Periodic & Piecewise I
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Activity Collection
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Featuring real-world contexts:
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Cell Phone Plan Pricing -Verizon
Credit Card Balance Transfer #2
Credit Card Balance Transfer #3
Custom Greeting Cards
Hours of Daylight
Hours of Daylight - Anchorage
Hours of Daylight - Easter Island
Hours of Daylight - Perth
Hours of Daylight - Phoenix
The Cost of Entertainment
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www.MakeItRealLearning.com
Frank C. Wilson
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Cover art: Blaine C. Wilson
©2009 by Make It Real Learning Company
With the purchase of this workbook, license is granted for one (1) teacher to copy the
activities in this workbook for use in classes and professional development
workshops. Copying pages in this workbook for any other use is prohibited without
written consent from Make It Real Learning Company. For permissions, visit
www.MakeItRealLearning.com and complete the Contact Us form.
Table of Contents
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Introduction ........................................................................................................................... 4
Activity Objectives ............................................................................................................... 5
Cell Phone Plan Pricing -Verizon: Investigating Piecewise Functions ......................... 6
Solutions ......................................................................................................................... 8
Credit Card Balance Transfer #2: Working with Financial Formulas .......................... 10
Solutions ........................................................................................................................ 12
Credit Card Balance Transfer #3: Working with Financial Formulas .......................... 14
Solutions ........................................................................................................................ 16
Custom Greeting Cards: Working with Piecewise Functions ........................................ 18
Solutions ........................................................................................................................ 20
Hours of Daylight - Anchorage: Working with Sinusoidal Models ............................. 22
Solutions ........................................................................................................................ 24
Hours of Daylight - Easter Island: Working with Sinusoidal Models .......................... 26
Solutions ........................................................................................................................ 28
Hours of Daylight - Perth: Working with Sinusoidal Models ....................................... 30
Solutions ........................................................................................................................ 32
Hours of Daylight - Phoenix: Working with Sinusoidal Models .................................. 34
Solutions ........................................................................................................................ 36
Hours of Daylight: Using Inverse Trigonometric Functions .......................................... 38
Solutions ........................................................................................................................ 40
The Cost of Entertainment: Working with Piecewise Functions ................................... 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 Periodic and Piecewise Functions I workbook focuses on real-world situations
that may be effectively analyzed using periodic and piecewise functions. From
determining whether it makes sense to do a credit card balance transfer to
forecasting the hours of daylight at locations around the world, learners get to use
mathematics in meaningful ways. Rest assured that each activity integrates real
world information not just “realistic” data. These are real organizations (e.g. Verizon,
Disneyland) and real world issues.
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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.
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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
Periodic and Piecewise Functions I Activity Objectives
Activity Title
Mathematical Objectives
Custom Greeting Cards:
Working with Piecewise Functions (p. 18)
Hours of Daylight - Anchorage:
Working with Sinusoidal Models (p. 22)
Hours of Daylight - Easter Island:
Working with Sinusoidal Models (p. 26)
Hours of Daylight - Perth:
Working with Sinusoidal Models (p. 30)
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Hours of Daylight - Phoenix:
Working with Sinusoidal Models (p. 34)
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Credit Card Balance Transfer #3:
Working with Financial Formulas (p. 14)
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Credit Card Balance Transfer #2:
Working with Financial Formulas (p. 10)
Create and graph a piecewise function model
Evaluate a function at a given value from an equation or graph
