, and y is the final value.

•

nother formula for exponential decay is y = a(1 – r) t, where a is the initial value, (1 – r) is the A decay rate, t is time, and y is the final value.

U1-133 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Introducing the Compound Interest Formula: •

nt

r The general form of the compounding interest formula is A = P 1 + , where A is the n initial value, r is the interest rate, n is the number of times the investment is compounded in a year, and t is the number of years the investment is left in the account to grow.

•

Use this chart for reference: Compounded…

n (number of times per year)

Yearly/annually

1

Semi-annually

2

Quarterly

4

Monthly

12

Weekly

52

Daily

365

•

Remember to change the percentage rate into a decimal by dividing the percentage by 100.

•

Apply the order of operations and divide r by n, then add 1. Raise that value to the power of the product of nt. Multiply that value by the principal, P.

Graphing Exponential Equations Using a Table of Values 1. Create a table of values by choosing x-values and substituting them in and solving for y. 2. D etermine the labels by reading the context. The x-axis will most likely be time and the y-axis will be the units of the final value. 3. D etermine the scales. The scale on the y-axis will need to be large since the values will grow or decline quickly. The value on the x-axis needs to be large enough to show the growth rate or the decay rate.

U1-134 CCSS IP Math I Teacher Resource 1.3.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Graphing Equations Using a TI-83/84: Step 1: Press [Y=] and key in the equation using [^] for the exponent and [X, T, θ, n] for x. Step 2: Press [WINDOW] to change the viewing window, if necessary. Step 3: Enter in appropriate values for Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl, using the arrow keys to navigate. Step 4: Press [GRAPH]. Graphing Equations Using a TI-Nspire: Step 1: Press the home key. Step 2: Arrow over to the graphing icon (the picture of the parabola or the U-shaped curve) and press [enter]. Step 3: At the blinking cursor at the bottom of the screen, enter in the equation using [^] before entering the exponents, and press [enter]. Step 4: To change the viewing window: press [menu], arrow down to number 4: Window/Zoom and click the center button of the navigation pad. Step 5: Choose 1: Window settings by pressing the center button. Step 6: Enter in the appropriate XMin, Xmax, YMin, and YMax fields. Step 7: Leave the XScale and YScale set to auto. Step 8: Use [tab] to navigate among the fields. Step 9: Press [tab] to “OK” when done and press [enter].

Common Errors/Misconceptions •

i ncorrectly applying the order of operations: multiplying a and b before raising b to the exponent in y = ab x

•

incorrectly identifying the rate—forgetting to add 1 or subtract from 1

•

using the exponential growth model instead of exponential decay

•

f orgetting to calculate the number of time periods it takes for a given rate of growth or decay and simply substituting in the time given

U1-135 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Guided Practice 1.3.2 Example 1 If a pendulum swings to 90% of its height on each swing and starts out at a height of 60 cm, what is the equation that models this scenario? What is its graph? 1. R ead the problem statement and then reread the scenario, identifying the known quantities. Initial height = 60 cm Decay rate = 90% or 0.90 2. S ubstitute the known quantities into the general form of the exponential equation y = ab x, where a is the initial value, b is the rate of decay, x is time (in this case swings), and y is the final value. a = 60 b = 0.90 y = ab x y = 60(0.90) x 3. Create a table of values. x 0 1 2 3 5 10 20 40

y 60 54 48.6 43.74 35.43 20.92 7.29 0.89

U1-136 CCSS IP Math I Teacher Resource 1.3.2

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 4. Set up the coordinate plane. Determine the labels by reading the problem again. The independent variable is the number of swings. That will be the label of the x-axis. The y-axis label will be the height. The height is the dependent variable because it depends on the number of swings.

Height in cm

To determine the scales, examine the table of values. The x-axis needs a scale that goes from 0 to 40. Counting to 40 in increments of 1 would cause the axis to be very long. Use increments of 5. For the y-axis, start with 0 and go to 60 in increments of 5. This will make plotting numbers like 43.74 a little easier than if you chose increments of 10.

Number of swings

U1-137 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 5. P lot the points on the coordinate plane and connect the points with a line (curve).

Height in cm

When the points do not lie on a grid line, use estimation to approximate where the point should be plotted. Add an arrow to the right end of the line to show that the curve continues in that direction toward infinity.

y = 60(0.90)x

Number of swings

U1-138 CCSS IP Math I Teacher Resource 1.3.2

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Example 2 The bacteria Streptococcus lactis doubles every 26 minutes in milk. If a container of milk contains 4 bacteria, write an equation that models this scenario and then graph the equation. 1. R ead the problem statement and then reread the scenario, identifying the known quantities. Initial bacteria count = 4 Base = 2 Time period = 26 minutes 2. Substitute the known quantities into the general form of the exponential equation y = ab x, for which a is the initial value, b is the base, x is time (in this case, 1 time period is 26 minutes), and y is the final value. Since the base is repeating in units other than 1, use the x

equation y = ab t , where t = 26. a=4 b=2 x

y = ab

26 x

y = 4(2)

26

3. Create a table of values. x y 0 4 26 8 52 16 78 32 104 64

U1-139 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 4. Set up the coordinate plane. Determine the labels by reading the problem again. The independent variable is the number of time periods. The time periods are in number of minutes. Therefore, “Minutes” will be the x-axis label. The y-axis label will be the “Number of bacteria.” The number of bacteria is the dependent variable because it depends on the number of minutes that have passed.

Number of bacteria

The x-axis needs a scale that reflects the time period of 26 minutes and the table of values. The table of values showed 4 time periods. One time period = 26 minutes and so 4 time periods = 4(26) = 104 minutes. This means the x-axis scale needs to go from 0 to 104. Use increments of 26 for easy plotting of the points. For the y-axis, start with 0 and go to 65 in increments of 5. This will make plotting numbers like 32 a little easier than if you chose increments of 10.

Minutes

U1-140 CCSS IP Math I Teacher Resource 1.3.2

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 5. P lot the points on the coordinate plane and connect the points with a line (curve).

Number of bacteria

When the points do not lie on a grid line, use estimation to approximate where the point should be plotted. Add an arrow to the right end of the line to show that the curve continues in that direction toward infinity.

x

y = 4(2) 26

Minutes

U1-141 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Example 3 An investment of $500 is compounded monthly at a rate of 3%. What is the equation that models this situation? Graph the equation. 1. R ead the problem statement and then reread the scenario, identifying the known quantities. Initial investment = $500 r = 3% Compounded monthly = 12 times a year 2. Substitute the known quantitiesntinto the general form of the compound r interest formula, A = P 1 + , for which P is the initial value, r is the n interest rate, n is the number of times the investment is compounded in a year, and t is the number of years the investment is left in the account to grow. P = 500 r = 3% = 0.03 n = 12 r A = P1+ n

nt

0.03 A = 500 1 + 12

12 t

A = 500(1.0025)12 t Notice that, after simplifying, this form is similar to y = ab x. To graph on the x- and y-axes, put the compounded interest formula into this r form, where A = y, P = a, 1+ = b, and t = x. n A = 500(1.0025)12t becomes y = 500(1.0025)12x.

U1-142 CCSS IP Math I Teacher Resource 1.3.2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 3. Graph the equation using a graphing calculator. On a TI-83/84: Step 1: Press [Y=]. Step 2: Type in the equation as follows: [500][×][1.0025][^][12][X, T, θ, n] Step 3: Press [WINDOW] to change the viewing window. Step 4: At Xmin, enter [0] and arrow down 1 level to Xmax. Step 5: At Xmax, enter [10] and arrow down 1 level to Xscl. Step 6: At Xscl, enter [1] and arrow down 1 level to Ymin. Step 7: At Ymin, enter [500] and arrow down 1 level to Ymax. Step 8: At Ymax, enter [700] and arrow down 1 level to Yscl. Step 9: At Yscl, enter [15]. Step 10: Press [GRAPH]. On a TI-Nspire: Step 1: Press the [home] key. Step 2: Arrow over to the graphing icon and press [enter]. Step 3: At the blinking cursor at the bottom of the screen, enter in the equation [500][×][1.0025][^][12x] and press [enter]. Step 4: To change the viewing window: press [menu], arrow down to number 4: Window/Zoom, and click the center button of the navigation pad. Step 5: Choose 1: Window settings by pressing the center button. Step 6: Enter in the appropriate XMin value, [0], and press [tab]. Step 7: Enter in the appropriate XMax value, [10], and press [tab]. Step 8: Leave the XScale set to “Auto.” Press [tab] twice to navigate to YMin and enter [500]. Step 9: Press [tab] to navigate to YMax. Enter [700]. Press [tab] twice to leave YScale set to “Auto” and to navigate to “OK.” Step 10: Press [enter]. Step 11: Press [menu] and select 2: View and 5: Show Grid. Note: To determine the y-axis scale, show the table to get an idea of the values for y. To show the table, press [ctrl] and then [T]. To turn the table off, press [ctrl][tab] to navigate back to the graphing window and then press [ctrl][T] to turn off the table. U1-143 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction 4. Transfer your graph from the screen to graph paper. Use the same scales that you set for your viewing window. The x-axis scale goes from 0 to 10 years in increments of 1 year.

Investment in dollars ($)

The y-axis scale goes from $500 to $700 in increments of $15. You’ll need to show a break in the graph from 0 to 500 with a zigzag line. 710 695 680 665 650 635 620 605 590 575 560 545 530 515 500

y

0

y = 500(1.0025)12x

1

2

3

4

5

6

7

8

9

x

10

Years

U1-144 CCSS IP Math I Teacher Resource 1.3.2

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Problem-Based Task 1.3.2: Investing Money You want to invest some money in a savings account. One bank offers an account that compounds the money annually at a rate of 3%. You have $2,000 to invest. As you are about to sign the papers, your friend texts you that a different bank offers a rate of 3.2% and this bank will compound the interest monthly. You decide to check out the second bank, but on your way there you spend $100. You end up choosing the second bank with the higher interest rate, but you want to know how spending $100 along the way affected your investment. Create a graph showing how much interest you would have earned on $2,000 at the first bank, then create another graph showing how much interest you will earn on the money you invested in the second bank. Use the graphs to help you determine about how long it will take to earn back the $100 you spent. How long will it take before the two graphs are equal? How would your investment have changed if you hadn’t spent the $100? What can you conclude about investing?

How long will it take before the two graphs are equal? How would your investment have changed if you hadn’t spent the $1 00? What can you conclude about investing?

U1-145 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Problem-Based Task 1.3.2: Investing Money Coaching a. What is the equation for the investment at the first bank?

b. W hat is the equation for the investment at the second bank? Keep in mind that you spent $100 of the money you initially planned to invest.

c. Graph the equations on the same set of axes, and be sure to label each equation.

d. L ooking at the graph of the investment you actually made, how many years does it take to earn back the $100 you spent?

e. H ow many years does it take before the investment you made is equal to the investment you almost made?

f. What would be the equation of the investment at the second bank if you had not spent the $100?

g. Graph the equation from part f on the same set of axes as the equation from part b.

h. L ook at various points along the graph and use the equations. What is the difference in investments after 10 years? 20 years?

i. C ompare the investments of all 3 graphs and make observations. What conclusions can you draw about the amount you invest initially or the principal amount? What can you conclude about the number of times the interest is compounded in a year? What effect does this have on the investment?

U1-146 CCSS IP Math I Teacher Resource 1.3.2

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction Problem-Based Task 1.3.2: Investing Money Coaching Sample Responses a. What is the equation for the investment at the first bank? r A= P 1+ n

nt

0.03 A = 2000 1 + 1

1t

A = 2000(1.03) t b. W hat is the equation for the investment at the second bank? Keep in mind that you spent $100 of the money you initially planned to invest. r A= P 1+ n

nt

0.032 A = 1900 1 + 12

12 t

A = 1900(1.00267)12 t c. Graph the equations on the same set of axes, and be sure to label each equation. To do this, first rewrite each equation in the form y = ab x. A = 2000(1.03) t becomes y = 2000(1.03) x.

Investment in dollars ($)

A = 1900(1.00267)12t becomes y = 1900(1.00267)12x.

y = 2000(1.03)x y = (1.0267)12x Years U1-147

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CCSS IP Math I Teacher Resource 1.3.2

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction d. L ooking at the graph of the investment you actually made, how many years does it take to earn back the $100 you spent? It looks like the investment earns back $100 and reaches $2,000 after a little more than a year and a half, or about 19 months. e. H ow many years does it take before the investment you made is equal to the investment you almost made? The graphs intersect at about 21 years, so the investments will be equal in about 21 years. f. What would be the equation of the investment at the second bank if you had not spent the $100? r A= P 1+ n

nt

0.032 A = 2000 1 + 12

12 t

A = 2000(1.00267)12 t g. Graph the equation from part f on the same set of axes as the equation from part b. Before graphing, rewrite the equation in the form y = abx.

Investment in dollars ($)

A = 2000(1.00267)12t becomes y = 2000(1.00267)12x.

y = 2000(1.00267)12x

y = 1900(1.00267)12x Years

U1-148 CCSS IP Math I Teacher Resource 1.3.2

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables Instruction h. L ook at various points along the graph and use the equations. What is the difference in investments after 10 years? 20 years? The investment of the principal amount of $2,000 will always be greater than the investment with the principal amount of $1,900. After 10 years, the investment of $2,000 grows to $2,754.18, and the investment of $1,900 grows to $2,616.47, a difference of $137.71. After 20 years, the investment of $2,000 grows to $3,792.76, and the investment of $1,900 grows to $3,603.12. The difference is $189.64. The gap between the larger and smaller investments is slowly widening. i. C ompare the investments of all 3 graphs and make observations. What conclusions can you draw about the amount you invest initially or the principal amount? What can you conclude about the number of times the interest is compounded in a year? What effect does this have on the investment? The more you invest to begin with, the more your investment will grow. The more times the interest is compounded in a year, the faster the investment will grow. If two banks are offering the same rate but one bank is compounding the interest more frequently, invest in the bank that compounds more often. If the rates are different, draw graphs to compare the investments.

