# JBL Topics In Algebra I Guidebook For TI 83 Plus / 84 (English) Alg1 Book

User Manual: JBL Topics in Algebra I guidebook for TI-83 Plus / TI-84 Plus (English) Topics in Algebra Guidebook for TI-83 Plus / TI-84 Plus

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ti Topics in Algebra 1 Software Application for the TI-83 Plus and the TI-73 Student and Teacher Classroom Materials Chapters 1 – 5 Important notice regarding book materials Texas Instruments makes no warranty, either expressed or implied, including but not limited to any implied warranties of merchantability and fitness for a particular purpose, regarding any programs or book materials and makes such materials available solely on an “as-is” basis. In no event shall Texas Instruments be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the purchase or use of these materials, and the sole and exclusive liability of Texas Instruments, regardless of the form of action, shall not exceed the purchase price of this book. Moreover, Texas Instruments shall not be liable for any claim of any kind whatsoever against the use of these materials by any other party. Permission is hereby granted to teachers to reprint or photocopy in classroom, workshop, or seminar quantities the pages or sheets in this work that carry a Texas Instruments copyright notice. These pages are designed to be reproduced by teachers for use in their classes, workshops, or seminars, provided each copy made shows the copyright notice. Such copies may not be sold, and further distribution is expressly prohibited. Except as authorized above, prior written permission must be obtained from Texas Instruments Incorporated to reproduce or transmit this work or portions thereof in any other form or by any other electronic or mechanical means, including any information storage or retrieval system, unless expressly permitted by federal copyright law. Send inquiries to this address: Texas Instruments Incorporated 7800 Banner Drive, M/S 3918 Dallas, TX 75251 Attention: Manager, Business Services Copyright © 2001, 2002 Texas Instruments Incorporated. Except for the specific rights granted herein, all rights are reserved. Printed in the United States of America. ³ TIp and ³ Try-It! are trademarks of Texas Instruments Incorporated. Table of Contents Introduction to Topics in Algebra 1 Note to Teachers Organization of Topics in Algebra 1 Classroom Materials NCTM Principles and Standards for School Mathematics Acknowledgements Installing This Application Deleting an Application Navigating Topics in Algebra 1 i ii iii iv iv v v vi Chapter 1: Number Sense Section 1: Integers Student Worksheet Teacher Notes 1-1 1-6 Section 2: Rational Numbers Student Worksheet Teacher Notes 1-11 1-18 Section 3: Real Numbers Student Worksheet Teacher Notes 1-23 1-30 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Student Worksheet Teacher Notes 2-1 2-6 Section 2: Using Algebra Student Worksheet Teacher Notes 2-10 2-16 Chapter 3: Linear Functions Section 1: Slope with Grid Student Worksheet Teacher Notes 3-1 3-6 Section 2: Slope Using Coordinates Student Worksheet Teacher Notes 3-10 3-14 Section 3: Slope Rate of Change Student Worksheet Teacher Notes 3-18 3-24 Section 4: Slope-Intercept Form Student Worksheet Teacher Notes 3-28 3-33 Chapter 4: Linear Inequalities: 1-Variable Section 1: Using Graphs & Tables Student Worksheet Teacher Notes 4-1 4-6 Section 2: Using Algebra Student Worksheet Teacher Notes 4-11 4-17 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Student Worksheet Teacher Notes 5-1 5-8 Section 2: Using Algebra Student Worksheet Teacher Notes 5-15 5-28 ³ TIpsé ³ TIpé 1: Resetting Your Calculator ³ TIpé 2: Adjusting Your Calculator Settings ³ TIpé 3: Graphing a Function in the Standard Window ³ TIpé 4: Creating a Table ³ TIpé 5: Adjusting the Viewing Window ³ TIpé 6: Using Lists ³ TIpé 7: Creating a Statistical Plot ³ TIpé 8: Finding the Best Line of Fit for a Set of Data ³ TIpé 9: Sending and Receiving Data between Calculators ³ TIpé 10: Managing Your Calculator’s Memory 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1 9-1 10-1 Introduction to Topics in Algebra 1 Note to Teachers Welcome to the Topics in Algebra 1 software application for the TI-83 Plus and TI-73 graphing calculators. The application and Classroom Materials were designed to help students review and reinforce selected concepts taught at the Algebra 1 level. Topics in Algebra 1 is easy to use, even for inexperienced calculator users, and it encourages students to explore concepts on their own. Navigating Topics in Algebra 1 (pages vi–ix) explains how to move around the application. You may wish to copy these pages for your students. We hope Topics in Algebra 1 proves useful to you and your students. All of your comments and suggestions are appreciated. You can contact TI: • Phone 1-800-TI-CARES (1-800-842-2737) • E-mail ti-cares@ti.com • TI website education.ti.com Topics in Algebra 1 © 2001 Texas Instruments Introduction i Organization of Topics in Algebra 1 Topics in Algebra 1 is organized in textbook fashion with chapters and sections arranged in table-of-contents form. Chapters Sections 1: Number Sense 1: Integers 2: Rational Numbers 3: Real Numbers 2: Linear Equations 1: Using Graphs & Tables 2: Using Algebra 3: Linear Functions 1: Slope with Grid 2: Slope Using Coordinates 3: Slope Rate of Change 4: Slope-Intercept Form 4: Linear Inequalities: 1-Variable 1: Using Graphs & Tables 2: Using Algebra Note: Chapters can be installed and deleted individually. This provides flexibility, allowing calculators to have only the applications students currently need. Each section in the application contains three subsections. • The Overview presents definitions and concepts for teachers to use in class discussions and for students to use for study and review. They contain animation and real-world examples. • The Observations show concepts or examples, followed by Write An Observation screens. Students are asked to write in the space provided on the Student Worksheet their observations about the content presented. Screens entitled Did you know display additional information and facts about the current concept. • The Activities include interactive activities that reinforce the concepts covered in the section. Many of the activities are in the form of games. Students can practice calculating equations with integers, rational numbers, and real numbers, or they can find the equation of a line. The application checks answers, gives a score, and provides students with the correct answer if they are unable to answer a question correctly. In the Linear Functions chapter, students can play a game called Screen Cross where they link to each other’s calculators and race to see who finishes first. Topics in Algebra 1 © 2001 Texas Instruments Introduction ii Classroom Materials The Topics in Algebra 1 Classroom Materials include Student Worksheets, Teacher Notes (with answers), and ³ TIpsé. • Student Worksheets provide explanations and instructions to students about using the application to review the concepts in the section. Each worksheet is divided into four parts— Overview, Observations, Activities (which correspond to the subsections of the application), and ³ Try-It!é on Your TI-83 Plus or TI-73. ³ Try-It!é activities let students investigate specific features on the calculator related to the section concepts. This includes step-by-step instructions with the exact keys to press to complete each step and corresponding screen pictures. The screens shown in the ³ Try-It! activities are for the TI-83 Plus. Usually, the TI-73 display varies only slightly, and the students should have no problems following the instructions. However, in the cases where the two calculators vary, the ³ Try-It! activities contain two separate exercises, one for each of the two calculators. Therefore, you only need to copy the section appropriate for the calculator that you use in your classroom. • Teacher Notes give a brief explanation of the concepts covered in the section, some of the common student errors that might be encountered, and answers to the questions on the Student Worksheets. There are Teacher Notes for each Student Worksheet. • ³ TIpsé provide keystroke examples for some of the more common tasks that you and your students need for Algebra 1 and beyond. The ³ TIps are intended to help students learn how to use the features of the calculator. The ³ TIps topics are: ³ TIp 1: Resetting Your Calculator ³ TIp 2: Adjusting Your Calculator Settings ³ TIp 3: Graphing a Function in the Standard Window ³ TIp 4: Creating a Table ³ TIp 5: Adjusting the Viewing Window ³ TIp 6: Using Lists ³ TIp 7: Creating a Statistical Plot ³ TIp 8: Finding the Best Line of Fit for a Set of Data ³ TIp 9: Sending and Receiving Data between Calculators ³ TIp 10: Managing Your Calculator’s Memory When students need to learn how to use the feature being covered, you can distribute the ³ TIps to them to make sure they are prepared for the lesson. Topics in Algebra 1 © 2001 Texas Instruments Introduction iii NCTM Principles and Standards for School Mathematics The Topics in Algebra 1 application and the Classroom Materials were written with the guidelines of the NCTM Principles and Standards for School Mathematics in mind. Attention was paid to the expectations laid out for the Algebra Content Standard for the Algebra 1 level that crosses the 6–8 and 9–12 grade bands. Some examples include: NCTM Standards and Expectations On the Application To understand patterns, relations, and functions. Students see how battery voltage values in a series can be represented by a linear function. To model and solve contextualized problems using various representations, such as graphs, tables, and equations. Students experience how to work with tables of values, graphs, and linear equations to see the multiple approaches to problem solving. To analyze the nature of changes in quantities in linear relationships. Students determine the constant rate of change of a diving submarine along each segment of its dive. In addition, scored, interactive activities at the end of each section help reinforce the concepts reviewed in the Overview and Observations subsections. You can find the complete expectations for the Algebra Content Standard and for the various grade bands set out by the NCTM at: http://standards.nctm.org/document/appendix/alg.htm You can view the NCTM Principles and Standards for School Mathematics online at: http://standards.nctm.org/ Acknowledgements Texas Instruments would like to thank these individuals for their support, ideas, and suggestions. Sharon Cichocki Michael Corrigan Cathy Jahr John LaMaster Brenda Levert Nicol Reiner Melissa Rowe Bill Stiggers Bob Tower Topics in Algebra 1 Hamburg High School Assumption Catholic School Westview High School Indiana University-Purdue University Fort Wayne Academy for Academics and Arts Roosevelt High School South Grand Prairie High School Cleveland Public Schools Roosevelt High School © 2001 Texas Instruments Hamburg, NY Jacksonville, FL Martin, TN Fort Wayne, IN Huntsville, AL Sioux Falls, SD Grand Prairie, TX Cleveland, OH Sioux Falls, SD Introduction iv Installing This Application Students may install all four chapters as a unit (ALG1PRT1) or one or more of the chapters individually (ALG1CH1, ALG1CH2, ALG1CH3, ALG1CH4). This provides flexibility, allowing calculators to have only the applications students currently need. When two or more chapters are installed separately, they each must be accessed directly from the Œ menu. Installing this application requires TI-GRAPH LINKé software and link cable. A link cable can be purchased from the online store: http://epsstore.ti.com If an Archive Full error message appears while installing Algebra 1 or one of the chapters, the calculator does not have sufficient memory for the application. Applications and/or archived variables must be deleted to make room (see below). Deleting an Application Deleting an application completely removes the application from the calculator. The space then becomes available for a different application. The deleted application may be reloaded at a later date. Before deleting an application from the calculator, it can be backed up to a PC using the Link > Receive Flash Application menu in TI-GRAPH LINK. You can reload it to the calculator later using the Link > Send Flash Software menu in TI-GRAPH LINK. To delete an application or archived variable: 1. Press y L to display the MEMORY menu. 2. Select 2:Mem Mgmt/Del (4:Delete… on the TI-73). 3. Select A:Apps... or B:AppVars... (8:Apps... or 9:AppVars... on the TI-73). 4. Press † until the 4 indicator is next to the item you wish to delete. 5. Press { (Í on the TI-73). 6. Select 2:Yes when asked Are You Sure? 7. Press y 5 to return to the Home screen. Topics in Algebra 1 © 2001 Texas Instruments Introduction v Navigating Topics in Algebra 1 Starting the Application 1. Press 9 to display the APPLICATIONS menu. 2. Press # until ALG1PRT1 (or a chapter such as ALG1CH1) is highlighted, and then press b to select it. The application title page is displayed. Topics in Algebra 1 title page 3. Press any key to start the application. A Table of Contents page is displayed. It is the Table of Contents from which you last exited Algebra 1. You may select one of the items on the menu or press áUPâ as often as necessary to go to the Table of Contents screen that you need. Table of Contents Screens There are three types of Table of Contents screens—chapters, sections, and subsections. Algebra Chapters Note: If you have installed the chapters individually, only the current chapter is shown on this screen. Press Í to select the chapter. When you select a chapter, the Algebra Sections screen is displayed. The number of sections listed varies according to chapter. Algebra Sections When you select a section, the Algebra Subsections screen is displayed. The items listed are always the same—Overview, Observations, and Activities. Algebra Subsections Note: When you exit and re-enter Topics in Algebra 1, you return to the Table of Contents screen from which you exited. Topics in Algebra 1 © 2001 Texas Instruments Introduction vi Navigating Topics in Algebra 1 (continued) Selecting the Chapter, Section, and Subsection You Want To select from a Table of Contents screen, press # if necessary to highlight the name of the chapter, section, or subsection, and then press b. When the Overview, Observations, or Activities opening screen appears, press ~ to begin. Returning to the Table of Contents From most Overview, Observations, or Activities screens, you can press - l as many times as necessary to return to the Table of Contents. You may then select one of the items on the menu or press áUPâ to go to higher levels of the Table of Contents. Note: If you see áCONTâ on the current screen, you must first press any key to continue. Leaving the Application To leave the application, press - l as many times as needed to return to a Table of Contents screen, and then press áEXITâ. When you re-enter the application, you return to this Table of Contents screen. Topics in Algebra 1 © 2001 Texas Instruments Introduction vii Navigating Topics in Algebra 1 (continued) Navigation Star The navigation star is located on the bottom right of most Overview and Observations screens. When the right and left arrows of the navigation star are flashing, press | and ~ on the calculator to page back and forward between screens. When the up and down arrows are flashing, } and † are used to navigate vertical menus (see below). Press ~ to display the next screen. Navigation star Horizontal Menus A horizontal menu may be displayed at the bottom of the screen. The menu items help you move between screens in the application. The menu items change from one screen to another. The following are the most frequently displayed menu items. Menu Item Press To Do This áEXITâ o Exit the application. áUPâ q Move up a level in the menu structure. áHELPâ s Display the HELP MENU. áMAINâ o Return to the previous screen from the HELP MENU. áBACKâ p Return to the HELP MENU from a HELP explanation screen. áMENUâ o Return to the previous menu screen. áQUITâ o Quit an activity. To select a menu item, press the calculator key that corresponds to the menu item on the screen. * In these materials, the items in the horizontal menus are shown in angle brackets; for example áEXITâ, áUPâ, and áHELPâ. Horizontal menu These menu items map to &'()* Topics in Algebra 1 these calculator keys. * © 2001 Texas Instruments The menu items change from one screen to another. On some screens, different menu items are mapped to the calculator keys. Introduction viii Navigating Topics in Algebra 1 (continued) Vertical Menus When the up and down arrows of the navigation star are flashing, press † and } on the calculator to highlight an item in a vertical menu. Vertical menu First item is highlighted. As you press † and } to move through the menu, additional information is displayed to the right of the menu. Some vertical menus only provide information. Other vertical menus are used for navigation. If Select or Select & Press [Enter] is displayed, you can press Í to select an item or to see more information. Press Í to select the second item on the vertical menu and display the first in a sequence of screens for this selection. Press áMENUâ (&) to return to the menu screen. áCONTâ When you see áCONTâ (continue), you can press any key to go to the next screen. Continue HELP From a Table of Contents screen, press áHELPâ to view information about features of the application. To select an item, press # to highlight it, and then press Í. Topics in Algebra 1 © 2001 Texas Instruments Introduction ix Chapter 1: Number Sense Section 1: Integers Name Date Number Sense: Integers Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate number sense concepts relating to the set of integers. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 How far apart are the bird and the fish? Should your answer be positive or negative? Write an explanation for your answers here. Observation 2 What is the sign of the answer to the problem L3 Q L11? In your own words, state the rules for multiplying and dividing signed numbers (shown in the grids on the next screens). Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-1 Chapter 1: Number Sense Section 1: Integers Name Date Activities The Activities help you practice integer concepts. You can select from two different activities— What Is My Sign? and Integer Smash. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. What Is My Sign? 1. In your head, quickly solve the sliding expression to determine if the result is positive, negative, or zero. Scoring: Every correct response earns 2 points. The game automatically ends if you have answered incorrectly four times (shown in the top right corner), or you press áQUITâ to stop. 2. As soon as you know the sign of the result, press # and $ to move the expression into the proper category on the left (+, 0, or M). Once the expression is in the correct row, you can press ! to slide it quickly to the left. If the answer is incorrect, the correct answer is displayed; press any key to resume play. 3. Follow your teacher’s instructions for how long to play the activity. 4. What was your score? 5. How many incorrect answers did you have? (Shown in top right corner of the screen.) Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-2 Chapter 1: Number Sense Section 1: Integers Name Date Activities (continued) Integer Smash 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You can earn up to 10 points. 2. For bronze or silver levels only, press á+â, áNâ, á¦â, or á÷â to select the operation you want to practice. 3. Enter the missing number (press Ì for negative numbers) and press b. If the answer is incorrect on the second attempt, the correct answer is displayed; press any key to resume play. As you play the activity, write each number sentence and solution. Show your work here. 4. What level and operation (bronze and silver only) did you play? 5. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-3 Chapter 1: Number Sense Section 1: Integers Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 Learn to use the subtraction key (¹), the negation key (Ì), and the absolute value function (abs(). Use the negation and the subtraction keys to calculate M5 N M4. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Calculate M5 N M4. Ì5¹Ì4 Í Note: The negation key (Ì) and the subtraction key (¹) are different. Display (TI.83 Plus shown) Enter M5 N M3 Q M2, first without using parentheses and then using parentheses. To Do This Press 1. Calculate M5 N M3 Q M2. Ì 5 ¹ Ì 3 M Ì2 Í Note: On the TI-73, notice the difference between the x variable key (I) and the multiplication key (M). 2. Calculate M5 N (M3 Q M2). Ì5¹ DÌ3MÌ2E Í 3. Calculate (M5 N M3) Q M2. DÌ5¹Ì3E MÌ2 Í Display (TI.83 Plus shown) The calculator uses the Order of Operations rules, which say that multiplication and division are performed from left to right, and then addition and subtraction are performed from left to right. Notice that multiplication was calculated before subtraction in the expression M5 N M3 Q M2. Notice that the expression M5 N (M3 Q M2) has the same answer. The calculator performs operations inside the parentheses before operations outside the parentheses. Notice that the expression (M5 N M3) Q M2 has a different answer than the expressions M5 N M3 Q M2 and M5 N (M3 Q M2). Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-4 Chapter 1: Number Sense Section 1: Integers Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Use the absolute value (abs() function on the calculator to find the absolute value of M12. To Do This Press 1. Find the absolute value of M12, which is written as |M12 |. 1" Display (TI.83 Plus shown) Note: The 1 NUM menu varies slightly from the TI-73 to the TI-83 Plus. 2. Select the absolute value (abs() function. It is copied to the Home screen. 1:abs( 3. Calculate the result. a 12 E Í Additional problems—Calculate the following problems by hand, then check your answers using the calculator. Remember to use the Order of Operations rules. 1. M4 + M12 Q M10 = 2. 4 Q M8 − M10 Q 2 = 3. M30 ÷ M5 − 6 = 4. | M12 +M28 | = 5. | M4 Q M8 N M10 Q 2 | = 6. M| M3 N 14 N M10 | = Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-5 Chapter 1: Number Sense Section 1: Integers Number Sense: Integers Teacher Notes Objectives • To illustrate the set of integers in a Venn diagram. • To give an overview of the following definitions: the sign of a number, numbers of opposite sign, and absolute value. • To give illustrations of the sets of positive and negative integers, and zero. • To review ordering and the additive inverse property. • To show examples of addition, subtraction, multiplication, and division of integers. Math Highlights This section begins with a Venn diagram and follows with an illustration of the use of integers on the Fahrenheit and Celsius scales. The temperature equivalencies shown are integer values. Students are reminded that the absolute value operation is defined as the distance of the number from zero. Examples of addition, subtraction, multiplication, and division operations are given. A number line model is used for addition and subtraction. Subtraction uses the add-the-opposite rule. Common Student Errors Students may have trouble identifying rules, such as the add-the-opposite rule for subtraction. Note: The number line model for addition and subtraction is shown on the calculator when the Topics in Algebra 1 application is installed. Following are some activities to help students construct the rules for multiplication and division. These activities are not part of the Topics in Algebra 1 application. Pattern Development for Multiplication and Division Investigation 1: What is the product of a positive and negative number? Begin by writing the product of two positive numbers, for example, 3 Q 3 = 9. Keep decreasing the second term by 1 to create the sequence of multiplication values shown to the right. Notice the values on the right side of the number sentences decrease by 3. 3Q3= 3Q2= 3Q1= 3Q0= 3 Q M1 = 3 Q M2 = 3 Q M3 = ... 9 6 3 0 M3 M6 M9 Observation: The product of a positive and negative number is negative. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 1-6 Chapter 1: Number Sense Section 1: Integers Common Student Errors (continued) Pattern Development for Multiplication and Division (continued) Investigation 2: What is the product of two negative numbers? Start with the last number sentence from Investigation 1 (3 × M3 = M9). Keep decreasing the first term by 1 to create the sequence of multiplication values shown to the right. Notice that the values on the right side of the number sentences increase by 3. Observation: The product of two negative numbers is positive. Investigation 3: What are the division rules for multiplying signed numbers? Observation: Division rules are developed using multiplication as the inverse operation. The rules are similar. 3 × M3 = 2 × M3 = 1 × M3 = 0 × M3 = M1 × M3 = M2 × M3 = M3 × M3 = … M9 M6 M3 0 3 6 9 M6 ÷ 3 = M2 since 3 × M2 = M6 M6 ÷ M2 = 3 since M2 × 3 = M6 Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, if necessary. 2. How to navigate the application, if necessary. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 1-7 Chapter 1: Number Sense Section 1: Integers Observations The Observations help students understand operations with signed integers. If necessary, tell students how to find the Observations. Observation 1 How far apart are the bird and the fish? Should the answer be positive or negative? Students write an explanation of their answers. The bird and the fish are +125 ft. apart. The distance is positive. (Students will see the answers on the next three screens. Signed numbers give a perspective as shown on the screens.) Explanations will vary. Students see the answers on the following three screens. Observation 2 What is the sign of the answer to the problem M3 Q M11? The answer is positive. M3 Q M11 = 33. Students see grids explaining the rules. What are the rules for multiplying and dividing signed numbers? The product of a positive and negative number is negative. The product of two numbers with the same sign is positive. Rules for division are similar to the rules for multiplication. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 1-8 Chapter 1: Number Sense Section 1: Integers Activities What Is My Sign? Tell students to: 1. Quickly solve the expression in their heads before it slides all the way to the left. Scoring: Students earn 2 points for each correct response. Unless you specify point or time limits for this activity, students can play the activity until they have answered incorrectly four times or they press áQUITâ to stop. There is no time limit. 2. Move the expression into the correct category (+, 0, or M) using $ and #. Once the expression is in the correct row, they can press ! to slide it quickly to the left. If the answer is incorrect, the correct answer is displayed; press any key to resume play. 3. Follow your instructions. For example, students can play: • Until they have answered incorrectly four times (no time limit). • Until a certain amount of time has expired (highest score with the fewest misses wins). • Until a certain score has been reached (first student to reach the score with the fewest misses wins). • Repeatedly over a period of time (days, weeks, etc.) for tracking improvement of high scores. 4. Record their score. 5. Record how many incorrect answers they had. (Shown in top right corner of the screen.) Integer Smash Tell students to: 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. Students can earn up to 10 points. Tip: You may want to remind students playing at the gold level about the Order of Operations rules. 2. Bronze or silver levels only: Press á+â, áNâ, á¦â, or á÷â to select the operation that they want to practice. 3. Enter the missing number (press Ì for negative numbers) and press b. As they play the activity, write each number sentence and solution, showing all of their work on the worksheet. 4. Record the level and operation (bronze and silver only) they played. 5. Record their score. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 1-9 Chapter 1: Number Sense Section 1: Integers ³ Try-It!é on Your TI.83 Plus or TI.73 These keystroke exercises let students practice using the basic operation keys (\, T, M, F), the negation key (a), the parentheses keys (D E), and the absolute value function (1 NUM 1:abs( ). Tell students to follow the steps exactly on the calculators. Example screens are displayed on the worksheets for students to compare with the calculator screens. Additional problems—These problems give students additional practice using the subtraction key (T), the negation key (a), and the absolute value function (1 NUM 1:abs( ). Remind them to follow the Order of Operations rules. Tell students to do the following calculations by hand, and then check the answers using the calculator. 1. M4 + M12 Q M10 = 116 2. 4 Q M8 − M10 Q 2 = M12 3. M30 ÷ M5 − 6 = 0 4. | M12 +M28 | = 40 5. | M4 Q M8 N M10 Q 2 | = 52 6. M| M3 N 14 N M10 | = M7 Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 1-10 Chapter 1: Number Sense Section 2: Rational Numbers Name Date Number Sense: Rational Numbers Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate number sense concepts relating to rational numbers. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 Write 4 2 as an improper fraction. 3 Write your answer here. Show all of your work. Observation 2 Write 2 5 and as decimals. 8 11 Write your answer here. Show all of your work. Observation 3 Write .1875 and 1 as percentages. 4 Write your answer here. Show all of your work. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-11 Chapter 1: Number Sense Section 2: Rational Numbers Name Date Activities The Activities help you practice rational number concepts. You can select from two activities— Slide and Number Smash. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and then press b. Slide 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. Scoring: When a row or column adds up to 1, it disappears, and you score 2 points. The game automatically ends if the screen is full or if you press áQUITâ to stop. 2. Line up the fractions so they add to 1 (horizontally or vertically). As a fraction slides across the screen, press # and $ to move it up or down. Once a fraction is in the correct row, you can press ! to slide it quickly to the left. 3. Follow your teacher’s instructions for how long to play the activity. 4. What level did you play? 5. What was your score? 6. Write a paragraph describing the Slide activity. What was your strategy for playing the game? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-12 Chapter 1: Number Sense Section 2: Rational Numbers Name Date Activities (continued) Number Smash 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. 2. Press á+â, áNâ, á¦â, or á÷â to select the operation you want to practice. 3. Enter the missing number (press Ì for negative numbers), and press b. As you play the activity, write each number sentence and solution. Show your work below. • To enter a mixed number, enter the whole number and press áUNITâ. Then enter the fraction. Tip: To change an answer before you press b, press : and re-enter the answer. • To enter a fraction, press án/dâ and enter the numerator. Press án/dâ again and enter the denominator. Tip: Press án/dâ to move between the numerator and denominator. If the answer is incorrect, the correct answer is displayed; press any key to resume play. You can earn up to 10 points. 4. What level and operation did you play? 5. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-13 Chapter 1: Number Sense Section 2: Rational Numbers Name Date ³ Try-It!é on Your TI.83 Plus Investigate how the calculator computes addition expressions. Solve To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Enter the expression on the Home screen. 2¥3Ã5¥6 3. To specify that you would like the result to be shown in fraction form, select 4Frac. It is copied to the Home screen. 1:4Frac 4. Evaluate the answer. Í 2 5 + . 3 6 Display (TI.83 Plus shown) Notice that the answer is in simplified form. The calculator follows the Order of Operations rules. Division is performed before addition. Solve this problem by hand. Show all of your work. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-14 Chapter 1: Number Sense Section 2: Rational Numbers Name Date ³ Try-It!é on Your TI.83 Plus (continued) Investigate how the calculator computes division expressions. Solve To Do This Press 1. Solve without parentheses and specify that you want the result in fraction form. 1¥2¥2¥3 1:4Frac 2. Solve using parentheses and specify that you want the result in fraction form. 1 2 ÷ . 2 3 Display (TI.83 Plus shown) Í £1¥2¤¥ £2¥3¤ 1:4Frac Í The calculator gives two different answers, depending on how you entered the expression. 1 2 Which one is the answer for the problem P ? 2 3 Solve this problem by hand. Show all of your work here. Explain which answer from the calculator is the desired answer and why. Additional problems—Calculate the following problems by hand. Simplify your answers. Then check your answers using the calculator. Remember to use the Order of Operation rules. 1. 3. M 3 5 5. M 7. M 1 3 P 2 5 1 6 1 + 4 5 6 + N = 2. = 1 4 1 12 4. Q + Topics in Algebra 1 2 3 = 3 4 = 6. 8. M 1 2 1 8 M 2 3 N Q 1 2 Q 5 6 M2 = 5 3 16 N © 2001 Texas Instruments = P 3 4 1 2 = P (M3) = Student Worksheet 1-15 Chapter 1: Number Sense Section 2: Rational Numbers Name Date ³ Try-It!é on Your TI.73 Investigate how the calculator computes addition expressions. Solve To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Select the b/c and Mansimp mode settings. . # # # " to highlight Note: See ³ TIpé 2: Adjusting Your Calculator Settings for more information. 2 5 + . 3 6 Display (TI.73 shown) b/c b # " to highlight Mansimp b 3. Calculate the result. -l 2=3"\ 5=6b 4. Simplify the fraction. Bb Simplified by a factor of 3. Notice that the answer is in simplified form. The calculator follows the Order of Operations rules. Division is performed before addition. Solve this problem by hand. Show all of your work. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-16 Chapter 1: Number Sense Section 2: Rational Numbers Name Date ³ Try-It!é on Your TI.73 (continued) Investigate how the calculator computes division expressions. Solve To Do This Press 1. Solve without parentheses and specify that you want the result in fraction form. 1F 2¥ 2F 3 2. Solve using parentheses and specify that you want the result in fraction form. £1F 2¤¥ £2F 3¤ >Í 3. Simplify the result by a factor of 25. B 25 1 2 ÷ . 2 3 Display (TI.73 shown) >Í The calculator gives two different answers, depending on how you entered the expression. Which one is the answer for the problem: 1 2 ÷ 2 3 Solve this problem by hand. Show all of your work here. Explain which answer from the calculator is the desired answer and why. Additional problems—Calculate the following problems by hand. Simplify your answers. Then check your answers using the calculator. Remember to use the Order of Operation rules. 1. 3. M 3 5 5. M 7. M 1 3 P 2 5 1 6 1 + 4 5 6 + N = 2. = 1 4 1 12 4. Q + Topics in Algebra 1 2 3 = 3 4 = 6. 8. M 1 2 1 8 M 2 3 N Q 1 2 Q 5 6 M2 = 5 3 16 N © 2001 Texas Instruments = P 3 4 1 2 = P (M3) = Student Worksheet 1-17 Chapter 1: Number Sense Section 2: Rational Numbers Number Sense: Rational Numbers Teacher Notes Objectives • To review the definition of rational numbers as ratios and as terminating and repeating decimals. • To review ordering and the reciprocal property of rational numbers. • To review operations with rational numbers. Math Highlights Students review rational numbers. This includes rational numbers as ratios, terminating decimals, repeating decimals, and integers, as well as operations with fractions. In the Observations, students are reminded of the connection between fractions, decimals, and percents. Common Student Errors • Students might confuse the algorithms for addition, subtraction, multiplication, and division of fractions. • Students may have trouble identifying whether a fraction is positive or negative. For example: M1 M3 • = 1 3 1 or M3 = M1 3 = M 1 3 Students may have problems because they use short cuts to change the decimal representation of a number to a percent representation. Using short cuts does not provide an understanding of why the representations are equal. Students should understand that the quantity stays the same. Using the multiplicative identity, 1=100/100, is the key to the change in the representation. For example, students change .1875 to a percent. A shift of the decimal point gives the correct answer, but without any connection to the math they know. However, multiplying by 1 in the form 100/100 gives the same result and makes the connection to the math as well. .1875 Q 100 100 Topics in Algebra 1 = 18.75 100 = 18.75% © 2001 Texas Instruments Teacher Notes 1-18 Chapter 1: Number Sense Section 2: Rational Numbers Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students understand number sense concepts relating to rational numbers. If necessary, tell students how to find the Observations. Observation 1 2 as an improper fraction. 3 14 Answer: 3 Write 4 Remind students to write the answer on the worksheet and to show all of their work. Students see the answer on the next two screens. Tell students to check the answers on the worksheet. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 1-19 Chapter 1: Number Sense Section 2: Rational Numbers Observations (continued) Observation 2 Write 5 2 and as decimals. 8 11 Answer: .625 and .18 . Remind students to write the answers on the worksheet and to show all of their work. Students see the answers on the next two screens. Tell students to check the answers on the worksheet. Observation 3 Write .1875 and 1 as percentages. 4 Answer: 18.75% and 25%. Remind students to write the answers on the worksheet and to show all of their work. Students see the answers on the next two screens. Tell students to check the answers on the worksheet. Activities Slide Tell students to: 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. Scoring: When a row or column adds up to 1, it disappears, and the player scores 2 points. The game automatically ends if the screen is full, or you press áQUITâ to stop. Topics in Algebra 1 2. Line up the fractions so they add to 1 (horizontally or vertically). As a fraction slides across the screen, press # and $ to move it up or down. Once a fraction is in the correct row, they can press ! to slide it quickly to the left. © 2001 Texas Instruments Teacher Notes 1-20 Chapter 1: Number Sense Section 2: Rational Numbers Activities (continued) Slide (continued) 3. Follow your instructions. For example, students can play: • Until the screen fills up (no time limit). Scoring: When a row or column adds up to 1, it disappears, and the player scores 2 points. The game automatically ends if the screen is full, or you press áQUITâ to stop. • Until a certain amount of time has expired (highest score wins). • Until a certain score has been reached (first student to reach the score wins). • Repeatedly over a period of time (days, weeks, etc.) for tracking improvement of high scores. 4. Record the level they played. 5. Record their scores. 6. Write on the worksheet a paragraph in which they describe the Slide activity and the strategy for playing. Number Smash Tell students to: 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. They can earn up to 10 points. Tips: Tell students that they can: • Change the answer before they press b by pressing : and re-entering the answer. • Press án/dâ to move between the numerator and denominator. Note: Unsimplified fractions are counted as correct. 2. Press á+â, áNâ, á¦â, or á÷â to select the operation that they want to practice. 3. Enter the missing number (press Ì for negative numbers), and press b. As they play the activity, they should write each number sentence and its solution on the worksheet, showing all their work. If the missing number is a mixed number or fraction, tell them: • To enter a mixed number, enter the whole number, press áUNITâ, and then enter the fraction. • To enter a fraction, press án/dâ and enter the numerator. Press án/dâ again and enter the denominator. If the answer is incorrect, the correct answer is displayed; press any key to resume play. 4. Record the level and operation they played. 5. Record their scores. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 1-21 Chapter 1: Number Sense Section 2: Rational Numbers ³ Try-It!é on Your TI.83 Plus or TI.73 Review the Order of Operations rules with students, if necessary. Explain to them that the calculator uses the Order of Operations rules to simplify expressions. Discuss with them how parentheses are used. Note: The ³ Try-It! activities are repeated for each of the two calculators—the TI-83 Plus and then the TI-73. The problems are the same, but they vary due to the differences in the two calculators. The Additional problems, which are identical, can be performed on either calculator. They are repeated for your convenience when you copy the activities. Tell students to do the two calculator ³ Try-It! investigations. • • 2 5 + . 3 6 1 2 Investigate how the calculator computes division expressions. Solve ÷ . 2 3 Investigate how the calculator computes addition expressions. Solve Ask students to explain the difference in the two results in second investigation. Although each of the two results is correct based on how the problem was entered, to make sure that the division is performed correctly, the problem must be entered as (1 à 2) à (2 à 3). The answer is 3/4, not 1/12. The calculator uses the Order of Operations rules. Operations inside parentheses are performed before operations outside parentheses. On the TI-73, students must simplify the fraction 75/100 to get 3/4. They may either specify the factor to use, as shown, or let the calculator simplify the fraction, one factor at a time by repeatedly pressing B. Additional problems—Make sure that students understand and use the Order of Operation rules so they can determine when to use parentheses. 1. 3. M 3 5 5. M 7. M 1 3 P 2 5 1 6 1 + 4 5 6 + N = 1 4 1 12 1 =M 2. 12 18 4. 25 Q + Topics in Algebra 1 2 3 =M 3 4 = 7 30 1 2 6. 8. M 1 2 1 8 M 2 3 N Q 1 2 Q 5 6 M2 = 5 3 16 N © 2001 Texas Instruments P 3 4 5 =M 9 9 10 1 2 = 3 64 P (M3) = M 1 4 Teacher Notes 1-22 Chapter 1: Number Sense Section 3: Real Numbers Name Date Number Sense: Real Numbers Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate number sense concepts relating to real numbers. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 Write three different irrational numbers. Show your work. Observation 2 Try these problems . . . Use the real number properties to solve the following problems quickly. Show your work. 25 Q 24 = 8 Q 102 = 6 Q 46 = Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-23 Chapter 1: Number Sense Section 3: Real Numbers Name Date Activities The Activities help you practice real number concepts. You can select from two activities— Raining Reals and What Is My Property? Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Raining Reals 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: Every correct placement earns 2 points. The game automatically ends if you have answered incorrectly four times (shown in the top right corner), or you press áQUITâ to stop. 2. As the numbers fall on your screen, quickly determine if the “raining” number is rational or irrational. 3. Press ! to move the number into the RATIONAL set, or press " to move the number into the IRRATIONAL set. If the answer is incorrect, the correct answer is displayed; press any key to resume play. 4. Follow your teacher’s instructions for how long to play the activity. 5. What level did you play? 6. What was your final score? 7. How many incorrect answers did you have? (Shown in top right corner of the screen.) 8. Write a paragraph describing the activity. Describe your strategy for playing. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-24 Chapter 1: Number Sense Section 3: Real Numbers Name Date Activities (continued) What Is My Property? 1. Look at the equation and decide which single property, out of these six, it represents. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You can earn up to 16 points. • Commutative + • Commutative … • Associative + • Associative … • Distributive … Over + • Distributive … Over N 2. Scroll through the property choices with # and/or $. To select a property, press b. If the answer is incorrect, the correct answer is displayed; press any key to resume play. As you play the activity, record each equation and its property. 3. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-25 Chapter 1: Number Sense Section 3: Real Numbers Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 Investigate how the calculator deals with irrational numbers. In the Overview, you used the Pythagorean theorem to find the length of À 2 . Draw a right triangle so that the hypotenuse has length À 2 . Pythagoras (569–475 B.C.), a great Greek mathematician, discovered irrational numbers (numbers that are not rational and therefore are not ratios). There is a proof that, for example, À 2 cannot be written as a fraction. Hint: Draw each leg with length 1 inch. The hypotenuse is Á1 2+1 2 = À1+1 = À 2 Remember that À 2 Q À 2 = 2. Picture this by envisioning a square whose sides measure À 2 units. You created this length in your picture above. Look at the square whose side has a length of À 2 on the Geoboard screen below. Can you see that the area is À 2 Q À 2 = 2 square units? Count it up! Shade in the area on the screen shown. From the TI-73 Geoboard application Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-26 Chapter 1: Number Sense Section 3: Real Numbers Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Find the calculator decimal approximation for À 2 . To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Select the Float setting from the mode screen. . # until Float is highlighted b Notes: See ³ TIpé 2: Adjusting Your Calculator Settings for details. Display (TI.83 Plus shown) The TI-83 Plus mode screen varies slightly from the TI-73. 3. Return to the Home screen. -l 4. Calculate the decimal approximation for À 2. -z2E b 5. Square your result. 6 b Note: Ans = previous answer. The calculator remembers that you entered ‡(2). Is À 2 equal to the decimal 1.414213562? It looks like the calculator says this is true. Calculate 2 1.414213562 to see. 6. Calculate the square of 1.414213562. 1`4142135626 b The answer shows 1.999999999, but you know you should get the answer 2. Be careful! When you use your calculator, you have to know your math. The calculator can do amazing math, but it only shows you an approximation for many answers. You have to understand the problem before you use the calculator. It is up to you to determine if calculator answers are reasonable and how you will use them. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-27 Chapter 1: Number Sense Section 3: Real Numbers Name Date ³ Try-It!é on Your TI.83 Plus or TI-73 (continued) Word Problem: Missy’s Garden Missy wants to build a small fence around a garden in her backyard. The garden is in the shape of a right triangle. One leg is 2 meters and the other leg is 1 meter. The store sells fencing in tenths of a meter. 2m 1m 1. What is the exact perimeter of Missy’s garden? Show all your work. Math Hint: Use the Pythagorean theorem to find the length of the third side of the garden, and then find the perimeter of the garden. (See the Real Numbers Overview on the calculator.) 2. What length of fencing should Missy buy if the store only sells the fencing in tenths of a meter? Use the calculator. Show all your work. Calculator Hint: After you calculate the approximate answer on your calculator, try setting . so that Float=1. See ³ TIpé 2: Adjusting Your Calculator Settings for details. This will give you one decimal place or tenths. Observe how the calculator displays the results! Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-28 Chapter 1: Number Sense Section 3: Real Numbers Name Date ³ Try-It!é on Your TI.83 Plus or TI-73 (continued) Word Problem: Jose and Maria’s Backyard Pool Jose and Maria have a circular pool in their backyard. Their parents would like to make a cover for the pool. They bought a square piece of material whose sides are the same length as the diameter of the pool. The diameter of the pool is 3.5 meters. 3.5 meters 1. How much material will they have left over? Find the exact and approximate answers. Use the calculator. Show all your work. (See Hints below.) 2. Exact answer: 3. Approximate answer (to 3 decimal places): Calculator Hints: • Use Float = 3 on your calculator to display 3 decimal places. • Press - „ to find the calculator's approximation for p. Math Hints: • Area of a square: A = s 2, where s is the length of the sides of the square. • Area of a circle: A = p r2, where r is the radius of the circle (diameter = 2r). Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 1-29 Chapter 1: Number Sense Section 3: Real Numbers Number Sense: Real Numbers Teacher Notes Objectives • To illustrate the real number system in a Venn diagram. • To identify real numbers as rational numbers ∪ irrational numbers. • To review writing rational numbers as terminating or repeating decimals. • To review writing irrational numbers as nonterminating, nonrepeating decimals. • To show physical representations of the irrational numbers, À 2 and p, and to review the Pythagorean theorem and the formula for finding the circumference of a circle. • To state the real number system properties—commutative, associative, and distributive—as well as the identity and inverse properties. Math Highlights This section starts with the building of the Venn diagram of the real number system. Definitions of rational and irrational numbers are given. Two examples of irrational numbers, À 2 and p, are developed. À 2 is shown as the length of the hypotenuse of a right triangle with legs of 1 unit. p is shown as the circumference divided by the diameter for any circle. The statements of the properties of the real numbers follow. Common Student Errors • Many students may not have developed a solid understanding of number sets. Remind them that using rational and irrational numbers, they can name every location on a number line. Later in their studies of mathematics, this will be referred to as the Completeness Property of Real Numbers, which was an important discovery in mathematics. Later, they will also extend the real numbers to the complex numbers, a + bÀ M1 = a + bi, which are numbers used, for example, in the study of the relationship between electricity and magnetism. • Students probably have used 22/7 or 3.14 as an approximation of p. They may think that these values are exactly p, but they are not equal to p. This provides an opportunity to talk about approximations to several decimal places in real problems. The worksheet problems give students an opportunity to find exact and approximate answers. There are wonderful web sites that show p to millions of places. Mathematicians are still searching for more place values. This study of p requires the use of computers to assist the search. • Some students may not be aware that the ratio of the circumference of a circle C divided by the diameter d is p. C/d= p. This may be confusing because they have been told that p is irrational and is not a ratio. Yet p came from a ratio of circumference to diameter. It turns out that either C or d is also irrational. The mathematics to prove this is not given at this level; therefore, students have to accept this without much explanation. This is a deep discussion that will not be of interest to some students, but other students may find it fascinating. Topics In Algebra 1 © 2001 Texas Instruments Teacher Notes 1-30 Chapter 1: Number Sense Section 3: Real Numbers Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students understand number sense concepts relating to real numbers. Tell students how to find the Observations. Observation 1 Write three different irrational numbers. Students show their work. Answers will vary. Students see this screen with three possible answers. Observation 2 Try these problems . . . Students use the real number properties to solve the problems, showing their work. Students should show that they know how to use the properties of the real number computations as shortcuts without the calculator. • • • associative … property: 25 … 24 = 25 … (4 … 6) = (25 … 4) … 6 = 100 … 6 = 600 distributive … over + property: 8 … 102 = 8 … (100 +2) = (8 … 100) + (8 … 2) = 800 +16 = 816 distributive … over N property: 6 … 46 = 6 … (50 N 4) = (6 … 50) N (6 … 4) = 300 N 24 = 276 Students see the answers on the next screen. Topics In Algebra 1 © 2001 Texas Instruments Teacher Notes 1-31 Chapter 1: Number Sense Section 3: Real Numbers Activities Raining Reals Tell students to: 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: Every correct placement earns 2 points. When students give an incorrect answer, the correct answer displays. The game automatically ends when they have answered incorrectly four times (shown in the top right corner), or they press áQUITâ to stop. 2. Determine if the “raining” number is rational or irrational. 3. Press ! to move the number into the RATIONAL set, or press " to move the number into the IRRATIONAL set. If the answer is incorrect, the correct answer is displayed; press any key to resume play. 4. Follow your instructions. For example, students can play: • until they have answered incorrectly four times (no time limit) • until a certain amount of time has expired (high score wins) • until a certain score has been reached (first student to reach the score with the fewest misses wins) • over a period of time (days, weeks, etc.) for tracking improvement of high scores 5. Record the level they played. 6. Record their final scores. 7. Record how many incorrect answers they had. (Shown in top right corner of the screen.) 8. Write their strategy for playing the game. What Is My Property? Tell students to: 1. Look at the equation and decide which one property, out of these six, it represents: Scoring: Students get two attempts to answer. They earn 2 points for a correct answer, 1 point for a correct answer on the second try. They can earn up to 16 points. • • • • • • Commutative + Commutative … Associative + Associative … Distributive … Over + Distributive … Over N 2. Scroll through the choices with # and/or $ and press b to select the correct property. If the answer is incorrect, the correct answer is displayed; press any key to resume play. As they play the activity, record each equation and its property. 3. Record their scores. Topics In Algebra 1 © 2001 Texas Instruments Teacher Notes 1-32 Chapter 1: Number Sense Section 3: Real Numbers ³ Try-It!é on Your TI.73 or TI.83 Plus Tell students to: • Use the Pythagorean theorem to draw a triangle whose two legs = 1 unit and whose hypotenuse = À 2 units. • Shade in the area on the Geoboard screen on the worksheets. • Go through the keystroke example to: - Find the calculator decimal approximation for À 2 . - Understand how the calculator approximates numbers. - Understand that they must be conscientious about the mathematics involved. Word Problem: Missy’s Garden Remind students to: • Use the Pythagorean theorem to find the third side of the garden and then the perimeter. • Set the decimal mode notation (.) to Float and then to 1 (answer rounded to tenths), so they can see how the calculator displays answers. 1. Exact perimeter: 1 + 2 + À 5 meters 2. Length of fencing rounded to tenths: 5.2 meters Note: The number rounds down. This is mathematically correct, but impractical in the real world, where Missy would need to purchase 5.3 meters in order to fence the garden. You may want to discuss meaningful interpretation of word problems with the students. Word Problem: Jose and Maria’s Backyard Pool If necessary, review the formulas for area of a square and area of a circle. They are shown on the worksheet. Covering material left over: 1. Exact: Area of square—area of circle = (3.5)2 N p(3.5/2)2= 12.25 N 3.0625 p square meters 2. Approximate: 2.629 square meters (3 decimal places; use Float=3) Topics In Algebra 1 © 2001 Texas Instruments Teacher Notes 1-33 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Name Date Linear Equations: Using Graphs & Tables Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate linear equation concepts relating to graphs and tables. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 Using a table, find the solution of MxN6 = M3.5. First, find the solution by hand, making a table with at least five x and y values. You can start with any x value you want. Show your work. x y Next, graph your solution on a number line. Be sure to label all points on the number line. ←→ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-1 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Name Date Activities The Activities help you practice graphs and tables. You can select from two activities—Beam Dale Up or Worksheet Activity. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Beam Dale Up 1. Look at the problem on the space ship and determine how to solve the equation for x from the four choices given. The values are ordered as they are on a number line. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. 2. Press ! and " to move Dale the Martian over the correct x value, and then press b. If the x value you pick is correct, Dale is beamed up to his ship! If the answer is incorrect on the second attempt, the correct answer is displayed; press any key to resume play. As you play the activity, solve each equation in the space below. Show all steps and work. You can earn up to 10 points. 3. What was your score? Worksheet Activity 1. Look at the table of values of equations Y1=2X+4.5 and Y2=7. Note: Press | or ~ to leave this screen. 2. When is 2X+4.5=7 (or, when is Y1=Y2)? Look at the table carefully. The solution is between which two X values? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-2 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 Use the table feature on your calculator to search for the solution (when Y1=Y2). To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. First, enter 2X+4.5 as Y1 and 7 as Y2 in the Y= editor. &‘ 2„\4`5 #‘ Note: See ³ TIpé 3: Graphing a Function in the Standard Window for more information. Display (TI.83 Plus shown) 7 Note: On the TI-73, use I rather than „. 3. Change the table settings to those used in the Worksheet Activity. yf 0 # .5 #Í #Í 4. Display the table. y0 yf 5. Change the table settings to focus on the range you found in the Worksheet Activity. Begin the table at the lower value and make the increments smaller. # .1 6. Display the table with the new settings. y0 1 7. The table still does not display the exact solution. What table settings would you choose in order to display the exact solution? TblStart= @Tbl= 8. What is the solution (when is Y1=Y2)? _________________________________________ 9. How do you know this is the solution? ________________________________________ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-3 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Solution Search Using X-Y Graphs: Find the solution for the equation X+3=1.7. First, use your number sense to estimate the solution for X+3=1.7. Record your estimate here. Use the table feature on your calculator to search for the solution (when Y1=Y2). To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Set the window format as shown. yg Display (TI.83 Plus shown) Note: See ³ TIpé 2: Adjusting Your Calculator Settings for more information. Screens are shown for the TI-83 Plus. There are slight differences on the TI-73. 3. Enter both sides of the equation into the Y= editor as shown. Note: See ³ TIpé 3: Graphing a Function in the Standard Window for more information. &‘ 1`7 #‘ „Ã3 Note: On the TI-73, use I rather than „. 4. Select ZDecimal to set the viewing window and automatically graph the functions. TI-83 Plus: ( 4:ZDecimal TI-73: ( 8:ZDecimal 5. Trace Y1=1.7. 6. Trace Y2=X+3. Note: The function displays in the upper left corner of the screen; the X and Y values are displayed on the bottom of the screen. r | or ~ to trace a function } and † to move between functions 7. Find where X+3=1.7. What is the solution? 8. How do you know? 9. Did you notice that the value of X+3 is 1.7 when X=M1.3? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-4 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Solution Search: Write the solution and explain how you found the solution using graphs and tables for each of the problems below. For each problem: • Before you start, estimate the solution so you have an idea of where the solution is located. • Search for the solution of the equation on the calculator using graphs and table. Note: See ³ TIpé 4: Creating a Table for additional help with the calculator. • Remember to change your viewing window or your table setting to do your search. Note: See ³ TIpé 5: Adjusting the Viewing Window for additional help with the calculator. • Explain how you found the solution. Calculator Fact: The calculator only uses the variables X and Y for graphs and tables. If an equation uses letters other than X, you will have to enter the letter for the unknown variable as X in the calculator in order to make the graphs and tables for your solution search. 