Linear Functions I by Frank C. Wilson Activity Collection Featuring the following real-world contexts: Choosing a Cell Phone Plan - T-Mobile Choosing a Cell Phone Plan - Verizon College Graduates Michigan Converting Temperatures Government Nutrition Program Alaska Government Nutrition Program Hawaii Government Nutrition Program Longer Life Spans Making Money Yahoo! Music Downloads www.makeitreallearning.com
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Table of Contents Introduction... 4 Activity Objectives... 5 Choosing a Cell Phone Plan T-Mobile: Investigating Linear Functions... 6 Solutions... 8 Choosing a Cell Phone Plan: Verizon: Investigating Linear Functions... 10 Solutions... 12 College Graduates Michigan: Using Linear Function Models... 14 Solutions... 16 Converting Temperatures: Solving Linear Equations... 18 Solutions... 20 Government Nutrition Program - Alaska: Linear Function Modeling... 22 Solutions... 24 Government Nutrition Program - Hawaii: Linear Function Modeling... 26 Solutions... 28 Government Nutrition Program: Linear Function Modeling... 30 Solutions... 32 Longer Life Spans: Using Linear Function Models... 34 Solutions... 36 Making Money: Working with Direct Proportionality... 38 Solutions... 40 Yahoo! Music Downloads: Working with Linear Equations... 42 Solutions... 44 About the Author... 46 Other Books in the Make It Real Learning Series... 46
Introduction When am I ever going to use this? It is a question that has plagued teachers and learners for decades. Now, with the help of the Make It Real Learning workbook series, you can answer the question. The Linear Functions I workbook focuses on real-world situations that may be effectively modeled by linear equations. Whether choosing a cell phone plan or downloading music, learners will discover how to use linear functions to help them become better informed consumers. Rest assured that each activity integrates real world information not just realistic data. These are real companies (e.g. Yahoo!, T-Mobile) and real world issues. The mathematical objectives of each activity are clearly specified on the Activity Objectives page following this introduction. Through the workbook series, we have consistently sought to address the content and process standards of the National Council of Teachers of Mathematics. There are multiple ways to use the activities in a teaching environment. Many teachers find that the activities are an excellent tool for stimulating mathematical discussions in a small group setting. Due to the challenging nature of each activity, group members are motivated to brainstorm problem solving strategies together. The interesting real world contexts motivate them to want to solve the problems. The activities may also be used for individual projects and class-wide discussions. As a ready-resource for teachers, the workbook also includes completely worked out solutions for each activity. To make it easier for teachers to assess student work, the solutions are included on a duplicate copy of each activity. We hope you enjoy the activities! We continue to increase the number of workbooks in the Make It Real Learning workbook series. Please visit www.makeitreallearning.com for the most current list of activities. Thanks! Frank C. Wilson Author 4
Linear II Activity Objectives Activity Title Counting Carbohydrates: Working with Linear Systems (p. 6) Investing in Entertainment: Using Linear Programming (p. 10) Investing in Fast Food: Using Linear Programming (p. 14) Owning Part of a Clothing Company: Using Linear Programming (p. 18) Selling Nuts: Using Linear Inequalities (p. 22) Travel Options - California: Working with Linear Systems (p. 26) Travel Options - Florida: Working with Linear Systems (p. 30) Travel Options - Utah: Working with Linear Systems (p. 34) Using Resources Wisely: Investigating Linear Programming (p. 38) Using Resources Wisely #2: Investigating Linear Programming (p. 42) Mathematical Objectives Create a linear system of equations for a real-world context Graph linear functions Find the point of intersection of two lines Solve a system of linear equations Create a system of linear inequalities from a verbal description Graph the solution region of a system of linear inequalities Create a system of linear inequalities from a verbal description Graph the solution region of a system of linear inequalities Create a system of linear inequalities from a verbal description Graph the solution region of a system of linear inequalities Create and graph linear inequalities Find the point of intersection of two lines Create and graph a linear function model Estimate a point of intersection from a graph of two lines Solve a system of linear equations algebraically Create and graph a linear function model Estimate a point of intersection from a graph of two lines Solve a system of linear equations algebraically Create and graph a linear function model Estimate a point of intersection from a graph of two lines Solve a system of linear equations algebraically Create a linear programming problem from a verbal description Graph the feasible region of a linear programming problem Find the corner points of a solution region Create a linear programming problem from a verbal description Graph the feasible region of a linear programming problem Find the corner points of a solution region 5