Investment in dollars ($)

y = 2000(1.00267)12x

y = 2000(1.03)x

y = 1900(1.00267)12x Years

Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

U1-149 © Walch Education

CCSS IP Math I Teacher Resource 1.3.2

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Practice 1.3.2: Creating and Graphing Exponential Equations Use a table of values to graph the following exponential equations. 1. y = 2(3)x

2. y = 1000(0.25) x

Write an equation to model each scenario, and then graph the equation. 3. A population of insects doubles every month. This particular population started out with 20 insects.

4. The half-life of rhodium, Rh-106, is about 30 seconds. You start with 500 grams.

5. A stock is declining at a rate of 75% of its value every 2 weeks. The stock started at $225.

6. A weed species triples in 6 days. A field started with 12 weeds in the early spring.

7. T he population of a big city is increasing at a rate of 2.5% per year. The city’s current population is 67,000.

8. An investment of $1,000 earns 3.7% interest and is compounded semi-annually.

9. An investment of $600 earns 2.9% interest and is compounded quarterly.

10. An investment of $3,000 earns 1.4% interest and is compounded weekly.

U1-150 CCSS IP Math I Teacher Resource 1.3.2

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment Progress Assessment Circle the letter of the best answer. 1. T he cost of having your car towed is $45 to hook up the car and then $3.50 per mile towed. Which equation models this scenario? a. y = 40x + 3.50 b. y = 3.50x + 45 c. y = 48.50x d. x + y = 48.50 2. A store is giving away 150 gift cards each valued at $20 for every hour that the store is open. What equation models this scenario? a. y = –20x + 3000 b. y = –x + 150 c. either a or b d. neither a nor b

continued U1-151 © Walch Education

CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 3. A company rents personal watercraft for $70 per hour plus an initial $15 fee. What is the graph of the equation that models this scenario?

Rental cost in dollars ($)

a.

Rental cost in dollars ($)

b.

250 240 230 220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

y

0.5

1

0.5

1

x

1.5

2

2.5

3

1.5

2

2.5

3

Hours

y

x Hours

continued

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment c.

Rental cost in dollars ($)

280

y

210

140

70

x

0

d.

Rental cost in dollars ($)

280

0.5

1

0.5

1

1.5

2

2.5

3

1.5

2

2.5

3

Hours

y

210

140

70

0

x Hours

continued U1-153 © Walch Education

CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 4. A cable company charges $80 a month for service and $4 for each on-demand movie watched. What is the graph of the equation for this scenario? a.

40

y

36

Cable cost in dollars ($)

32 28 24 20 16 12 8 4

x

0

1

2

3

4

5

6

7

8

9

10

7

8

9

10

Number of movies rented

Cable cost in dollars ($)

b.

80 76 72 68 64 60 56 52 48 44 40 36 32 28 24 20 16 12 8 4 0

y

x 1

2

3

4

5

6

Number of movies rented

continued

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment c.

100

y

Cable cost in dollars ($)

96

92

88

84

80

x

0

d.

120

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

Number of movies rented

y

116

Cable cost in dollars ($)

112 108 104 100 96 92 88 84 80 0

x 1

2

3

4

5

Number of movies rented

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CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 5. Y ou are starting your own business making websites. You spent $525 to get started, and will charge each customer $150 to build their website. Which graph represents the equation of your profit?

Profit in dollars ($)

a.

Number of websites

Profit in dollars ($)

b.

Number of websites

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment

Profit in dollars ($)

c.

Number of websites

Profit in dollars ($)

d.

Number of websites

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CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 6. T he half-life of niobium-97m is 1 minute. If an experiment started with 200 grams, which equation represents this scenario? a. y = 200(2) x b. y = 200(0.5) x c. y = –0.5x + 200 d. y = –2x + 200 7. An investment of $750 earns 3.3% interest compounded monthly. What is the equation? a. A = 750(1.033) x b. A = 750(1.033)12 x c. A = 750(1.00275)12 x d. A = 750(1.00275) x

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 8. A mole population doubles every month. If you start with 2 moles, what is the graph of the equation?

Number of moles

a.

Months

Number of moles

b.

Months

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CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment

Number of moles

c.

Months

Number of moles

d.

Months

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 9. A wildflower population triples every 2 months. If a meadow starts out with 12 wildflowers, what is the graph of the equation?

Number of wildflowers

a.

Number of months

Number of wildflowers

b.

Number of months

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CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment

Number of wildflowers

c.

Number of months

Number of wildflowers

d.

Number of months

continued U1-162 CCSS IP Math I Teacher Resource 1.3

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment 10. A hockey tournament starts out with 128 teams. Half the teams are eliminated after each round. What is the graph of the equation?

Number of teams

a.

Number of rounds

Number of teams

b.

Number of rounds

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CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment

Number of teams

c.

Number of rounds

Number of teams

d.

Number of rounds

continued U1-164 CCSS IP Math I Teacher Resource 1.3

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 3: Creating and Graphing Equations in Two Variables

Date:

Assessment Use what you’ve learned about comparing graphs of different interest rates to solve. 11. C ompare two investments of $500 that each earn 3% interest. The first investment earns simple interest. (Remember, simple interest earns money only on the principal amount.) The second investment is compounded quarterly. Compare the two investments using equations and graphs. Which is the better investment and why?

U1-165 © Walch Education

CCSS IP Math I Teacher Resource 1.3

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Assessment Pre-Assessment Circle the letter of the best answer. 1. Given the equation y = 5x – 7, which point is a solution? a. (1, 2)

c. (–1, 2)

b. (0, 7)

d. (–2, –17)

2. Given the inequality y ≤ –3x + 6, which point is NOT a solution? a. (1, –3)

c. (–1, –9)

b. (0, –2)

d. (2, 3)

3. J ulia has $6.50 to spend on peaches and apples at the farmer’s market. She bought 4 peaches at $0.75 each. How much money can she spend on apples? Determine which system of inequalities represents this situation. a. a + 4(0.75) ≤ 6.50 c. a + 4(0.75) ≥ 6.50 a ≥ 0 a ≤ 0 b. a + 4(0.75) ≤ 6.50 a ≤ 0

d. a + 4(0.75) ≥ 6.50 a ≥ 0

4. Y our cell phone company charges $29.99 a month plus $0.25 for each text message sent. You have budgeted no more than $35.00 for cell phone service each month. Given this situation, determine the minimum and maximum number of texts you can send without going over budget. Let x represent the number of texts. a. x < 20.04

c. x > 0 and x < 20

b. x ≥ 0 and x ≤ 20.04

d. x ≥ 0 and x ≤ 20

5. Y our doctor recommends that you eat at least 46 grams of protein each day. One serving of peanuts contains 9 grams of protein, while one egg contains 6 grams of protein. Determine which system of inequalities represents the number of servings of eggs and peanuts you must eat in order to reach the minimum recommendation. 9 x + 6 y ≤ 46 a. x ≤ 0 y≤0

9 x + 6 y ≥ 46 c. x ≥ 0 y≥0

9 x + 6 y ≤ 46 b. x ≥ 0 y≥0

9 x + 6 y ≥ 46 d. x ≤ 0 y≤0

U1-166 CCSS IP Math I Teacher Resource 1.4

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES

Lesson 4: Representing Constraints Instruction Common Core State Standard A–CED.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.★

Essential Questions 1. How can you model real-world applications using equations? 2. How can you solve real-world applications by graphing systems of equations? 3. How can you model real-world applications using inequalities? 4. Why are there constraints when solving and graphing real-world applications? WORDS TO KNOW algebraic inequality

an inequality that has one or more variables and contains at least one of the following symbols: <, >, ≤, ≥, or ≠

constraint

a restriction or limitation on either the input or output values

inequality

a mathematical sentence that shows the relationship between quantities that are not equivalent

solution set

the value or values that make a sentence or statement true

system of equations

a set of equations with the same unknowns

system of inequalities

a set of inequalities with the same unknowns

Recommended Resources •

NCTM Illuminations. “Dirt Bike Dilemma.” http://walch.com/rr/CAU1L4SysEquations Students use a system of equations to maximize profits for a dirt bike manufacturer.

•

Purplemath.com. “Linear Programming: Word Problems.” http://walch.com/rr/CAU1L4SysInequalities This site offers a review of systems of inequalities and constraints associated with real-world situations. It also includes graphs of feasible regions. U1-167

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CCSS IP Math I Teacher Resource 1.4

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Lesson 1.4.1: Representing Constraints Warm-Up 1.4.1 Read the scenario and answer the questions that follow. Roshanda pays $5 in tolls and uses 3 gallons of gasoline each day she drives to work. In one day, Roshanda spent a total of $15.23 on tolls and gasoline. 1. How much did each gallon of gas cost? Explain how you found your answer.

2. R oshanda needs to work 5 days next week and has set aside $75 for tolls and gas. Will Roshanda have enough money for her workweek? Explain how you found your answer.

U1-168 CCSS IP Math I Teacher Resource 1.4.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction Lesson 1.4.1: Representing Constraints Common Core State Standard A–CED.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.★

Warm-Up Debrief Roshanda pays $5 in tolls and uses 3 gallons of gasoline each day she drives to work. In one day, Roshanda spent a total of $15.23 on tolls and gasoline. 1. How much did each gallon of gas cost? Explain how you found your answer. Write an equation to represent this situation. Let x represent the cost of a gallon of gas. 5 + 3x = 15.23 Solve for x. 5 + 3x = 15.23

Subtract 5 from both sides of the equation.

3x = 10.23

Divide both sides by 3.

x = 3.41 Interpret the solution. The cost of 1 gallon of gas is $3.41. 2. R oshanda needs to work 5 days next week and has set aside $75 for tolls and gas. Will Roshanda have enough money for her workweek? Explain how you found your answer. Multiply the amount of money Roshanda spends on her travels for 1 day of work by the number of days she will be working. 15.23(5) = 76.15 Compare this amount to the amount that Roshanda has saved. Roshanda has saved $75 and will need $76.15. She will not have enough money. Connection to the Lesson •

he upcoming lesson will have students examining limitations and constraints on equations T and inequalities. Students will need to think about their answers and determine if they are appropriate given the constraints. U1-169

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CCSS IP Math I Teacher Resource 1.4.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction Prerequisite Skills This lesson requires the use of the following skills: •

reading and writing inequalities

•

creating and evaluating inputs and outputs of equations and inequalities

Introduction Situations in the real world often determine the types of values we would expect as answers to equations and inequalities. When an inequality has one or more variables and contains at least one inequality symbol (<, >, ≤, ≥, or ≠), it is called an algebraic inequality. Sometimes there are limits or restrictions on the values that can be substituted into an equation or inequality; other times, limits or restrictions are placed on answers to problems involving equations or inequalities. These limits or restrictions are called constraints. Key Concepts •

any real-world situations can be modeled using an equation, an inequality, or a system M of equations or inequalities. A system is a set of equations or inequalities with the same unknowns.

•

hen creating a system of equations or inequalities, it is important to understand that the W solution set is the value or values that make each sentence in the system a true statement.

•

eing able to translate real-world situations into algebraic sentences will help with the B understanding of constraints.

Common Errors/Misconceptions •

incorrectly translating verbal descriptions to algebraic sentences

•

not including appropriate constraints as related to the situation

U1-170 CCSS IP Math I Teacher Resource 1.4.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction Guided Practice 1.4.1 Example 1 Determine whether the coordinate (–2, 9) is a solution to the inequality y ≤ 5x + 6. 1. Substitute the values for x and y into the original inequality. y ≤ 5x + 6 9 ≤ 5(–2) + 6 2. Simplify the sentence. 9 ≤ 5(–2) + 6 9 ≤ –10 + 6 9 ≤ –4

Multiply 5 and –2. Add –10 and 6.

3. Interpret the results. 9 is NOT less than or equal to –4; therefore, (–2, 9) is not a solution to the inequality y ≤ 5x + 6. Example 2 A taxi company charges $2.50 plus $1.10 for each mile driven. Write an equation to represent this situation. Use this equation to determine how far you can travel if you have $10.00. What is the minimum amount of money you will spend? 1. T ranslate the verbal description into an algebraic equation. Let m represent the number of miles driven and let C represent the total cost of the trip. 2.50 + 1.10m = C 2. T he total cost of the trip can’t be more than $10.00 because that is all you have to spend. Substitute this amount in for C. 2.50 + 1.10m = 10.00

U1-171 © Walch Education

CCSS IP Math I Teacher Resource 1.4.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction 3. A lthough you have $10.00 to spend, you could also spend less than that. Change the equal sign to a less than or equal to sign (≤). 2.50 + 1.10m ≤ 10.00 4. Solve the inequality by isolating the variable. 2.50 + 1.10m ≤ 10.00

Subtract 2.50 from both sides.

1.10m ≤ 7.50

Divide both sides by 1.10.

m ≤ 6.82 You can travel up to 6.82 miles and not pay more than $10.00. Because the company charges by the mile, you can travel no more than 6 miles. 5. T he minimum amount the taxi driver charges is $2.50, but it is unlikely that he or she will charge you if you get in the cab and get right back out without going anywhere. You will pay $1.10 if you travel 1 mile or less; add this to the minimum charge of $2.50 to arrive at $3.60. 6. You will spend a minimum of $3.60, but no more than $10.00. Example 3 A school supply company produces wooden rulers and plastic rulers. The rulers must first be made, and then painted. •

I t takes 20 minutes to make a wooden ruler. It takes 15 minutes to make a plastic ruler. There is a maximum amount of 480 minutes per day set aside for making rulers.

•

I t takes 5 minutes to paint a wooden ruler. It takes 2 minutes to paint a plastic ruler. There is a maximum amount of 180 minutes per day set aside for painting rulers.

U1-172 CCSS IP Math I Teacher Resource 1.4.1

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction Write a system of inequalities that models the making and then painting of wooden and plastic rulers. 1. Identify the information you know. There is a maximum of 480 minutes for making rulers. •

It takes 20 minutes to make a wooden ruler.