1. p+3=1 2. 3x − 2 = 1.6 3. 4. 1 2 x+ 5 8 = 7 8 4 + 0.5C = 7 Calculator Hint: Use ZStandard to change the graph window so that: M10X10 and M10Y10. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-5 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Linear Equations: Using Graphs & Tables Teacher Notes Objectives • To illustrate how to locate the real number solution of a linear equation using tables. • To illustrate how to locate the real number solution of a linear equation using a graphical method on a Cartesian (x-y) graph. Math Highlights In the table of values method, students see a table of values for the left and right side of the equation. They see that the x value, which makes the two sides of the equation equal, is the solution. They also see that they may need to refine the table of values to search for the solution. In the x-y graphical method, students graph both sides of the equation and find the intersection of the lines. If students have not graphed lines, they can first make a table of values and then plot the points to graph the lines. The x coordinate of the intersection of the lines is the solution. Common Student Errors • Using graphs and tables can mislead a student to think that they can always find the exact solution using these methods. Although they often will find exact solutions using these methods, using algebra will give exact answers for these equations. Have students try to search for the solution to x + À 2 = À 7 . The exact answer is x = À 7 N À 2 = 1.2315377 . . . • At times, introducing the algebraic solution of equations gives students just the mechanics of doing a problem. Algebraic methods alone usually do not invite the student to reason out the solution using number sense. The graphs and tables method gives students the opportunity to see the values of each side of the equation so they can see when the right side and left side of the equation are equal. Visual learners benefit by seeing the numbers and graphs first, and then by using these as the tool to find the solution. • Some students are able to see the solution to an equation using their number sense and may have difficulty taking the time to show their work. This may also be an issue in Chapter 2: Linear Equations, Section 2: Using Algebra. Encourage the use of written mathematics and drawing graphs and tables as a communication tool. Have students look in newspapers for graphs and tables of information to show real examples for the need for this communication skill. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 2-6 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students understand linear equation concepts relating to graphs and tables. If necessary, tell students how to find the Observations section for this section. Observation 1 Using a table, find the solution of MxN6 = M3.5. First, find the solution by hand, making a table with at least five x and y values. The students can start with any x value they want. Students show their work. Next, they graph the solution on a number line. Remind them to label all points on the number line. Students make a table by hand on the worksheet to search for the solution of Mx N 6 = M3.5. Then they transfer these x and y values to a number line. Look for correct labeling. Students see the table and graph screens as they finish viewing the observation. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 2-7 Chapter 2: Linear Equations Section 1: Using Graphs & Tables Activities Beam Dale Up Tell students to: Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. Students can earn up to 10 points. 1. Look at the problem on the space ship and determine how to solve the equation for x from the four choices given. The values are ordered as they are on a number line. 2. Press ! and " to move Dale the Martian over the correct x value, and then press b. If the x value they pick is correct, Dale is beamed up to his ship! As they play the activity, students record the solution to each equation, showing all steps and work. If the answer is incorrect on the second attempt, the correct answer is displayed; press any key to resume play. 3. Record their scores. Worksheet Activity Tell students to: 1. Look at the table of values of the expressions, Y1=2X+4.5 and Y2=7. Note: Press | or ~ to leave this screen. 2. Answer the questions: When is 2X+4.5=7 (or, when is Y1=Y2)? The solution is between which two X values? The solution for 2X+4.5=7 is found between 1 and 1.5. Students will continue to search for the solution in the ³ Try-It! section. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 2-8 Chapter 2: Linear Equations Section 1: Using Graphs & Tables ³ Try-It!é on Your TI.83 Plus or TI.73 Tell students to: • Follow the steps exactly on the calculator, using the table feature to find the solution. Example screen pictures are displayed on the worksheet for students to compare with the calculator screen. This keystroke exercise lets students define Y1=2X+4.5 and Y2=7. • Determine what the table settings should be for an exact solution and record them on their worksheet. Answer: ∆Tbl=0.25 and TblStart=1. • Find the solution and record it. Answer: Y1=Y2 when X=1.25. Solution Search Using X-Y Graphs: Find the solution for the equation X+3=1.7. Tell students to: • Estimate the solution and record it on their worksheets. • Follow the steps exactly on the calculator. Example screen pictures are displayed on the worksheet for students to compare with the calculator screen. This keystroke exercise lets students find the solution where X+3=1.7. • Record the solution and explain their answers on the worksheet after step 6. Answers will vary. Student estimates using number sense should be close to the answer M1.3. They see that the two lines intersect at x=M1.3. Note: See the TI-83 Plus Guidebook, for information on the TI-83 Plus automatic feature “Intersection,” which is not covered in this section. This feature is not on the TI-73. Solution Search: Students write the solution and explain how they found the solution using graphs and tables for each of the problems below. Tell students to: • Review how to use the graphing features of the calculator to search for a solution. • Find the intersection of the lines by tracing the graph. Remind students that the viewing window must be set appropriately to find the intersection, and finding the exact solution is not guaranteed; they must combine the two methods to find the solutions. • Use number sense to complete the search. 1. 3. when p = M2 p+3=1 1 2 x+ 5 8 = 7 8 Topics in Algebra 1 when x = 1 2 2. 3x − 2 = 1.6 when x = 1.2 4. 4 + 0.5C = 7 when C = 6 © 2001 Texas Instruments Teacher Notes 2-9 Chapter 2: Linear Equations Section 2: Using Algebra Name Date Linear Equations: Using Algebra Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate linear equation concepts relating to algebra. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 xN5 = 8 Solve for x. Show all of your work below. Observation 2 3 2x=9 Solve for x. Show all of your work below. Observation 3 4Nx = 10 Topics in Algebra 1 Solve for x. Show all of your work below. © 2001 Texas Instruments Student Worksheet 2-10 Chapter 2: Linear Equations Section 2: Using Algebra Name Date Activities The Activities help you practice algebraic concepts. You can select from three activities— Solve It!, Beam Dale Up, and Free Fall. Follow these steps to play an activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Solve It! 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: You get two attempts to pick the correct step or steps. You get 2 points for a correct choice on the first try, and 1 point for a correct choice on the second try. You get an additional 2 points for the correct solution. 2. Look at the algebraic expression at the top of the screen and decide what step to take to solve the equation for x. 3. Press # or $ to cycle through steps to choose from, and then press b to select the correct step (some problems require two steps). If your second choice is incorrect, the correct step is displayed; press any key to continue play. As you play the activity, write the equations, the steps required to solve them, and the solutions to the equations. The total number of points available varies. 4. What level did you play? 5. What was your score? Topics in Algebra 1 © 2001 Texas Instruments out of Student Worksheet 2-11 Chapter 2: Linear Equations Section 2: Using Algebra Name Date Activities (continued) Beam Dale Up 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You can earn up to 10 points. 2. Look at the problem on the space ship and determine how to solve the equation for x from the four choices given. 3. Press ! and " to move Dale the Martian over the correct x value, and then press b. If the x value you pick is correct, Dale is beamed up to his ship! If the answer is incorrect on the second attempt, the correct answer is displayed; press any key to resume play. As you play the activity, solve each equation in the space below. Show all steps and work. 4. What level did you play? 5. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-12 Chapter 2: Linear Equations Section 2: Using Algebra Name Date Activities (continued) Free Fall 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. 2. When you are ready to start, press any key. Scoring: Points are based on how quickly you solve each equation. The game automatically ends if four missed equations stack up, or you press áQUITâ to stop. 3. Watch the equation as it falls, and quickly solve for x. Enter the solution (press Ì for negative numbers), and press b before the equation hits bottom. If you give an incorrect answer, the correct answer is displayed; press any key to resume play. The incorrect equation stacks up at the bottom of the screen, giving you less time to solve the next equation. 4. Follow your teacher’s instructions for how long to play the activity. 5. What level did you play? 6. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-13 Chapter 2: Linear Equations Section 2: Using Algebra Name Date ³ Try-It!é on Your TI.83 Plus or TI-73 Can your calculator check your work? Is X=2 the solution to the equation X+3.1=5.5? To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : Display (TI.83 Plus shown) From the Home screen, the calculator can tell you if a sentence is TRUE or FALSE (1=TRUE and 0=FALSE). 2. First, check for the value currently stored in X. „ Í Note: On the TI-73, use I rather than „. 3. What is the value of X stored in your calculator? Check around the class. Most likely, there are many different values stored to X. 4. Next, find out: Is X+3.1=5.5 true or false when X=2? When you tell the calculator that X=2, this is called storing a value in X. 2X„ Í Note: On the TI-73, use I rather than „. 5. Enter the expression X+3.1=5.5. This takes three steps. „\3`1 a. Enter the first part of the expression: X+3.1. Note: On the TI-73, use I rather than „. b. Enter the equal sign (=) for the test. - | $ until = is highlighted Í c. Now, complete the sentence and see the result. Since the result is 0 (1=TRUE and 0=FALSE), the sentence X+3.1=5.5 is false when X=2. 5`5 Í 6. Test another point. Try X=3. Is the sentence true or false? What value do you think makes the sentence true? Hint: You do not have to type in the expressions again. • On the TI-73, press $ to highlight previous entries, and press b to paste an entry on the current line. You can edit the line and use it again. • On the TI-83 Plus: y [ displays previous entries. When you get to the one you want, you can edit the line and use it again. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-14 Chapter 2: Linear Equations Section 2: Using Algebra Name Date ³ Try-It!é on Your TI.83 Plus or TI-73 (continued) Additional Problems—Solve the following equations by hand. Show all of your work. Then test your solution using the calculator as shown in the ³ Try-It! example on the previous page. 1. x − (M4) = 12 2. 3 1 5 WN =M 4 4 8 3. 5.25 + P = M3.5 Hint: To find = and other relations: • TI-83 Plus: Press y :, and then select the relation you want. • TI-73: Press - t to enter the text editor, select the relation you want, press b, select Done, and press b. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 2-15 Chapter 2: Linear Equations Section 2: Using Algebra Linear Equations: Using Algebra Teacher Notes Objectives • To review one-step and two-step linear equations. • To review the idea of isolating the variable, balancing equations and checking solutions. • To review the properties of equality. Math Highlights This section opens with an explanation of the idea of balancing equations using a pan balance. The properties of equality are displayed. Then, examples of solving linear equations of the forms x + a = b, ax = b, and ax + b = c are shown. These examples start with word problems and are then modeled with linear equations, solved, and the solution is checked. Common Student Errors • Students may have a hard time deciding which steps to follow to solve an equation. In particular, if they are given an equation in the form x + a = b, they may choose the wrong step to take if the equation is given as a + x = b. Students might make sign errors as they add or subtract from both sides of the equation. • Students should connect the idea of a zero model from working with integers; for example, they should connect 7 + M7 = 0 with the concept of an additive inverse. They use the additive inverse to create the zero model. The term zero model is not discussed. • Students should notice that they are using the multiplicative inverse to isolate the variable when they multiply or divide both sides of equations. • Although this section deals with the mechanical way of finding the solution set, students should be reminded that they should check to see if the solution is reasonable. They need to keep using number sense. • Many students are able to see the answer using number sense without the written work. Learning how to write mathematics correctly is part of the communication skill and needs to be encouraged. This can cause frustration for students who find the problems easy to solve “in their heads.” Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 2-16 Chapter 2: Linear Equations Section 2: Using Algebra Observations The Observations help students understand algebraic concepts relating to linear equations. If necessary, tell students how to find the Observations. Observation 1 x−5 = 8 Solve for x. Students show all their work. Students solve for x. The answer is displayed on this screen. Observation 2 3 2x=9 Solve for x. Students show all their work. Students solve for x. The answer is displayed on this screen. Observation 3 4−x = 10 Solve for x. Students show all their work. Students solve for x. The answer is displayed on these screens. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 2-17 Chapter 2: Linear Equations Section 2: Using Algebra Activities Solve It! Tell students to: 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: Students get two attempts to pick the correct step or steps. They get 2 points for a correct choice on the first try, and 1 point for a correct choice on the second try. They get an additional 2 points for the correct solution. The total number of points available varies. 2. Look at the algebraic expression at the top of the screen and decide what must be done to solve the equation for x. Students must select from the choices offered; this activity presents only one sequence of steps (to first isolate x and then change the coefficient of x to 1), although other sequences may be correct. 3. Press # or $ to cycle through steps to choose from, and then press b to select the correct step (some problems require two steps). If the second choice is incorrect, the correct step is displayed. They must press a key to continue play. As they play the activity, students should write the equations, the steps required to solve them, and the solutions. 4. Record the level they played. 5. Record their scores. Beam Dale Up Tell students to: 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. Students can earn up to 10 points. 2. Look at the problem on the space ship and determine how to solve the equation for x from the four choices given. 3. Press ! and " to move Dale the Martian over the correct x value, and then press b. If the x value they pick is correct, Dale is beamed up to his ship! As they play the activity, students record the solution to each equation, showing all steps and work. If the answer is incorrect on the second attempt, the correct answer is displayed; press any key to resume play. 4. Record the level they played. 5. Record their scores. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 2-18 Chapter 2: Linear Equations Section 2: Using Algebra Activities (continued) Free Fall Tell students to: 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: Points are based on how quickly students solve each equation. Unless you specify point or time limits for this activity, students can play the activity when four missed equations stack up, or they press áQUITâ to stop. There is no time limit. 2. When they are ready to start, press any key. 3. Watch the equation as it falls, and quickly solve for x. Enter the solution (press Ì for negative numbers), and press b before the equation hits bottom. If students give an incorrect answer, the correct answer is displayed; press any key to resume play. The incorrect equation stacks up at the bottom of the screen, giving them less time to solve the next equation. 4. Follow your instructions. For example, students can play: • Until they have answered incorrectly four times (no time limit). • Until a certain amount of time has expired (highest score with the fewest misses wins). • Until a certain score has been reached (first student to reach the score with the fewest misses wins). • Repeatedly over a period of time (days, weeks, etc.) for tracking improvement of high scores. 5. Record the level they played. 6. Record their scores. ³ Try-It!é on Your TI.83 Plus or TI.73 Tell students to: • Solve the following equations by hand and show all of their work. • Test their solutions using the calculator as shown in the ³ Try-It! example on the previous page. 1. x N (M4) = 12 when x=8 2. 1 5 3 WN =M 4 8 4 when W=M 3. 5.25 + P = M3.5 when P = M8.75 Topics in Algebra 1 1 2 © 2001 Texas Instruments Teacher Notes 2-19 Chapter 3: Linear Functions Section 1: Slope with Grid Name Date Linear Functions: Slope with Grid Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate the slope of a straight line. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 What do you notice about the slope of straight lines? Write your thoughts. Activities The Activities help you practice determining the slope of a line using a grid. You can select from two activities—Screen Cross or Linked Calculators Screen Cross. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-1 Chapter 3: Linear Functions Section 1: Slope with Grid Name Date Activities (continued) Screen Cross 1. Read the directions on the screen, and then press any key to continue. Note: Repeat steps 2 and 3 for line segments 2 through 5. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. 2. Calculate the slope between the two points. Record your work below. 3. Press $ or # to view the choices for the slope (UNDEF = undefined slope). When you think the correct slope is displayed, press b. You can earn up to 10 points. segment 1: segment 2: segment 3: segment 4: segment 5: 4. What was your score? Linked Calculators Screen Cross Play Screen Cross against another student. Race another student across the screen by calculating the slopes more quickly than your opponent. Scoring: There are no points awarded in the linked version. You must answer correctly before the next line segment is shown. The player who reaches the right side of the grid first wins. Topics in Algebra 1 1. Connect two similar calculators using a unit-to-unit cable. 2. Use the grid to help determine the slope of the line segment as quickly as possible. 3. Select the slope as you did in Screen Cross. 4. Who won? © 2001 Texas Instruments Student Worksheet 3-2 Chapter 3: Linear Functions Section 1: Slope with Grid Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 Draw a line segment with endpoints at (M2,2) and (3,M2). Use the Line( command, which draws line segments on the graph screen. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Check to see if any functions or statistical plots are turned on. o 3. Turn off statistical plots or functions, if necessary. !, ", #, or $ b to deselect Display (TI.83 Plus shown) Note: See ³ TIpé 8: Creating a Statistical Plot for more information. 4. Display the window format screen. -g 5. Select GridOn. TI-83 Plus: ##"b TI-73: #"b 6. Display the ZOOM menu. These settings set the viewing window automatically. ( Note: For more information, see ³ TIp 5: Adjusting the Viewing Window. 7. Select ZDecimal. TI-83 Plus: Note: This also displays the graph screen automatically. 4:ZDecimal TI-73: 8:ZDecimal Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-3 Chapter 3: Linear Functions Section 1: Slope with Grid Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) To Do This Press 8. Select the DRAW menu. TI-83 Plus: -< Display (TI.83 Plus shown) TI-73: 2 9. Select the Line( command, which draws line segments on the graph screen. 2:Line( 10. Move the cursor to (M2,2). !, ", #, or $ 11. Set the endpoint (M2,2). b 12. Move the cursor to (3,M2). !, ", #, or $ 13. Set the other endpoint (3,M2). The line segment is drawn. b 14. Clear your drawing with ClrDraw, if desired. TI-83 Plus: - 2 1:ClrDraw TI-73: 2 1:ClrDraw Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-4 Chapter 3: Linear Functions Section 1: Slope with Grid Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Draw the Slopes! 1. On your calculator use Line to draw segments that have the following slopes (m), and then copy the graph onto the screens below. a. m = 1/2 b. m = 2/4 c. m = 3/2 d. m = M2/3 2. Compare your graphs with others in your class. Do you all get the same graph? If not, why? ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ 3. Compare the four segments drawn. Record your observations. ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-5 Chapter 3: Linear Functions Section 1: Slope with Grid Linear Functions: Slope with Grid Teacher Notes Objectives • To impose a grid on a line to quantify the steepness of a line. • To introduce slope as the ratio of the vertical change divided by the horizontal change. • To illustrate the slope characteristics of lines as positive, zero, negative, or undefined (no slope). • To associate the terms with the appropriate graph of a line: increasing, horizontal, decreasing, or vertical. • To observe that the slope of a straight line is a constant. Math Highlights This section defines slope as the steepness of a line. It begins with a bike riding along a piecewise linear path. Students see the definition of slope as a ratio: Slope = m= vertical change horizontal change = rise run A grid is imposed on the bike path to help illustrate how students can quantify the slope of each line segment. Next, lines with positive, zero, negative, and no slope are illustrated. Common Student Errors • Students may have difficulties counting the spaces between grid points; therefore, they may count the grid points instead of the spaces. For example, students who count grid points may think that the slope shown on the screen below is 7à4 instead of 6à3. Using a Geoboard or grid paper, redraw the line segment to help these students count the distance between the grid points rather than the points themselves. • In the screen below, the rise/run of the fourth segment is calculated as M6à5. For some students, associating “down 6 units, or M6” with the word rise can be confusing. The use of the terms vertical change or horizontal change helps students understand that change could be up or down, right or left. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-6 Chapter 3: Linear Functions Section 1: Slope with Grid Common Student Errors (continued) • Students can calculate the slope for the fourth segment in four ways. With the animation to show direction, students usually will begin counting with the upper left-hand point on the segment. or 5 spaces to the right (+5) and 6 spaces down (M6) to get M6à5 6 spaces down (M6) and 5 spaces to the right (+5) to get M6à5 Without the animation to show direction, students may be as likely to begin counting from the lower right-hand point on the segment. or 6 spaces up (+6) and 5 points to the left (M5) to get 6àM5 Slope = m= 6 M5 = M6 5 =M 5 points to the left (M5) and 6 spaces up (+6) to get 6àM5 6 5 All four ways of counting are correct, resulting in the same slope. The Observations reinforce this, stating, “The slope between any two points on a line is always the same ratio.” Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out the new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-7 Chapter 3: Linear Functions Section 1: Slope with Grid Observations The Observations help students understand the slope of a straight line. If necessary, tell students how to find the Observations for this section. Observation 1 Students write what they observe about the slope of straight lines. Answers may vary. Students observe that a line has constant slope. Activities Screen Cross Tell students to: 1. Read the directions on the screen, and then press any key to continue. Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. Students can earn up to 10 points. Note: Repeat steps 2 and 3 for line segments 2 through 5. 2. Calculate the slope between the two endpoints. Record their work. 3. Press $ or # to view the choices for the slope (UNDEF = undefined slope). When they think the correct slope is displayed, press b. 4. Record the endpoints and slopes for each segment and their final score. The segments are different on each calculator and each time the game is played. Linked Calculators Screen Cross Tell students to: 1. Connect two similar calculators using a unit-to-unit cable. Scoring: There are no points awarded in the linked version. Students must answer correctly before the next line segment is shown. The player who reaches the right side of the grid first wins. Topics in Algebra 1 2. Use the grid to help determine as quickly as possible the slope of the line segment. 3. Select the slope as they did in Screen Cross. 4. Record which player won. © 2001 Texas Instruments Teacher Notes 3-8 Chapter 3: Linear Functions Section 1: Slope with Grid ³ Try-It!é on Your TI.83 Plus or TI.73 Students follow the example, which covers drawing a line on the graph screen from specified endpoints using the Draw menu. Draw the Slopes! 1. These are sample answers only. Segment location varies depending on the starting point used. a. and b. c. and d. 2. Students should notice that their segments are not necessarily the same as others depending on the starting point used. Later, they will see that the slope and a point determine a line. 3. Students should notice that 1/2 and 2/4 have the same steepness. These segments are parallel (or on the same line). They should also observe that segments with slopes, M2/3 and 3/2, are perpendicular line segments. See graphs above. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-9 Chapter 3: Linear Functions Section 2: Slope Using Coordinates Name Date Linear Functions: Slope Using Coordinates Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate mathematical concepts relating to finding the slope using graph coordinates. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Is every set of three coordinate points collinear? Write your thoughts below. Which is the correct way to calculate the given slope? Write your thoughts below. Activities The Activities section helps you practice finding a slope using graph coordinates. You can select from two activities—Screen Cross or Linked Calculators Screen Cross. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-10 Chapter 3: Linear Functions Section 2: Slope Using Coordinates Name Date Activities (continued) Screen Cross This activity is similar to the Screen Cross in the Slope with Grid section. There is no grid displayed, but the endpoints are given. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You can earn up to 10 points. 1. Read the directions on the screen, and then press any key to continue. Note: Repeat steps 2 and 3 for line segments 2 through 5. 2. Calculate the slope between the two points. Record your work below. 3. Press $ or # to view the choices for the slope (UNDEF = undefined slope). When you think the correct slope is displayed, press b. point 1 (x 1,y1) point 2 (x 2,y 2) slope segment #1 segment #2 segment #3 segment #4 segment #5 4. What was your score? Linked Calculators Screen Cross Play Screen Cross against another student. Race another student across the screen by calculating the slopes more quickly than your opponent. Scoring: There are no points awarded in the linked version. You must answer correctly before the next line segment is shown. The player who reaches the right side of the grid first wins. 1. Connect two similar calculators using a unit-to-unit cable. 2. Use the grid to help determine as quickly as possible the slope of the line segment. 3. Select the slope as you did in Screen Cross. 4. The player who reaches the right side of the grid first wins. 5. Who won? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-11 Chapter 3: Linear Functions Section 2: Slope Using Coordinates Name Date ³ Try-It!é on Your TI.83 Plus or TI-73 Use the calculator to find the slope of a line through the points (2,6) and (7,4). The slope is: m = 6N4 2N7 To Do This on the TI.83 Plus Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Calculate the slope. £6¹4¤¥ £2¹7¤ b 3. Change the fraction to a decimal. Use the 4Frac function on the MATH menu. 1:4Frac b To Do This on the TI.73 Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Calculate the slope. £6¹4¤¥ £2¹7¤ b 3. >b Change the decimal to a fraction. 4. Simplify the fraction. Note: The mode setting ManSimp must be selected first. For more information, see ³ TIp 2: Adjusting Your Calculator Settings. 5. Use = to enter the problem. Topics in Algebra 1 Display (TI.83 Plus shown) Display (TI.73 shown) .#### " to ManSimp b Bb =6T4 =2T7b © 2001 Texas Instruments Student Worksheet 3-12 Chapter 3: Linear Functions Section 2: Slope Using Coordinates Name Date ³ Try-It!