Choosing a Cell Phone Plan T-Mobile Investigating Linear Functions I n 2008, T-Mobile offered the following cell phone plans to consumers. T Mobile Individual Plans Name Monthly Anytime Minutes Monthly Fee Charge for Extra Minutes Basic 300 minutes $29.99 $0.40 per minute Value 600 minutes $39.99 $0.40 per minute Plus 1000 minutes $49.99 $0.40 per minute Source: www.t-mobile.com 1. If a customer expects to use 400 minutes per month, which plan is the best value? Basic Plan: $29.99 + (100 extra minutes)($0.40 per minute) = $69.99 Value Plan: $39.99 The Value Plan costs less that the Basic Plan for a customer who uses 400 minutes. 2. A cell phone plan consists of a fixed cost (the monthly fee) and a variable cost (charge for extra minutes). Find a linear function that gives the total cost of each plan when x extra minutes are used. Total Cost = Fixed Cost + Variable Cost = Monthly Fee + ( Cost per Extra Minute) ( Extra Minute) = Monthly Fee + ( Cost per Extra Minute) ( x ) Basic Plan: C = 29.99 + 0.40x Value Plan: C = 39.99 + 0.40x Plus Plan: C = 49.99 + 0.40x 3. Using the cost functions from (2) above, determine the cost of using 600 total minutes and 1000 total minutes with each plan. Then determine which plan is the best deal for each level. For the Basic Plan, 600 minutes requires 300 minutes more than the included minutes and 1000 minutes requires 700 minutes more than the included minutes. For the Value Plan, 0 and 400 extra minutes are needed, respectively. For the Plus Plan, no extra minutes are needed. Plan 600 minutes 1000 minutes Basic C = 29.99 + 0.40 ( 300 ) = $149.99 C = 29.99 + 0.40 ( 700 ) = $309.99 Value C = $39.99 ** C = 39.99 + 0.40 ( 400 ) = $199.99 Plus C = $49.99 C = $49.99 ** The Value Plan is best for 600 minutes. The Plus Plan is best for 1000 minutes. 6
4. The number of minutes a person uses each month often varies. Suppose that a T Mobile customer uses the total number of monthly minutes shown below. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 531 599 615 627 578 670 712 623 598 637 601 654 Which plan would cost the consumer the least amount money over the year? From (3) we learned that using more than the included minutes is very costly. Since more than 300 minutes we used every month, we throw out the Basic Plan. We need only determine if the Value Plan or the Plus Plan is the better value. This consumer used more that 600 minutes in every month except January, February, May and September. To determine the total number of extra minutes used, we add up the minutes for the other eight months and subtract the included minutes. ( 615 + 627 + 670 + 712 + 623 + 637 + 601 + 654) 8( 600) = 339 In total, the person used 339 extra minutes throughout the year. The basic fee for the 600 Plan is $39.99 per month so the person s annual cost is AnnualCost = ( Monthly fee) ( months) + ( cost per minute) ( extra minutes) = ( 39.99)( 12) + ( 0.40)( 339) = 479.88 + 135.60 = 615.48 The annual cost for the Value Plan is $615.48. The Plus Plan costs $49.99 per month for all 12 months for a total cost of $599.88. The Plus Plan is the better value for this consumer. 5. At what number of extra minutes does the annual cost of the Plus Plan become less expensive than the Value Plan? The annual cost of the Value Plan if no extra minutes are used is ( 39.99) ( 12 ) = $479.88. Extra minutes are $0.40 each. The annual cost of the Plus Plan if no extra minutes is used is ( 49.99)( 12 ) = $599.88. We need to solve the following equation. Value Plan Cost = Lowest Cost for Plus Plan 479.88 + 0.40x = 599.88 0.40x = 120 x = 300 If the total number of extra minutes per year remains at 300 minutes or below, the Value Plan is the least expensive of the two plans. Above 300 minutes, the Plus Plan is the better value. 6. Which plan would you recommend to a consumer who expects to use between 500 and 650 minutes each month? Explain your reasoning. From (5) we learned that as long as the total number of extra minutes used in a year remains below 300 minutes, the Value Plan is the best value. (This is equivalent to 25 extra minutes per month.) However, if the consumer uses 650 minutes each month, the number of extra minutes for the year is 600. These 600 extra minutes will cost $240. We advise the consumer to choose the Value Plan and try to keep the number of minutes used monthly between 500 and 625 minutes. 7