•

It takes 15 minutes to make a plastic ruler.

There is a maximum of 180 minutes for painting rulers. •

It takes 5 minutes to paint a wooden ruler.

•

It takes 2 minutes to paint a plastic ruler.

2. W rite an inequality to represent the amount of time needed to make the rulers. Let w represent the wooden rulers and p represent the plastic rulers. 20w + 15p ≤ 480 3. W rite an inequality to represent the amount of time needed to paint the rulers. Use the same variables to represent wooden and plastic rulers. 5w + 2p ≤ 180 4. N ow consider the constraints on this situation. It is not possible to produce a negative amount of either wooden rulers or plastic rulers; therefore, you need to limit the values of w and p to values that are greater than or equal to 0. w≥0 p≥0 5. C ombine all the inequalities related to the situation and list them in a brace, {. These are the constraints of your scenario. 20 w + 15 p ≤ 480 5 w + 2 p ≤ 180 w ≥ 0 p ≥ 0 U1-173 © Walch Education

CCSS IP Math I Teacher Resource 1.4.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction Example 4 Use the system of inequalities created in Example 3 to give a possible solution to the system. 1. W e know from this situation that you cannot produce a negative amount of rulers, so none of our solutions can be negative. 2. I n future lessons, we discuss more precise ways of determining the solution set to a system. For now, we can use our knowledge of numbers and ability to solve algebraic sentences to find possible solutions. 3. Choose a value for w. Let w = 0. Substitute 0 for each occurrence of w in the system and solve for p. 20w + 15p ≤ 480 20(0) + 15p ≤ 480

Substitute 0 for w.

15p ≤ 480

Divide both sides by 15.

For the first inequality, p ≤ 32. 5w + 2p ≤ 180 5(0) + 2p ≤ 180

Substitute 0 for w.

2p ≤ 180

Divide both sides by 2.

For the second inequality, p ≤ 90. 4. Interpret the results. In 480 minutes, the company can make no more than 32 plastic rulers if 0 wooden rulers are produced. In 180 minutes, the company can paint no more than 90 plastic rulers if there are no wooden rulers to paint.

U1-174 CCSS IP Math I Teacher Resource 1.4.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Problem-Based Task 1.4.1: Skate Constraints A sporting goods company produces figure skates and hockey skates. One group of workers makes the blades for both types of skates. Another group makes the boots for both types of skates. •

I t takes 2 hours to make the blade of a figure skate. It takes 3 hours to make the blade of a hockey skate. There is a maximum of 40 hours per week in which the blades can be made for both types of skates.

•

I t takes 3 hours to make the boot of a figure skate. It takes 1 hour to make the boot for a hockey skate. There is a maximum of 20 hours per week in which boots can be made for both types of skates.

What are possible combinations of the number of figure skates and hockey skates that can be produced given the constraints of this situation?

What are possible combinations of the number of figure skates and hockey skates that can be produced given the constraints of this situation?

U1-175 © Walch Education

CCSS IP Math I Teacher Resource 1.4.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Problem-Based Task 1.4.1: Skate Constraints Coaching a. W hat information do you know about the amount of time needed to make the blade of a figure skate?

b. W hat information do you know about the amount of time needed to make the blade of a hockey skate?

c. How many hours each week can be spent making skate blades?

d. W hat inequality can be used to represent the amount of time it takes to make blades for both figure skates and hockey skates?

e. W hat information do you know about the amount of time needed to make the boot of a figure skate?

f. W hat information do you know about the amount of time needed to make the boot of a hockey skate?

continued U1-176 CCSS IP Math I Teacher Resource 1.4.1

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

g. How many hours each week can be spent making skate boots?

h. W hat inequality can be used to represent the amount of time it takes to make the boots for both figure skates and hockey skates?

i. What other constraints are needed in this situation?

j. What is the system of inequalities that represents this situation?

k. I s it possible to construct 3 figure skates and 4 hockey skates given the constraints of this situation?

l. I s it possible to construct 8 figure skates and 5 hockey skates given the constraints of this situation?

m. W hat is another possible combination of the number of figure skates and hockey skates that can be produced given the constraints of this situation?

U1-177 © Walch Education

CCSS IP Math I Teacher Resource 1.4.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction Problem-Based Task 1.4.1: Skate Constraints Coaching Sample Responses a. W hat information do you know about the amount of time needed to make the blade of a figure skate? It takes 2 hours to make the blade of a figure skate. b. W hat information do you know about the amount of time needed to make the blade of a hockey skate? It takes 3 hours to make the blade of a hockey skate. c. How many hours each week can be spent making skate blades? No more than 40 hours can be spent making skate blades. d. W hat inequality can be used to represent the amount of time it takes to make blades for both figure skates and hockey skates? The total amount of time needed to make the blades can be represented by the sum of the time needed for each type of blade. Let f represent figure skates and h represent hockey skates. The total number of hours needed for making the blades can’t be more than 40 hours. 2f + 3h ≤ 40 e. W hat information do you know about the amount of time needed to make the boot of a figure skate? It takes 3 hours to make the boot of a figure skate. f. W hat information do you know about the amount of time needed to make the boot of a hockey skate? It takes 1 hour to make the boot of a hockey skate. g. How many hours each week can be spent making skate boots? No more than 20 hours can be spent making skate boots. h. W hat inequality can be used to represent the amount of time it takes to make the boots for both figure skates and hockey skates? The total amount of time needed to make the boots can be represented by the sum of the time needed for each type of boot. The total number of hours needed for making the boots can’t be more than 20 hours. 3f + h ≤ 20 U1-178 CCSS IP Math I Teacher Resource 1.4.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction i. What other constraints are needed in this situation? The number of figure skates can’t be less than 0. f ≥ 0 The number of hockey skates can’t be less than 0. h≥0 j. What is the system of inequalities that represents this situation? To create the system of inequalities, group all of the related inequalities. 2 f + 3 h ≤ 40 3 f + h ≤ 20 f ≥0 h ≥ 0 k. Is it possible to make 3 figure skates and 4 hockey skates given the constraints of this situation? Substitute 3 for f and 4 for h in each inequality of the system. 2f + 3h ≤ 40

First inequality in the system

2(3) + 3(4) ≤ 40

Substitute values for f and h, then multiply.

6 + 12 ≤ 40

Simplify.

18 ≤ 40

This is a true statement.

3f + h ≤ 20

Second inequality in the system

3(3) + (4) ≤ 20

Substitute values for f and h, then multiply.

9 + 4 ≤ 20

Simplify.

13 ≤ 20

This is also a true statement.

f ≥ 0

Third inequality; substitute 3 for f.

3 ≥ 0

This is a true statement.

h ≥ 0

Last inequality; substitute 4 for h.

4 ≥ 0

This is a true statement.

The substitution of 3 for f and 4 for h results in true statements for each inequality of the system. Therefore, it is possible to make 3 figure skates and 4 hockey skates given the constraints of this situation. U1-179 © Walch Education

CCSS IP Math I Teacher Resource 1.4.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints Instruction l. Is it possible to make 8 figure skates and 5 hockey skates given the constraints of this situation? Substitute 8 for f and 5 for h in each inequality of the system. 2f + 3h ≤ 40

First inequality in the system

2(8) + 3(5) ≤ 40

Substitute values for f and h, then multiply.

16 + 15 ≤ 40

Simplify.

31 ≤ 40

This is a true statement.

3f + h ≤ 20

Second inequality in the system

3(8) + (5) ≤ 20

Substitute values for f and h, then multiply.

24 + 5 ≤ 20

Simplify.

29 ≤ 20

This is NOT a true statement.

f ≥ 0

Third inequality; substitute 8 for f.

8 ≥ 0

This is a true statement.

h ≥ 0

Last inequality; substitute 5 for h.

5 ≥ 0

This is a true statement.

The substitution of 8 for f and 5 for h results in true statements for 3 of the 4 inequalities of the system. Because one of the inequalities is not true, it is not possible to make 8 figure skates and 5 hockey skates given the constraints of this situation. m. W hat is another possible combination of the number of figure skates and hockey skates that can be made given the constraints of this situation? Students can use trial and error to find other possible combinations that will satisfy each of the inequalities in the system. Possible solutions: 4 figure skates and 5 hockey skates; 2 figure skates and 10 hockey skates Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

U1-180 CCSS IP Math I Teacher Resource 1.4.1

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Practice 1.4.1: Representing Constraints Determine whether each coordinate listed below is a solution to the given algebraic sentence. 1. Is the coordinate (–2, –4) a solution to the equation y = 3x – 2? 2. Is the coordinate (1, –3) a solution to the inequality y ≤ –4x + 6? Read each scenario and use it to complete the parts that follow. 3. Given the inequalities y > 5x – 8 and y ≥ 3x + 4, find a point that a. satisfies both inequalities. b. satisfies neither inequality. c. satisfies one inequality, but not the other. 4. You pay $12 to get into the fair, plus $3 per ticket for x ride tickets. a. Write an equation to find the total cost of attending the fair. b. N ow write an inequality and solve it to determine the maximum number of tickets you can buy if you have $24 to spend. c. What is the minimum amount of money you will spend? 5. C harlie borrowed $500 from his aunt. He has already paid back $75. His aunt doesn’t charge any interest and he is planning on making $15 payments each Friday. a. W rite an equation that represents the number of weeks it will take Charlie to repay his aunt if he pays $15 each Friday. b. I s the solution to the equation the actual number of weeks it will take Charlie to repay his aunt? Explain your answer.

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CCSS IP Math I Teacher Resource 1.4.1

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Use the information in each scenario to complete problems 6–10. 6. T he concession stand at the football game sells cans of soda for $0.75 and bottles of water for $1.25. You have $10.00. Write an inequality to represent this situation. What can you buy?

7. A stained glass artist has a fixed cost of $150. It costs the artist $15 to produce each piece, but each piece sells for $35. The equation C = 150 + 15n represents the total cost, C, for producing n pieces. The total revenue for n pieces is determined by the equation R = 35n. What constraint is necessary to include when modeling this situation?

8. Y our dad needs to rent a chain saw to cut down trees in your yard. The rental company charges $20 plus $6.50 per hour to rent the chain saw. Your dad wants to spend no more than $50. What constraints apply to this situation? What is the maximum number of hours your dad can rent the chain saw?

9. J ermaine has $10.00 to spend on ice cream. Three scoops cost $5.99, plus $0.75 for each topping. He always leaves a 20% tip for the cashier. Write an inequality and use it to determine if Jermaine can afford to buy a three-scoop ice cream with three toppings plus tip the cashier.

10. T he local florist never has more than a combined total of 40 daisy and carnation bouquets and never more than 12 carnation bouquets. Write a system of inequalities that represents this situation. Be sure to include all constraints.

U1-182 CCSS IP Math I Teacher Resource 1.4.1

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Assessment Progress Assessment Circle the letter of the best answer. 1. Which point is a solution of the equation y = –7x + 3? a. (–2, 11)

c. (0, –3)

b. (–1, 10)

d. (1, 4)

2. Given the inequality y ≥ 2x – 9, which point is NOT a solution? a. (–3, –15)

c. (0, –9)

b. (2, –7)

d. (3, –3)

3. Given the inequalities y < 2x – 6 and y ≤ 4x + 5, the point (2, 3) is: a. a solution to both inequalities b. a solution to y < 2x – 6 only c. a solution to y ≤ 4x + 5 only d. not a solution of either of the inequalities 4. A n online company is advertising a subscription service for downloads of e-books. The monthly fee is $6.50, plus $1.99 for each downloaded book. You can afford to spend no more than $15.00 each month on e-books. What is the maximum number of e-books you can download? a. 8

c. 5

b. 7

d. 4

5. U sed video games are advertised for $12.00 and new video games are advertised for $20.00. You have a gift card for $100.00. Which of the following game combinations can you NOT buy? a. 7 used video games and 1 new video game b. 4 used video games and 2 new video games c. 1 used video game and 4 new video games d. 0 used video games and 5 new video games

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CCSS IP Math I Teacher Resource 1.4

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Assessment 6. A n office manager needs to buy 25 laptops and has planned on spending $30,000. Two models are available; one costs $1,100, and the other costs $1,800. Determine which system of inequalities represents this situation. 1100 x + 1800 y ≤ 30, 000 a. x ≥ 0 y≥0

1100 x + 1800 y ≥ 30, 000 c. x + y = 25 x ≤ 0 y ≤ 0

1100 x + 1800 y ≥ 30, 000 b. x ≤ 0 y≤0

1100 x + 1800 y ≤ 30, 000 d. x + y = 25 x ≥ 0 y ≥ 0

7. C arlos has a monthly budget of $250.00 for groceries. He spent $57.63 the first week, $46.89 the second week, and $68.13 the third week. Using g for groceries, how much can Carlos spend on groceries during the final week of the month and still stay within budget? a. g ≤ 77.35

c. g < 77.35

b. g ≥ 0 and g ≤ 77.35

d. g > 0 and g < 77.35

8. A science test worth 100 points has 10 questions. The test consists of fill-in-the-blank questions worth 6 points each and short-answer questions worth 16 points each. How many fill-in-theblank questions are on the test? a. 4 b. 6 c. 10 d. There is not enough information to determine the correct answer.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 4: Representing Constraints

Date:

Assessment 9. M adison works part time at a local farm. She makes $10.00 an hour for cleaning the animals and $7.00 for feeding them. Because Madison is a student, she cannot work more than 15 hours a week. Which system of inequalities represents the amount of hours Madison will work if she wants to earn at least $150 a week? x + y ≤ 15 a. 10.00 x + 7.00 y ≤ 150 x ≥ 0 y ≥ 0

c. x + y ≤ 15 10.00 x + 7.00 y ≤ 150

x + y < 15 b. 10.00 x + 7.00 y < 150 x ≥ 0 y ≥ 0

x + y ≤ 15 d. 10.00 x + 7.00 y ≥ 150 x ≥ 0 y ≥ 0

10. J ason loves going to the movies. Sometimes he is able to catch a matinee for $6.00, but otherwise he has to go at night for $9.50. If Jason went to see 22 movies this past month, what is the minimum and maximum amount of money he could have spent? a. minimum: $0; maximum: $209

c. minimum: $132; maximum: $209

b. minimum: $0; maximum: $132

d. minimum: $60; maximum: $114

Use the information that follows to complete problem 11. A computer company produces desktop and laptop computers. It takes 3 hours to build a desktop computer. It takes 2 hours to build a laptop computer. The maximum time allotted to this task is 60 hours per week. The computer company then packages the computers to mail them to customers. It takes 2 hours to package a desktop computer. It takes 1 hour to package a laptop. The maximum time allotted for this task is 45 hours per week. 11. W rite a system of inequalities to represent the number of computers that can be made and packaged in one week. Be sure to include all constraints.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Assessment Pre-Assessment Circle the letter of the best answer. 1. Solve the equation 8x + 4y = 12 for y. a. y = 2x – 3

c. y = –3x + 2

b. y = –2x + 3

d. y = 3x – 2

1 2. Solve the equation − y + 3 x = 7 for y. 5 a. y = 15x – 35 b. y = –15x + 35

c. y = 35 – 15x d. y = 35 + 15x

3. The formula P = 2l + 2w is used to calculate the perimeter of a rectangle. Solve this formula for l. 2w − P 2 a. l = c. l = 2 2w − P P − 2w 2 b. l = d. l = 2 P − 2w 4. The formula V = lwh is used to calculate the volume of a prism. Solve this formula for l. wh V a. l = c. l = V wh b. l = V – wh

d. l = wh – V

2π r 5. T he speed, v, of a point on the edge of a spinning disk is found using the formula v = . T Solve this formula for T. v a. T = c. T = 2πr – v 2π r 2π r b. T = d. T = v – 2πr v

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES

Lesson 5: Rearranging Formulas Instruction Common Core State Standard A–CED.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.★

Essential Questions 1. H ow is solving a literal equation or formula for a specific variable similar to solving an equation with one variable? 2. How could solving a literal equation or formula for a specific variable be helpful? 3. How do you determine for which variable a literal equation or formula should be solved? WORDS TO KNOW formula

a literal equation that states a specific rule or relationship among quantities

inverse

a number that when multiplied by the original number has a product of 1

literal equation

an equation that involves two or more variables

reciprocal

a number that when multiplied by the original number has a product of 1

Recommended Resources •

CRCTLessons.com. “Solving Equations Game.” http://walch.com/rr/CAU1L5SolvingEquations Practice solving equations for a given variable with this online basketball game.

•

Purplemath.com. “Solving Literal Equations.” http://walch.com/rr/CAU1L5LitEquations This site has an overview of literal equations, with worked examples on how to solve equations and formulas for a given variable.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Lesson 1.5.1: Rearranging Formulas Warm-Up 1.5.1 Read the scenario below. Write an equation and use it to answer the questions that follow. In January 2011, the national average for 5 gallons of gasoline was $17.20. 1. What was the national average for 1 gallon of gas?

2. What was the price for 18 gallons of gas?

U1-188 CCSS IP Math I Teacher Resource 1.5.1

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction Lesson 1.5.1: Rearranging Formulas Common Core State Standard A–CED.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.★

Warm-Up 1.5.1 Debrief 1. What was the national average for 1 gallon of gas? Set up the equation. Let x represent the cost per gallon of gasoline. 5x = 17.20 Using inverse operations, solve this equation for x. 5 x = 17.20 5x

=

Original equation

17.20

Divide each side by 5. 5 5 x = 3.44 Interpret the solution. The average cost of 1 gallon of gasoline was $3.44. 2. What was the price for 18 gallons of gas? To find the cost of 18 gallons of gasoline, multiply the number of gallons by the cost per gallon. 18 • 3.44 = $61.92 Interpret the solution. The cost of 18 gallons of gasoline at $3.44 a gallon is $61.92. Connection to the Lesson •

S tudents are asked to solve equations, much like they will be asked to solve a literal equation or formula for a specified variable.

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CCSS IP Math I Teacher Resource 1.5.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction Prerequisite Skills This lesson requires the use of the following skills: •

order of operations

•

solving multi-step equations

Introduction Literal equations are equations that involve two or more variables. Sometimes it is useful to rearrange or solve literal equations for a specific variable in order to find a solution to a given problem. In this lesson, literal equations and formulas, or literal equations that state specific rules or relationships among quantities, will be examined. Key Concepts •

I t is important to remember that both literal equations and formulas contain an equal sign indicating that both sides of the equation must remain equal.

•

Literal equations and formulas can be solved for a specific variable by isolating that variable.

•

To isolate the specified variable, use inverse operations. When coefficients are fractions, multiply both sides of the equation by the reciprocal. The reciprocal of a number, also known as the inverse of a number, can be found by flipping a number. Think of an integer as a fraction with a denominator of 1. To find the reciprocal of the number, flip the fraction. 2 2 The number 2 can be thought of as the fraction . To find the reciprocal, flip the fraction: 1 1 1 becomes . You can check if you have the correct reciprocal because the product of a number 2 and its reciprocal is always 1.

Common Errors/Misconceptions •

solving for the wrong variable

•

improperly isolating the specified variable by using the opposite inverse operation

•

incorrectly simplifying terms

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction Guided Practice 1.5.1 Example 1 Solve 6y – 12x = 18 for y. 1. Begin isolating y by adding 12x to both sides. 6y – 12x = 18 + 12x

+ 12x

6y = 18 + 12x 2. Divide each term by 6. 6y 6

=

18 6

+

12 x 6

y = 3 + 2x Example 2 Solve 15x – 5y = 25 for y. 1. Begin isolating y by subtracting 15x from both sides of the equation. 15 x − 5 y = 25 −15 x − 15 x − 5 y = 25 − 15 x 2. T o further isolate y, divide both sides of the equation by the coefficient of y. The coefficient of y is –5. Be sure that each term of the equation is divided by –5. −5 y −5 −5 y

=

25 − 15 x

−5 25 15 x = − −5 −5 −5 y = −5 + 3 x

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CCSS IP Math I Teacher Resource 1.5.1

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction Example 3 Solve 4y + 3x = 16 for y. 1. Begin isolating y by subtracting 3x from both sides of the equation. 4 y + 3 x = 16 − 3x − 3x 4 y = 16 − 3 x 2. T o further isolate y, divide both sides of the equation by the coefficient of y. The coefficient of y is 4. Be sure that each term of the equation is divided by 4. 4 y 16 − 3 x = 4 4 16 3 x y= − 4 4 3 y= 4− x 4

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction Example 4

1 The formula for finding the area of a triangle is A = bh , where b is the length of the base and h is 2 the height of the triangle. Suppose you know the area and height of the triangle, but need to find the length of the base. In this case, solving the formula for b would be helpful. 1. Begin isolating b by multiplying both sides of the equation by the 1 reciprocal of , or 2. 2 1 A = bh 2 1 2 • A = 2 • bh 2 2 A = bh Multiplying both sides of the equation by the reciprocal is the same as 1 dividing both sides of the equation by . The result will be the same. 2 2. To further isolate b, divide both sides of the equation by h. 2 A bh 2A = or b = h h h 2A =b h 3. T he formula for finding the length of the base of a triangle can be found by doubling the area and dividing the result by the height of the triangle.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction Example 5 The distance, d, that a train can travel is found by multiplying the rate of speed, r, by the amount of time that it is travelling, t, or d = rt. Solve this formula for t to find the amount of time the train will travel given a specific distance and rate of speed. 1. Isolate t by dividing both sides of the equation by r. d rt = r r d t= r 2. The formula for finding the amount of time it will take a train d to travel a given distance at a given speed is t = . r

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Problem-Based Task 1.5.1: Bricklayers The formula N = 7LH is used to determine N, the number of bricks needed to build a wall that is L feet in length and H feet high. A customer would like a wall constructed that is 4 feet high. If the bricklayer wants to use all of the 1,820 bricks that he has readily available, how long will the wall be?

If the bricklayer wants to use all of the 1 ,820 bricks that he has readily available, how long will the wall be?

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Problem-Based Task 1.5.1: Bricklayers Coaching a. What does each of the variables represent?

b. What variable is the given formula solved for?

c. Which variable does the formula need to be solved for?

d. Solve the given formula for the unknown variable.

e. What values are given in the problem statement?

f. How long can the wall be if the bricklayer has 1,820 bricks and the wall must be 4 feet high?

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction Problem-Based Task 1.5.1: Bricklayers Coaching Sample Responses a. What does each of the variables represent? N represents the number of bricks. L represents the length of the wall in feet. H represents the height of the wall.

b. What variable is the given formula solved for? The formula is solved for N, the number of bricks.

c. Which variable does the formula need to be solved for? The formula needs to be solved for L, the length of the wall.

d. Solve the given formula for the unknown variable. N = 7 LH N 7 LH = 7H 7H N L= 7H

Original equation Divide both sides by 7H.

e. What values are given in the problem statement? The number of bricks, N, is 1,820. The height of the wall, H, is 4 feet.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas Instruction f. How long can the wall be if the bricklayer has 1,820 bricks and the wall must be 4 feet high? Substitute the known values into the formula. L= = =

N 7H 1820 7(4) 1820

28 = 65

Original equation Substitute 4 for H. Multiply 7 and 4. Simplify.

The length of the wall, L, can be 65 feet. Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Practice 1.5.1: Rearranging Equations and Formulas For problems 1–4, solve each equation for y. 1. 9y + 18 = 27x

2. 6y + 24x = 66

3. 10x – 77 = 7y

4. 44 – 4y = 20x

Read each scenario and solve for the given variable. 5. To convert degrees Celsius to Kelvin, the formula K = C + 273.15 is used. Solve this formula for C.

6. The formula C = 2πr is used to calculate the circumference of a circle. Solve this formula for r.

7. The formula V = lwh is used to calculate the volume of a prism. Solve this formula for w.

8. The formula S = 2πr2 + 2πrh is used to find the surface area of a cylinder. Solve this formula for h. 9 9. T he formula for converting degrees Celsius to degrees Fahrenheit is F = C + 32 . Solve this 5 formula for C. 1 10. The formula for calculating the volume of a cone is V = π r 2 h . Solve this formula for h. 3

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Assessment Progress Assessment Circle the letter of the best answer. 1. In the equation 4x – 2y = 19, the first step to isolating y would likely be: a. Divide both sides of the equation by 2y. b. Subtract 4x from both sides of the equation. c. Divide both sides of the equation by 4x. d. Subtract y from both sides of the equation.

2. Solve the equation a. y =

10 3

b. y = x −

2 3

y + x = 4 for y.

−x 10 3

3 c. y = ( x − 4) 2 3 d. y = (4 − x ) 2

3. Solve the equation 7x – 3y = 16 for y. a. y = b. y =

7 x − 16 3 16 − 7 x 3

c. y = d. y =

3 x − 16 7 16 − 3 x 7

4. Solve the equation 6y + 12x = 18 for y. a. y = –3x + 2

c. y = –2x + 3

b. y = 2x – 3

d. y = 3x – 2

continued U1-200 CCSS IP Math I Teacher Resource 1.5

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Assessment 3 5. Solve the equation 6 x + y = 9 for y. 4 27 x − 18 a. y = 4 b. y = 8x – 12

c. y =

−27 x + 18 4

d. y = –8x + 12

6. The formula for calculating simple interest is I = prt. Solve this formula for r. pt a. r = I – pt c. r = I I b. r = d. r = pt – I pt

7. The formula for calculating the volume of a cylinder is V = π r 2 h . Solve this formula for h. V πr2 a. h = c. h = 2 πr V 2 b. h = V − π r d. h = π r 2 − V

8. The formula for calculating the surface area of a cone is A = π r 2 + π rl . Solve this formula for l. A A−πr2 a. l = c. l = 2 = π r πr πr b. l =

πr A−πr2

d. l =

πr2 A

=πr

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Lesson 5: Rearranging Formulas

Date:

Assessment 9. Solve the formula B =

(R + M ) 2

a. R = 2B + M 2B b. R = M

10. Solve the formula r 3 = a. V = b. V =

4π r 3 3 3r 3 4π

for R. c. R =

M

2B d. R = 2B – M

3V 4π

for V. c. V = 4π r 3 − 3 d. V = 12r 3π

Use your knowledge of rearranging formulas to answer the question that follows. 11. T he formula for calculating the number of calories burned while swimming at a moderate rate is C = 2.9wt, where C represents the number of calories burned, w represents your weight, and t represents the time you swim in hours. How could you use this formula to help you determine how long you should swim in order to burn a specific number of calories?

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Unit Assessment

Date:

Assessment Unit 1 Assessment Circle the letter of the best answer. 1. How many terms are in the simplified expression 22 x 3 + 14 x 2 − 10 x 2 + 3 x + 7 ? a. 5

c. 3

b. 4

d. 2

2. T he product of –3, a, and b is represented by the expression –3ab. If the value of a is negative, what must be said about the value of b in order for the product to remain negative? a. b must be 0.

c. b must be negative.

b. b must be positive.

d. The value of b does not matter.