é on Your TI.83 Plus or TI-73 (continued) Additional Problems 1. Calculate the same problem, m = 6N4 2N7 , on the calculator from the Home screen, using the following keystrokes: 6T4F2T7b Did you get the same answer? If not, why not? Which answer is correct? 2. Find the slope of the line containing the given points using the calculator. Who can do these problems faster, the calculator or you? You can also pair with another student and challenge each other to see who is faster at the calculations. Write your calculations and answers. a. (3,M4) and (M2,8) b. (M12,17) and (5,M6) c. (36,6) and (M25,6) d. (1.36,2.54) and (1.36,5.72) Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-13 Chapter 3: Linear Functions Section 2: Slope Using Coordinates Linear Functions: Slope Using Coordinates Teacher Notes Objectives • To introduce slope as a ratio as the change in y to the change in x. • To find the slope of a line using the slope formula. • To illustrate the characteristics of lines with positive, zero, negative, or undefined slope. • To associate increasing, horizontal, decreasing, or vertical with the appropriate slope. Math Highlights This section again highlights a review of slope as the steepness of a line. The section opens with a bike riding along a piecewise linear path. Students see the definition of slope as a ratio. Slope = m = rise run = change in y change in x y2 N y1 =x Nx 2 1 Several examples of the calculations are shown. Next, the characteristics of lines with positive, zero, negative and undefined slopes are summarized. Common Student Errors • Many students are concerned that they have to know which point is (x1, y1) and which point is (x2,y2). Show that both calculations give the same answer. In the second example below, some students are confused when they see division with two negative numbers results in a positive slope. Discuss this. For the line containing the points (0,0) and (10,2), the slope is calculated by: 2N0 10 N 0 • = 1 5 or 0N2 0 N 10 = M1 M5 = 1 5 Watch for an incorrect substitution where students switch the order in the numerator or denominator. For the example above, Incorrect: 0N2 10 N 0 or 2N0 0 N 10 • This is a good opportunity to open a discussion on different representations of the same number. Students might write a calculator answer without thinking about whether or not the representation is the best for the problem. Ask when it is most useful to have the slope represented in decimal form versus in fraction form. • Remind students that they need to use the Order of Operations rules. • In the ³ Try-It!é section, the correct calculation is (6 − 4) ÷ (2 − 7), not 6 − 4 ÷ 2 − 7 (where division would be performed before subtraction). Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-14 Chapter 3: Linear Functions Section 2: Slope Using Coordinates Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students understand mathematical concepts relating to finding the slope using graph coordinates. If necessary, tell students how to find the Observations for this section. Observation 1 Is any set of three points collinear? Three points may be collinear, but are not necessarily collinear. Observation 2 Find the slope of the line through (2,3) and (L4,8). Students can pick either point as (x1,y1) and (x2,y2) to compute the slope using the formula. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-15 Chapter 3: Linear Functions Section 2: Slope Using Coordinates Activities Screen Cross Tell students to: 1. Read the directions on the screen, and then press any key to continue. Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. Students can earn up to 10 points. Note: Repeat steps 2 and 3 for line segments 2 through 5. 2. Calculate the slope between the two points. Record their work. 3. Press $ or # to view the choices for the slope (UNDEF = undefined slope). When they think the correct slope is displayed, press b. Record the endpoints and slopes for each segment. The segments are different on each calculator and each time the game is played. 4. Record their final score. Linked Calculators Screen Cross Tell students to: 1. Connect two similar calculators using a unit-to-unit cable. Scoring: There are no points awarded in the linked version. Students must answer correctly before the next line segment is shown. 2. Try to determine as quickly as possible the slope of the line segment. The player who reaches the right side of the screen first wins. 4. The player who reaches the right side of the screen first wins. Topics in Algebra 1 3. Select the slope as they did in Screen Cross. © 2001 Texas Instruments Teacher Notes 3-16 Chapter 3: Linear Functions Section 2: Slope Using Coordinates ³ Try-It!é on Your TI.83 Plus or TI.73 Students investigate how to input the calculation for slope on the Home screen. See the Student Worksheet for instructions. Warn students about the correct use of parentheses. They need to be reminded of the Order of Operations rules. Notice that the TI-73 has stacked fraction capabilities; therefore, the calculation is performed correctly without parentheses. Additional Problems 1. Students input 6−4à2−7 in the calculator. The calculator computes 6 − (4 ÷ 2) − 7 = M3 since division is performed before subtraction according to the Order of Operation rules. This is not the correct answer. Students need to be aware that parentheses, such as (6 − 4) ÷ (2 − 7), override the Order of Operation rules. 2. Students practice calculating the slope using the calculator, but remind students that they can compute these answers faster than the calculator can! Pair students and challenge them to try to beat the calculator. 12 a. (3,M4) and (M2,8) slope = M b. (M12,17) and (5,M6) slope = M c. (36,6) and (M25,6) slope = 0 d. (1.36,2.54) and (1.36,5.72) slope = undefined Topics in Algebra 1 5 23 17 © 2001 Texas Instruments Teacher Notes 3-17 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Name Date Linear Functions: Slope as Rate of Change Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate mathematical concepts relating to finding the slope using graph coordinates. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. What kinds of graphs have a nonconstant rate of change? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-18 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Name Date Activities The Activities section helps you practice finding a slope as rate of change. You can select from two activities—Balloon Speed or Dive. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Balloon Speed 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. 2. Select whether you want to play during Nighttime or Daytime. Scoring: You get two attempts to solve each problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You can earn up to 10 points. 3. Take a trip in the balloon. Record the times and positions in the table. 4. Calculate the rate of change (the slope) of each trip, showing all work, including the correct units. Enter the rate and press Í. If the answer is incorrect, the correct answer is displayed; press any key to resume play. 5. What level did you play? 6. What was your score? Trip Start Time Start Position Stop Time Stop Position Speed (include units) 1 2 3 4 5 Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-19 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Name Date Activities (continued) Dive 1. Take a dive in the submarine. Record the times and positions in the table below. Scoring: You get one attempt to select the correct row. If you select an incorrect row, the correct row is identified. 2. Calculate the rate of change (slope) of each segment in the table below. Show all your work. Include the correct units. 3. After the third segment, the next screen displays three rows of three rates. Use your rate answers from the table to determine the correct row. To select a row, press # or $ to move the submarine cursor, and then press b. 4. Did you select the correct row? Segment Start Time Start Position Stop Time Stop Position Speed (include units) 1 2 3 Car Trip Word Problem A car takes a 20-minute trip. The car starts at home, (0,0). It travels 6 miles in 10 minutes. The car stops in traffic for 5 minutes. The driver forgot something at home so the car returns in 5 minutes. Draw a straight-line graph of the trip. Write the rate of change for each leg of the trip on the graph to the right. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-20 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Name Date ³ Try-It!é on Your TI.83 Plus See the section on Slope Using Coordinates to find out how to use your calculator to find the slope of a line containing two given points. Review the keystrokes. Use these keystrokes to compute the rate of change for each pair of data points given. Include the correct units with your answer. 1. x (seconds) y (feet) x (hour) y (phone calls) 10 6.72 2 18 20 8.05 4 25 2. Rate of Change _______________________ Rate of Change _____________________ 3. A space shuttle has an average velocity of 25,405 ft/sec. The velocity of a car on a highway can be 60 mi/hr. To find out how many times faster the space shuttle travels compared to the car, you must convert the speed of the space shuttle to miles per hour. First, look at the conversion by hand, and then do it on the calculator. When set to Float, the calculator shows the answer to five decimal places. 25,405 ft sec Q 60 sec 1 min Q 60 min 1 hr Q 1 mi 5280 ft = 17321.59091 mi hr To Do This Press 1. If necessary, exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Calculate 25,405 Q 60 Q 60 à 5,280. Display (TI.83 Plus shown) 25405¯60¯60¥ 5280Í 3. How many times faster does the space shuttle travel compared to the car? ____________ Additional Problems Try these by hand, and show your work below. Then check the answers on your calculator. 5280 feet = 1 mile 60 seconds = 1 minute 1000 meters = 1 kilometer 60 minutes = 1 hour 1. 55 miles/hour " feet/second 2. 35 meters/second " kilometers/hour Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-21 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Name Date ³ Try-It!é on Your TI.73 See the section on Slope Using Coordinates to find out how to use your calculator to find the slope of a line containing two given points. Review the keystrokes. Use these keystrokes to compute the rate of change for each pair of data points given. Include the correct units with your answer. 1. x (seconds) y (feet) x (hour) y (phone calls) 10 6.72 2 18 20 8.05 4 25 2. Rate of Change _______________________ Rate of Change _____________________ 3. A space shuttle has an average velocity of 25,405 ft/sec. The velocity of a car on a highway can be 60 mi/hr. To find out how many times faster the space shuttle travels compared to the car, you must convert the speed of the space shuttle to miles per hour. First, look at the conversion by hand, and then do it on the calculator. When set to Float, the calculator shows the answer to five decimal places. 25,405 ft sec Q 60 sec 1 min Q 60 min 1 hr Q 1 mi 5280 ft = 17321.59091 mi hr To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Enter the velocity, 25,405 on the Home Screen. 25405 3. Open the CONVERSIONS menu, and select the category, Speed. 4. Select the units you are converting from, ft/sec. 5. Select the units you are converting to, mi/hr. Display (TI.73 shown) -‚ 7:Speed 1:ft/sec 3:mi/hr b 25,405 ft/s rounded to tenths is 17321.6 mi/hr. 6. How many times faster does the space shuttle travel compared to the car?_______________ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-22 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Name Date ³ Try-It!é on Your TI.73 (continued) Additional Problems Try these by hand, and show your work below. Then check the answers on your calculator. 5280 feet = 1 mile 60 seconds = 1 minute 1000 meters = 1 kilometer 60 minutes = 1 hour 1. 55 miles/hour " feet/second 2. 35 meters/second " kilometers/hour Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-23 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Linear Functions: Slope as Rate of Change Teacher Notes Objectives • To associate the slope of a straight line with a constant rate of change. • To calculate the rate of change from data points on a line, using the correct units. • To read from a linear graph: the rate of change, the scale of the axes, and the correct units. Math Highlights This section highlights the use of the slope formula to find the rate of change from graphs and data. Students may need to review the slope formula; see previous sections on the calculator. y 2 N y1 rate of change = m= x N x units 2 1 Emphasize the use of appropriate units. For example, in the Fill the Pool example in the Overview, the students see a pool filling at a constant rate of change. Two data points from the growth graph or table shown are (4,2) and (6,3). To find the slope, students should calculate: rate of change = m = 3 N 2 ft 6 N 4 hr = 1 ft 2 hr = .5 ft hr Mention to students that if the graph of a real problem were nonlinear, the calculation of the rate of change using two data points gives the average rate of change over the interval chosen. Common Student Errors • Neglecting to write the appropriate units. • Misunderstanding how to use the position with respect to a starting place rather than distance. In the slope formula, y2−y1 gives the signed distance traveled. • Specifying which point is (x1,y1) and which point is (x2,y2). Show students that both calculations give the same answer. Some students are confused when they see a division by two negative numbers that results in a positive growth. Discuss this with the students. Watch for an incorrect substitution where students switch the order in the numerator or denominator. For example, if the data points (0,0) and (10,2) give the growth of a plant in cm per day, the rate of change is calculated: Correct: Incorrect: • 2 N 0 cm 10 N 0 day = 1 cm 5 day or 0 N 2 cm M1 cm = M5 0 N 10 day day 0 N 2 cm 10 N 0 day Students need to be exposed to other variables besides x and y. For example, it is useful to use t for time and d for distance. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-24 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students understand mathematical concepts relating to finding the slope using graph coordinates. If necessary, tell students how to find the Observations for this section. Observation 1 What kinds of graphs have a nonconstant rate of change? Students write their answer on their worksheet. Nonlinear graphs have a nonconstant rate of change. Students see the answer on the screen at the right. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-25 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Activities Balloon Speed Tell students to: 1. Highlight a level (bronze = least difficult; gold = most difficult), and press b to select it. Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. Students can earn up to 10 points. 2. Select whether they want to play during Nighttime or Daytime. 3. Take a trip in the balloon. Record the times and positions in the table. 4. Calculate the rate of change (the slope) of each trip, showing all work, including the correct units. Enter the rate and press Í. If the answer is incorrect, the correct answer is displayed; press any key to resume play. 5. Record the level they played. 6. Record their score. Dive Tell students to: 1. Take a dive in the submarine. Record the times and positions in the table below. Scoring: You get one attempt to select the correct row. If you select an incorrect row, the correct row is identified. 2. Calculate the rate of change (slope) of each segment in the table on the worksheet, showing all their work, including the correct units. 3. After the third segment, the next screen displays three rows of three rates. Students use their rate answers from the table to determine the correct row. To select a row, press # or $ to move the submarine cursor, and then press b. 4. Record whether they selected the correct row. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-26 Chapter 3: Linear Functions Section 3: Slope as Rate of Change Activities (continued) Car Trip Word Problem Distance Rate (0,0) to (10,6) .6 mi/min (10,6) to (15,6) 0 mi/min (15,6) to (20,0) M1.2 mi/min Notice that the rate is positive going away from home. The rate is negative on the return part of the trip indicating the direction of going back home. ³ Try-It!é on Your TI.83 Plus or TI.73 Note: There is a ³ Try-It for the TI-83 Plus and one for the TI-73; they are on separate pages. Students review using the calculator to compute the slope of a line containing two points. They also see how to use the calculator to do conversions. The TI-73 has conversion functionality whereas users of the TI-83 Plus have to know how to set up the conversion. See the Student Worksheet for instructions. 1. 0.133 ft per sec 2. 3.5 phone calls per hour Have students discuss the meaning of the rates and their choice of representation. For example, is 3.5 phone calls per hour the most meaningful way of communicating the rate of change or is 7 phone calls every two hours a better way of describing the rate? Have students justify their reasoning. 3. (TI-83 Plus) or 6. (TI-73) The space shuttle travels 288.7 times faster than a car (17321.59091 màhr P 60 miàhr). Calculator Problems Tell students to follow the steps exactly on the calculators. Example screens are displayed on the worksheets for students to compare with the calculator screens. Additional Problems Students work problems by hand and check on calculator. 2 ft 1. 80 2. 126 3 sec = 80.6 ft sec km hr Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-27 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Name Date Linear Functions: Slope-Intercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate mathematical concepts relating to writing slope in slope-intercept form. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation Use your algebraic knowledge to change 3x + 4y = 8 to the form y = mx + b. Show your work. Remember: 3x + 4y = 8 is Standard Form. Activities The Activities section helps you practice using the slope-intercept form of lines. You can select from two activities—Match It! or Line Soccer. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-28 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Name Date Activities (continued) Match It! 1. Select the correct answer to the question. Questions include: Scoring: You get two attempts to answer the problem. You earn 2 points for a correct answer, 1 point for a correct answer on the second try. • Selecting the graph that correctly illustrates an equation. • Selecting the equation that correctly describes a graph. • Selecting the equation that goes with the table. 2. What was your score? You can earn up to 12 points. Line Soccer 1. Pass the ball by answering a question correctly. Possible slope-intercept questions include: Scoring: You get two attempts to answer the problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You score a goal when you answer a question correctly against Alex, the goalkeeper. You also get to keep any points you earned while passing the ball. • Selecting the (x,y) coordinate that solves an equation. • Using an (x,y) coordinate and b to solve for m in an equation. • Using an (x,y) coordinate and m to solve for b. • Using two (x,y) coordinates to calculate m and b. 2. To enter your answer: • As a negative number, press Ì. • As a mixed number, enter the whole number and press áUNITâ, and then enter the fraction. • As a fraction, press án/dâ and enter the numerator. Press án/dâ again and enter the denominator. 3. Show your work for each problem: 4. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-29 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Name Date Activities (continued) Additional Problems 1. Make a table and draw the graph for the following linear functions. a. Y = MX + 2 X Y b. Y = 2X N 1 X Y 2. Think of the graph below as a map. You start to walk from the point (M1,1). You can walk only along a path that is the line with the slope: m = M2 a. Label two points, A and B, that you can walk to on the graph if the slope of your path =M2. Each tick mark is 1 unit. b. Find the equation of the line that describes your walking path. Show your work below. c. Record the coordinate points of A and B below. Use the equation of your path to check to see if your points A and B are on really on your line! Show your work below. Point Check It A= B= Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-30 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 Parent and Family of Functions Compare graphs of Y1=X and Y2=X+1. If you know the graph of Y1=X, can you draw the graph of any line in slope-intercept form? Investigate using your calculator! Note: See ³ TIpé 3: Graphing a Function and ³ TIpé 5: Adjusting the Viewing Window for instructions if you need help. Set up your calculator so that GridOn on the window format (y .) screen and ZDecimal on the zoom menu (q) are selected. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Define Y1=X and Y2=X+1. o: „ #: „\1 Note: On the TI-73, use I rather than „. 3. Set the viewing window. Display (TI.83 Plus shown) TI-83 Plus: ( 4:ZDecimal TI-73: ( 8:ZDecimal 4. Trace both Y1 and Y2. r | or ~ to trace a function } and † to move between functions 5. Write a description of what you see. How does the graph of Y=X compare to the graph of Y=X+1? ________________________________________________________________________________ ________________________________________________________________________________ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-31 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) To Do This Press 6. Enter and trace Y3=XN1. & ##: „T1 r | or ~ to trace a function } and † to move between functions Note: On the TI-73, use I rather than „. Display (TI.83 Plus shown) 7. Write a description of what you see. How does the graph of Y=X compare to the graph of Y=XN1? ________________________________________________________________________________ 8. Where will the line, Y=X+2 be located? Draw in your prediction and then use the calculator to check your work. Was your graph correct? If not, why not? 9. Now predict and draw the following lines on the graph to your right. Draw your own axes. Do not use the calculator. Y=X+3 Y=XN3 Y=X+8 Y=XN8 10. Check your work! Graph each line on the calculator. Could you see all of the lines? Record your observations. Note: On the TI-73, you need to clear a line in the Y= editor to make room for your new lines. ________________________________________________________________________________ 11. You cannot see Y=X+8 or Y=X.8 on your graphing calculator. Why do you think that the line is not showing? Change your WINDOW (') settings so that you are able to see these lines. Record the changes you made. ________________________________________________________________________________ 12. Compare the lines: Y=X, Y=2X, and Y=3X. What do you notice? Try other lines in the form Y = AX. Next, try to graph Y=2X+3 without the calculator. Explain your strategy. ________________________________________________________________________________ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 3-32 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Linear Functions: Slope-Intercept Form Teacher Notes Objectives • To review the form of the equation y = mx + b, where m is slope and b is the y-intercept. • To view graphing lines and developing equations from data. • To view tables of values and graphing lines from an equation of the form y = mx + b. • To emphasize three ways of looking at lines, tables, graphs and equations. • To develop the equation of a line from information given, such as slope and y-intercept, point and slope, and two points. • To review the slope of parallel and perpendicular lines. Math Highlights This section highlights a review of the slope-intercept form of a line, y = mx + b. Students will review graphing a line using one point and the slope of the line. They will also see the equation of a line developed inductively from data by simulating the amount of volts produced by lining up batteries in series. Students will also review how to find a table and graph a line using the equation y = mx + b. A review of calculations follow, which show the step-by-step procedures needed to find the equation of a line given, the slope and y-intercept, the slope and a point on the line, and finally two points. Common Student Errors • When students perform the calculations to find the equation of a line given the slope and y-intercept, the slope and a point on the line, or two points, they often have problems following the steps. For example, given two points, the student needs to perform three steps. First, the student needs to calculate the slope of the line. Students may forget the formula for slope, and then be careless about substituting the correct values in the formula. Also, they may make errors in sign. In the second step, they need to find b using the calculated slope and one of the points. Students need to understand that they could pick either point for this calculation. If the slope is incorrect from the first step, the error cascades into the rest of the solution. In the third step, they need to place all of the information into the final answer as y = mx + b. • Students become comfortable using the variables x and y. They should be exposed to the use of other letters for the independent and dependent variables. Students should be able to recognize, for example, D = 2T + 4 as a linear relationship with independent variable T, dependent variable D, a slope of 2 and the D-intercept as 4. They need to learn that they will need to change the variables so the function reads y = 2x + 4 in order to graph it on the calculator. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-33 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students understand using algebra to change a linear equation in standard form to slope-intercept form. If necessary, tell students how to find the Observations for this section. Observation Students use algebra to write 3x+4y=8 in y=mx+b form. Answers may vary, especially the step order. One possible answer: Step 1: Subtract 3x from both sides of the equation. 3x - 3x + 4y = 8 - 3x Step 2: Simplify the equation. 4y = M3x + 8 Step 3: Divide both sides of the equation by 4. 8 4y M3 4 = 4 x+4 Step 4: Simplify to get the answer. M3 y= 4 x+2 Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-34 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Activities Match It! Tell students to: 1. Select the correct answer to the question. Questions include: Scoring: Students get two attempts to answer the problem. They earn 2 points for a correct answer, 1 point for a correct answer on the second try. 2. • Selecting the graph that correctly illustrates an equation. • Selecting the equation that correctly describes a graph. • Selecting the equation that goes with the table. Record their scores. Students can earn up to 12 points. Line Soccer Tell students to: 1. Pass the ball by answering a question correctly. Possible slope-intercept questions include: Scoring: Students get two attempts to answer the problem. They earn 2 points for a correct answer, 1 point for a correct answer on the second try. Students score a goal when they answer a question correctly against Alex, the goalkeeper. They also get to keep any points they earned while passing the ball. Note: Unsimplified fractions are counted as correct. • Selecting the (x,y) coordinate that solves an equation. • Using an (x,y) coordinate and b to solve for m in an equation. • Using an (x,y) coordinate and m to solve for b. • Using two (x,y) coordinates to calculate m and b. 2. Enter their answers in the following ways, as necessary: • as a negative number, press Ì. • as a mixed number, enter the whole number and press áUNITâ, and then enter the fraction. • as a fraction, press án/dâ and enter the numerator. Press án/dâ again and enter the denominator. 3. Show their work for each problem. 4. Record their scores. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-35 Chapter 3: Linear Functions Section 4: Slope-Intercept Form Activities (continued) Additional Problems 1. Students make a table and draw the graph for Y=MX+2 and Y=2XN1. Answers may vary. Make sure that students correctly label the axes and all points from the corresponding table. 2. Students label two points, A and B, where the line segment between the two has a slope that is M2. a. Possible points students can walk to on the screen shown are (M2,3), (0,M1), and (1,M3). b. Students should use m=M2 and the point; for example, (M1,1) to find the y-intercept, b. The answer is y=M2xN1. c. Students should substitute their point in the line to see if the equation is satisfied. ³ Try-It!é on Your TI.83 Plus or TI.73 Students need to know the features of the calculator covered in ³ TIpé 3: Graphing a Function and ³ TIpé 5: Adjusting the Viewing Window. They work through an investigation of parent and family of functions of Y=X. See Student Worksheet for details. Tell students to follow the steps exactly on the calculators. Example screens are displayed on the worksheets for students to compare with the calculator screens. This keystroke exercise lets students discover the connection between the parent and family of functions so they can quickly graph functions of the form y=mx+b. 5. Possible answers for comparing the graph of Y=X to Y=X+1: parallel, equivalent slope, offset by 1, Y=X crosses the origin (0,0), etc. 7. Possible answers for comparing the graph of Y=X to Y=XN1: same as above. 8. Students should draw the function on the worksheet first before graphing it on the calculator. Y=X+2 intersects the X-axis at (M2,0) and the Y-axis at (0,2). 10. Students should draw the function on the worksheet first before graphing it on the calculator. If they graph it without changing the window values, the screen looks like this: Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-36 Chapter 3: Linear Functions Section 4: Slope-Intercept Form ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Parent and Family of Functions 11. Students cannot see all functions when they don’t change their window values, which are: Xmin=M4.7, Xmax=4.7, Xmin=M3.1, Xmax=3.1. To fit all functions on the screen, Xmin>8 and Xmax>8. Functions in order from top to bottom: Y=X+8, Y=X+3, Y=XN3, Y=XN8. 12. In order from top to bottom: Y=3X, Y=2X, Y=X (ZFit window values). Graph of Y=2X+3 (ZStandard window values). To graph the function Y=2X+3, students should think that the graph is parallel to Y=2X, and then shift Y=2X up 3 units on the Y-axis. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 3-37 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Name Date Linear Inequalities: Using Graphs & Tables Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate mathematical concepts relating to inequalities. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 If x is a real number, draw the number line graph of each inequality. Remember to use open and closed notation and to label each number line correctly. 1. x<3 ←→ 2. x≤3 ←→ 3. x>6 ←→ 4. x≥6 ←→ Draw the number line graphs for these inequalities (which are not in the Topics in Algebra 1 application). 5. 1 x>M2 ←→ 6. x 1.25 ←→ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-1 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Name Date Activities The Activities help you practice using graphs and tables to solve inequalities. You can select from three different activities—Build the Solution Set!, Worksheet Activity 1, and Worksheet Activity 2. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Build the Solution Set! 1. Use ! or " to move the cursor along the number line. Press b to select the first point to test. The point is displayed along with the result of the test. Scoring: There are five problems in each set. You get two attempts to solve for x for each problem. You get 2 points for a correct choice on the first try, and 1 point for a correct choice on the second try. You can earn up to 10 points. 2. Choose another point to test. After the point and result of the test for the second point are displayed, four possible solutions for x are displayed. 3. Press ! or " to highlight the correct solution, and then press b to select it. If you choose an incorrect solution on the first try, you get another try. If you choose an incorrect solution on the second try, the correct answer is displayed; press any key to go to the next problem. As you play the activity, write the inequalities and their solutions. 4. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-2 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Name Date Activities (continued) Worksheet Activity 1 Note: Press | or ~ to leave this screen. Use the number lines below to estimate the solution set of the following inequalities. Use number sense to place a scale on each number line so that you are able to show the solution set. Check several points as shown in the previous activity. Show all of your work. 1. x + 2.5 < 7 ←→ 2. x N 3.6 M2 ←→ 3. 3 1 x + 4 > 48 ←→ Worksheet Activity 2 Use the table of the expression 2.5X−1 at the left to find out when 2.5X−1<3. First estimate the answer, and then use number sense to determine the exact answer. Write your strategy for the solution, and draw the solution set on a number line. 2.5X−1<3 ←→ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-3 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 Solution search using X-Y graphs: Find the solution set for the inequality X−1<1. Before you start, estimate the solution using number sense. Write your estimate here. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. Set your window format as shown. TI-83: -g ##"b Note: See ³ TIpé 5: Adjusting the Viewing Window for more information. Display (TI.83 Plus shown) TI-73: -g #"b 3. Enter both sides of the inequality into the Y= editor as shown. Note: On the TI-73, use I rather than „. Note: You may need to deselect the other Y= functions. See ³ TIpé 3: Graphing a Function in the Standard Window. 4. Select ZDecimal to set the viewing window and graph the functions. &: „T1 #: 1 TI-83: ( 4:ZDecimal TI-73: ( 8:ZDecimal 5. Trace the functions. Note: The function displays in the upper left corner of the screen; the X and Y values are displayed on the bottom of the screen. r | or ~ to trace a function } and † to move between functions 6. Compare Y1 and Y2 for the same X values. For example, notice that when X = M1, Y1= M2 and Y2=1 as shown on the screens. Where is X −1 < 1? Hint: Trace Y1 to find out when Y1< Y2. 7. Why? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-4 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Additional Problems Solution Search: On the following problems, first estimate the solution using number sense. Then search for the solution set for each inequality using graphs as shown in the previous example. Finally, draw your graphs from your calculator and show your work. Don’t forget to set an appropriate viewing window in order to see your graphs. Note: See ³ TIpé 5: Adjusting the Viewing Window for more information. 1. 2X+3 7 2. X+4 > M3 3. 0.5XN1 < 2 4. XN2 ‚ M3 Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-5 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Linear Inequalities: Using Graphs & Tables Teacher Notes Objectives • To illustrate how to estimate the solution set, in the real numbers, of a linear inequality using graphical methods on a number line and on a Cartesian (x-y) graph. • To illustrate how to estimate the solution set, in the real numbers, of a linear inequality using tables. Math Highlights In the number line method, students use a guess-and-test approach to search for the solution on a number line. A point is chosen to test, the substitution is shown and students see whether the statement is true or false. A point is plotted for a true statement. Students see an estimate of the solution set built on the number line. In the table of values method, students see a table of values for each side of the inequality. They see where the inequality is satisfied, and then they see how to refine the estimate of the solution set. In the x-y graphical method, students plot both sides of the inequality and use the graph to determine the solution set for the inequality by testing points. In the 2-D graph, students are able to see which graph is higher or lower than the other graph. This helps them estimate the solution set. Note: The inequalities of the form ax + b < c (for <, ≤, >, ≥) with a < 0 are not discussed in this section. Common Student Errors • Using graphs and tables can mislead students. They may think that they can always find the exact solution using graphs and table. Although they will often find exact solutions using these methods, using algebra will always give exact solutions for inequalities. For the calculator example, x + 2 1, only integer values are tested. If the example had been x + 2 < 1, the students would need to test points closer and closer to x = M1 to see that the solution set contains values strictly less than M1. The endpoint (M1) is not included in the solution. Encourage students to pick many points. Remind them that they would have to test all points with these methods to get the exact solution, in the real numbers and that is physically impossible to test all points! • At times, introducing the algebraic solution of inequalities gives students just the mechanics of doing a problem. Algebraic methods alone usually do not invite students to reason out the solution using number sense. The graphs and tables method gives students the opportunity to see the values of each side of the inequality as a graph or table so they can compare the size of the numbers, thus helping them create the solution set. • Visual learners can benefit by seeing the graphs and numbers and using them as the tool to find the solution set. • Some students may still have difficulty remembering the meaning of the symbols, <, ≤, >, and ≥. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 4-6 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students understand concepts about linear inequalities relating to graphs and tables. If necessary, tell students how to find the Observations. Observation 1 1. x<3 2. x3 3. x > 6 4. x ≥ 6 1 5. x > M 2 (Not in the Topics in Algebra 1 application.) 6. x 1.25 (Not in the Topics in Algebra 1 application.) Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 4-7 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Activities Build the Solution Set! Tell students to: Scoring: There are five problems in each set. Students get two attempts to solve for x for each problem. You get 2 points for a correct choice on the first try, and 1 point for a correct choice on the second try. Students can earn up to 10 points. 1. Use ! or " to move the cursor along the number line. Press b to select the first point to test. The point is displayed along with the result of the test. 2. Choose another point to test. After the point and result of the test for the second point are displayed, four possible solutions for x are displayed. 3. Press ! or " to highlight the correct solution, and then press b to select it. If students choose an incorrect solution on the first try, they get another try. If they choose an incorrect solution on the second try, the correct answer is displayed. Students can press any key to go to the next problem. As they play the activity, they should write the inequalities and their solutions. 4. Record their scores. Worksheet Activity 1 Tell students to: 1. Use number sense to estimate the solution set for the following three inequalities on the number lines provided. Note: Students can press | or ~ to leave this screen. 2. Place the appropriate scale on each number line so that they are able to show the solution set appropriately. 3. Check several points as shown in the activity above. 4. Show all of their work. Topics in Algebra 1 1. x + 2.5 < 7 when x < 4.5 2. x N 3.6 M2 when x 1.6 3. 3 1 x + 4 > 48 when 3 x > 38 © 2001 Texas Instruments Teacher Notes 4-8 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables Activities (continued) Worksheet Activity 2 Tell students to: Use the table of the expression 2.5X−1 shown on their worksheets (shown at left) to find out when 2.5X−1<3 by first estimating the answer, and then using number sense to determine the exact answer. Remind them to write their strategy for the solution and to draw the solution set on a number line. From the table, students should see that they need to test more values between X=1 and X=2 to find where 2.5XN1<3. If you wish the students to continue the activity on their calculators, tell them to enter the function in the Y= editor. Then on the TABLE SETUP screen (- f), they can refine their search as shown in the screens below. The students see that 2.5XN1 is equal to 3 at X=1.6, but they need to figure out when 2.5XN1< 3. Students can go to the TABLE SETUP screen and change the settings to TblStart=1 and ∆Tbl=.1, and then investigate further. This problem should be discussed to show that the search should still continue because the endpoint of 1.6 is not in the solution set. The answer is X<1.6. To investigate further, refine the table values by changing TblStart=1.5, and ∆Tbl=.01. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 4-9 Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables ³ Try-It!é on Your TI.83 Plus or TI.73 Solution Search Using X-Y Graphs: Tell students to first estimate the solution for the inequality X−1<1 using number sense and then find the solution set using a calculator. 6. X−1 < 1 when X< 2. 7. Answers may vary. Students can trace the graph to see the result. Additional Problems Students investigate inequalities using X-Y graphs to compare numbers in order to create the solution set. See the student worksheet for calculator details. Students must set appropriate graphing windows in order to see the graphs. Graphing windows shown use ( ZStandard, which sets X and Y values so that M10 ≤ value ≤ 10. 1. 2X+3 7 2. X+4 > M3 3. 0.5XN1 < 2 4. XN2 ‚ M3 Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 4-10 Chapter 4: Linear Inequalities Section 2: Using Algebra Name Date Linear Inequalities: Using Algebra Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate mathematical concepts relating to using algebra in linear inequalities. Scroll through the Observations using " (! to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation Solve the inequality. Show your work. Draw the solution set on the number line. ←→ Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-11 Chapter 4: Linear Inequalities Section 2: Using Algebra Name Date Activities The Activities help you practice using algebra to solve inequalities. You can select from two different activities—Solve It! and Free Fall. Follow these steps to play the activity and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. Solve It! 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: You get two attempts to pick the step. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You get an additional 2 points for the correct solution. The total number of points available varies. 2. Look at the algebraic expression at the top of the screen and decide what must be done to solve the inequality for x. 3. Press # or $ to cycle through steps to choose from, and then press b to select the correct step (some problems require two steps). If your second choice is incorrect, the correct step is displayed; press any key to continue play. If the activity prompts you to select the correct result, highlight it with " or !, and then press b to select it. As you play the activity, write each inequality. What was your strategy for finding the solution? 4. What level did you play? 5. What was your score? Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-12 Chapter 4: Linear Inequalities Section 2: Using Algebra Name Date Activities (continued) Free Fall 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. 2. When you are ready to start, press any key. Scoring: Points are based on how quickly you solve each equation. Unless you specify point or time limits for this activity, students can play the activity when four missed equations stack up, the game is over.or they press áQUITâ to stop. There is no time limit. 3. Watch the equation as it falls, and quickly solve for x. Enter the solution (press Ì for negative numbers), and press b before the equation hits bottom. If you give an incorrect answer, the correct answer is displayed; press any key to resume play. The incorrect equation stacks up at the bottom of the screen, giving you less time to solve the next equation. 4. Follow your teacher’s instructions for how long to play the activity. 5. What level did you play? 6. What was your score? Additional Problems Solve the following inequalities. Show your work. 1. 6.4x > 16 3. 2x N 2.1 < M3.5 2. 1 2 x N 2 M2 4. A cell phone company charges a flat fee of $15 per month to use the phone. They also charge 25 cents for each phone call you make no matter how long you are on the phone. Your budget allows you to spend at most $30 per month for the phone. How many phone calls will you be able to make each month? (Tax is included in the prices.) Show your work. Write a short paragraph explaining why would you choose this company or why not. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-13 Chapter 4: Linear Inequalities Section 2: Using Algebra Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 Testing points and graphing solution sets: If X=3, is 2X+4>7 true or false? Try it by hand first. Show your work below. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : Display (TI.83 Plus shown) 2. From the Home screen, the calculator can tell you if a sentence is TRUE or FALSE (1=TRUE and 0=FALSE). 3. First, check what value is currently stored in X. „ Í Note: On the TI-73, use I rather than „. 4. What is the value of X stored in your calculator? Check around the class. Most likely, there are many different values of X. 5. First, if you want to find out if 2X+4 > 7 is true or false when X=3, you must tell the calculator that X=3, which is called storing a value in X. : 3X„ Í Note: On the TI-73, use I rather than „. 6. Next, input the number sentence 2X+4>7. This takes two steps. 2„\4 Note: On the TI-73, use I rather than „. 7. You can find the > (greater than) sign in the catalog (- |). Shortcuts: On the TI-83 Plus, press y : 3. On the TI-73, press - t # # # " " b to select > and # b to select Done. 8. Now, complete the sentence and see the result! -| $ until the cursor is next to > Í 7 Í Remember, 1=TRUE and 0=FALSE. If you substitute X=3, you get 10>7 which is TRUE. Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-14 Chapter 4: Linear Inequalities Section 2: Using Algebra Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) To Do This Press 9. Test another point! (Try M1.) a1X„ Í Note: On the TI-73, use I rather than „. 10. You do not have to type in the number sentence 2X+4>7 again. Note: You also can edit an expression after it is pasted to the line. Since 1=TRUE and 0=FALSE, 2X+4>7 is FALSE when X=M1. Display (TI.83 Plus shown) TI-83 Plus: y[ y[ Í TI-73: $ $ $ $ to highlight expression Í to copy Í Graph the inequality 2X + 4 > 7. To Do This Press 1. Graph this solution set. &: 2„\4 -| $ until the cursor is next to > Í Note: On the TI-73, use I rather than „. Display (TI.83 Plus shown) 7 2. Set the graph style to the animate graph style, which lets you see the graph as it is being plotted, even when it is on the axis at Y=0. ! until the cursor is in the left column Í until the ë style is indicated 3. Set the viewing window and graph the inequality by selecting ZDecimal. TI-83 Plus: ( 4:ZDecimal TI-73: ( 8:ZDecimal Topics in Algebra 1 © 2001 Texas Instruments Student Worksheet 4-15 Chapter 4: Linear Inequalities Section 2: Using Algebra Name Date ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) To Do This Press 4. Trace the function. Notice that where the graph is FALSE, Y1=0; where the graph is TRUE, Y1=1. ) | or ~ to trace the function 5. Notice that Y1=0 (FALSE) up through X=1.5 (when X1.5). ! or " to trace the function 6. Keep tracing and you see that the graph jumps to Y1=1 (TRUE) when X>1.5. ! or " to trace the function Display (TI.83 Plus shown) False True Find the exact set using algebra. Solve 2X+4>7. Does your answer agree with this graph? Additional Problems Above, you solved the following inequalities using algebra. Now, graph the solution set of these inequalities using your calculator. Compare the graph to your calculated answers above. 1. 6.4X>16 2. 1 2XN2M2 3. 2XN2.1, and ≥. Word problems about an amusement park are included for students to solve. The students also see the need to interpret the answers obtained using the appropriate number set. For example, they find that they have to knock down at least 9 3/8 bottles to earn enough points for the best prize. They see that they need to interpret this answer as at least 10 bottles in order to win the prize. The inequality properties of addition, subtraction, multiplication and division are given. When multiplying or dividing by a negative number, students are reminded about the reversal of the inequality sign. They also are reminded of how to translate phrases such as “at least” and “at most” to the appropriate relation in the Observations subsection. Common Student Errors • Students may have a hard time deciding which steps to follow to solve the inequality. They should connect this work back to the methods of solving linear equations. Students might make sign errors as they add or subtract from both sides of the inequality or reverse the inequality when multiplying or dividing by a negative number. • Students may have difficulty making the connection that C > 0 means C is positive and C < 0 means C is negative. • Although this section deals with the mechanical way of finding the solution set, students should be reminded that they should check to see if the solution set is reasonable. They need to keep using number sense. • Many students are able to find the answer using number sense without the written work. Learning how to write mathematics correctly is part of the communication skill and needs to be encouraged. This can cause frustration for students who find the problems easy to solve “in their heads.” • Some students have difficulty remembering the meaning of the symbols, <, ≤, >, and ≥. They also have trouble translating the phrases such as “at least” and “at most.” Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 4-17 Chapter 4: Linear Inequalities Section 2: Using Algebra Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, or tell them to review the instructions on the worksheet. 2. How to navigate the application, if they are not yet familiar with the application. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations give students an opportunity to practice finding the solution set to given inequalities. They are reminded about reversing the inequality sign when multiplying or dividing an inequality by a negative number. They also see a reminder of how to translate a phrase such as more than to the symbol >. If necessary, tell students how to find the Observations for this section. Observation 1 Solve for x: M2x+3 <11 x>M4. Students will see the answers on the next screen. Students’ work may vary with the step order shown on the screen. Students label the result on a number line. A possible number line looks like this: M6 Topics in Algebra 1 M4 M2 0 © 2001 Texas Instruments 2 Teacher Notes 4-18 Chapter 4: Linear Inequalities Section 2: Using Algebra Activities Solve It! Tell students to: 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: Students get two attempts to solve each problem. They earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. They get an additional 2 points for the correct solution. The total number of points available varies. 2. Look at the algebraic expression at the top of the screen and decide what must be done to solve the inequality for x. Students must select from the choices offered; this activity presents only one sequence of steps (to first isolate x and then change the coefficient of x to 1), although other sequences may be correct. 3. Press # or $ to cycle through steps to choose from, and then press b to select the correct step (some problems require two steps). If their second choice is incorrect, the correct step is displayed; press any key to continue play. If the activity prompts them to select the correct result, highlight it with " or !, and then press b to select it. As they play the activity, write each inequality and their strategy for finding the solution. 4. Record the level they played. 5. Record their scores. Free Fall Tell students to: 1. Highlight a level (silver = less difficult; gold = more difficult), and press b to select it. Scoring: Points are based on how quickly students solve each equation. If they give an incorrect answer, the equation stacks up at the bottom of the screen, giving them less time to solve the next equation. Unless you specify point or time limits for this activity, students can play the activity when four missed equations stack up, the game is over.or they press áQUITâ to stop. There is no time limit. 2. When they are ready to start, press any key. 3. Watch the equation as it falls, and quickly solve for x. Enter the solution (press Ì for negative numbers), and press b before the equation hits bottom. If they give an incorrect answer, the correct answer is displayed; press any key to resume play. 4. Follow your instructions. For example, students can play: • Until they have answered incorrectly four times (no time limit). • Until a certain amount of time has expired (highest score with the fewest misses wins). • Until a certain score has been reached (first student to reach the score with the fewest misses wins). • Repeatedly over a period of time (days, weeks, etc.) for tracking improvement of high scores. 5. Record the level they played. 6. Record their scores. Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 4-19 Chapter 4: Linear Inequalities Section 2: Using Algebra Activities (continued) Additional Problems 1. 6.4x > 16 2. 1 2 x N 2 M2 3. 2x N 2.1 < M3.5 when x>2.5 when x 0 when x < M0.7 4. Let C = the number of phone calls per month. Therefore, 15 + 0.25C ≤ 30. The solution is C ≤ 60. Students could discuss that this is about 2 calls per day. They should write whether or not this phone plan would be adequate for them. ³ Try-It!é on Your TI.83 Plus or TI.73 This keystroke exercise lets students learn about the logic functionality built in the calculator. They see how to store a variable, test an inequality and finally graph a TRUE(1)/FALSE(0) graph. By graphing the solution set using the calculator’s logic, they can verify their algebraic work on the inequalities. Students first solve the problem by hand. They should show all their work. If X=3, is 2X+4>7 true or false? Answer: 2(3) + 4 >7 6 + 4 >7 10>7 Therefore, the statement is true. Tell students to follow the steps exactly on the calculators. Example screens are displayed on the worksheets for students to compare with the calculator screens. Additional Problems Tell students to do the following calculation by hand, and then check the answers using the calculator. They should show all their work. The graphs are shown below using ZDecimal (() window values. Students should discuss how they determine the endpoint of the solution set. Remind them that using algebra gives the exact answer. The graph helps them see where the solution is approximately located. 1. Topics in Algebra 1 2. 3. © 2001 Texas Instruments Teacher Notes 4-20 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Name Date Linear Systems: Using Graphs & Tables Student Worksheet Overview The Overview introduces topics covered in Observations and Activities. Scroll through the Overview using ~ ( | to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations The Observations illustrate that two lines in a plane intersect, are parallel or are the same line. By graphing the lines of a system of two linear equations with two variables, you can determine the possible number of solutions. Scroll through the Observations using ~ ( | to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answer to the question on your worksheet. Observation 1 How many solutions does this system of linear equations have? Observation 2 How many solutions does this system of linear equations have? Observation 3 How many solutions does this system of linear equations have? Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-1 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Name Date Activities The Activities help you practice using graphs and tables. You can select from two different activities—System Match It! and a worksheet activity. Follow these steps to play the activities and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using $ or #, and press b. System Match It! 1. Select the correct answer to the question. Questions include: Scoring: You get one attempt to answer the problem. You earn 2 points for a correct answer. You can earn up to 12 points. • Selecting the graph that correctly illustrates the system. • Selecting the system that correctly describes a graph. • Selecting the correct solution of a system given a table. 2. What was your score? Tip: Be careful! The graphs may appear close together. Worksheet Activity Solve a system using tables. Notes: See ³ TIp™ 4: Creating a Table to help you with tables. See also ³ Try-It!™ in Chapter 2 Sections 1 and 2 for working with tables and linear equations. a. Blue Lake is Pat’s favorite place to swim during the summer. There is an entry fee of $12.00 per car per day. Pat notices a sign as they approach the entrance. His family could join the Blue Lake Club for the summer! It costs $48.00 to join and then the entry fee becomes $6.00 per car per day. Fill in the table to help you develop the equations. Days at the Lake Cost Without Membership Cost With Membership 1 2 3 4 … D $12(1) = $12 $12(2) = $24 $48 + $6(1) = $54 $48 + $6(2) = $60 … … b. Let C = the entry cost to the lake. Let D = the number of days Pat’s family goes to the lake during one summer. Write the system of two equations that describe the cost of the entry fee without membership and with membership. Use the table above (in part a) to help you write the equations. Cost Without Membership: _______________________________ Cost With Membership: Topics in Algebra 1 ________________________________ © 2001, 2002 Texas Instruments Student Worksheet 5-2 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Name Date Activities (continued) c. Check that the equations are written in slope-intercept form, y = mx + b. This prepares you to enter the system into your calculator. You will also have to change the variable. Notice that the cost, C, is the dependent variable (Y) and the number of days, D, is the independent variable (X). Write the equations so you can enter them in your calculator. Y1= _________________ Y2= _________________ d. Use a table to find the cost of the entry to the lake with and without membership. When is the entry cost without membership equal to the entry cost with membership? e. If Pat’s family goes to the lake 6 times this summer, should they join the Blue Lake Club to save money? Why? f. If Pat’s family goes to the lake 10 times this summer, should they join the Blue Lake Club to save money? Why? Challenge: Learn how to take a picture of a screen on your calculator using TI Connect™ software or TI-GRAPH LINK™ software and cable. You can paste a screen into a word processor and then print it out to hand in for your homework! Go to http://education.ti.com and search for TI Connect or TI-GRAPH LINK. Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-3 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Name Date ³ Try-It!é on Your TI-83 Plus or TI-73 You will: • Graph two lines of a system of equations. • Use r to locate the intersection of the lines. • Use y 0 and y - to locate the exact solution of the system of equations. Y1 = L4X + 2 Find the solution of the system: Y2 = 2X N 2.5 Notice that these equations are already in the form of y = mx + b. They are in the form to enter into your calculator. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. y5 áEXITâ ‘ 2. First, enter L4X + 2 as Y1 and 2X N 2.5 as Y2 in the Y= editor. o‘ Ì4„Ã2 Note: See ³TIp 3: Graphing a Function in the Standard Window for more information. Display (TI.83 Plus shown) †‘ 2„¹2Ë5 Note: On the TI-73, use I rather than „. 3. Select the Zoom Decimal viewing window. The graph displays. Remember: You have to adjust the viewing window depending on the system of equations. You can see the intersection of the lines in the ZDecimal window for this example. Topics in Algebra 1 TI-83 Plus: q 4:ZDecimal TI-73: q 8:ZDecimal © 2001, 2002 Texas Instruments Student Worksheet 5-4 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Name Date ³ Try-It!é on Your TI-83 Plus or TI-73 (continued) To Do This Press 4. Trace close to the intersection of the graphs to find a value close to the solution. Display (TI.83 Plus shown) r }, †, ~, or | Since Zoom Decimal traces by tenths, you can get close to the answer but this is not the exact answer. Notice that the point (0.8, L1.2) on the screen is on Y1. Is this point on Y2? Check it out! For this particular calculator setup, you don’t get the exact answer. Read on to see how to refine the answer. 