Use numerical results to make informed consumer decisions
Use annuity formulas to find the future value of an investment
Create a piecewise function to model a real-world context
Use numerical results to make informed consumer decisions
Use annuity formulas to find the future value of an investment
Create and graph piecewise functions to model real-world data
Use graphical results to make informed consumer decisions
Create and graph a piecewise function model
Evaluate a function at a given value from an equation or graph
Use numerical results to make informed consumer decisions
Determine the periodic function that will best fit a scatter plot
Calculate the midline and amplitude of a sinusoidal model
Calculate the period of a periodic data set
Create a sinusoidal model for a periodic data set algebraically
Determine the periodic function that will best fit a scatter plot
Calculate the midline and amplitude of a sinusoidal model
Calculate the period of a periodic data set
Create a sinusoidal model for a periodic data set algebraically
Determine the periodic function that will best fit a scatter plot
Calculate the midline and amplitude of a sinusoidal model
Calculate the period of a periodic data set
Create a sinusoidal model for a periodic data set algebraically
Determine the periodic function that will best fit a scatter plot
Calculate the midline and amplitude of a sinusoidal model
Calculate the period of a periodic data set
Create a sinusoidal model for a periodic data set algebraically
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Cell Phone Plan Pricing -Verizon:
Investigating Piecewise Functions (p. 6)
Hours of Daylight: Using Inverse
Trigonometric Functions (p. 38)
Use inverse trig functions to solve trigonometric equations
Create a sinusoidal function model for a periodic data set
The Cost of Entertainment:
Working with Piecewise Functions (p. 42)
Create and graph a piecewise function from a verbal description
Evaluate a piecewise function at a given value
Use numerical results to make informed consumer decisions
Cell Phone Plan Pricing - Verizon
Investigating Piecewise Functions
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n 2008, Verizon offered the following cell phone plans to consumers. (Source: www.verizon.com)
Verizon: Nationwide Basic
Plan
Monthly Anytime Minutes
Monthly Fee Charge for Extra Minutes
Plan 1
450 minutes
$39.99
$0.45 per minute
Plan 2
900 minutes
$59.99
$0.40 per minute
Plan 3
1350 minutes
$79.99
$0.30 per minute
1. Create a piecewise function model for the monthly cell phone cost as a function of the total number of
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minutes used in the month for each of the three plans.
39.99
0 ≤ t ≤ 450
⎧
Plan1: c (t ) = ⎨
t > 450
⎩39.99 + 0.45 (t − 450 )
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59.99
0 ≤ t ≤ 900
⎧
Plan 2: c (t ) = ⎨
t > 900
⎩59.99 + 0.40 (t − 900 )
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79.99
0 ≤ t ≤ 1350
⎧
Plan 3: c (t ) = ⎨
t > 1350
⎩79.99 + 0.30 (t − 1350 )
2. Graph each of the functions from (1) on the axes below.
3. Referring to the graph in (2), determine which of the three plans is least expensive for each of the four
usage levels: 475 minutes, 800 minutes, and 1100 minutes.
For 475 minutes, Plan 1 is the least expensive.
For 800 minutes, Plan 2 is the least expensive.
For 1100 minutes, Plan 3 is the least expensive.
4. Use the function equations in (1) to verify the accuracy of the results in (3).
Minutes
475
Plan 1
39.99 + 0.45 ( 475 − 450 ) = $51.24
Plan 2
$59.99
Plan 3
$79.99
800
39.99 + 0.45 (800 − 450 ) = $197.49
$59.99
$79.99
1100
39.99 + 0.45 (1100 − 450 ) = $332.49
59.99 + 0.40 (1100 − 900 ) = $139.99
$79.99
5. The table below shows a person’s cell phone usage (in minutes) over a year.
Nov
820
Dec
938
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Jan
Feb
Mar
Apr May Jun
Jul
Aug
Sep
Oct
512
598
627
702
639
821
915
979
815
911
Which plan would cost the consumer the least amount money over the year?
Mar
627
177
0
0
Apr
702
252
0
0
May
639
189
0
0
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Feb
598
148
0
0
Sa
Jan
Minutes 512
450
62
900
0
1350
0
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We must determine the total number of extra minutes used over the course of the year. To do this, we
subtract the number of included minutes from the total minutes used and record the results in the table.
Jun
821
371
0
0
Jul
915
465
15
0
Aug
979
529
79
0
Sep
815
365
0
0
Oct
911
461
11
0
Nov
820
370
0
0
Dec
938
488
38
0
The total number of extra minutes for the 450-minute plan is 3877. Each of these extra minutes costs
$0.45. The monthly cost is $39.99. Therefore, the annual cost is
annual cost = ( $39.99 per month ) (12 months ) + ( $0.45 per minute ) ( 3877 minutes )
= $2224.53
The total number of extra minutes for the 900-minute plan is 143. Each of these extra minutes costs
$0.40. The monthly cost is $59.99. Therefore, the annual cost is
annual cost = ( $59.99 per month ) (12 months ) + ( $0.40 per minute ) (143 minutes )
= $777.08
There are no extra minutes for the 1350-minute plan. Therefore, the annual cost is
annual cost = ( $79.99 per month ) (12 months )
= $959.88
Of the three plans, the 900-minute plan (Plan 2) is the best value.