Choosing a Cell Phone Plan T-Mobile Investigating Linear Functions I n 2008, T-Mobile offered the following cell phone plans to consumers. T Mobile Individual Plans Name Monthly Anytime Minutes Monthly Fee Charge for Extra Minutes Basic 300 minutes $29.99 $0.40 per minute Value 600 minutes $39.99 $0.40 per minute Plus 1000 minutes $49.99 $0.40 per minute Source: www.t-mobile.com 1. If a customer expects to use 400 minutes per month, which plan is the best value? Basic Plan: $29.99 + (100 extra minutes)($0.40 per minute) = $69.99 Value Plan: $39.99 The Value Plan costs less that the Basic Plan for a customer who uses 400 minutes. 2. A cell phone plan consists of a fixed cost (the monthly fee) and a variable cost (charge for extra minutes). Find a linear function that gives the total cost of each plan when x extra minutes are used. Total Cost = Fixed Cost + Variable Cost = Monthly Fee + ( Cost per Extra Minute) ( Extra Minute) = Monthly Fee + ( Cost per Extra Minute) ( x ) Basic Plan: C = 29.99 + 0.40x Value Plan: C = 39.99 + 0.40x Plus Plan: C = 49.99 + 0.40x 3. Using the cost functions from (2) above, determine the cost of using 600 total minutes and 1000 total minutes with each plan. Then determine which plan is the best deal for each level. For the Basic Plan, 600 minutes requires 300 minutes more than the included minutes and 1000 minutes requires 700 minutes more than the included minutes. For the Value Plan, 0 and 400 extra minutes are needed, respectively. For the Plus Plan, no extra minutes are needed. Plan 600 minutes 1000 minutes Basic C = 29.99 + 0.40 ( 300 ) = $149.99 C = 29.99 + 0.40 ( 700 ) = $309.99 Value C = $39.99 ** C = 39.99 + 0.40 ( 400 ) = $199.99 Plus C = $49.99 C = $49.99 ** The Value Plan is best for 600 minutes. The Plus Plan is best for 1000 minutes. 8
4. The number of minutes a person uses each month often varies. Suppose that a T Mobile customer uses the total number of monthly minutes shown below. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 531 599 615 627 578 670 712 623 598 637 601 654 Which plan would cost the consumer the least amount money over the year? From (3) we learned that using more than the included minutes is very costly. Since more than 300 minutes we used every month, we throw out the Basic Plan. We need only determine if the Value Plan or the Plus Plan is the better value. This consumer used more that 600 minutes in every month except January, February, May and September. To determine the total number of extra minutes used, we add up the minutes for the other eight months and subtract the included minutes. ( 615 + 627 + 670 + 712 + 623 + 637 + 601 + 654) 8( 600) = 339 In total, the person used 339 extra minutes throughout the year. The basic fee for the 600 Plan is $39.99 per month so the person s annual cost is AnnualCost = ( Monthly fee) ( months) + ( cost per minute) ( extra minutes) = ( 39.99)( 12) + ( 0.40)( 339) = 479.88 + 135.60 = 615.48 The annual cost for the Value Plan is $615.48. The Plus Plan costs $49.99 per month for all 12 months for a total cost of $599.88. The Plus Plan is the better value for this consumer. 5. At what number of extra minutes does the annual cost of the Plus Plan become less expensive than the Value Plan? The annual cost of the Value Plan if no extra minutes are used is ( 39.99) ( 12 ) = $479.88. Extra minutes are $0.40 each. The annual cost of the Plus Plan if no extra minutes is used is ( 49.99)( 12 ) = $599.88. We need to solve the following equation. Value Plan Cost = Lowest Cost for Plus Plan 479.88 + 0.40x = 599.88 0.40x = 120 x = 300 If the total number of extra minutes per year remains at 300 minutes or below, the Value Plan is the least expensive of the two plans. Above 300 minutes, the Plus Plan is the better value. 6. Which plan would you recommend to a consumer who expects to use between 500 and 650 minutes each month? Explain your reasoning. From (5) we learned that as long as the total number of extra minutes used in a year remains below 300 minutes, the Value Plan is the best value. (This is equivalent to 25 extra minutes per month.) However, if the consumer uses 650 minutes each month, the number of extra minutes for the year is 600. These 600 extra minutes will cost $240. We advise the consumer to choose the Value Plan and try to keep the number of minutes used monthly between 500 and 625 minutes. 9