5. Use the table setup to refine the solution. y‘ 0 † 0.25 ‘ Set up your table to show values close to X = 0.8. As shown here, you can use a starting value of 0 with increments of 0.25. 6. Search through the table to see that the lines intersect at (0.75, L1). Notice that both Y1 and Y2 are L1 when X = 0.75. Verify by hand that the solution is (X, Y) = (0.75, L1). y0 You also know that there is only one solution because the lines intersect at one point, so your search is complete. Note: Learn more about the calculator features intersect and Solver in the ³ Try-It!™ section in Chapter 5: Linear Systems, Section 2: Using Algebra. Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-5 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Name Date Solution Search Write the solution and explain how you found the solution using graphs and a table for each of the problems below. Do the following for each problem. • Rewrite the system in slope-intercept form, y = mx + b, if necessary. • Use the slope-intercept form of the equations to draw a rough sketch of the lines. You can verify your graph on your calculator. Estimate the solution so that you have an idea of how many solutions there are and where the solution is located. • Search for the solution of the equation on the calculator using graphs and a table. • See ³ TIp 4: Creating a Table and ³ TIp 5: Adjusting the Viewing Window for additional help with the calculator. • Remember to change your viewing window ( p ) or your table setting ( y - ) to do your search. • Explain how you found the solution. • Write out how you checked the solution. Remember: The TI-73 and the TI-83 Plus only use the variables X and Y for graphs and tables. If an equation uses letters other than X and Y, you have to change the variables in the problem to X and Y on the TI-73 and the TI-83 Plus. Use parentheses, if needed, when entering the equations in the Y= editor. 1. y = 2x + 4 3x + y = L11 2. Lx + 3y = 4 10 1 x+y= 3 3 Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-6 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Name Date Solution Search (continued) 3. 2w + t = 35 L2 1 w + t = 19 5 5 4. 4x + 7y = 8 4x + 7y = 14 5. x N 9y = 7 2x N 18y = 14 Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-7 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Linear Systems: Using Graphs & Tables Teacher Notes Objectives • To illustrate how to locate the real number solution of a system of linear equations (two equations and two variables) using tables. • To illustrate how to locate the real number solution of a system of linear equations (two equations and two variables) using graphs. • To graphically illustrate the types of solutions expected for a system of linear equations. Math Highlights Students work with a system of linear equations that has two equations in two variables. They begin the Overview by setting up an analysis of two different cell phone plans. The two plans can be modeled by linear equations. They investigate when the two plans cost the same amount of money. In the table of values example, students see a table of values for each equation. To create the table, the equations are in the form y = mx + b. They see that the x value that gives the same y value for both equations is the solution. They also see that they may need to refine the table of values to search for the solution. In the x-y graphical example, students graph both equations and locate the intersection of the lines. The (x,y) coordinate of the intersection of the lines is the solution. Since the graphs of the linear equations in the system can intersect, be parallel, or be the same line, students also see that they may find a unique solution, no solution, or an infinite number of solutions to the system. In Observations, students associate the graphs of the lines of a system with the number of solutions of the system. This is covered again at a higher level in Section 2: Using Algebra. Common Student Errors • Students have to rewrite the system in slope-intercept form in order to enter the equations into the calculator. Many students tend to make sign errors and division errors when they rewrite equations. For example, given 2T + 3S = 57 students would first have to rewrite the equation as S = (L2/3)T + (57/3), assuming S is the dependent variable. Then, the students have to rewrite this equation as Y1 = (L2/3)X + (57/3). A common division error is to write the equation as Y1 = (L2/3)X + 57, which is incorrect. • Students forget to enter fractions into the Y= editor using parentheses. Remind students about the order of operation. If they enter M2/3X, the calculator interprets this as L2 ÷ (3 Q X) following the order of operation rules. The correct entry is (M2/3) Q X. Note: TI-73 users can use = to enter the fractions. However, you should still remind them how to use parentheses and about the order of operation rules. • When solving by graphing using the graphing calculator, some students trace along one function to what appears to be the intersection point without verifying that that point is also on the other line. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-8 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Common Student Errors (continued) • After students have practiced using graphs and tables to solve a system of equations, they may think that they can always find the exact solution for a system using these methods. Although they can often find exact solutions using these methods, using algebra always gives exact answers for these equations. To help them understand this idea, have students try to search for the solution to the system y = 2x + 3 and y = 2x + 4. Using a table, they could search forever since these lines are parallel. Using a graph, they might think that the lines are parallel, but they are only looking at a few viewing windows. Ask them how they can know if there is a window where the lines intersect. Open a discussion with your class to see if they think they can verify that this system has no solution using tables or graphs. • Algebraic methods alone usually do not invite the student to reason out the solution using their knowledge of number sense and geometry. Many students learn the mechanics of solving a problem without understanding the problem or the solution. The graphs and tables method gives students the opportunity to see the values and graphs of the equations so they can see when two equations have the same value. Ask students to look at the equations y = 2x +3 and y = 2x + 4 again, and use their number sense. Ask them if 2x + 3 could ever be the same value as 2x + 4 for a given x? Encourage students to first look at the equations to see if their knowledge of geometry or their number sense can tell them something about the system before they start their method of solution. • Some visual learners benefit by seeing the numbers and graphs first, and then by using these as the tool to find the solution. However, many students can see the solution to some systems using their number sense. These students may have difficulty taking the time to show and write about their work. This may also be an issue in Chapter 5: Linear Systems, Section 2: Using Algebra. Encourage student to use written mathematics as well as drawing graphs and tables as a communication tool. Have students look in newspapers and on the web for graphs and tables of information to show real examples for the need for this communication skill. Require students to write out complete solutions to problems, including the mathematics and the interpretation of what the numbers mean in the problem. For example, the cell phone problem in the Overview subsection requires not only the numeric answer but also an explanation about what the numbers mean with respect to the cell phone users. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-9 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, if necessary. 2. How to navigate the application, if necessary. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-10 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Observations The Observations help students start to uncover the types of solutions that arise in systems of linear equations. If necessary, tell students how to find the Observations section. Students are asked to observe the number of solutions from the given graph. They should question whether they are seeing enough of the graph to make a conjecture about the number of solutions. Observation 1 Students see the answer after the third observation question. Observation 2 Observation 3 Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-11 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Activities System Match It! Tell students to: 1. Select the correct answer to the question. Questions include: Scoring: You get two attempts to answer the problem. You earn 2 points for a correct answer on the first try, 1 point for a correct answer on the second try. You can earn up to 12 points. • Selecting the graph that correctly illustrates the system. • Selecting the system that correctly describes a graph. • Selecting the correct solution of a system given a table. 2. Record their scores. Remind students that the graphs might appear very close together on the screen. They need to use their knowledge about both the functions and the graph to determine the correct answer. Worksheet Activity Students investigate the entry fee to Blue Lake. They compare the entry fees with and without a membership fee. Notes: See ³ TIps™ 4: Creating a Table to help you with tables. See ³ Try-It!™ in Chapter 2 Sections 1 and 2 for working with tables and linear equations. a. Students should gather information from the problem to write a system of equations for the investigation. Filling in the table with all of the calculations written out helps students develop the equations inductively. Days at the Lake Cost Without Membership Cost With Membership 1 2 3 4 … D $12(1) = $12 $12(2) = $24 $12(3) = $36 $12(4) = $48 … 12D $48 + $6(1) = $54 $48 + $6(2) = $60 $48 + $6(3) = $66 $48 + $6(2) = $72 … 48 + 6D b. Variables are suggested. Review the concept of independent and dependent variables with the students. Let C = the entry cost to the lake. Let D = the number of days Pat’s family goes to the lake during one summer. C = 12D C = 6D + 48 (Students could also enter 48 + 6D.) Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-12 Chapter 5: Linear Systems Section 1: Using Graphs & Tables Activities (continued) c. Rewrite the problem in terms of Y1, Y2 and X to prepare to enter the system into the Y= editor. Y1 = 12X —or— Y1 = 6X + 48 Y2 = 6X + 48 Y2 = 12X The independent variable (days at the lake) and the dependent variable (entry cost) are stated in the problem. Remind students that the calculator treats X as the independent variable and Y as the dependent variable. Enter equations in Y= editor. d. Students need to enter the equations in the Y= editor and should set up the table. Discuss that the domain of the system should be whole numbers starting at 0 since x counts the number of trips to the lake. Notice at x = 8, Y1 = Y2 which is the breakeven point. If Pat’s family goes to the lake up to and including 8 times, they might not choose to join the Blue Lake club. e. If Pat’s family goes to the lake only 6 times during the summer, they will spend more money if they join the club for $48. f. Set up the table. If Pat’s family goes to the lake 10 times during the summer, they will save money if they join the club. They will save $120 N $108 = $16. Discuss the savings if the family goes to the lake more than 10 times. Pose questions such as, when will they save $50? Have students find other situations that set fees in this manner. One source is the national and state park services web pages. Search for where the equations are equal. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-13 Chapter 5: Linear Systems Section 1: Using Graphs & Tables ³ Try-It!é on Your TI-83 Plus or TI-73 Students search for the solution of a system of two linear equations in two variables using graphing and tracing, and a table. The problem has been chosen so that students do not trace to the exact solution and need to use the table to search. They could also choose to change the window so that they could trace to the exact solution. This is not an efficient choice, but could be pursued, and the investigation would be enriching. The students will: • Graph two lines of a system of equations. • Use r to locate the intersection of the lines. • Use y - and y 0 to locate the exact solution of the system of equations. Note: Students will learn more about the calculator features intersect (TI-83 Plus) and Solver (TI-83 Plus and TI-73) in the ³ Try-It! section in Chapter 5: Linear Systems, Section 2: Using Algebra. Tell students to follow the steps exactly on the calculators. Example screens are displayed on the worksheets for students to compare with the calculator screens. Solution Search Tell students to: • Rewrite the system in slope-intercept form, y = mx + b, if necessary. • Use the slope-intercept form of the equations to draw a rough sketch of the lines. Verify the graphs on the calculator. Estimate the solution so they have an idea of how many solutions there are and where the solution is located. • Search for the solution of the equation on the calculator using graphs and a table. Notes: Since the calculator only uses the variables X and Y for graphs, tables and some other features, students must decide which variable in the problem should be X and which one should be Y when the problem uses other variables. Discuss independent and dependent variables, emphasizing that the calculator is set up to treat X as the independent variable and Y as the dependent variable. Remind students to use parentheses correctly when they enter equations into the Y= editor. For example, (1/3)X is not the same as 1/3X which is 1/(3X) when the order of operation rules are applied by the calculator. However, when TI-73 users enter 1/3 using the = key, their entry is calculated correctly. Remind students to change the viewing window ( p ) or table setting ( y - ) to do the search. ³ TIp and ³ TIp 5: Adjusting the Viewing Window provide additional help with the calculator. • Explain how they found the solution. • Write out the check of their solution. 4: Creating a Table Answers: 1. (x, y) = (M3, M2) 2. (x, y) = (3, 7/3) 3. (w, t) = (M15, 65) Students are not given which variable is dependent and which is independent. They may very well write the solution as (t, w) = (65, M15). 4. Lines are parallel, which implies that there are no solutions. 5. Lines are the same, which implies that there are an infinite number of solutions. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-14 Chapter 5: Linear Systems Section 2: Using Algebra Name Date Linear Systems: Using Algebra Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using ~ ( | to review, if necessary). Read each screen carefully. Look for new terms, definitions, and concepts. Observations Scroll through the Observations using ~ ( | to review, if necessary). Read each screen carefully. When you come to a Write an Observation screen, stop and write the answers to the questions on your worksheet. Observation 1 Find the solution of each system shown on the screen. Use either the substitution or elimination method. Observation 2 For the system of equations shown on the screen, can the expression L2x + y equal 2 and 5 for the same (x, y)? Observation 3 Solve the given system using the elimination method. Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-15 Chapter 5: Linear Systems Section 2: Using Algebra Name Date Activities The Activities help you practice using algebra to solve linear systems. You can select from two different activities—What Am I? and Balloon Ride. Follow these steps to play the activities and complete your worksheet. 1. Make sure you are in the Activities for this section. 2. Highlight an activity using } or † and press Í. What Am I? 1. Highlight a level (silver = less difficult; gold = more difficult), and press Í to select it. Scoring: You get one attempt to pick the correct classification of the system. You get 2 points for a correct choice and 1 point for a correct choice if you press áHINTâ to see the graph. Four systems are given for a maximum score of 8 points. 2. Look at the system of equations and decide if the system is consistent & independent, consistent & dependent, or inconsistent. Press áHINTâ if you need to see the graph. You only get 1 point for the problem if you press áHINTâ. Press } or † to cycle through the choices and then press Í to select the correct answer. The correct answer and graph are displayed if the incorrect answer is chosen. You must press a key to continue play. 3. In the space below, write out the algebraic steps (using elimination or substitution) for each problem or explain why you knew the correct answer. 4. What level did you play? 5. What was your score? Problem 1: Problem 2: Problem 3: Problem 4: Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-16 Chapter 5: Linear Systems Section 2: Using Algebra Name Date Activities (continued) Balloon Ride 1. Highlight a level (silver = less difficult; gold = more difficult), and press Í to select it. Scoring: You get two attempts to pick or input the correct solution to the system of equations. You get 2 points for a correct choice or input on the first try, and 1 point for a correct choice or input on the second try. There are 4 problems for a maximum score of 8 points. 2. Look at the system of equations and solve using the algebraic methods of substitution or elimination. Silver level: Press } or † to cycle through the solutions to choose from, and then press Í to select the solution. You must press a key to continue play. Gold level: Use } or † to select an answer or to get to the input box. Select or input your answer and press Í. (Press Ì to enter negative numbers.) You must press a key to continue play. 3. As you play, write out the algebraic steps (using elimination or substitution) for each problem or explain why you knew the correct answer in the space below. 4. What level did you play? 5. What was your score? Problem 1: Problem 2: Problem 3: Problem 4: Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-17 Chapter 5: Linear Systems Section 2: Using Algebra Name Date Extra Practice: Using Substitution or Elimination 1. Cathy found a part-time job for the summer. Each week, when she works up to 10 hours, she earns a regular hourly wage. If she works more than 10 hours each week, she earns more money per hour for the overtime hours. She worked 14.5 hours during the first week. Her paycheck, before taxes and deductions, was $93.75. The second week, she worked 12 hours and her paycheck, before taxes and deductions, was $75.00. Cathy’s boss had told her what her hourly and overtime rates were, but Cathy was so excited to get the job that she couldn’t remember what she was told. • Write a system of equations for Cathy’s hourly and overtime pay rates. Clearly define the variables and their meanings. • Solve the system of equations using substitution. Show your work and explain the steps you used to solve the system of equations. Show the check of your solution. • Write a sentence explaining Cathy’s pay rate. ___________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-18 Chapter 5: Linear Systems Section 2: Using Algebra Name Date 2. Cathy decides to spend some of her earnings from her new job, so she and her friend Brenda go to the mall. Their favorite store has blouses and jeans on sale. A variety of blouses is on one rack and are all the same price. A variety of jeans is on another rack and are all the same price. The price tags are not on the clothes but Cathy and Brenda know that the store’s prices are usually within their budgets. Cathy picks out 3 blouses and 2 pairs of jeans from the sale racks. At the checkout, she sees that her total bill is $57.00. Brenda picks out 4 blouses and 3 pairs of jeans from the sale racks. Her total bill is $81.00. They are shopping in the state of Delaware where there is no sales tax. To find the cost of one blouse or one pair of jeans, Cathy and Brenda could just look at the sales receipt. Instead, they try to figure out the prices themselves. • Write the system of equations that Cathy and Brenda need to solve. Clearly define the variables and their meaning. • Solve the system of equations using elimination. Show your work and explain the steps you used to solve the system of equations. Show the check of your solution. Hint: You can multiply both equations by a factor to avoid working with fractions! • Write a sentence explaining the price Cathy and Brenda paid for each top and each pair of jeans. Remember: You always have a choice of picking the method that you think will be easiest to perform when solving systems of equations. In the problems above, you are asked to use a specific method. __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-19 Chapter 5: Linear Systems Section 2: Using Algebra Name Date ³ Try-It!é on Your TI.83 Plus You will: • Graph two lines of a system of equations. • Use ) to locate the intersection of the lines. • Use the intersect feature to find the solution to a system of equations. • Check the solution on the home screen using ¿. Find the solution of the system: Y1 = L4X + 2 Y2 = 2X N2.5 Notice that these equations are already in the form of y = mx + b. They are in the form to enter into your calculator. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. yl áEXITâ : 2. First, enter M4X + 2 as Y1and 2X N 2.5 as Y2in the Y= editor. o‘ Ì4„Ã2 †‘ 2„¹2Ë5 Note: See TIp 3: Graphing a Function in the Standard Window for more information. 3. Select the Zoom Decimal viewing window. Display q 4:ZDecimal Remember: You have to adjust the viewing window depending on the system of equations. You can see the intersection of the lines in the ZDecimal window for this example. 4. The graph displays. Since this system has one solution, you can find the numerical solution using the intersect feature. Important Reminder: Be sure to use your knowledge about the equations of lines to determine if the lines are parallel or the same line. When you use the intersect feature, an error displays if the lines are parallel. If the lines are the same line, the calculator shows ONLY one answer. 5. Use the CALCULATE menu item called intersect to find the calculated numeric solution. Topics in Algebra 1 y / 5:intersect © 2001, 2002 Texas Instruments Student Worksheet 5-20 Chapter 5: Linear Systems Section 2: Using Algebra Name Date To Do This Press 6. You have to select a First Curve in order for the calculator to calculate the numerical solution. Notice the cursor is on the line Y1 = L4X + 2. Í 7. You have to select a Second Curve in order for the calculator to calculate the solution. Notice the cursor is on the line Y2 = 2X N 2.5. Í 8. The calculator now needs a close guess at the solution. Move the cursor closer to the intersection. ~ or | 9. The calculator uses a program to calculate the numerical solution. The numerical solution given is at (0.75, L1). Check to see if this solution is the exact solution or an approximate value. b 10. Check the solution on the home screen. The value 0.75 is stored in the X variable in the calculator. X is fixed at 0.75 after you follow these steps. -5‘ Ë 75 ¿„Í Display Important Reminder: The calculator always has a value stored in each variable. You must store the value you want in order to understand how the calculator interprets a variable expression. 11. Enter L4X + 2 to find the Y1 value at X = 0.75. Notice that the output is L1. Ì4„Ã2Í 12. Enter 2X N 2.5 to find the Y2 value at X = 0.75. Notice that the output is L1 and Y1 = Y2 at X = 0.75. 2„¹2Ë5 Í The solution is (0.75, L1). Press * to see if this agrees with the graph of the lines! Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-21 Chapter 5: Linear Systems Section 2: Using Algebra Name Date Extra Practice: Using Your Calculator to Find Solutions 1. Use your calculator to find the solution to each system of equations. Write the solution and a description of how you used your calculator to find and verify the solution. Remember: You need to rewrite the equations in the form y = mx + b to work with your calculator. Note: If the calculator gives the value .6666666667 or X=.66666666666666 in Solver the exact answer is most likely X = 2 . 3 Explain how you found the exact answer if the calculator only gave an approximate answer. Verify your answer! a. y = L1.2x + 3.725 b. 3x + y = 6 1 1 x N y = L1 3 2 3x N y = 8.875 2. Solve each of the systems you found in Extra Practice: Using Substitution or Elimination, using your calculator. Write an explanation of how you used your calculator. a. Cathy’s pay rate b. Cathy and Brenda’s shopping trip Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-22 Chapter 5: Linear Systems Section 2: Using Algebra Name Date Challenge Investigate using the Solver feature on the TI-83 Plus to find the solution! Find this feature in 0:Solver. It is best to use this feature if the system has only one solution. Hint: Press and select 0:Solver.You have to enter the equation in the Solver as 0 = L 4X + 2 N (2XN2.5) to solve for the X value. You have to input a guess for the X solution. bound = {M1åå99, 1åå99} represents the real number line for the calculator. You can make the set smaller to find solutions in a particular interval. Then, use ƒ \ to find the calculator’s numerical solution. See the TI-83 Plus guidebook for more details about the Solver feature. Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-23 Chapter 5: Linear Systems Section 2: Using Algebra Name Date ³ Try-It!é on Your TI.73 You will: • Graph two lines of a system of equations. • If the lines intersect, use the 1 feature Solver to find the X value of the solution to the system. • Find the Y value by using X on the home screen. Find the solution of the system: Y1 = L4X + 2 Y2 = 2X N 2.5 Notice that these equations are already in the form of y = mx + b. They are in the form to enter into your calculator. To Do This Press 1. Exit the Topics in Algebra 1 application and clear the Home screen. -l áEXITâ : 2. It is good practice to look at the graph of the system before you use the Solver feature. Enter L4X + 2 as Y1 and 2X N 2.5 as Y2 in the Y= editor. &: a4I\2 #‘ 2IT2`5 Display Note: See TIp 3: Graphing a Function in the Standard Window for more information. 3. Select the Zoom Decimal viewing window. ( 8:ZDecimal Remember: You will have to adjust the viewing window depending on the system of equations. You can see the intersection of the lines in the ZDecimal window for this example. 4. The graph displays. Since this system has one solution, find the numerical solution next using the Solver feature on the TI-73. Important Reminder: Before you use the intersect feature, use your knowledge about the equations of lines to determine if the lines are parallel or the same line. 5. Use the 1 menu item called Solver to find the calculated numerical solution. Topics in Algebra 1 1 6:Solver © 2001, 2002 Texas Instruments Student Worksheet 5-24 Chapter 5: Linear Systems Section 2: Using Algebra Name Date To Do This Press 6. To find the solution, you want to find when the two equations are equal. Find Y1 = Y2. You need to enter L4X + 2 = 2x N 2.5 on the eqn: line. a4I\2 7. Find the equal sign (=) in - t. -t ###b # 8. Finish entering the equation. b2IT2`5 9. Enter a guess of X=1 as the solution. The calculator needs a starting value for its computation. b:1 Display Hint: bound = {L1å99, 1å99} represents the real number line for the calculator. You can make the set smaller to find solutions in a particular interval. See the TI-73 guidebook for more details. 10. Highlight X on the Solve:X line and the solution is given as $X=.75. Notice that $ appears after the calculator has computed the numerical solution. ##b 11. Find the Y1 and Y2 values on the home screen using X. The value 0.75 will be stored in the X variable in the calculator. X is fixed at 0.75 once you follow these steps. -l: ` 75 XIb Important Reminder: The calculator always has some value stored in each variable. You must store the value you want in order to understand how the calculator interprets a variable expression. 12. Enter L4X + 2 to find the Y1 value. Notice that the output is L1. Topics in Algebra 1 a4I\2b © 2001, 2002 Texas Instruments Student Worksheet 5-25 Chapter 5: Linear Systems Section 2: Using Algebra Name Date To Do This Press 13. Enter 2X N 2.5 to find the Y2 value. Notice that the output is L1 and Y1 = Y2 at X = 0.75. 2IT2`5b Display The solution is (0.75, L1). Press * to see if this agrees with the graph of the lines! Extra Practice: Using Your Calculator to Find Solutions 1. Use your calculator to find the solution to each system of equations. Write the solution and a description of how you used your calculator to find and verify the solution. Remember: You need to rewrite the equations in the form y = mx + b to work with your calculator. Note: If the calculator gives the value .6666666667 or X=.66666666666666 in Solver, the exact answer is most likely X = 2 . 