3. A family’s cell phone plan costs $70 per month for 1,300 minutes and 40 cents per minute over the limit. This month, the family paid $118.40. By how much time did they exceed their plan? a. 121 minutes

c. 20 minutes

b. 471 minutes

d. 76 minutes

4. Y ou have no more than $60 to spend. You want a drink that costs $1.50 including tax, and you want to buy a pair of pants, which will have 4% sales tax. What is the inequality that represents the amount of money you have to spend? a. x + 0.04x + 1.50 > 60

c. x + 0.04x + 1.50 < 60

b. x + 0.04x + 1.50 ≥ 60

d. x + 0.04x + 1.50 ≤ 60

5. A store has a display with pencils that are for sale. The owner typically sells 6 pencils a day. The display holds 50 pencils. The owner insists that there be no fewer than 32 pencils in the display. When should the owner restock the display? a. in more than 3 days

c. in 3 days or less

b. in less than 3 days

d. in 3 days or more

6. A type of bacteria doubles every 7 hours. If you started with 16 bacteria in a Petri dish, how many bacteria would you have after 56 hours? a. 2,048 bacteria

c. 7.2 • 1016 bacteria

b. 4,096 bacteria

d. 1.15 • 1018 bacteria

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Unit Assessment

Date:

Assessment 7. A form of the element actinium, Ac-225, has a half-life of 10 days. At the end of an experiment that lasted 40 days, there were 2 grams of Ac-225 left. How much Ac-225 was there at the beginning of the experiment? a. 2.2 • 1012 grams

c. 0.125 grams

b. 32 grams

d. 16 grams

8. A photo service charges $25.00 a year as well as $0.05 for each photo ordered. Which graph models the total cost of ordering photos? a.

y

c.

y

30

29.5

29.5

29

29

28.5

28.5

Cost in dollars ($)

Cost in dollars ($)

30

28 27.5 27 26.5 26 25.5

28 27.5 27 26.5 26 25.5 25

25

24.5

24.5 0

x 10 20 30 40 50 60 70 80 90 100

0

1

2

b.

30

d.

5

29.5

5

6

7

8

9

x 10

y

4.5

29

4

28.5

Cost in dollars ($)

Cost in dollars ($)

4

Pictures ordered

Pictures ordered y

3

28 27.5 27 26.5 26 25.5

3 2.5 2 1.5 1

25

0.5

24.5 0

3.5

x 10 20 30 40 50 60 70 80 90 100

Pictures ordered

0

0

10

20

30

40

50

60

70

80

x 90 100

Pictures ordered

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Unit Assessment

Date:

Assessment 9. A 4-door sedan holds 17 gallons of gas and the tank averages 0.04 gallons per mile. Which graph models the amount of gas left in the tank? y 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0

c.

20

y

18 16

Gas in gallons

Gas in gallons

a.

14 12 10 8 6 4 2

5

10

15

20

25

30

35

x 40

0 0

d.

20

15

20

25

30

35

x 40

y

18 16

Gas in gallons

Gas in gallons

y 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0 40

10

Distance traveled in miles

Distance traveled in miles

b.

5

14 12 10 8 6 4 2

x 80 120 160 200 240 280 320 360 400

Distance traveled in miles

0 0

x 40

80 120 160 200 240 280 320 360 400

Distance traveled in miles

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Unit Assessment

Date:

Assessment 10. A n investment of $1,000 is compounded monthly at a rate of 2.5%. Which graph models the change of the investment over time? Investment value in dollars ($)

1200

y

c.

1100

Investment value in dollars ($)

a.

1000 900 800 700 600 500 400 300 200 100 0

0

.5

1

1.5

2

2.5

3

3.5

4

4.5

x 5

2500 2400 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900

y

0

1

2

y

0

d.

2000

Investment value in dollars ($)

Investment value in dollars ($)

2500 2400 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900

1

2

3

Years

4

x 5

Years

Years

b.

3

4

x 5

y

1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 0

5

10

15

20

25

x 30

Years

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Unit Assessment

Date:

Assessment 11. Y our doctor told you to eat at least 70 milligrams of vitamin C each day. One tomato contains 16 milligrams of vitamin C, while one potato contains 17 milligrams. Determine which system of inequalities represents the number of tomatoes and potatoes you must eat in order to reach your minimum recommended amount of vitamin C. 16 x + 17 y ≥ 70 16 x + 17 y ≤ 70 a. x ≤ 0 c. x ≥ 0 y≥0 y≤0 16 x + 17 y ≤ 70 b. x ≥ 0 y≥0

16 x + 17 y ≥ 70 d. x ≤ 0 y≤0

12. The formula for calculating a person’s body mass index is B =

w

, for which w represents h2 weight in kilograms and h represents height in meters. Solve this formula for w. B 2 a. w = Bh c. w = 2 h 2 b. w = B − h

2 d. w = ( Bh)

Read each scenario and answer the questions that follow. Write your answers below each lettered part and show your work in the space provided. 13. K eisha bought 6 tickets to an indoor water park. She paid a 5% service charge for buying them online. Her total cost was $252. a. What equation can be used to model the total cost of the tickets?

b. What was the price of each ticket?

c. How much did Keisha pay in service charges?

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Unit Assessment

Date:

Assessment 14. M in-Ji injured her elbow during a varsity volleyball game. Her doctor has recommended physical therapy several times a week. Min-Ji’s parents want to plan for the potential cost of therapy over the course of a month. They pay $160 a month for insurance and then another $20 fee each time Min-Ji goes to physical therapy. a. What equation models the total fees for physical therapy?

b. W hat does the graph of the equation look like? Graph the equation below. Be sure to label the axes.

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Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Unit Assessment

Date:

Assessment 15. T he population of Georgia is growing at a yearly rate of about 1.3%. The current population is about 9,815,210 people. The population of North Carolina is growing at a yearly rate of about 1.25%. North Carolina’s current population is about 9,656,401 people. a. What is the equation that models Georgia’s population growth?

b. What is the equation that models North Carolina’s population growth?

c. W hat do the graphs of the equations look like? Graph the equations below. Be sure to label the axes.

d. W rite a few sentences comparing the population models of Georgia and North Carolina. What can you conclude based on your models?

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES

Answer Key

Lesson 1: Interpreting Structure in Expressions Pre-Assessment, pp. 1–2 1. c 2. a 3. d

4. c 5. b

Warm-Up 1.1.1, p. 5 1. 2d 2. 2d + 3c 3. 2d + 3c + 5.60

Practice 1.1.1: Identifying Terms, Factors, and Coefficients, pp. 15–16 1. terms: 12a , 16a, 4 factors: 12 and a3, 16 and a coefficients: 12, 6 constant: 4 2. terms: 6x2, 3x, 9 factors: 6 and x2, 3 and x coefficient: 6, 3 constant: 9 3. expression: ((x + y)/2) – y/3 = (1/2)x + (1/6)y terms: (1/2)x, (1/6)y factors: 1/2 and x, 1/6 and y coefficients: 1/2, 1/6 constants: there are none 4. expression: 5x3 + (6 – x3) = 4x3 + 6 terms: 4x3, 6 factors: 4 and x3 coefficient: 4 constant: 6 5. Answers may vary; possible response: 3x3 + 6x2 + 9x + 4 6. expression: 6x – 0.15(6x) = 5.1x term: 5.1x factors: 5.1 and x coefficient: 5.1 constants: there are none 7. expression: 10x – 0.25(10x) + 5.99 = 7.5x + 5.99 terms: 7.5x, 5.99 factors: 7.5 and x coefficient: 7.5 constant: 5.99 8. expression: (30.24 – 2.24)/(x + 1) = 28/(x + 1) term: 28/(x + 1) factors: 28 and 1/(x + 1) coefficient: 28 constants: there are none 3

9. expression: 1/2(b1 + b2 )h or 1/2(b1)h + 1/2(b2 )h terms: 1/2(b1)h, 1/2(b2 )h factors: 1/2, b1, h and 1/2, b2 , h coefficients: 1/2, 1/2 constants: there are none 10. expression: 2πr2 + 2πrh terms: 2πr2, 2πrh factors: 2, π, r2 and 2, π, r, h coefficients: 2, 2 constants: there are none

Warm-Up 1.1.2, p. 17 1. $112.50

2. $862.50

Practice 1.1.2: Interpreting Complicated Expressions, pp. 27–28 1. No, the only time the expressions will be equal is when x = 1. You must follow the order of operations when evaluating expressions. 2. No, exponents must be applied before carrying out multiplication. 3. Yes, the parentheses must be cleared before applying exponents. 4. The cost of the permit is not affected by the number of cubic yards. The cost of the permit remains the same and is described using a constant in the expression given. 5. A lower service fee will result in a constant less than 34.75. 6. The value of y must be greater than or equal to 0 as it represents the number of weeks. 7. The value of y must also be negative in order for the product of 7, x, and y to be positive. 8. The amount will be decreased. 9. x must be greater than 0 10. Changing the value of r does not have an effect on the value of C because the amount of air the tire can hold is not affected by the rate at which it loses air.

Progress Assessment, pp. 29–31 1. a 6. c 2. c 7. a 3. c 8. c 4. d 9. b 5. b 10. d 11. The order of operations must be followed in each expression. In the expression a xb, you must first apply the exponent to a and then multiply by b. In the expression ab x, you must first apply the exponent to b and then multiply by a. In the expression (ab) x, you must first find the product of a and b and then apply the exponent. U1-211

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Lesson 2: Creating Equations and Inequalities in One Variable Pre-Assessment, p. 32 1. b 2. d 3. c

4. a 5. b

Warm-Up 1.2.1, p. 35 1. 10x 2. 10x – 0.25(10x) 3. 10x – 0.25(10x) + 5.99

Practice 1.2.1: Creating Linear Equations in One Variable, pp. 54–55 1. Answers may vary. Possible answers: a. miles per hour b. inches per hour c. gallons per second d. calories per gram 2. 99.4 ft2 3. 6 hours per day 4. $3.50 per gallon 5. $25 per ticket 6. $39 per month 7. $103 8. 48 ft3 9. 35t = 40(0.75 – t); t = 0.4; d = (35)(0.4) = 14 miles. The distance from your house to your friend’s house is 14 miles. 10. 400 = ((15 + r) + r)5; r = 32.5 mph. One train is traveling at 32.5 miles per hour and the other is traveling at 47.5 miles per hour.

Warm-Up 1.2.2, p. 56 1. Eric weighs about 104 pounds. 2. If the seesaw is 5 feet or longer, then Eric can balance the seesaw.

Practice 1.2.2: Creating Linear Inequalities in One Variable, pp. 68–69 1. x ≤ 55 2. x ≤ 60 3. 19 + x ≥ 39; x ≥ 20; She needs to save $20 or more. 4. 20x ≥ 1800; x ≥ 90; You must work 90 hours or more. 5. 83 + x ≥ 180; x ≥ 97; Hiram must get a score of 97 points or more. 6. 15x + 10 ≤ 100; x ≤ 6; Claire can purchase up to 6 DVDs. 7. 53 – 1.5h ≤ 41; h ≥ 8 hours. The rain will likely change over at or after 8 hours from 8:30 a.m., which means the changeover will occur at or after 4:30 p.m. 8. 17 – 3x ≥ 5, where x is time in 5-minute increments; x ≤ 4, so 5(4) = 20 minutes. After the first hour, the players will be able to continue for 20 minutes or less. 9. 15,000 – 2000x ≥ 3000; x ≤ 6 days; The giveaway can last for 6 days or less. 10. Answers may vary. Sample answer: You have no money left in your wallet but found a gift card for the local deli that has $9. You want to make sure you spend less than that for 2 lunch specials for you and your friend. Your friend gave you $5 toward the bill. In this scenario, each lunch special must cost $7 or less; x ≤ 7.

Warm-Up 1.2.3, p. 70

Inequality: 70x + 100 > 500 Solution: x > 5 School started more than 5 months ago.

Practice 1.2.3: Creating Exponential Equations, pp. 85–86 1. a. linear; b. exponential 2. a. linear; b. exponential 3. y = 100(2)7; 12,800 insects 4. 64 bacteria 5. y = 125(0.5) 4 ; $7.81 6. 75 = a(0.5) 3 ; $600 7. 5000 = a(0.5)2; $20,000 8. y = 15(3) 7 ; 32,805 dandelions 9. y = 12,000(1.02) 4 ; about 12,990 people 10. y = 4(2) 4 ; 64,000,000 people if rush hour continued in the morning, and y = 4(0.5) 4 ; 250,000 people if rush hour continued in the evening

Progress Assessment, pp. 87–89 1. d 6. d 2. c 7. b 3. d 8. b 4. a 9. a 5. a 10. a 11. 65h + 99 = 261.50; h = 2.5 hours. The mechanic worked on your car for 2 1/2 hours.

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Lesson 3: Creating and Graphing Equations in Two Variables

y

2. 5

Pre-Assessment, pp. 90–92 1. d 2. a 3. c

4

4. b 5. d

3 2

Warm-Up 1.3.1, p. 95

1

1. Table of values:

–5 –4 –3 –2 –1

0

Minutes used

Total amount charged ($)

0

20 + 0(0.05) = 20.00

10

20 + 10(0.05) = 20.50

–2

20

20 + 20(0.05) = 21.00

–3

30

20 + 30(0.05) = 21.50

40

20 + 40(0.05) = 22.00

–4

50

20 + 50(0.05) = 22.50

60

20 + 60(0.05) = 23.00

Number of revolutions

4 3

10 8 6 4 2 0

2

4

6

8

10

x

4. y = (9/5)x + 32; points could include (0, 32) and (5, 41)

1

y

1

2

3

4

5

x

–1

Degrees Fahrenheit

–2

–5

x

Time in seconds

2

–4

5

y

5

–3

4

3. y = 2x. The two plotted points (0, 0) and (1, 2) would be in a table of values and shown on the graph.

y

0

3

–5

Practice 1.3.1: Creating and Graphing Linear Equations in Two Variables, p. 129

–5 –4 –3 –2 –1

2

–1

2. y = 0.05x + 20 3. x represents the number of minutes used, and y represents the total amount charged.

1.

1

50 45 40 35 30 25 20 15 10 5 –45–40–35–30–25–20–15–10 –5 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50

5 10 15 20 25 30 35 40 45 50

x

Degrees Celsius

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5. y = 0.4x + 2.5; points could include (0, 2.5) and (5, 4.5)

7. y = 2.6x + 6.7; points could include (0, 6.7) and (10, 32.7)

y

10 9

7

Fee in dollars ($)

Cab fare in dollars ($)

8

6 5 4 3 2 1 x 0

y

75 70 65 60 55 50 45 40 35 30 25 20 15 10 5

x

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of gallons in thousands

Time in minutes

6. y = 50x + 300; points could include (0, 300) and (1, 350) 1000

8. y = –40x + 1250; points could include (0, 1250) and (3, 1130)

y

900

700 600 500 400 300 200 100 x 0

y

Amount owed in dollars ($)

Weekly earnings in dollars ($)

800

1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100

1

2

3

4

5

6

Number of recruits

7

8

9

10

0

x 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 Time in months

Employees

9. y = (16/8) x + 3 = 2x + 3

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3. y = 20(2) x

Pounds of food

Number of insects

10. y = –8x + 70

0

Months

Days

Warm-Up 1.3.2, p. 130

4. y = 500(0.5) 2x, for which x is in minutes

Practice 1.3.2: Creating and Graphing Exponential Equations, p. 150 1.

