WCPSS Math 1 Linear & Exponential Functions · LINEAR & EXPONETIAL FUNCTIONS Mathematics Vision...

The Mathematics Vision Project

Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

WCPSS Math 1 Unit 2: MVP Module 2

Linear & Exponential Functions

SECONDARY

MATH ONE

An Integrated Approach

SECONDARY MATH 1 // MODULE 2

LINEAR & EXPONETIAL FUNCTIONS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 2 - TABLE OF CONTENTS

LINEAR AND EXPONENTIAL FUNCTIONS

2.1 Piggies and Pools – A Develop Understanding Task Page 9

Introducing continuous linear and exponential functions

(NC.M1.F-IF.3, NC.M1.F-BF.1a, NC.M1.F-LE.1)

READY, SET, GO Homework: Linear and Exponential Functions 2.1

2.2 Shh! Please Be Discreet (Discrete!) – A Solidify Understanding Task Page 15

Connecting context with domain and distinctions between discrete and continuous functions

(NC.M1.F-IF.3, NC.M1.F-BF.1a, NC.M1.F-LE.1, NC.M1.A-REI.10)


2.3 Linear Exponential or Neither – A Practice Understanding Task Page 23

Distinguishing between linear and exponential functions using various representations

(NC.M1.F-BF.1a, NC.M1.F-LE.1)


2.4 Getting Down to Business – A Solidify Understanding Task 3

Comparing growth of linear and exponential models

(NC.M1.F-BF.1a, NC.M1.F-BF.2, NC.M1.F-LE.3, NC.M1.F-LE.5, NC.M1.F-IF.7, NC.M1.F-IF.9)


2.5 Making My Point – A Solidify Understanding Task Page 39

Interpreting equations that model linear and exponential functions

(NC.M1.A-SSE.1a, NC.M1.A-CED.2, NC.M1.F-LE.5)


2.0 Do You Have the Power - A Develop Understanding Task Page 1

Introducing continuous linear and exponential functions (NC.M1.N-RN.2)



LINEAR & EXPONETIAL FUNCTIONS


2.7 I Can See-Can’t You? Page 65

Calculating and interpreting the average rate of change of a function in a given interval

(NC.M1.F-IF.6)


2.6 Form Follows Function – A Solidify Understanding Task 7

Building fluency and efficiency in working with linear and exponential functions in their various forms

(NC.M1.BF-1a, NC.M1.F-LE.5, NC.M1.F-IF.7, NC.M1.F-IF.9)


Page 572.6b Up a Little, Down a Little – A Solidify Understanding Task

Compound Interest

(NC.M1.F-LE.5, NC.M1.F-IF.7, NC.M1.F-IF.9, NC.MA.A-SSE.1a, NC.M1.A-CED.2)

READY, SET, GO Homework: Linear and Exponential Functions 2.6b

2.6C What Makes a Population Change – A Solidify Understanding Task Page 61

Interpreting rates of growth and decay

(NC.M1.F-LE.5, NC.M1.A-SSE.1a, NC.M1.F-IF.8b)

READY, SET, GO Homework: Linear and Exponential Functions 2.6 c

2.0 “Do you have the power?” A Develop Understanding Task

In Unit 1, you explored many examples of geometric sequences such as the one shown below:

fn = fn-1 2 ; f(1) = 5

1. Show how you would use this recursive formula to determine the 9th term in the sequence.

2. Write the explicit formula for this sequence. Use the explicit formula to determine the 9th

term in sequence.

3. Which form is more efficient for determining the 9th term in the sequence? Explain.

4. As you saw from the difference in your explicit and recursive rules for the sequence above,exponents can be used to write repeated multiplication more concisely. It is much moreefficient to write 35 3. On the other hand, the expanded form can bebeneficial in understanding combinations of exponential expressions, so both forms havetheir value.a. Fill in the table below in order to learn more about products of exponential expressions.