Cell Phone Plan Pricing - Verizon
Investigating Piecewise Functions
I
n 2008, Verizon offered the following cell phone plans to consumers. (Source: www.verizon.com)
Verizon: Nationwide Basic
Plan
Monthly Anytime Minutes
Monthly Fee Charge for Extra Minutes
Plan 1
450 minutes
$39.99
$0.45 per minute
Plan 2
900 minutes
$59.99
$0.40 per minute
Plan 3
1350 minutes
$79.99
$0.30 per minute
1. Create a piecewise function model for the monthly cell phone cost as a function of the total number of
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minutes used in the month for each of the three plans.
39.99
0 ≤ t ≤ 450
⎧
Plan1: c (t ) = ⎨
t > 450
⎩39.99 + 0.45 (t − 450 )
m
59.99
0 ≤ t ≤ 900
⎧
Plan 2: c (t ) = ⎨
t > 900
⎩59.99 + 0.40 (t − 900 )
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79.99
0 ≤ t ≤ 1350
⎧
Plan 3: c (t ) = ⎨
t > 1350
⎩79.99 + 0.30 (t − 1350 )
2. Graph each of the functions from (1) on the axes below.
3. Referring to the graph in (2), determine which of the three plans is least expensive for each of the four
usage levels: 475 minutes, 800 minutes, and 1100 minutes.
For 475 minutes, Plan 1 is the least expensive.
For 800 minutes, Plan 2 is the least expensive.
For 1100 minutes, Plan 3 is the least expensive.
4. Use the function equations in (1) to verify the accuracy of the results in (3).
Minutes
475
Plan 1
39.99 + 0.45 ( 475 − 450 ) = $51.24
Plan 2
$59.99
Plan 3
$79.99
800
39.99 + 0.45 (800 − 450 ) = $197.49
$59.99
$79.99
1100
39.99 + 0.45 (1100 − 450 ) = $332.49
59.99 + 0.40 (1100 − 900 ) = $139.99
$79.99
5. The table below shows a person’s cell phone usage (in minutes) over a year.
Nov
820
Dec
938
file
Jan
Feb
Mar
Apr May Jun
Jul
Aug
Sep
Oct
512
598
627
702
639
821
915
979
815
911
Which plan would cost the consumer the least amount money over the year?
Mar
627
177
0
0
Apr
702
252
0
0
May
639
189
0
0
m
Feb
598
148
0
0
Sa
Jan
Minutes 512
450
62
900
0
1350
0
ple
We must determine the total number of extra minutes used over the course of the year. To do this, we
subtract the number of included minutes from the total minutes used and record the results in the table.
Jun
821
371
0
0
Jul
915
465
15
0
Aug
979
529
79
0
Sep
815
365
0
0
Oct
911
461
11
0
Nov
820
370
0
0
Dec
938
488
38
0
The total number of extra minutes for the 450-minute plan is 3877. Each of these extra minutes costs
$0.45. The monthly cost is $39.99. Therefore, the annual cost is
annual cost = ( $39.99 per month ) (12 months ) + ( $0.45 per minute ) ( 3877 minutes )
= $2224.53
The total number of extra minutes for the 900-minute plan is 143. Each of these extra minutes costs
$0.40. The monthly cost is $59.99. Therefore, the annual cost is
annual cost = ( $59.99 per month ) (12 months ) + ( $0.40 per minute ) (143 minutes )
= $777.08
There are no extra minutes for the 1350-minute plan. Therefore, the annual cost is
annual cost = ( $79.99 per month ) (12 months )
= $959.88
Of the three plans, the 900-minute plan (Plan 2) is the best value.