Grams of rhodium

1. y = 128(0.5)5 2. y = 4 grams

Minutes

5. y = 225(.75) , for which x is in weeks

0

2.

Stock worth in dollars ($)

x/2

Weeks

6. y = 12(3) x/6, for which x is in days

Weeds

0

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7. y = 67,000(1.025) x

Population

Progress Assessment, pp. 151–165

Years

8. y = 1000(1.0185)

1. b 6. b 2. c 7. c 3. a 8. a 4. d 9. c 5. d 10. b 11. The first investment equation is y = 500 + 0.03(500) x. The second investment equation is y = 500(1 + 0.03/4)4x = 500(1.0075)4x. The first investment has a linear rate of growth, while the second equation has an exponential rate of growth. Since the second equation has the variable in the exponent, the investment should grow more quickly, but the rate is very small, 3%. Graphing the equations illustrates the growth.

2x

Years

Investment worth in dollars ($)

9. y = 600(1.00725)4x

Investment worth in dollars ($)

Investment worth in dollars ($)

From the graph, the investments are about the same for the first 3 years, but after the fourth year the second investment starts to take off and grow much more rapidly. If this is going to be a long-term investment, then the better choice is the second investment option that compounds quarterly. If it will be a short-term investment (less than 3 years), then the two options are about the same.

Years

Lesson 4: Representing Constraints Pre-Assessment, p. 166 Years

10. y = 3000(1.00027)52x

1. d 2. d 3. a

4. d 5. c

Warm-Up 1.4.1, p. 168

Investment worth in dollars ($)

1. $3.41 per gallon 2. No, she will not have enough money.

Practice 1.4.1: Representing Constraints, pp. 181–182

Years

1. no 2. yes 3. Answers will vary. Sample responses: a. (–1, 2); b. (4, 2); c. (1, 3) 4. a. y = 3x + 12; b. 4 tickets; c. You will spend no less than $12. 5. a. 425/x = 15; b. The solution to the equation (x = 28.33 ) is not the actual number of weeks it will take Charlie to repay his aunt. Charlie makes payments each Friday, so he will have to pay her the remaining balance on the 29th week.

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x + y ≤ 40 x ≤ 12 10. x ≥ 0 y ≥ 0

Progress Assessment, pp. 183–185 1. b 2. b 3. c 4. d 5. a

6. d 7. b 8. b 9. d 10. c

3 x + 2 y ≤ 60 11. 2 x + y ≤ 45 x ≥ 0 y ≥ 0

11. Rearrange the formula so that it is solved for t, the number of hours you should swim. Your weight, w, will not change in the equation, leaving just the number of calories burned, C, and time.

Unit Assessment pp. 203–209

1. b 7. b 2. c 8. b 3. a 9. d 4. d 10. d 5. c 11. c 6. b 12. a 13. a. 252 = 6x + 0.05(6x); b. $40; c. $12 14. a. y = 20x + 160; b. Graph:

Cost in dollars ($)

6. Inequality: 0.75s + 1.25w ≤ 10.00. Answers will vary. You can buy a maximum of 13 cans of soda if you don’t buy any water. You can buy a maximum of 8 bottles of water if you don’t buy any soda. There are many combinations of cans and bottles that can be purchased. 7. n must be greater than or equal to 0 because items created cannot be negative. 8. Your dad can spend no more than $50. The number of hours must be greater than 0. Your dad can rent the chain saw for no more than 4 hours. 9. 1.2(5.99 + 0.75x) ≤ 10.00. After solving for x = 3, he will have $0.11 remaining.

Physical therapy visits

15. a. y = 9,815,210(1.013) x; b. y = 9,656,401(1.0125) x; c. Graphed equations:

Lesson 5: Rearranging Formulas Pre-Assessment, p. 186 4. c 5. b

Warm-Up 1.5.1, p. 188

GA Population

1. b 2. a 3. d

NC

1. $3.44 2. $61.92

Practice 1.5.1: Rearranging Equations and Formulas, p. 199 1. y = 3x – 2 2. y = 11 – 4x 3. y = 10/7x – 11 4. y = –5x + 11 5. C = K – 273.15

6. r = C/2π 7. w = V/lh 8. h = (S – 2πr2)/(2πr) 9. C = 5/9(F – 32) 10. h = 3V/2πr2

Progress Assessment, pp. 200–202 1. b 2. d 3. a 4. c 5. d

6. b 7. c 8. a 9. d 10. a

Time in years

d. Georgia starts out with more people than North Carolina. This is seen in the y-intercepts. Georgia starts out higher on the y-axis. Georgia also has a higher growth rate, 1.3% versus 1.25%. The different growth rates cause the graphs to grow farther apart as time goes on. If the current growth rates continue, Georgia’s population will always be greater than North Carolina’s population.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES

Student Book Answer Key Practice 1.1.1: Identifying Terms, Factors, and Coefficients, pp. 7–8

Practice 1.1.2: Interpreting Complicated Expressions, pp. 12–13

1. terms: 14x2, 2x, –9 factors: 14 and x2, 2 and x coefficients: 14, 2 constant: –9 2. terms: 13x, 20 factors: 13 and x coefficient: 13 constant: 20 3. terms: (4x3)/5, 9x factors: 4/5 and x3, 9 and x coefficient: 4/5 constants: there are none 4. expression: (x2)/3 + 4 terms: (x2)/3, 4 factors: x2, 1/3 coefficient: 1/3 constant: 4 5. expression: x6 + 3x terms: x6, 3x factors: 3 and x coefficient: 3 constants: there are none 6. Answers may vary; possible response: 12x4 + 15x3 + 18x2 – 21x + 3 7. expression: 2x + 0.05(x) = 2.05x terms: 2.05x factors: 2.05 and x coefficient: 2.05 constants: there are none 8. expression: 4x – 0.15(4x) + 4.85 = 3.4x + 4.85 terms: 3.4x, 4.85 factors: 3.4 and x coefficient: 3.4 constant: 4.85 9. expression: x + x + (x – 4) + (x – 4) = 2(x) + 2(x – 4) = 4x – 8 terms: 4x, –8 factors: 4 and x coefficient: 4 constant: –8 10. expression: 5/9(F – 32) = (5/9F) – (160/9) terms: 5/9(F), –160/9 factors: 5/9 and F coefficient: 5/9 constant: –160/9

1. The order of operations indicates that exponents must be applied before multiplying. 2. The order of operations indicates that the parentheses must be cleared before applying exponents. 3. The number of books does not affect the value of m; the number of books is a constant and remains unchanged by the number of magazines. 4. Lowering the service fee will result in a constant less than 23.90. 5. The value of the expression will be greater than 9. 6. x must be less than 0 7. The amount will be increased. 8. Changing the value of r has no effect on the value of d. d represents the initial dose; changing the rate at which it loses effectiveness will not change the initial amount taken. 9. The values of (1 + r) would be less than 1. 10. The number of inactive months has no effect on the rate. The rate will still be 1% for each month that the card is inactive.

Practice 1.2.1: Creating Linear Equations in One Variable, pp. 28–29 1. Answers may vary. Possible answers: a. miles per hour b. inches per minute or miles per hour c. meters per second or feet per second d. dollars per pound 2. 40.49π cm3 or 127.22 cm3 3. 287.88 ft2 4. 960 calories per day 5. $15 6. 6 pairs 7. $17 per lunch 8. 25 minutes 9. 76,800 square feet 10. 1.5(r – 5) = 1r ; r = 15. Brian’s rate is 15 mph, while Alex’s rate is 10 mph.

Practice 1.2.2: Creating Linear Inequalities in One Variable, pp. 36–37 1. x ≤ 8 2. x ≤ 250 3. w ≤ 2400 4. 32 + x ≥ 88; x ≥ 56; Jeff needs $56 or more. 5. 15x ≥ 950; x ≥ 63 1/3; You must work at least 63 1/3 hours. 6. 79 + x ≥ 160; x ≥ 81; Mackenzie needs 81 points or more.

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7. 22x + 9 ≤ 75; x ≤ 3; Arianna can buy up to 3 computer games. 8. 16 – 2x ≥ 6, where x is time in 7-minute increments; x ≤ 5, so 5(7) = 35 minutes. After the first hour, the team can play for no more than 35 minutes before there are too few players. 9. 25,000 – 3000x ≥ 4000; x ≤ 7 days. The giveaway should end in 7 days or less in order to have $4,000 or more to give away for the grand prize. 10. Answers may vary. Sample answer: You have a gift card for $6 to the office supply store and you want to use it all up. You want to buy 3 separate notebooks that cost x dollars and you have a coupon for $3 off your purchase. To satisfy these criteria, the notebooks must cost $3 or more each; x > 3.

Practice 1.2.3: Creating Exponential Equations, pp. 47–48

5 4 3 2 1 –5 –4 –3 –2 –1

0

1

2

3

4

5

2

4

6

8

10

x

–1 –2 –3 –4 –5 y

3.

1. a. linear; b. exponential 2. a. linear; b. exponential 3. a. exponential; b. linear 4. 1920 = a(2)6; 30 bacteria 5. 4800 = a(2)3; $600 6. y = 24(3)4; 1,944 insects 7. y = 100(0.5)8; 0.39 grams 8. 8 = a(0.5)3; 64 teams 9. y = 63,000(1.01)4; about 65,559 people 10. y = 3000(0.99)5; about 2,853 people and y = 3000(0.99)10; about 2,714 people

10 8 6 4 2 –10 –8 –6 –4 –2

0

x

–2 –4

Practice 1.3.1: Creating and Graphing Linear Equations in Two Variables, p. 79

–6

y

1.

y

2.

–8

5

–10

4 3 2 1 –5 –4 –3 –2 –1

0

1

2

3

4

5

x

–1 –2 –3 –4 –5

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4. y = 1/2x; slope = 1/2; y-intercept: (0, 0)

6. y = 75x + 50; slope = 75; y-intercept: (0, 50)

y

500

10

450

8

400 350

4 2 –8

–6

–4

0

–2

2

4

6

8

10

–2

x

Fare in dollars ($)

Number of revolutions

6

–10

y

300 250 200 150

–4

100

–6

50

–8

x

0

1

–10

2

50

30

Degrees Celsius

20 10 -20

-10

0

10

-10 -20 -30 -40 -50

Degrees Fahrenheit

20

30

40

50

Weekly earnings in dollars ($)

40

-30

5

7. y = 65x + 100; slope = 65; y-intercept: (0, 100)

5. y = 5/9(x – 32); slope = 5/9; y-intercept: (0, –17 7/9)

-40

4

Time in minutes

Time in seconds

-50

3

1000 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 50 0

y

x 2

4

6

8

10

12

14

16

18

20

Time in hours

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Practice 1.3.2: Creating and Graphing Exponential Equations, p. 92

Fee in dollars ($)

8. y = 8x + 59; slope = 8; y-intercept: (0, 59) y

150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

1.

2. x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of on-demand movies

9. y = –15x + 500; slope = –15; y-intercept: (0, 500) 500

y

450

Amount owed in dollars ($)

400 350

3.

300 250 200 150 100 50 0

x 5

10

15

20

25

30

35

40

45

50

Time in months

4. y = 64(0.5)x

Employees

Number of teams

10. y = –36/12 x + 65 = –3x + 65

Months

Rounds U1-221

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9. y = 300(1.0006)52x

Bacteria

Investment worth in dollars ($)

5. y = 16(2)24x/36 = 16(2)2x/3, for which x is in days

Days

6. y = 9000(1.017)

Years

10. y = 500(1.00005)

x

Population

Investment worth in dollars ($)

365x

Years

7. y = 15,000(0.978)

Years

x

Population

Practice 1.4.1: Representing Constraints, pp. 99–100

Years

8. y = 2500(1.00192) Investment worth in dollars ($)

12x

Years U1-222 CCSS IP Math I Teacher Resource

1. no 2. yes 3. no 4. Answers will vary. Sample responses: a. (–2, 3); b. (5, 5); c. (2, 1) 5. a. y = 2.60x + 7.90; b. x = 8.5; maximum = 8,500 gallons 6. Inequality: 7.50g + 5.00s ≤ 30.00. Answers will vary. You can buy a maximum of 4 graphic tees if you don’t buy any solid tees. You can buy a maximum of 6 solid tees if you don’t buy any graphic tees. There are many other combinations of graphic and solid tees that you can purchase. y = 7 x + 15 7. y = 5 x + 20 x ≥ 1 8. The length must be greater than 0, but less than or equal to 80 feet. x + y ≤ 32 y≤6 9. x ≥ 0 y ≥ 0 x + y ≤ 15 10. x ≥ 4 y≥0 © Walch Education

Practice 1.5.1: Rearranging Equations and Formulas, p. 107 1. y = 10x – 6 2. y = 9 – 2x 3. y = 3x – 9 4. y = –5x + 13 5. d = C/π

6. r = d/t 7. t = I/pr 8. l = (A – s2)/2s 9. F = 9/5C + 32 10. h = 3V/b2

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions Instruction Goal: To provide opportunities for students to develop concepts and skills related to unit conversion, finding percents, simplifying algebraic ratios, and solving algebraic proportions Common Core State Standards N–Q.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.★

A–CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

Student Activities Overview and Answer Key Station 1 Students will be given 12 index cards with pairs of equivalent units of measurement written on them. They will work together to match the cards that are an equivalent unit of measurement. Then they will perform unit conversion. Answers 1. 1 0 mm = 1 cm; 12 in. = 1 ft; 3 ft = 1 yd; 2 pints = 1 quart; 4 quarts = 1 gallon; 1 ton = 2,000 pounds 2. 8 pints in a gallon; 2 pints = 1 quart and 4 quarts = 1 gallon, so 2(4) = 8 pints 3. 18 inches; 1/2 yard = 1.5 feet and 12 inches = 1 foot, so 12(1.5) = 18 inches 4. 5,000 pounds 5. 850 mm 6. 13.5 feet 7. 3 quarts = 0.75 gallons 8. A nswers will vary. Possible answers include: cooking, when modifying recipes for more or fewer people; carpentry, when creating custom-size cabinetry Station 2 Students will be given a calculator to help them solve the problems. They work as a group to solve real-world applications of unit conversions.