Product Expanded Form of Product Simplified Form

24 22 26

35 2

(- - - - - -4)

512

b. Using your observations from the table above, how could you write x20 7 in simplifiedform?

Page 1

c. Write a rule that you think will always work to find a simplified form of an exponentialproduct like ax y.

5. a. Fill in the table below in order to learn more about quotients of exponential expressions.

Quotient Expanded Form of Quotient Simplified Form22 7 7 7 7 7 77 7 785

b. Using your observations from the table above, how could you write the expression belowin simplified form?

c. Write a rule that will always work to find a simplified form of an exponential quotient like

.

6. a. Use your calculator to evaluate each of the following exponential expressions:

50 = _____ 70= _____ -30 = _____ 1,566,7920 = ______

b. You should have noticed that any base raised to a power of 0 is equal to 1 ( 0 = 1). Whydo you think that is the case? You may want to reference the rule you made in problem 5c.

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7. a. Use your rule from problem 5c to write the following expression in simplified form:

b. Write the expanded form of the expression , then simplify.

c. How do your simplified expressions from part a and part b compare?

8. Stanley, after completing problems 6 and 7, decided to create a general rule to explain whatto do with negative exponents. For each step in Stanley’s reasoning, state whether you agreeor disagree and why.= (1)= (2) = (3)

Therefore, =

Page 3

9. Fill in the table below in order to learn about powers of products and powers of quotients ofexponential expressions.

Power of Product/Quotient Expanded Form Simplified Form(2 ) 16( ) ( )( )( )3

10. Describe a general approach for finding powers of products such as (x2y3)3.

11. Describe a general approach for finding powers of quotients such as .

Page 4

2.0

READYTopic: notation for sequences

1. The sequence below shows the number of trees that a nursery plants each year.2, 8, 32, 128 …

Let represent the current term in the sequence and represent the previous term in thesequence. Which formula could be used to determine the number of trees the nursery willplant in year ?

A) = 4B) =C) = 2 + 4D) = + 6

2. Given the sequence defined by the function = + 12 with = 4, write an explicitfunction rule.

3. Given the sequence defined by the function = with = 424, write an explicit function rule.

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SETTopic: Simplifying algebraic expressions using properties of exponents

1. · 2. 7 · 9 3. 5 5

4. 5. 6.

7. 8. 9. ( )

10. 12 11.( ) 12.( ) ( )

13. 14. 15.

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GO!

Topic: Solving two-step equations

1. 4 + 3 = 15 2. 2 6 = 12

3. 4 = 3 4. 3 + 8 = 7

5. + 5 = 1 6. Create a two-step equation whose solutionis 8.

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LINEAR & EXPONENTIAL FUNCTIONS – 2.1


2.1 Connecting the Dots: Piggies and Pools

A Develop Understanding Task

Page 9




Page 10




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SECONDARY MATH I // MODULE 2


Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

2.1

READY

SET

READY, SET, GO!

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2.1

GO

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2.2 Shh! Please Be Discreet (Discrete)!

A Solidify Understanding Task

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•

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2.2

READY

SET

READY, SET, GO!

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2.2

GO

Topic: Solving one-step equations

Solve the following equations. Remember that what you do to one side of the equation must also be done to the other side. (Show your work, even if you can do these in your head.)

Example: Solve for x .

Example: Solve for x.

Note that multiplying by gives the same result as dividing everything by 9.

11. 12. 13.

14. 15. 16.

17. 18. 19.

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2.3 Linear, Exponential or Neither?

A Practice Understanding Task

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235

-13-31-49

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Page 27




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2.3

READY

READY, SET, GO!

-5

-5

5

5

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2.3

SET

GO

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2.3

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2.4 Getting Down to Business


Page 33




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2.4

READY

SET

READY, SET, GO!

x 3 x 3 x 3 x 3

+80 +80 +80 +80

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2.4

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2.4

GO

x f (x)

2

3

4

5

-4

-11

-18

-25

x f (x)

-1

0

1

2

2/5

2

10

50

x f (x)

2

3

4

5

-24

-48

-96

-192

x f (x)

-4

-3

-2

-1

81

27

9

3

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2.5 Making My Point


Page 39






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Page 41






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2.5

READY

READY, SET, GO!