3 Explain how you found the exact answer if the calculator only gave an approximate answer. Verify your answer! a. y = L1.2x + 3.725 3x N y = 8.875 Topics in Algebra 1 b. 3x + y = 6 1 1 x N y = L1 2 3 © 2001, 2002 Texas Instruments Student Worksheet 5-26 Chapter 5: Linear Systems Section 2: Using Algebra Name Date 2. Solve each of the systems you found in Extra Practice: Using Substitution or Elimination using your calculator. Write an explanation of how you used your calculator. a. Cathy’s pay rate b. Cathy and Brenda’s shopping trip Topics in Algebra 1 © 2001, 2002 Texas Instruments Student Worksheet 5-27 Chapter 5: Linear Systems Section 2: Using Algebra Linear Systems: Using Algebra Teacher Notes Objectives • To review the substitution method of solving a system of two linear equations in two variables. • To review the elimination method of solving a system of two linear equations in two variables. • To review the definitions of consistent (independent and dependent) and inconsistent systems. • To associate the number of solutions of a system with the classification of consistent (independent and dependent) and inconsistent systems. Math Highlights In this section, students work with a linear system of equations with two equations and two variables and review the methods of substitution and elimination. In the Overview, students associate the graphs of the lines of a system with the number of solutions of the system and the classification of the system as consistent and inconsistent (dependent and independent) systems. This was also covered at a lower level in Section 1: Using Graphs & Tables. In the substitution example, caramel corn is sold as a class fundraiser, and the students need to know how many bags of caramel corn they need to sell to make a profit. A system of linear equations is written to model the costs of producing the caramel corn and the revenue earned from selling bags of caramel corn. Solving the system of equations gives the number of bags of caramel corn the students need to sell to make a profit. The term profit is used and should be discussed in the class. The terms breakeven and loss are not covered, but students would benefit from a complete classroom discussion of the problem, not just the profit point. In the elimination example, Jon and Mia earn money by recycling cans and glass. The recycling center gives each of them one payment for both the cans and the glass. Jon and Mia want to know how much money they earned for each pound of cans and each pound of glass. A system of linear equations is written to model the amount of money that Jon and Mia earned for recycling. Solving the system of equations gives the prices per pound that they were paid for cans and glass. Students can eliminate either variable in the example. The variables used in these problems are x and y; however, you should encourage students to use variables that make sense in the problem. The Try-It! examples for the TI-83 Plus and the TI-73 are slightly different. They are printed on separate pages so that you can make copies of only the pages you need. Students can use either calculator to complete the problems in the Student Worksheet. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-28 Chapter 5: Linear Systems Section 2: Using Algebra Common Student Errors • When they use the substitution method, students may make sign and division errors when they solve for one of the variables. They may also need to be reminded to use parentheses when needed. Students might incorrectly distribute expressions. • When solving by substitution, students might solve one equation for y (or x) and substitute back into the same equation getting a result of 0 = 0, when they should have substituted back into the other equation. • When they use the elimination method, students may miss multiplying every term in an equation by the appropriate constant. • As they continue using the elimination method, many students subtract incorrectly. They usually subtract the first term correctly, but often forget to subtract the other terms. Encourage students to choose the multiplier so that they add the equations for the elimination rather than subtract them. • When an algebraic solution results in a statement that is always true, such as 2 = 2 in a dependent system (coincident lines), or a statement that is never true, such as 2 = 4 in an inconsistent system (parallel lines), students may be unsure how to state the solution. • Students often skip checking their solution. In addition to checking their solution by substituting it back into the original equations, they should also make sure that the solution is reasonable. For example, in the Student Worksheet problem where Cathy and Brenda are shopping for tops and jeans, the variable must be positive. • Students should practice rewriting the system after each step to keep track of their manipulations. Encourage students to create good math habits by doing the following. • Pick the method—graphs, tables, substitution, or elimination—which is best for the system. First, look at the physical problem, if appropriate, or use geometry and number sense to analyze the system. • Remember that for real problems (word problems), some solutions may need to be omitted. For example, if you need to find the quantity of an item, the solution must be positive. • Notice when a system obviously has no solution because the lines have the same slope (i.e., parallel lines), and when a system has an infinite number of solutions because the lines are the same line. • Rewrite the system after each step to keep track of the manipulations. Note: Students who can think through the steps in their mind tend to be impatient with recording each step. However, forming this habit now will help them in the future, especially when they learn how to solve systems using matrices. • Write out the mathematics you use to solve the system. Also, write phrases or sentences that explain your steps. Draw graphs and tables to use as aids. • Interpret what the solution means in real problems. For example, the recycling problem in the Overview subsection requires not only the numeric answer but also an explanation about what the numbers mean with respect to the price per pound. Tip: Have students look in newspapers and on the web for graphs and tables of information to show real examples of the importance of both computations and explaining what the computations mean. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-29 Chapter 5: Linear Systems Section 2: Using Algebra Student Worksheet Notes with Answers Overview Tell students: 1. How to find the Overview, if necessary. 2. How to navigate the application, if necessary. 3. To scroll through the Overview on the calculator. Point out new terms, definitions, and concepts, and tell students to look for them as they go through the Overview. Observations The Observations help students review substitution and elimination methods. If necessary, tell students how to find the Observations. Observation 1 Students are asked to solve both systems using either the substitution method or the elimination method. The systems are identical except for the choice of variables. Students should become comfortable with a change in variables and should recognize that they do not have to do this calculation twice. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-30 Chapter 5: Linear Systems Section 2: Using Algebra Observations (continued) Observation 2 Students are asked if the same left-hand sides of the equations can equal different right-hand sides. They should be able to reason this out using their number sense. They can then verify their answer using algebra. They will see the verification using the elimination method on the following screens in the application. The second equation is multiplied by L1 and then the equations are added. Multiplying by L1 was chosen to avoid sign errors. Some students will want to immediately subtract the two equations. This is also correct, but they need to be very careful to subtract each term! Observation 3 Students see the elimination steps on the following screens in the application. The result screen is given below. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-31 Chapter 5: Linear Systems Section 2: Using Algebra Activities What Am I? Tell students to: Scoring: Students get one attempt to pick the correct classification of the system. They earn 2 points for each correct choice, and 1 point for a correct choice if they press áHINTâ to see the graph. Four systems are given for a maximum score of 8 points. 1. Highlight a level (silver = less difficult; form y = mx + b; gold = more difficult; mixed slope-intercept and standard forms), and press Í to select it. 2. Look at the system of equations and decide if the system is consistent & independent, consistent & dependent, or inconsistent. 3. Write out the algebraic steps to make this determination, except possibly at the silver level. Students could use their knowledge of the graph of y = mx + b in order to answer some problems. 4. Press áHINTâ if they want to see a graph of the system. They will only get 1 point for the problem if they use the áHINTâ. 5. Press } or † to cycle through the choices and then press Í to select the correct answer. The correct answer and graph will show if the incorrect answer is chosen. They must press a key to continue play. As they play the activity, students should write an algebraic solution to the system on their worksheet. 6. Record the level they played. 7. Record their scores. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-32 Chapter 5: Linear Systems Section 2: Using Algebra Activities (continued) Balloon Ride Tell students to: 1. Read the introduction screen and then highlight a level (silver = less difficult; gold = more difficult), and press Í to select it. Students answer multiple-choice problems in the silver level and they input their solution in the gold level. Scoring: Students get two attempts to pick or input the correct solution to the system of equations. They earn 2 points for a correct choice or input on the first try, and 1 point for a correct choice or input on the second try. Silver Level: Press } or † to cycle through the solutions to choose from, and then press Í to select the solution. Students must press a key to continue play. Gold Level: Use } or † to select an answer or to get to the input box. Select or input your answer and press Í. (Press ¹ to enter negative numbers.) Students must press a key to continue play. There are 4 problems for a maximum score of 8 points. 3. As they play the activity, students should write their work using either elimination or substitution. 4. Record the level they played. 5. Record their scores. Extra Practice: Using Substitution or Elimination 1. Cathy’s pay rate can be determined by the following. Let S = hourly pay rate Let T = overtime pay rate Week 1: 10S + 4.5T = 93.75 Week 2: 10S + 2T = 75 10S = 75N2T To use substitution, the student could solve the Week 2 equation for 10S, 10S = 75N2T. Then, the Week 1 equation becomes the following. (75 N 2T) + 4.5T = 93.75 2.5T = 18.75 T = 7.5 Once T is known, substitute T into either equation and solve for S. 10S + 4.5(7.5) = 93.75 10S = 60 S=6 Interpret the answer: S = $6.00 per hour T = $7.50 per hour Cathy earns $6.00 per hour for the first 10 hours and $7.50 per hour for any overtime. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-33 Chapter 5: Linear Systems Section 2: Using Algebra 2. Brenda and Cathy can figure out the cost of each top and pair of jeans purchased by the following. Let T = price of one top Let J = price of one pair of jeans Cathy: 3T + 2J = 57 Brenda: 4T + 3J = 81 To use elimination, multiply Cathy’s equation by 4 and Brenda’s equation by M3. Cathy: 12T + 8J = 228 Brenda: -12T N 9J = M243 Add the two equations to get: LJ = L15 J = 15 Substitute this value for J into either equation to find T. 3T + 2(15) = 57 3T = 27 T=9 Interpret the answer: T = $9.00 per top J = $15.00 per pair of jeans Cathy and Brenda paid $9.00 for each top and $15.00 for each pair of jeans. ³ Try-It!é on Your TI.83 Plus and TI.73 Note: The Try-It! exercises cover different functionality available to solve a system of equations. The TI-83 Plus exercise uses the intersect feature that the TI-73 does not have. The TI-73 exercise uses the Solver to solve an independent system. The TI-83 Plus also has a Solver feature. Using the Solver feature on the TI-83 Plus is given as a challenge investigation for the students. (Hints are given below.) The linear system to be solved is the same for both calculators. The problem set, Using Your Calculator to Find Solutions, given after the Try-It!, is identical and is repeated after each Try-It! for your copying convenience. On the TI-83 Plus, students will: • Graph two lines of a system of equations. • Use r to locate the intersection of the lines. • Use the y / intersect feature to find the solution to a system of equations. Check the solution on the home screen using ¿. On the TI-73, students will: • Graph two lines of a system of equations. • If the lines intersect, use the 1 Solver feature to find the X value of the solution to the system. • Find the Y value by using the X on the home screen. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-34 Chapter 5: Linear Systems Reminders for the Section 2: Using Algebra Try-It!™ Exercises TI-83 Plus and TI-73: • Only the variables X and Y are used for graphs and tables. If an equation uses letters other than X and Y, you have to change the variables in the problem to X and Y in order to use some features on the calculators. • Some value is always stored in each variable. You must store the value you want in order to understand how the calculator interprets a variable expression. The calculator does not perform symbolic manipulations. • The intersect feature on the TI-83 Plus and the Solver feature on both the TI-83 Plus and the TI-73 are best used to find the solution to a system with one solution. Students should use their knowledge about the equations of lines to determine if lines are parallel or the same line before working the problem. • In the Solver, bound = {M1åå99, 1åå99} represents the real number line for the calculator. You can make the set smaller to find solutions in a particular interval. See the TI-83 Plus or TI-73 guidebook for more details. TI-83 Plus: • When the intersect feature is used, an error is displayed if the lines are parallel. If the lines are the same line, the calculator will show only one answer. • In the challenge problem, students are asked to investigate using the Solver feature on the TI-83 Plus to find the solution. Find this feature in 0:Solver. See the TI-83 Plus guidebook for details about the Solver feature. You have to enter the equation in the Solver as 0 = L4x + 2 N (2x N 2.5). You have to input a guess for the X solution. Then, use ƒ \ to find the calculator’s numerical solution. Extra Practice: Using Your Calculator to Find Solutions Note: These problems are identical for both the TI-83 Plus and the TI-73. 1a. (3, 0.125) 1b. (2/3, 4) 2a. Answers will vary. Students use their choice of calculator feature to solve for Cathy’s pay rate. 2b. Answers will vary. Students use their choice of calculator feature to solve for Cathy and Brenda’s shopping trip. Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-35 ³ TIp 1: Resetting Your Calculator ³ TIpé 1: Resetting Your Calculator You can easily reset all default, or factory, settings on your calculator from the MEMORY menu. Resetting the default settings ensures that all calculators show exactly the same results in the same form. Resetting defaults does not erase any lists, programs, or variables stored in memory. Note: You should reset your calculator’s default settings to ensure that the results you see on your calculator will match the results in all example screens in the ³ TIps. Resetting your calculator to the defaults For more information Resets all mode and window format defaults. ³ TIp 2: Adjusting Your Calculator Settings Turns off function plots. ³ TIp 3: Graphing a Function in the Standard Window Resets window values to ZStandard. ³ TIp 5: Adjusting the Viewing Window Turns off statistical plots. ³ TIp 7: Creating a Statistical Plot Calculator Keys Used in This ³ TIpé y L 7:Reset 2:Defaults 2:Reset ³ TIpé Highlights Your calculator contains many settings that control the interpretation of results and graphs. For example, you can control how the calculator: • • Displays mathematical results (for example, with floating or fixed decimal notation). Interprets and displays graphs and statistical plots (for example, scale of the X-axis and Y-axis). Note: Types of settings vary between the TI-83 Plus and the TI-73. For more information, see the calculator guidebook. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 1-1 ³ TIp 1: Resetting Your Calculator ³ Try-It!é on Your TI.83 Plus or TI.73 Reset the calculator defaults. To Do This Press 1. Display the MEMORY menu and select Reset. -Ÿ 2. Select Defaults. 2:Defaults 3. Select Reset to reset the calculator. 2:Reset 4. Clear the screen and return to the Home screen. : Display (TI.83 Plus shown) 7:Reset TI-83 Plus TI-73 You are now ready to work through the ³ TIps. Remember, resetting the calculator ensures that you get the same results shown in all ³ TIps examples. After you become familiar with your calculator, it will not be necessary to reset it every time because you will know how you have changed the settings, and you will understand the results the calculator displays in light of those changes. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 1-2 ³ TIp 2: Adjusting Your Calculator Settings ³ TIpé 2: Adjusting Your Calculator Settings You can control how the calculator displays results and graphs by changing settings. For example, the mode settings screen is displayed below (varies slightly for the TI-73). The highlighted settings are the ones that are selected. All highlighted settings are selected. Calculator Keys Used in This ³ TIpé • . • y. ³ TIpé Highlights This ³ TIp shows how to change settings on two frequently-used screens—the mode screen and the window format screen. For more information about settings, see the calculator guidebook. • Mode (.) settings determine how the calculator interprets and/or displays numbers and results. • Window format (y .) settings determine how the calculator interprets and/or displays graphs. Note: The TI-83 Plus and the TI-73 mode and window format screens are slightly different. TI-83 Plus Mode screen TI-73 Window Format Screen Mode screen Window Format Screen In the ³ Try-It! example, you will select new calculator settings following these main steps. 1. Display the setting screen. 2. Press #, $, !, or ", as necessary, to highlight the setting you want to select. 3. Press b to select it. Note: The examples in the next section assume that the calculator has been reset to the default settings as described in ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 2-1 ³ TIp 2: Adjusting Your Calculator Settings ³ Try-It!é on Your TI.83 Plus or TI.73 Change a Mode Setting Change the mode setting so that results display to 2 decimal places. To Do This Press 1. Display the mode settings screen. . Display (TI.83 Plus shown) Note: The TI-73 mode screen varies slightly from the TI-83 Plus. 2. Change the decimal notation setting from Float to 2. # until Float is highlighted " until 2 is highlighted b to select 3. Press - l to return to the Home screen, and clear the current line, if needed. -l : Tip: Pressing - l always takes you back to the Home screen, except in the Topics in Algebra 1 application. 4. 1`479\ 2`897 Calculate 1.479 + 2.897. b The calculator rounds the result to 2 decimal places. 5. Change the decimal notation setting back to Float. Note: Float displays a number up to 10 digits. . # until Float is highlighted b to select 6. Return to the Home screen. -l 7. Calculate 1.479 + 2.897 again. - £ to display the previous entry b Tip: On the TI-73, you can press $ $ to highlight a previous entry on the Home screen, and then press b to paste it to the current line. Topics in Algebra 1 © 2001 Texas Instruments The calculator displays up to 10 digits. ³ TIp 2-2 ³ TIp 2: Adjusting Your Calculator Settings ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Change a Window Format Setting To display a grid on the graph screen, select GridOn from the window format screen. To Do This Press 1. Display the window format screen. -g Display (TI.83 Plus shown) Note: The TI-73 mode screen varies slightly from the TI-83 Plus. 2. Change the grid setting from GridOff to GridOn. # until GridOff is highlighted " until GridOn is highlighted b to select 3. Show the graph screen. Notice both the grid and axes are on. * On Your Own ³ Change your mode setting (.) to scientific notation (Sci). Do some calculations on the Home screen. Try 25¦63. Notice how the calculator displays scientific notation. Is this the way you write scientific notation? Explain how this setting affects your results. ³ Turn the grid off on the graph screen. Check this by pressing *. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 2-3 ³ TIp 3: Graphing a Function in the Standard Window ³ TIpé 3: Graphing a Function in the Standard Window You can graph a function on the graph screen. First, enter the function in the Y= editor, and then you can graph the function in the standard graphing window. Screens on the TI-73 may vary. Y1=2x+3 in the Y= editor Standard Graphing Window Graph of Y1=2x+3 Calculator Keys Used in This ³ TIpé • & • * • ) ³ TIpé Highlights In the ³ Try-It! example, you will graph a function following these main steps. 1. Enter the function in the Y= editor (&). 2. Graph the function on the graph screen (*). Note: If you reset your calculator defaults (³ TIp 1: Resetting Your Calculator), the graphing window sets up the following boundaries for the X and Y values: L10 { X { 10 and L10 { Y { 10. 3. Press ). Then press " or ! to move the cursor from one plotted point to another. Tracing the graph displays the (X,Y) values at the bottom of the screen. For some functions, the graphing window has to be adjusted so that you can see your graph. This ³ TIp uses the standard window settings (L10 { X { 10 and L10 { Y { 10). The function selected for the example on the next page displays the graph in this window. For more information on adjusting window settings on the WINDOW menu, see ³ TIp 5: Adjusting the Viewing Window. Note: The examples in the next section assume that the calculator has been reset to the default settings as described in ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 3-1 ³ TIp 3: Graphing a Function in the Standard Window ³ Try-It!é on Your TI.83 Plus or TI.73 Enter the Function Enter y = 2x + 3 as Y 1=2X+3 in the Y= editor. To Do This Press 1. Enter the Y= editor. & Display (TI-83 Plus shown) Note: The TI-73 Y= editor varies slightly from the TI-83 Plus. The TI-73 lets you enter up to 4 functions. 2. Clear Y1 and enter the function, 2X+3. Note: On the TI-73, use I rather than „. : 2„\3 Note: The equal sign next to the function is highlighted. This means that the function is selected or turned on and will be graphed. If other functions are selected, press # and ! as necessary to position the cursor over the = sign and press enter to turn off the function. Display the Graph Display the graph of Y 1=2X+3 on the graph screen. To Do This Press 1. Display the graph. * Topics in Algebra 1 © 2001 Texas Instruments Display (TI-83 Plus shown) ³ TIp 3-2 ³ TIp 3: Graphing a Function in the Standard Window ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Trace the Graph Trace the (X,Y) values for Y1=2X+3 on the graph screen. To Do This Press 1. Enter trace mode. ) 2. Trace the (X,Y) values along the graph. " and ! Display (TI-83 Plus shown) X and Y values display at the bottom of the screen. 3. Quickly find Y when X=M4. a4 4. Notice that when X=M4 then Y=M5. b Cursor moves to (L4, L5). On Your Own ³ Display a grid on the graph screen by selecting GridOn in the window format screen (- g). ³ Define Y2= MX+6. ³ Graph Y 1=2X+3 and Y2= MX+6 at the same time and compare. Now trace along each function to find the point of intersection. Record the point of intersection. Hint: Use # and $ to move between functions. ³ Graph Y2 only. Hint: You must deselect Y1 so that the calculator does not graph it. To deselect Y1, highlight its equal sign (=) in the Y= editor with the cursor, and then press b. Notice the equal sign is no longer highlighted. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 3-3 ³ TIp 4: Creating a Table ³ TIpé 4: Creating a Table You can automatically create a table of values (X,Y) based on functions in the Y= editor. For example, if Y=2X+3 is defined in the Y= editor, a corresponding table of (X,Y) values could look like this: Y1=2x+3 in the Y= editor Table for Y1=2x+3 Calculator Keys Used in This ³ TIpé • -i • -f ³ TIpé Highlights In the ³ Try-It! example, you will display the table values (X,Y) for a function defined in the Y= editor following these main steps. 1. Define a function (Y1=2X+3) in the Y= editor (&). 2. Set up your table in the TABLE SETUP screen (y -). 3. Display the table (- i). The following settings in the table setup (y -) screen determine how the table is displayed. • TblStart — First X value. • @Tbl — Amount by which X values increase and decrease (for example, if @Tbl=5, then X values increase or decrease by 5). • Auto or Ask — Allows you to choose whether or not to display automatically the independent (X) or dependent (Y) variable values. For more information about table settings, see the calculator guidebook. Note: The examples in the next section assume that the calculator has been reset to the default settings as described in ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 4-1 ³ TIp 4: Creating a Table ³ Try-It!é on Your TI.83 Plus or TI.73 Enter the Function Enter Y1=2X+3 in the Y= editor (&). See ³ TIp 3: Graphing a Function in the Standard Window for step-by-step instructions for defining functions. Your Y= editor should look like this. Change the Table Settings Define the following settings for your table: TblStart=50 and @Tbl=5. To Do This Press 1. Display the table setup screen. y- 2. Change the following settings: TblStart=50, @Tbl=5. : 50 # :5 Note: Indpnt:Auto and Depend:Auto are the defaults. Topics in Algebra 1 © 2001 Texas Instruments Display (TI-83 Plus shown) ³ TIp 4-2 ³ TIp 4: Creating a Table ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Display the Table Display the table for Y1=2X+3 on the graph screen. To Do This Press 1. Display the table. -i 2. Scroll through the table values with the arrow keys. !, ", #, or $ as necessary Display (TI-83 Plus shown) Note: Notice that when you press $ and move to the top of the Y1 column, the cursor moves to Y1 instead of displaying Y-values that are off the screen. The function is displayed on the edit line and can be changed. On Your Own ³ Try to display more (X,Y) values. Set @Tbl=1 (pronounced “delta table”) and display the table again. Then set @Tbl=.1 and display the table. Notice how the table values differ. ³ Enter Y2=MX+6 in the Y= editor. Display the table of values for both Y1 and Y2. Is Y1 ever equal to Y2? Hint: Set TblStart=0 and @Tbl =1 and search through the table. ³ Change the mode (.) setting from Normal to Sci. Display the table. Notice how this affects the table values. ³ Go to the table setup screen and figure out what the Depend: Ask setting does. Hint: To reveal an invisible Y value in the table, place the cursor on that space and press b. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 4-3 ³ TIp 5: Adjusting the Viewing Window ³ TIpé 5: Adjusting the Viewing Window You can adjust the viewing window for a specific graph. For example, with the ZStandard viewing window default values of L10 { X { 10 and L10 { Y { 10, the graph of Y1=50X appears as shown in the screen below on the left. Y increases so rapidly in relation to X that this line is not easily seen in this window. You can adjust the window (') values by scaling the X axis and Y axis to see the graph in a better perspective. Y1=50X viewed with the default window values. Y1=50X viewed with adjusted window values. Calculator Keys Used in This ³ TIpé • ' • ( ³ TIpé Highlights Window values put specific boundaries on the graph screen. ZOOM (q) functions automatically adjust window values for you; however, you can manually set window values using the window (p) menu. You can set minimum and maximum X and Y values (Xmin, Xmax, Ymin, and Ymax), and you can set the scale (distance between tick marks) of the X axis and Y axis (Xscl and Yscl). In the ³ Try-It! example, you will adjust the viewing window values in two ways. • Automatically adjust them by selecting a zoom function from the ZOOM (() menu. • Manually adjust them using the window (') menu. In the example that follows, you see how to change window (p) values to see a different perspective of the graph of Y1=50X. Note: The examples in the next section assume that the calculator has been reset to the default settings as described in ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 5-1 ³ TIp 5: Adjusting the Viewing Window ³ Try-It!é on Your TI.83 Plus or TI.73 Enter the Function Enter Y1=50X in the Y= editor (&). See ³ TIp 3: Graphing a Function in the Standard Window for step-by-step instructions for defining functions. Your Y= editor should look like this. Display the Function on the Graph Screen Note: The window values shown here are the default window values, also equivalent to ZStandard (() (L10 { X { 10; L10 { Y { 10). To Do This Press 1. Graph Y1. s 2. Display the window settings menu. ' Display (TI-83 Plus shown) Note: This window shows (X,Y) values so that L10 { X { 10 and L10 { Y { 10. This is equivalent to the default ZStandard setting. Define a Different Viewing Window for Your Graph View the table of (X,Y) values created by Y1=50X to help you redefine window values. Start the table at L3 and increment the values by 1. See ³ TIp 4: Creating a Table for step-by-step instructions for creating a table. Tip: You can choose any X values for the table. For this exercise, you view a table of values when L3 X 3. When you set the table to start at L3 with increments of 1, you can see the relationship between X and Y near the origin of the graph. Your table should look like this: Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 5-2 ³ TIp 5: Adjusting the Viewing Window ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Define a Different Viewing Window for Your Graph (continued) Notice in the table that when L3 X 3, the Y values range from L150 Y 150. Since Y increases by 50 each time X increases by 1, you can set the Yscl to 50 and leave Xscl set at 1. Now, create a window to see the graph for this domain and range by changing the window values. To Do This Press 1. Display the window settings screen. p Display (TI-83 Plus shown) Note: Window settings screen is slightly different on the TI.73. 2. Define the values so that Xmin = M3, Xmax = 3, Xscl = 1,Ymin = M150, Ymax = 150, and Yscl = 50. Note: This changes the settings so that: L3 { X { 3 and L150 { Y { 150. Since Xscl=1 and Yscl=50, there is one unit between each tick mark on the X axis and 50 units between each tick mark on the Y axis. a3 # (to Xmax) # (to Ymin) a 150 # (to Ymax) # (to Yscl) 150 3 50 TI-73: @X scales automatically, based on Xmin and Xmax. See the calculator guidebook for details. * 3. Graph Y1=50X in this different window. Note: Remember that each tick mark on the X axis represents 1 unit, while each one on the Y axis represents 50. It’s always important to know the scale of your graphs so that you understand the graph you are viewing. 4. Trace the graph. Note: Each tick mark on the X axis is 1 and on the Y axis is 50. 5. Display the Y value when X=2. 2 Tip: You can go directly to any coordinate pair on the graph by entering the value you want. Topics in Algebra 1 ) ! or " as necessary b © 2001 Texas Instruments ³ TIp 5-3 ³ TIp 5: Adjusting the Viewing Window ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Using ZDecimal—the Friendly Window • Display and trace the graph Y2=2x, using ZDecimal window values. ZDecimal lets you trace the X values by tenths (.1, .2, .3, …). • Multiply Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl by 100 in the window menu, and trace the function. This lets you trace the X values by tens. To Do This Press 1. Turn off the graph and table of Y1=50X and enter Y2=2x in the Y= editor. & !