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UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions Instruction Answers 1. His friend measures temperature in Celsius, and Evan measures it in Fahrenheit. F = 95° 2. P = 36.67 yards; P = 1,320 inches, A = 77.78 yds2; A = 100,800 in2 3.

Feet Tim Jeremy Martin

300 400 229.66

Yards 100 133.33 76.55

Meters 91.44 121.95 70

Time 12 seconds 12 seconds 12 seconds

Jeremy, Tim, Martin; Tim = 25 feet/sec, Jeremy = 33.33 feet/sec; Martin = 19.14 feet/sec Station 3 Students will be given a bag containing 24 green marbles and 16 yellow marbles. They will use the marbles to create ratios and percents. They will then solve percent problems. Answers 1. A nswers will vary. Possible answers include: green = 1; yellow = 7; total = 8. Find 1/8 = 0.125 = 12.5%; 12.5% were green. Subtract 12.5% from 100% to get 87.5% or 7/8 = 87.5%; 87.5% were yellow. 2. There are 40 marbles so 24/40 = 60% green marbles; 100% – 60% = 40% or 16/40 = 40% 3. 9 yellow marbles; student drawings should depict 9 yellow marbles and 12 green marbles. 4. 24(1/4) = 6 or 24(0.25) = 6 5. 17(2/1) = 34 or 17(2.0) = 34 6. 10(14) = 140 in2; increased dimensions by 200% then found the area of the photograph Station 4 Students will be given 8 large blue algebra tiles and 20 small yellow algebra tiles. Students visually depict ratios and proportions with the algebra tiles. They then solve proportions for a specified variable including a real-world application. Answers 8 blue 2 = 1. 20 yellow 5 2 blue 4 blue = 2. 3 yellow 6 yellow

U1-226 CCSS IP Math I Teacher Resource

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions Instruction 3. 8/20 = x/100, so x = 40 blue 4. 8/20 = x/15, so x = 6 blue 5. x = 4 6. x = 40 7.

blue 6 3 = = yellow 10 5 Let x = number of blue pencils and 24 – x = number of yellow pencils. 3 x = , so x = 9 blue pencils and 24 – x = 15 yellow pencils 5 ( 24 − x )

Materials List/Setup Station 1

12 index cards with the following written on them:

10 millimeters, 12 inches, 3 feet, 2 pints, 4 quarts, 1 ton, 1 centimeter, 1 foot, 1 yard, 1 quart, 1 gallon, 2,000 pounds Station 2

calculator

Station 3

24 green marbles; 16 yellow marbles

Station 4

8 large blue algebra tiles; 20 small yellow algebra tiles

U1-227 © Walch Education

CCSS IP Math I Teacher Resource

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions Instruction Discussion Guide To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to “debrief” the station activities. Prompts/Questions 1. How do you perform unit conversion? 2. When would you use unit conversion in the real world? 3. What are two ways to find the percent of a number? 4. What is a ratio? 5. How do you know if two ratios are equivalent? 6. What is a proportion? 7. When would you use ratios and proportions in the real world? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses 1. Use ratios and proportions to convert units. 2. A nswers will vary. Possible answers include: creating scale models of buildings; using the metric system instead of U.S. customary units; converting Celsius to degrees Fahrenheit and vice versa 3. Multiply the number by a decimal or fraction that represents the percentage. 4. A ratio is a comparison of two numbers by division. 5. Two ratios are equivalent if, when simplified, they are equal. 6. A proportion is when two ratios are set equal to each other. 7. A nswers will vary. Possible answers include: enlarging photos; scale models; modifying quantities of ingredients in a recipe

U1-228 CCSS IP Math I Teacher Resource

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions Instruction Possible Misunderstandings/Mistakes •

Not keeping track of units and using incorrect unit conversions

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Not recognizing that terms must have the same units in order to compare them

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S etting up proportions with one of the ratios written with the incorrect numbers in the numerator and denominator

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Not recognizing simplified forms of ratios in order to find equivalent ratios

U1-229 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

Station 1 You will be given 12 index cards with the following written on them: 10 millimeters, 12 inches, 3 feet, 2 pints, 4 quarts, 1 ton, 1 centimeter, 1 foot, 1 yard, 1 quart, 1 gallon, 2,000 pounds Shuffle the index cards and deal a card to each student in your group until all the cards are gone. As a group, show your cards to each other and match the cards that are an equivalent unit of measurement. 1. Write your answers on the lines below. The first match is shown: 10 mm = 1 cm

2. F ind the number of pints in a gallon. Explain how you can use your answers in problem 1 to find the number of pints in a gallon.

3. F ind the number of inches in half of a yard. Explain how you can use your answers in problem 1 to find the number of inches in half of a yard.

continued U1-230 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

Perform the following unit conversions by filling in the blanks. 4. 2.5 tons = _____ pounds

5. 85 cm = ______ mm

6. 4.5 yd = ______ ft

7. 6 pints = ______ quarts = ______ gallons

8. When would you use unit conversions in the real world?

U1-231 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

Station 2 You will be given a calculator to help you solve the problems. Work as a group to solve these realworld applications of unit conversions. 1. Evan has a friend in England. His friend said the temperature was very hot at 35°. Evan thought he heard his friend incorrectly since 35° is cold. What caused his misunderstanding? 5 (Hint: C = ( F − 32) ) 9

F ind the equivalent temperature in the United States that would make the claim of Evan’s friend valid. Write your answer in the space below.

2. A nna is going to build a patio. She wants the patio to be 20 feet by 35 feet. What is the perimeter of the patio in yards?

What is the perimeter of the patio in inches?

What is the area of the patio in yards?

continued U1-232 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

What is the area of the patio in inches?

3. T im claims he can run the 100-yard dash in 12 seconds. Jeremy claims he can run 400 feet in 12 seconds. Martin claims he can run 70 meters in 12 seconds. (Hint: 1 yard = 0.9144 meters and 1 yard = 3 feet.) Fill in the table below to create equivalent units of measure. Feet

Yards

Meters

Time (seconds)

Tim Jeremy Martin List the three boys in order of fastest to slowest:

How fast did each boy run in feet/second?

U1-233 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

Station 3 You will be given a bag containing 24 green marbles and 16 yellow marbles. You will use the marbles to create ratios and percents. You will then solve percent problems. Work together as a group to solve the following problems. 1. S hake the bag of green and yellow marbles so that the colors are mixed. Have each student select 2 marbles from the bag without looking. Group all your marbles together by color. How many green marbles did you draw?

How many yellow marbles did you draw?

What was the total number of marbles drawn?

How can you determine the percentage of marbles that were green?

Find the percentage of marbles you drew that were green.

Name two ways you can find the percentage of marbles you drew that were yellow.

Find the percentage of marbles you drew that were yellow.

continued U1-234 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

2. T ake all the marbles out of the bag. How can you determine what percentage of all the marbles are green?

How can you determine what percentage of all the marbles are yellow?

3. P lace 12 green marbles on the table. How many yellow marbles do you need to have 75% as many yellow marbles on the table?

Draw a picture of the number of green marbles and yellow marbles you have placed on the table.

4. Use equations to show two ways you can find 25% of 24.

5. Use equations to show two ways you can find 200% of 17.

6. R eal-world application: Bryan is a photographer. He has a 5 in. by 7 in. photo that he wants to enlarge by 200%. What is the area of the new photo? Explain your answer in the space below.

U1-235 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

Station 4 You will be given 8 large blue algebra tiles and 20 small yellow algebra tiles. Work as a group to arrange the algebra tiles so they visually depict the ratio of blue to yellow algebra tiles. 1. What is this ratio? Rearrange the tiles to visually depict the following ratios: 2 blue 3 yellow

1 blue 10 yellow

4 blue 6 yellow

1 blue 1 yellow

2. Which ratios are equivalent ratios? Explain your answer.

3. K eeping the same ratio of yellow to blue tiles, if there were 100 yellow algebra tiles, how many blue algebra tiles would there be? Use a proportion to solve this problem. Show your work in the space below. (Hint: A proportion is two ratios that are equal to each other.)

4. K eeping the same ratio of yellow to blue tiles, if there were 15 yellow algebra tiles, how many blue algebra tiles would there be? Use a proportion to solve this problem. Show your work in the space below.

continued U1-236 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 1: Ratios and Proportions

Date:

Work together to solve the following proportions for the variable. 5.

2 x = ;x= 7 14

6.

8 2 = ;x= x 10

Use the following information to answer problem 7: Allison has 6 blue pencils and 10 yellow pencils. Sadie has 24 pencils that are either blue or yellow. The ratio of blue pencils to yellow pencils is the same for both Allison and Sadie. 7. H ow many blue pencils and yellow pencils does Sadie have? Show your work in the space below by setting up a proportion using a variable, x.

U1-237 © Walch Education

CCSS IP Math I Teacher Resource

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving inequalities Common Core State Standard A–CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

Student Activities Overview and Answer Key Station 1 Students are given a series of inequalities and a number cube. For each inequality, they roll the number cube and then work together to decide if the number shown on the cube is a solution of the inequality. Students explain the strategies they used to decide whether each value was a solution. Answers: 1–3. Answers will depend upon numbers rolled. 4. Yes; 5. No; 6. Possible strategies: Substitute the value for the variable. Simplify and check to see if the resulting inequality is true. Station 2 In this activity, students work together to use number lines to help them solve inequalities. To do so, they test various values of the variable in the given inequalities, and check to see whether each value is a solution. They keep track of the values that are solutions by marking them on a number line. After testing enough values to see a pattern, students shade the values that represent all solutions of the inequality. Then they write the solution algebraically. Answers 1. x < 2 –5

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2. x ≥ –2

3. x > 1

U1-238 CCSS IP Math I Teacher Resource

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities Instruction 4. x ≤ 0 –5

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5. x < 2

Station 3 In this activity, students work together to match a set of given inequalities with a set of given solutions. Once students have paired each inequality with its correct solution, they discuss the strategies they used to solve the problem. Answers: The cards should be paired as follows: 5x + 2 < 12 and x < 2; 4x + 3 < –5 and x < –2; –3x < 6 and x > –2; – x ⁄ 2 > 2 and x < –4; 3x + 1 > –11 and x > –4; x ⁄ 4 + 1 > 2 and x > 4. Possible strategies: Solve each inequality using inverse operations and look for the solution among the given choices. Station 4 Students are given a set of inequalities and a set of real-world situations. They work together to match each situation to an inequality. Then they solve the inequality. At the end of the activity, students explain the strategies they used to match the inequalities to the situations. Answers 1. 10x + 5 ≤ 105, x ≤ 10 2. 5x + 10 < 105, x < 19 3. 10x – 5 ≥ 105, x ≥ 11 4. 5(x – 10) < 105, x < 31 Possible strategies: Use the words or phrases that refer to arithmetic operations as clues to identifying the corresponding inequalities. For example, a decrease corresponds to subtraction. Match words to inequalities. For example, “no more than” refers to a “less than or equal to” inequality (≤).

U1-239 © Walch Education

CCSS IP Math I Teacher Resource

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities Instruction Station 5 Students work together to match inequalities with their solution graphs. Then they discuss the strategies they used. Answers 2x – 5 > –3 –5

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4 – 3x ≤ 7

–2x + 7 ≥ 3 –5

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4x – 5 ≥ 3 –5

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–11 < 6x – 5

7 – 7x ≤ 0

5x + 12 ≤ 7 –5

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Possible strategies: Test points in the inequalities to determine the solution, or solve for the variable and then match the graph.

U1-240 CCSS IP Math I Teacher Resource

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities Instruction Materials List/Setup Station 1

number cube (numbers 1-6)

Station 2

none

Station 3

set of index cards with the following inequalities written on them: 5x + 2 < 12 – x ⁄ 2 > 2

4x + 3 < –5

3x + 1 > –11

–3x < 6

x⁄

set of index cards with the following solutions written on them: x < –4, x > –4, x < –2, x > –2, x < 2, x > 4

4+1>2

The two sets of cards should be placed in two piles, face-up, on a table or desk at the station. Station 4

none

Station 5

7 note cards with the following inequalities on them: 5x + 12 ≤ 7 2x – 5 > –3 –2x + 7 ≥ 3 4 – 3x ≤ 7 4x – 5 ≥ 3 –11 < 6x – 5 7 – 7x ≤ 0

7 note cards with the graphs in the Station 5 answers on the previous page

U1-241 © Walch Education

CCSS IP Math I Teacher Resource

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities Instruction Discussion Guide To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to “debrief” the station activities. Prompts/Questions 1. What is the difference between < and ≤? 2. How do you check to see if a value of the variable is a solution of an inequality? 3. How is the solution of an inequality different from the solution of an equation? 4. How do you solve an inequality using algebra? 5. W hat strategies can you use to prevent confusion when the variable of an inequality is on the right side of the inequality symbol? (Example: 2 < x) Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses 1. The symbol < means “less than.” The symbol ≤ means “less than or equal to.” 2. S ubstitute the value for the variable in the inequality. Check to see if the resulting inequality is true. If it is, the value is a solution. 3. I n general, the solution of an inequality is itself an inequality (a range of values). The solution of an equation is usually a single value (or several discrete values). 4. U se inverse operations, as when solving an equation, to isolate the variable on one side of the inequality. If you multiply or divide by a negative number, reverse the direction of the inequality. 5. R earrange the inequality so that the variable is on the left. (Using the example above, x > 2.) Students could also read the inequality from right to left. (Using the example above, “x is greater than 2.”) Possible Misunderstandings/Mistakes • Using an incorrect operation to solve an inequality (e.g., solving x + 2 < 5 by adding 2 to both sides) •

I ncorrectly translating verbal expressions to inequalities (e.g., representing the phrase “no more than” by < rather than ≤)

•

F orgetting to reverse the direction of the inequality when multiplying or dividing by a negative number

U1-242 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities

Date:

Station 1 You will find a number cube at this station. For each inequality, roll the number cube and write the number in the box. Then work together to decide if this value of the variable is a solution of the inequality. Write “yes” or “no” on the line provided. 1. 2x + 1 < 7

Solution? ________

2. 3x – 4 ≥ 5

Solution? ________

3. –3x < –12

Solution? ________

x + 1 < 5 2

Solution? ________

5. 1 – x > 0

Solution? ________

4.