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2.5

SET

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2.5

GO

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2.6 Form Follows Function

A Practice Understanding Task

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Page 50






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2.6

READY

20

18

16

14

12

10

8

6

4

2

5 10 15 20

20

18

16

14

12

10

8

6

4

2

5 10 15 20

READY, SET, GO!

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2.6

.

SET

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2.6

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2.6

GO

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Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license

Up a Little, Down a LittleA Solidify Understanding Task

One of the most common applications of exponential growth is compound interest. For example, Mama Bigbucks puts $20,000 in a bank savings account that pays 3% interest compounded annually. “Compounded annually” means that at the end of the first year, the bank pays Mama 3% of $20,000, so they add $600 to the account. Mama leaves her original money ($20000) and the interest ($600) in the account for a year. At the end of the second year the bank will pay interest on the entire amount, $20600. Since the bank is paying interest on a previous interest amount, this is called “compound interest”. Model the amount of money in Mama Bigbucks’ bank account after t years.

Use your model to find the amount of money that Mama has in her account after 20 years. A formula that is often used for calculating the amount of money in an account that is compounded annually is: Where: A = amount of money in the account after t years P = principal, the original amount of the investment r = the annual interest rate t = the time in years Apply this formula to Mama’s bank account and compare the result to the model that you created. Based upon the work that you did in creating your model, explain the (1 + r) part of the formula.

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Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license

Another common application of exponential functions is depreciation. When the value of something you buy goes down a certain percent each year, it is called depreciation. For example, Mama Bigbucks buys a car for $20,000 and it depreciates at a rate of 3% per year. At the end of the first year, the car loses 3% of its original value, so it is now worth $19,400. Model the value of Mama’s car after t years. Use your model to find how many years will it take for Mama’s car to be worth less than $500?

How is the situation of Mama’s car similar to Mama’s bank account?

What differences do you see in the two situations?

Consider your model for the value of Mama’s car and develop a general formula for depreciation.

Page 58

READY Topic: Evaluating equations

Fill out the table of values for the given equations.

1. y = 17x – 28

x y

-3

1

4

5

2. y = -8x – 3

x y

-10

-6

2

9

3. y = ½ x + 15

x y

-26

-14

-1

9

4. y = 6x

x y

-3

-1

1

2

5

5. y = 10x

x y

-3

-1

0

2

6

.

6. 𝑦 = (1

5)𝑥

x y

-4

-2

0

3

5

SET

Topic: Evaluate using the formulas for simple interest or compound interest.

Given the formula for simple interest: i = Prt, calculate the simple interest paid.

(Remember, i = interest, P = the principal, r = the interest rate per year as a decimal, t = time in years )

7. Find the simple interest you will pay on a 5 year loan of $7,000 at 11% per year.

READY, SET, GO! Name Period Date

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8. How much interest will you pay in 2 years on a loan of $1500 at 4.5% per year?

Use i = Prt to complete the table. All interest rates are annual.

i = P × r × t

9. $11,275 12% 3 years

10. $1428 $5100 4%

11. $93.75 $1250 6 months

12. $54 8% 9 months

Given the formula for compound interest: 𝐴 = 𝑃(1 + 𝑟)𝑡 , write a compound interest function to

model each situation. Then calculate the balance after the given number of years.

(Remember: A = the balance after t years, P = the principal, t =the time in years, r = the annual interest rate expressed as a decimal)

13. $22,000 invested at a rate of 3.5% compounded annually for 6 years.

14. $4300 invested at a rate of 2.8% compounded annually for 15 years.