Í #"‘ 2„ Note: On the TI-73, use I rather than „. 2. Graph Y2=2x using ZDecimal window values. ( # until ZDecimal is highlighted Í 3. Trace the graph. ) ! or " as necessary Note: Notice that the window is set up so that the X values are tracing by tenths. 4. Multiply Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl by 100. Tip: Press - " to move the cursor to the end of a window setting line. Note: TI-83 Plus: See the calculator guidebook for details about Xres. It is not discussed in this ³ TIp. TI-73: See the calculator guidebook for details about @X. It is not discussed in this ³ TIp. 5. Trace the graph. Note: Notice that the window is set up so that the X values are tracing by tens. Display (TI-83 Plus shown) p Repeat for each value: -" M 100 Í ) ! or " as necessary On Your Own ³ Graph Y1=.1X. How can you set an appropriate viewing window for this graph? Hint: Make a table of values for Y1=.1X and use the table information to change the window values. Try finding the best values to choose when L5X5. If X=5, Y=.1(5)=.5. Therefore, the Y values need to be very small. Try changing Ymin, Ymax, and Yscl. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 5-4 ³ TIp 6: Using Lists ³ TIpé 6: Using Lists You can enter lists of different kinds of data into the list editor. For example, let’s look at two students’ test scores. • Jamal’s test scores are 80, 85, 90, 75, and 85. • Jian’s test scores are 85, 70, 75, 95, and 100. You can create two separate lists in the list editor containing these scores. Then, on the Home screen, you can find the sum of the elements of both lists, and then divide by the number of elements (5) to calculate each student’s test average or mean. You can find the mean or median on the Home screen. L1 lists Jamal’s scores. Calculator Keys Used in This ³ TIpé • - | (to access the ClearAllLists and SetUpEditor commands) • TI-83 Plus: y 9, … • TI-73: 3, - v ³ TIpé Highlights Both the TI-83 Plus and the TI-73 have two menus—OPS and MATH—that contain various list commands that help you find mean, median, and much more. For more information on all menu items, see the calculator guidebook. TI-83 Plus: y9"" TI-73: -v"" In the ³ Try-It! example, you will use the ClearAllLists and SetUpEditor commands, which are located in the CATALOG (- |) on both the TI-83 Plus and the TI-73. These commands clear and set up the list editor as described below. • ClearAllLists erases all elements of all existing lists on your calculator. The list names are still in memory, but the lists are empty. You cannot get the list elements back. • SetUpEditor removes all list names from the list editor except the default list names, L1 through L6. It also creates one blank list after L6. Note: The examples in the next section assume that the calculator has been reset to the default settings as described in ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 6-1 ³ TIp 6: Using Lists ³ Try-It!é on Your TI.83 Plus or TI.73 Set Up the List Editor Use the commands ClearAllLists and SetUpEditor to reset the list editor. To Do This Press 1. Display the list editor. TI-83 Plus: … 1:Edit Note: Your list editor may vary. The following commands will clear and set up lists L1 through L6 in your list editor. 2. Return to the Home screen and clear it. Then clear all lists in the list editor by selecting ClearAllLists. Note: This command clears all elements from the lists. The list names are still in memory, but the lists are empty. Hint: In the CATALOG, you can go to the first item starting with a particular letter: Display (TI-83 Plus shown) TI-73: 3 -l: -| # until ClrAllLIsts is selected b b TI-83 Plus: Press C (). Notice that you are already in ALPHA mode since the Ø displays in the upper right corner. TI-73: Press - t, highlight C, and then press b. 3. Setup the list editor with SetUpEditor. Hint: You also can select SetUpEditor on the TI-83 Plus by pressing … 5. -| # until you highlight SetUpEditor b b 4. Display the list editor. Note: Lists L1 through L6 appear and are clear, and there is a blank list following L6. TI-83 Plus: … 1:Edit TI-73: 3 Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 6-2 ³ TIp 6: Using Lists ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Enter the Lists In the list editor, enter Jamal’s scores as L1={80,85,90,75,85} and Jian’s scores as L2={85,70,75,95,100}. To Do This Press 1. Enter data into L1. 80 # 85 # 90 # 75 # 85 # 2. Enter data into L2. " (to place the cursor under L2) 85 # 70 # 75 # 95 # 100 # Topics in Algebra 1 © 2001 Texas Instruments Display (TI-83 Plus shown) ³ TIp 6-3 ³ TIp 6: Using Lists ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Display the Sum and Average of All Elements in a List On the Home screen, find the sum of Jian’s test scores (L2). To Do This Press Display (TI-83 Plus shown) 1. Return to the Home screen and clear the current line, if needed. y5: Tip: Once you exit the application, you can press y 5 to return to the Home screen from any menu or command. 2. Select sum( from the MATH menu. TI-83 Plus: y9"" 5:sum( TI-73: -v"" 7:sum( 3. Select the list (L2) you want, and display the sum. TI-83 Plus: y9 2:L2 E b TI-73: -v 2:L2 E b 4. Divide L2 by 5 (the number of test scores) to find the average (or mean) of the list of the scores. F5 b Ans = 425 Note: Ans is the previous answer, 425. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 6-4 ³ TIp 6: Using Lists ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Find the Mean and Median of a Set of Data On the Home screen, find the mean and median of Jamal’s test scores (L1). To Do This Press 1. Return to the Home screen and clear the current line, if needed. y5‘ 2. Select mean( from the MATH menu. TI-83 Plus: y9"" Display (TI-83 Plus shown) 3:mean( TI-73: -v"" 3:mean( 3. Select the list (L1) that you want, and then calculate the mean. TI-83 Plus: y 9 1:L1 E b TI-73: - v 1:L1 E b 4. Select median( from the MATH menu. TI-83 Plus: y9"" 4:median( TI-73: -v"" 4:median( 5. Select the list (L1) that you want, and then display the median. TI-83 Plus: y 9 1:L1 E b TI-73: - v 1:L1 E b Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 6-5 ³ TIp 6: Using Lists ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) On Your Own ³ Calculate the 1-variable statistics analysis for L1 using the 1-Var Stats command on the Home screen. Use the arrow keys (# and $) to view all of the information about L1. Select the 1-Var Stats command from this menu: • TI-83 Plus: … CALC menu • TI-73: - v CALC menu Hint: Select the 1-Var Stats command first, and then select L1. For more information about 1-Var guidebook. Stats, see the calculator ³ If you are already familiar with statistical plots, find the quartile statistics in the list of data that was generated above, and draw a box plot of the data. For help, see ³ TIp 7: Creating a Statistical Plot. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 6-6 ³ TIp 7: Creating a Statistical Plot ³ TIpé 7: Creating a Statistical Plot Statistical plots are graphs of data values that have been stored in lists. You can create several types of statistical plots, such as scatter plots, histograms, box plots, and pie charts (TI-73 only). Scatter Plot Histogram Box Plot Pie Chart (TI.73 only) Calculator Keys Used in This ³ TIpé • TI-73: - e • TI-83 Plus: - , • ( ZoomStat ³ TIpé Highlights In the ³ Try-It! example, you will graph a statistical plot following these main steps. 1. Enter all necessary lists in the list editor. 2. Define your statistical plot in the stat plot editor. 3. Display the statistical plot by pressing ( (ZoomStat). Pressing ) allows you to move the cursor from one plotted point to another using " and !. It also displays the values (X,Y) at the bottom of the screen. For more information about statistical plot options, see the calculator guidebook. Note: The examples in the next section assume that the calculator has been reset to the default settings as described in ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 7-1 ³ TIp 7: Creating a Statistical Plot ³ Try-It!é on Your TI.83 Plus or TI.73 You have collected the measurements (in inches) of how far a boy from age 5 to age 14 could throw a ball above his head. The following data was recorded. Create a scatter plot (Ô) based on these lists, where L1 = Age in years and L2 = Distance in inches. Age in years (L1) Distance in inches (L2) 5 8 9 10 12 14 66.9 75.8 77.7 79.9 85.8 91.7 Enter the Lists in the List Editor See ³ TIp 6: Using Lists for step-by-step instructions for entering lists of data (remember to use ClearAllLists and SetUpEditor). Your list editor should look like this: Define the Statistical Plot Define Plot1 as a scatter (Ô) plot where Xlist=L1 and Ylist=L2. To Do This Press 1. Display the STAT PLOTS screen. TI-83 Plus: y, Display (TI-83 Plus shown) TI-73: -e 2. Display the Plot1 settings screen. 1:Plot1 3. Select On to turn on Plot1. b Note: Scatter (Ô), is already selected because you reset the calculator for this ³ TIp. Xlist and Ylist do not have to be changed because, by default, they already match our selected lists. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 7-2 ³ TIp 7: Creating a Statistical Plot ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Display the Statistical Plot on the Graph Screen Display the statistical plot on the graph screen using the ZoomStat command and trace it. To Do This Press 1. Select ZoomStat from the ZOOM menu. TI-83 Plus: ( Display (TI-83 Plus shown) 9:ZoomStat TI-73: ( 7:ZoomStat 2. Trace along the statistical plot. Note: P1:L1,L2 in the upper left corner shows that the lists L1 and L2 contain the data for the graph. The data points are displayed at the bottom of the screen. ) " or ! as necessary L1 and L2 contain the data for this graph. (X,Y) coordinates of the data point at the cursor location. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 7-3 ³ TIp 7: Creating a Statistical Plot ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) On Your Own 1. Enter the following math test scores in L1 and L2. Scores (L1) Frequency (L2) 99 96 92 88 84 78 74 70 66 64 4 4 3 2 3 2 2 1 1 1 2. Set up the histogram (Ò) where Xlist=L1 and Freq=L2 in the STAT PLOTS menu. 3. Graph the histogram using ZoomStat, and then trace ()) the histogram. Letter grades correspond to these test scores: A = 100–90; B = 89–80; C = 79–70; D = 69–60; F = 59–0. Did the graph display the way you expected? Why is it necessary to change the window values manually? 4. Change the window values manually to group the data in intervals of 10. Remember that you want the data grouped in intervals of 10 so you can see how many test scores fall into the grade categories given above. You can tell the calculator to do this by setting Xscl=10 on the graphing window ('). Xmin should be a little less than the lowest score and Xmax should be a little higher than the highest score to display a nice graph. Try setting the window as Xmin=50, Xmax=100, Ymin= M4, Ymax=15, and Yscl=10. 5. Graph and trace the histogram again to see why these settings show the grades grouped by scores that match the letter grades A, B, C, D, and F. Hint: Remember that if you have functions defined and turned on in the Y= editor, the calculator graphs these at the same time as the stat plot. To turn off a function, highlight the = sign next to it, and then press b. Or you can select FnOff. To do this: • On the TI-83 Plus: Press ~ 4:On/Off 2:FnOff • On the TI-73: Topics in Algebra 1 Press - } 2:Y-Vars 6: FnOff © 2001 Texas Instruments ³ TIp 7-4 TIp 8: Finding the Best Line of Fit for a Set of Data ³ TIpé 8: Finding the Best Line of Fit for a Set of Data Both the TI-83 Plus and the TI-73 have regression commands that automatically find the closest equation to your statistical plot data. The LinReg(ax+b) command finds the closest linear equation y=ax+b and displays the values for a (slope) and b (y-intercept) on the Home screen. You can save this equation in the Y= editor so that you can graph the regression equation and the statistical plot data at the same time, and then compare the two. Statistical plot data points Linear Regression line based on statistical plot data Calculator Keys Used in This ³ TIpé • TI-83 Plus: … ~ (CALC menu) LinReg(ax+b) • TI-73: - v ! (CALC menu) LinReg(ax+b) ³ TIpé Highlights This ³ TIp shows you how to use LinReg(ax+b) on the Home screen to find the linear regression of two list names, XList (L1) and YList (L2). You will use the same list values that you used to define the statistical plot. Note: For more information about statistical plots, see ³ TIp 7: Creating a Statistical Plot. In the ³ Try-It! example, you will find a linear regression following these main steps. 1. Select LinReg(ax+b) from the … CALC menu. 2. Enter the two list names that define the statistical plot. 3. Save the equation in the Y= editor. 4. Graph the statistical plot and the linear regression. Note: The examples in the next section assume that the calculator has been reset to the default settings as described in ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 8-1 TIp 8: Finding the Best Line of Fit for a Set of Data ³ Try-It!é on Your TI.83 Plus or TI.73 You have collected the measurements (in inches) of how far a boy could throw a ball above his head from age 5 to age 14. The following data was recorded. Create a scatter (Ô) plot based on these lists, where Xlist=L1 and YList= L2. Age in years (L1) Distance in inches (L2) 5 8 9 10 12 14 66.9 75.8 77.7 79.9 85.8 91.7 Enter the Lists for the Statistical Plot See ³ TIp 6: Using Lists for step-by-step instructions for entering lists of data. Your list editor should look like this: Create a Scatter Statistical Plot for L1 and L2 See ³ TIp 7: Creating a Statistical Plot for step-by-step instructions for creating a scatter statistical plot. Your statistical plot editor and scatter plot (using ZoomStat) should look like this: Stat Plot Editor Graph of Plot Note: The stat plot editor in the TI-73 looks slightly different. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 8-2 TIp 8: Finding the Best Line of Fit for a Set of Data ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Find the Linear Regression for the Scatter Plot Find the regression (the line which best fits the data) for L1 and L2 using LinReg(ax+b). To Do This Press 1. Return to the Home screen, and clear it. -l: 2. Select LinReg(ax+b) from the STAT CALC menu. TI-83 Plus: …" Display (TI-83 Plus shown) 4:LinReg(ax+b) TI-73: -v! 5:LinReg(ax+b) 3. Select the lists L1 and L2. TI-83 Plus: y 9 1:L1 ¡ y 9 2:L2 ¡ TI-73: - v 1:L1 ¡ - v 2:L2 ¡ 4. Display Y-VARS menu. Then select the Y variable (Y1) from the FUNCTION menu. TI-83 Plus: ~ 5. Calculate the regression equation. b 1:Function 1:Y1 TI-73: - } 2:Y-Vars 1:Y1 b & 6. Display the Y= editor. Note: Both Y1 and Plot1 are highlighted. This means that both graph and the plot are turned on. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 8-3 TIp 8: Finding the Best Line of Fit for a Set of Data ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Graph the Statistical Plot and the Linear Regression Graph and trace the statistical plot and linear regression at the same time and compare them. To Do This Press 1. Select the ZoomStat function to display the statistical plot and the regression on the graph screen at the same time. TI-83 Plus: q Display (TI-83 Plus shown) 9:ZoomStat TI-73: q 7:ZoomStat 2. Trace the function or the statistical plot. Notes: The function or plot being traced is displayed in the upper left corner of the screen. The X and Y coordinates display at the bottom of the screen. r | or ~ to trace a function } and † to move between functions Tracing along the stat plot Tracing along the line On Your Own ³ Use the function to predict approximately how far you think the boy can throw the ball above his head at 18 years old. Do you think the line will give a good idea of how high this person will be able to throw the ball when he is 35? Explain your thoughts. Hint: Use the table to determine the value of Y1 when X=18. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 8-4 ³ TIp 9: Sending and Receiving Data between Calculators ³ TIpé 9: Sending and Receiving Data between Calculators You can send and receive data between calculators using the SEND and RECEIVE menus. To connect two calculators using the unit-to-unit cable, which comes packaged with your calculator, use the I/O (input/output) port located at the center of the bottom edge of the calculator. • Insert either end of the unit-to-unit cable into the I/O calculator port. • Insert the other end of the cable into the I/O port of the other calculator. Tip: The cable must be firmly inserted into the I/O port. If you receive a “link error,” make sure the cable is completely inserted. Sending unit Receiving unit I/O ports for cable Calculator Keys Used in This ³ TIpé • TI-83 Plus: y 8 • TI-73: 9 1:Link ³ TIpé Highlights You can send or receive many types of data such as lists, programs, pictures, and applications that can be shared. To communicate between two calculators, you must set up one calculator to send the data and the other calculator to receive the data. Note: You can only link two TI-73 calculators or two TI-83 Plus calculators. You cannot link a TI-73 and a TI-83 Plus. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 9-1 ³ TIp 9: Sending and Receiving Data between Calculators ³ TIp Highlights (continued) In the ³ Try-It! example, you will link two calculators and send lists of data from one calculator to the other following these main steps. Receiving Data After you link the calculators using the unit-to-unit cable, set up one calculator to receive data by following these steps: 1. Display the link menu. • TI-83 Plus: y 8 • TI-73: 9 1:Link 2. Press " to display the RECEIVE menu. 3. Select 1:Receive. The message Waiting… and the busy indicator are displayed. The calculator is ready to receive transmitted items. Sending Data After you link the calculators using the unit-to-unit cable and have one calculator waiting to receive data, set up the other calculator to send data by following these steps: 1. Display the link menu. • TI-83 Plus: y 8 • TI-73: 9 1:Link 2. Select the type of data (for example, lists) that you want to send. The corresponding SELECT screen is displayed. Each SELECT screen is displayed initially with no items selected. Note: The All+ menu item selects all items on your calculator that can be transmitted. The All- menu item deselects all items that you have selected to transmit. Select the type of data. All lists on the calculator are displayed. 3. Press $ and # to move the selection cursor (4) to an item you want to select or deselect. 4. Press b to select or deselect an item. Selected names are marked with a box (0). You can select more than one if you want. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 9-2 ³ TIp 9: Sending and Receiving Data between Calculators ³ Try-It!é on Your TI.83 Plus or TI.73 Transfer L1 and L2 from one calculator to another one. Use L1 and L2 as defined in ³ TIp 6: Using Lists. • L1={80,85,90,75,85} • L2={85,70,75,95,100} To Do This Press 1. On the sending calculator, enter the lists above in the list editor. TI-83 Plus: … 1:Edit Display (TI-83 Plus shown) TI-73: 3 Sending unit 2. Link the two calculators using the unit-to-unit cable. Receiving unit Connect the calculators with the cable using the I/O ports. 3. On the receiving calculator, select Receive. Confirm that Waiting… is displayed on the screen. Your calculator is now ready to receive the lists. 4. On the sending calculator, display the SEND menu, and then select List. TI-83 Plus: y8 " 1:Receive TI-73: 9 1:Link " 1:Receive TI-83 Plus: y8 4: List TI-73: 9 1:Link 4: List 5. Select L1 and L2. Note: The selected lists are marked with a box (0). Topics in Algebra 1 b #b © 2001 Texas Instruments ³ TIp 9-3 ³ TIp 9: Sending and Receiving Data between Calculators ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) 6. Send the lists to the receiving calculator using the TRANSMIT menu. " 7. If the lists have been previously defined, the receiving calculator asks you if you want to: 2:Overwrite 1:Transmit 1: Rename 2: Overwrite 3: Omit 4: Quit Select Overwrite for each list. The name and type of each data item are displayed line by line on the sending unit as the item is transmitted, and then on the receiving unit as each item is accepted. Sending calculator After both lists are transmitted, the message, Done, is displayed on both calculators. Receiving calculator On Your Own ³ Enter a function in the Y= editor and send it to another calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 9-4 ³ TIp 10: Managing Your Calculator’s Memory ³ TIpé 10: Managing Your Calculator’s Memory You can check available memory, manage memory, and get information about your calculator by selecting items from the - Ÿ MEMORY menu. For example, you can find your calculator ID number, which is necessary for registering and installing some applications. A delete menu item lets you delete any type of data (variables, lists, programs, applications, etc.) so that you can set up your calculator with the information you need for your current classes. You can change the data for future classes. TI-73 TI-83 Plus Calculator Keys Used in This ³ TIpé yL ³ TIpé Highlights The About screen displays: • Your calculator operating system version number • Your calculator ID number, which is used to register and install or reinstall an application The operating system version number The calculator ID number TI-83 Plus TI-73 As new operating system versions become available, you can download them from the Texas Instruments website. To obtain some applications from Texas Instruments, you must provide your calculator ID number, which is unique to your calculator. The Mem Mgmt/Del menu item lets you delete variables, lists, programs, applications, etc., so that you can set up your calculator with the information you need for your current classes. The Reset menu item lets you reset your calculator default settings. Resetting all RAM on your calculator: • Restores memory to the factory settings. • Deletes all programs. Note: Resetting RAM does NOT erase applications. To find out more about resetting defaults, see ³ TIp 1: Resetting Your Calculator. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 10-1 ³ TIp 10: Managing Your Calculator’s Memory ³ Try-It!é on Your TI.83 Plus or TI.73 Displaying the About Screen Display your calculator’s About screen and find the operating system version number and the ID number. To Do This Press 1. Select About from the MEMORY menu. -Ÿ Display (TI-83 Plus shown) 1:About 2. Notice the operating system version number under the name of the calculator (1.12) and the ID number composed of 14 numbers and letters. Topics in Algebra 1 © 2001 Texas Instruments ³ TIp 10-2 ³ TIp 10: Managing Your Calculator’s Memory ³ Try-It!é on Your TI.83 Plus or TI.73 (continued) Deleting Items Delete L4 from your calculator’s memory. To Do This Press Display (TI-83 Plus shown) 1. Display the MEMORY menu and select TI-83 Plus: Mem Mgmt/Del. -Ÿ Note: The TI-73 screen looks somewhat different. 2:Mem Mgmt/Del TI-73: -Ÿ 4:Delete 2. Select the category, List. TI-83 Plus: 4:List All existing lists on your calculator display on this screen and the number TI-73: of bytes of RAM that they use (L1=66 3:List bytes, etc.). 3. Move the cursor so that it points to L4. # to L4 4. Delete L4. TI-83 Plus: / Note: Notice that no confirmation message displayed when you deleted this item. Confirmation messages only display when you delete an item from Flash ROM (an application) or when you delete an archived item. If an item has an asterisk (…) next to it, it is archived (TI-83 Plus only). 5. L4 no longer displays in the list editor. Use the arrow key (") to scroll through the lists. Tip: See ³ TIp 6: Using Lists for information on how to use the SetUpEditor to put L4 back into the list editor. Topics in Algebra 1 TI-73: b TI-83 Plus: … 1:Edit " TI-73: 3" © 2001 Texas Instruments ³ TIp 10-3 General Information Hardware and Software Requirements Hardware and Software Requirements Hardware and software Notes TI-73 with version 1.60 or higher of the Graph Explorer software —or— TI-83 Plus or TI-83 Plus Silver Edition with version 1.14 or higher of the operating system software You can download a free copy of the latest Graph Explorer or operating system software from education.ti.com/softwareupdates. Follow the link to Operating Systems. Computer with Microsoft ® Windows® 95/98/2000, Windows NT®, or Apple® Mac® OS 7.1 or higher installed TI-GRAPH LINK™ computer-to-device cable If you do not have this cable, call your distributor, or order the cable from TI’s online store. TI-GRAPH LINK software that is compatible with the TI-73 or the TI-83 Plus —or— TI™ Connect software, which works with all supported models of Flash-based TI graphing devices. You can download free copies of TI-GRAPH LINK and TI Connect software from education.ti.com/softwareupdates. Follow the link to Connectivity Software. Topics in Algebra 1 © 2002 Texas Instruments 1-1 General Information Deleting an Application Deleting an Application TI-73 1. Press y L to display the MEMORY menu. 2. Select Delete. 3. Select Apps. 4. Move the cursor to the application name. 5. Press Í. A confirmation message is displayed. 6. Select Yes to delete the application. TI-83 Plus 1. Press y L to display the MEMORY menu. 2. Select Mem Mgmt/Del. 3. Select Apps. 4. Move the cursor to the application name. 5. Press {. A confirmation message is displayed. 6. Select Yes to delete the application. Topics in Algebra 1 © 2002 Texas Instruments 1-2 General Information Installation Error Messages Installation Error Messages Low Battery Do not attempt to download a Flash application if the low-battery message appears on the home screen. Low battery indication is shown on the initial screen. If you receive this error during an installation, change the batteries before trying again. Invalid Signature or Certificate Either this device does not have a certificate to run the application, or electrical interference caused a link to fail. Try to install the application again. Error in Xmit This problem is usually associated with the unit-to-unit cable and its connection between the devices. Make sure the cable is firmly inserted in the link port of each device. Communication Error This error indicates that the TI Connect™ software (“Unable to communicate with device”) or TI-GRAPH LINK™ software (“Link Transmission Error”) is unable to communicate with the device. The problem is usually associated with the TI-GRAPH LINK cable and its connection to the device or to the computer. • Make sure the cable is firmly inserted in the device link port and the computer. • Verify that the correct cable type is selected in the software link settings. • Verify that the correct communications port (Com Port) is selected in the software link settings. (This does not apply if you use the USB port and TI Connect software.) Archive Full This error occurs when the TI-83 Plus does not have sufficient memory for the application. In order to make room for another application, you must delete an application or archived variables from the TI-83 Plus. Before you delete an application from the TI-83 Plus, you can save it on your computer using TI Connect or TI-GRAPH LINK™ software for the TI-83 Plus. You can reload it to the TI-83 Plus later using TI Connect or TI-GRAPH LINK software. Memory Error This error occurs when the TI-73 does not have sufficient memory for the application. In order to make room for another application, you must delete an application from the TI-73. Before you delete an application from the TI-73, you can save it on your computer using TI Connect software or TI-GRAPH LINK software for the TI-73. You can reload it to the TI-73 later using TI Connect or TI-GRAPH LINK software. Other Errors See Appendix B in the TI-73 guidebook or pages B-6 through B-10 in the TI-83 Plus guidebook for information about the specific error. Topics in Algebra 1 © 2002 Texas Instruments 1-3 General Information Checking Version Numbers and Free Space Checking Version Numbers and Free Space Verify Operating System Version and ID Number The Topics in Algebra 1 application is compatible with TI-73 Graph Explorer software version 1.60 and higher or the TI-83 Plus operating system 1.14 and higher. To verify your operating system version number: 1. From the home screen, press y L. 2. Select ABOUT. The operating system version number is displayed below the product name and has the format x.yy. The ID number appears on the line below the product number. Verify Flash Application Version The version number appears on the information screen below the application name. To display the information screen, do one of the following: • Press Œ, and then select MathHand. —or— • Select INFO from the application’s SELECT AN OPERATION menu. Check Amount of Flash Application Free Space TI-73 1. From the home screen, press y L. 2. Select Check Apps. The Math by Hand application requires one free space to load the application. For more information about memory and memory management, refer to the TI-73 guidebook. TI-83 Plus 1. From the home screen, press y L. 2. Select Mem Mgmt/Del. The Math by Hand application requires at least 16,384 bytes of ARC FREE (Flash), or one space, to load the application. For more information about memory and memory management, refer to the TI-83 Plus guidebook. Topics in Algebra 1 © 2002 Texas Instruments 1-4 General Information Support and Service Information Texas Instruments (TI) Support and Service Information For General Information E-mail: ti-cares@ti.com Phone: 1.800.TI.CARES (1.800.842.2737) For US, Canada, Mexico, Puerto Rico, and Virgin Islands only Home page: http://education.ti.com/ For Technical Questions Phone: 1.972.917.8324 For Product (Hardware) Service Customers in the US, Canada, Mexico, Puerto Rico, and Virgin Islands: Always contact TI Customer Support before returning a product for service. All other customers: Refer to the leaflet enclosed with your product (hardware) or contact your local TI retailer/distributor. 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Manufacturer is Texas Instruments Incorporated, 7800 Banner Drive, M/S 3962, Dallas, Texas 75251. 04/13/00 I:\Legal Information\Support&Serv, Warranty, & Licenses\Licenses\Calcapps\Standard\Sold_doc\sold-cal-app-eng.doc [Item #]

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