6. Explain the strategies you used to decide whether each value was a solution of the inequality.

U1-243 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities

Date:

Station 2 You can use number lines to help you solve inequalities. For each inequality, work together to test different values of the variable to see if they are solutions of the inequality. If a value is a solution, draw a solid dot at that value on the number line. Test at least five different values for each inequality. When you think you know what the solution set of an inequality looks like, shade the correct part of the number line to show all the solutions. Finally, write the solution in the space provided. 1. 2x – 3 < 1 –5 –4 –3 –2 –1 0 1 2 3

4

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5

Solution: ________ 2. 3x + 1 ≥ –5 –5 –4 –3 –2 –1 0 1 2 3 Solution: ________ 3. –2x < –2 –5

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x + 3 < 4 2 –5 –4 –3 –2 –1 0 1 2 3

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Solution: ________ 4. –3x ≥ 0 –5

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Solution: ________ 5.

Solution: ________

U1-244 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities

Date:

Station 3 At this station, you will work with other students to match inequalities to their solutions. You will find a set of cards with the following inequalities written on them: 5x + 2 < 12

4x + 3 < –5

–3x < 6

–

x >2 2

3x + 1 > –11

x +1>2 4

You will also find a set of cards with these solutions written on them: x < –4

x > –4

x < –2

x > –2

x<2

x>4

Work together to match each inequality with its solution. When everyone agrees on the answers, write the matching pairs below.

Explain the strategies you used to match up the cards.

U1-245 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities

Date:

Station 4 At this station, you will match inequalities to real-world situations and then solve the inequalities. Work with other students to match each situation to one of the following inequalities. When everyone agrees on the correct inequality, write it on the line provided. Then work together to solve it. 5x + 10 < 105

10x + 5 ≤ 105

5(x – 10) < 105

10x – 5 ≥ 105

1. M ai rents DVDs by mail. There is a one-time membership fee of $5 and the service costs $10 per month. Mai wants to spend no more than $105. For how many months can she rent DVDs with this service? Inequality: ____________________ Solution: ________

2. T yrone bought 5 trays of food for a party. The price of each tray of food was the same. He also spent $10 on paper plates, napkins, and utensils. Altogether, he spent less than $105. What was the price of each tray of food? Inequality: ____________________ Solution: ________

continued U1-246 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities

Date:

3. M r. Garcia ordered 10 copies of a novel for students in his English class. He had a coupon for $5 off the total price of the order. The total cost of the order, before tax, came to no less than $105. What was the price of each novel? Inequality: ____________________ Solution: ________

4. R achel bought 5 pairs of jeans. Each pair of jeans was the same price. She had a coupon for $10 off the price of each pair of jeans. The total cost of the jeans, before tax, came to less than $105. What was the price of each pair of jeans before the coupon was applied? Inequality: ____________________ Solution: ________

Explain the strategies you used to match the inequalities to the situations.

U1-247 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 2: Solving Inequalities

Date:

Station 5 At this station, you will match inequalities to their solution graphs. Write the correct inequality on the line next to each solution. At the end, discuss the strategies that you used. Inequalities: –11 < 6x – 5

5x + 12 ≤ 7

4x – 5 ≥ 3

–2x + 7 ≥ 3

2x – 5 > –3

4 – 3x ≤ 7

7 – 7x ≤ 0

Solutions: ___________________ –5

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___________________

___________________ –5

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___________________ –5

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___________________

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___________________ –5

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What strategies did you use to match the inequalities to their solution graphs?

U1-248 CCSS IP Math I Teacher Resource

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving equations and creating and interpreting graphs representing real-world situations Common Core State Standards A–CED.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

A–CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

Student Activities Overview and Answer Key Station 1 In this activity, students use cups and counters to model linear equations. In the given pictures, each cup is holding an unknown number of counters. Students use this idea to write the equation that is modeled by each picture. Then they use actual cups and counters, as well as logical reasoning, to help them find the unknown number of counters in each cup. This is equivalent to solving the corresponding equation. Answers 1. x + 1 = 10, x = 9 2. 2x = 12, x = 6 3. 2x + 3 = 7, x = 2 4. 10 = 3x + 1, x = 3 Station 2 Students are given a set of equations and a set of real-life situations. They work together to match each situation to an equation. Then they solve the equation. At the end of the activity, students explain the strategies they used to match the equations to the situations. Answers 1. 2x + 3 = 25, x = 11 2. 3x – 25 = 2, x = 9 3. 2x –3 = 25, x = 14 4. 3x + 2 = 25, x = 7 2 ⁄ 3

U1-249 © Walch Education

CCSS IP Math I Teacher Resource

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations Instruction Possible strategies: Use the words or phrases that refer to arithmetic operations as clues to identifying the corresponding equations. For example, gathering equal groups of objects corresponds to multiplication. Station 3 Students will be given a ruler and graph paper. They work together to graph the linear equation of two cell phone company plans. Then they use the graph to compare the two cell phone plans. Answers 1. y = 40 + 0 .5 x ; answers will vary, possible values include: Minutes (x) 5 Cost in $ (y) 42.5 65.00 62.50 60.00 57.50 55.00 52.50 50.00 47.50 45.00 42.50 40.00 37.50 35.00 32.50 30.00 0

10 45

20 50

35 57.50

45 62.50

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x 5

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65

U1-250 CCSS IP Math I Teacher Resource

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations Instruction 2. y = 60 + 0 .1 x ; answers will vary, possible values include: Minutes (x) 5 Cost in $ (y) 60.50 67.00 66.50 66.00 65.50 65.00 64.50 64.00 63.50 63.00 62.50 62.00 61.50 61.00 60.50 60.00 0

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3. They should choose Bell Phone’s plan because it only costs $55 versus $63. 4. They should choose Ring Phone’s plan because it only costs $68 versus $80. 5. At 50 minutes, it doesn’t matter which plan the customer chose because both plans cost $65. Station 4 Students will be given a ruler and graph paper. They will work together to complete a table of values given an equation, and then graph the equation. They will analyze the slope of the graph as it applies to a real-world value. Answers 1. y = 20 x + 10 2. Answers will vary. Possible table of values: Number of months 0 2 4 5 8 9

Account balance ($) 10 50 90 110 170 190 U1-251

© Walch Education

CCSS IP Math I Teacher Resource

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations Instruction 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

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3. H e will have $310 in his savings account after 15 months. This will allow him to buy the $300 stereo. 4. T he slope of the graph would be steeper because the amount he saves each month represents the slope. 5. T he slope of the graph would not be as steep because the amount he saves each month represents the slope.

Materials List/Setup Station 1

3 paper cups; 12 counters or other small objects, such as pennies or beans

Station 2

none

Station 3

graph paper; ruler

Station 4

graph paper; ruler

U1-252 CCSS IP Math I Teacher Resource

© Walch Education

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations Instruction Discussion Guide To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to “debrief” the station activities. Prompts/Questions 1. How do you know which operation to use first when you solve a two-step equation? 2. How can you check your solution to an equation? 3. What strategies can you use to create an equation in one variable? 4. What strategies can you use to create an equation in two variables? 5. How can you use an equation to plot its graph? 6. W hat are examples of real-world situations in which you could construct a graph to represent the data? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses 1. Y ou usually add or subtract on both sides of the equation before you multiply or divide on both sides of the equation. (You reverse the order of operations to “undo” the operations on the variable.) 2. S ubstitute the value for the variable in the equation and simplify. If the solution is correct, the two sides of the equation should be equal. 3. R ead the problem first before doing any writing. Identify the variable and the operations being performed with that variable. Look for keywords indicating the four operations. 4. I dentify the independent and dependent variables. Look for keywords indicating the rate or slope. Look for flat fees or constants. 5. C reate a table of values that are solutions to the equation. Graph these ordered pairs and draw a line through these points. 6. A nswers will vary. Possible answers: Business: yearly revenues; Biology: growth rate; Finance: savings account balance

U1-253 © Walch Education

CCSS IP Math I Teacher Resource

UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations Instruction Possible Misunderstandings/Mistakes •

sing an incorrect operation to solve an equation (e.g., solving x + 3 = 12 by adding 3 to U both sides)

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Attempting to solve an equation such as x + 4 = 9 by subtracting x from both sides

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pplying an operation that does not isolate the variable (e.g., solving 9 = 3x by dividing both A sides by 9)

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Reversing the x-values and the y-values when reading the graph

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Incorrectly reading the graph by matching up the wrong x- and y-values

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Reversing the x-values and y-values when constructing the graph

•

Incorrectly plugging x-values into the given equation to find the y-values

U1-254 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations

Date:

Station 1 In each picture, the cup is holding an unknown number of counters, x. If there is more than one cup, every cup is holding the same number of counters. Each picture shows an equation. This picture shows x + 5 = 7. To make the two sides equal, there must be 2 counters in the cup. This means x = 2. =

=

= = Work with other students to write = an equation for each picture. Then find the number of counters in each cup. You can use the cups and counters at the station to help you. =

1. =

Equation: __________________

=

Solution: __________________

= =

=

=

2.

Equation: __________________

= =

Solution: __________________

=

=

= =

3. =

Equation: __________________

Solution: __________________ =

=

= = = 4. =

Equation: __________________

Solution: __________________

U1-255 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations

Date:

Station 2 At this station, you will match equations to real-life situations and then solve the equations. Work with other students to match each situation to one of the following equations. When everyone agrees on the correct equation, write it on the line. Then work together to solve it. 2x – 3 = 25

2x + 3 = 25

3x + 2 = 25

3x – 25 = 2

1. R osa bought some notebooks that cost $2 each. She also bought a compass that cost $3. She spent a total of $25. How many notebooks did she buy? Equation: ________________________ Solution: ________________________ 2. M s. Chen brought 3 packages of pencils for her class. Each package contained the same number of pencils. The 25 students in her class each took one pencil. There were 2 pencils left over. How many pencils were in each package? Equation: ________________________ Solution: ________________________ 3. T yler bought two copies of a DVD to give as gifts. He had a coupon for $3 off his total purchase. The final cost of the DVDs was $25. How much did each DVD cost? Equation: ________________________ Solution: ________________________ 4. A bowl can hold 25 fluid ounces of liquid. Omar empties a full teacup of water into the bowl 3 times. Then he adds another 2 fluid ounces of water to fill the bowl. How many fluid ounces of liquid does the teacup hold? Equation: ________________________ Solution: ________________________ Explain the strategies you used to match the equations to the situations.

U1-256 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations

Date:

Station 3 You will be given a ruler and graph paper. Work together to analyze data from the real-world situation described below, then, as a group, answer the questions. You are going to get a new cell phone and need to choose between two cell phone companies. Bell Phone Company charges $40 per month. It costs $0.50 per minute after you have gone over the monthly number of minutes included in the plan. Ring Phone Company charges $60 per month. It costs $0.10 per minute after you go over the monthly number of minutes included in the plan. Let x = minutes used that exceeded the plan. Let y = cost of the plan. 1. Write an equation that represents the monthly cost of Bell Phone Company’s plan.

Complete the table by selecting values for x and calculating y. Minutes (x) Cost in $ (y) Use your graph paper to graph the ordered pairs. Use your ruler to draw a straight line through the points and complete the graph. 2. Write an equation that represents the monthly cost of Ring Phone Company’s plan.

Complete the table by selecting values for x and calculating y. Minutes (x) Cost in $ (y)

continued U1-257 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations

Date:

On the same graph, plot the ordered pairs. Use your ruler to draw a straight line through the points and complete the graph. Use your graphs to answer the following questions. 3. W hich plan should a customer choose if he or she uses 30 minutes of extra time each month? Explain.

4. W hich plan should a customer choose if he or she uses 80 minutes of extra time each month? Explain.

5. A t what number of extra minutes per month would it not matter which phone plan the customer chose since the cost would be the same? Explain.

U1-258 CCSS IP Math I Teacher Resource

© Walch Education

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations

Date:

Station 4 You will be given a ruler and graph paper. Use the information from the problem scenario below to answer the questions. Let x = months and y = savings account balance. Marcus is going to start saving $20 every month to buy a stereo. His parents gave him $10 for his birthday to open his savings account. 1. W rite an equation that represents Marcus’s savings account balance x months after he began saving. 2. Complete the table by selecting variables for x and calculating y. Number of months

Account balance ($)

Use your graph paper to define the scale of the x- and y-axis and graph the ordered pairs. Use your ruler to draw a straight line through the points and complete the graph. 3. U se your graph to estimate the number of months it will take Marcus to save enough money for a $300 stereo. Explain.

continued U1-259 © Walch Education

CCSS IP Math I Teacher Resource

Name: UNIT 1 • RELATIONSHIPS BETWEEN QUANTITIES Station Activities Set 3: Solving Equations

Date:

4. I f Marcus saved $40 per month instead of $20, how would you expect the slope of the graph to change? Explain.

5. I f Marcus saved $10 per month instead of $20, how would you expect the slope of the graph to change? Explain.

U1-260 CCSS IP Math I Teacher Resource

© Walch Education