15. Suppose that when you are 15 years old, a magic genie gives you the choice of investing $10,000

at a rate of 7% or $5,000 at a rate of 12%. Either choice will be compounded annually. The money

will be yours when you are 65 years old. Which investment would be the best? Justify your answer.

GO

Topic: Using order of operations when evaluating equations

Evaluate the equations for the given values of the variables.

16. pq ÷ 6 + 10; when p = 7 and q = -3 17. m + n(m – n); when m = 2, and n = 6

18. (b – 1)2 + ba2 ; when a = 5, and b = 3 19. y(x − (9 – 4y)); when x = 4, and y = -5

20. x – (x – (x – y3)); when x = 7, and y = 2 21. an4 + a(n – 7)2 + 2n; when a = -2, and n = 4

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2.6c What makes a population change?A Solidify Understanding Task

The following functions indicate the population change in Russia and Brazil. Both of the functions are in the form ( ) = (1 + ) where is the initial population in millions and ris the growth/decay rate in decimal form.

Brazil: ( ) = 164.4(1.013) is time since 1996

Russia: ( ) = 146.6(.9984) t is time since 2000

1. What are some similarities and differences between the population models for Russia andBrazil? What do those similarities and differences tell us about the populations of eachcountry?

2. By what percentage is the population of Brazil growing?

3. By what percentage is the population of Russia declining?

4. According to these models, what will the population of each nation be in 2040?

5. According to the model, when will Brazil’s population grow to 200 million?

6. According to the model, when will Russia’s population decline to 50 million?

7. Using personal knowledge and internet sources, what factors explain the differences in thepopulation models? Why is Russia’s population declining while Brazil’s population isgrowing? Consider factors like climate, political environment, infrastructure, medical quality,etc. Here is a link that might help: http://www.nationmaster.com/country-info/compare/Brazil/Russia/Economy

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2.6c

READY

Topic: Recursive and explicit forms of sequences

1. A concert hall has 58 seats in Row 1, 62 seats in Row 2, 66 seats in Row 3, and so on. Theconcert hall has 34 rows of seats.a. Write a recursive formula to find the number of seats in each row. How many seats are in

row 5?

b. Write the explicit equation for this sequence and use it to determine which row has 94seats.

c. How many seats are in the last row?

2. Given the sequence defined by the function = + 5 with = 2, write an explicitfunction rule.

3. Given the sequence defined by the function = with = 60, write an explicit function rule.

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SET

Topic: Multiple representations of exponential functions

In each of the problems below I share some of the Information that I know about Growth and Decay Functions. Your job is to add all the things that you know about the function form the information that I have given. Depending on the Function, some of the things you may be ableto figure out for the function are:

A table;A graph;A pattern;An explicit equation;A recursive formula;A story context.

1. Let’s talk about the Bacteria that forms in your mouth! Assume a minute after you brush your teeth that 3 bacteria find their way into your mouth. They are multiplying at a disturbing rate, such that each bacteria splits into 3 new cells each minute, tripling the number of bacteria present. How many bacteria will be in your mouth after 2 hours? Using the tools above, create multiple representations of the situation (at least 4).

2. The NCAA Basketball championship has decided to expand the number of teams thatparticipate in the March Madness tournament for the 2019-2020 season. They have decidedto allow 128 teams in, with only the winning team progressing to the next round. How manyteams are left after 4 rounds of the tournament? Using the tools above, create multiplerepresentations of the situation (at least 4).

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GO!

Topic: Solving two-step equations with justification

1. 2 + 3 = 15 Justification

2. 7 = 3 Justification

3. + 9 = 1 Justification

4. 6 + 5 = 41 Justification

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LINEAR & EXPONENTIAL FUNCTIONS – 2.7H




2.7 I Can See—Can’t You?


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2.7

READY

READY, SET, GO!

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2.

7

SET

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2.

7

GO

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WCPSS Math 1 Linear & Exponential Functions · LINEAR & EXPONETIAL FUNCTIONS Mathematics Vision...

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