A Different Kind of Change

ID: mivoh-gabud
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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35 questions

A Different Kind of Change

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Problem 1

Here are a few pairs of positive numbers whose sum is 50.


Find the product of each pair of numbers.

1) 1 and 49

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2) 2 and 48

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3) 10 and 40

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4) Find a pair of numbers that have a sum of 50 and will produce the largest possible product.

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5) Explain how you determined which pair of numbers have the largest product.

Problem 2

Here are some lengths and widths of a rectangle whose perimeter is 20 meters.

6) Determine the area, in square meters, given the length is 1 meter and the width is 9 meters.

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7) Determine the area, in square meters, given the length is 3 meter and the width is 7 meters.

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8) Determine the width, in meters, given the length is 5 meters.

9) Then, what is the area?

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10) Determine the width, in meters, given the length is 7 meters.

11) Then, what is the area?

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12) Determine the width, in meters, given the length is 9 meters.

13) Then, what is the area?

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14) What do you notice about the areas?

15) Without calculating, predict whether the area of the rectangle will be greater or less than 25 square meters if the length is 5.25 meters.

16) On the coordinate plane, plot the points for length and area from your table.

A template for answering this question. Ask your instructor for an alternative.

17) Do the values change in a linear way?

18) Do they change in an exponential way?

Problem 3

The table shows the relationship between xx and yy, the side lengths of a rectangle, and the area of the rectangle.


x( cm)y( cm)area (sq cm)248483261272816128\begin{array}{|c|c|c|} \hline \\[-1em] x \textbf{( cm)} & y \textbf{( cm)} & \textbf{area (sq cm)} \\[-1em] \\ \hline \\[-1em] 2 & 4 & 8 \\[-1em] \\ \hline \\[-1em] 4 & 8 & 32 \\[-1em] \\ \hline \\[-1em] 6 & 12 & 72 \\[-1em] \\ \hline \\[-1em] 8 & 16 & 128 \\[-1em] \\ \hline \end{array}

19) Explain why the relationship between the side lengths is linear.

20) Explain why the relationship between xx and the area is neither linear nor exponential.

Problem 4

21) Which statement best describes the relationship between a rectangle's side length and area as represented by the graph.

A template for answering this question. Ask your instructor for an alternative.
a)

As the side length increases by 1, the area increases and then decreases by an equal amount.

b)

As the side length increases by 1, the area increases and then decreases by an equal factor.

c)

As the side length increases by 1, the area does not increase or decrease by an equal amount.

d)

As the side length increases by 1, the area does not change.

Problem 5

Copies of a book are arranged in a stack. Each copy of a book is 2.1 cm thick.


Determine the stack height, in cm, given the stack has...

22) 0 copies of the book.

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23) 1 copy of the book.

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24) 2 copies of the book.

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25) 3 copies of the book.

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26) 4 copies of the book.

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27) What do you notice about the differences in the height of the stack of books when a new copy of the book is added?

28) What do you notice about the factor by which the height of the stack of books changes when a new copy is added?

29) How high is a stack of bb books?

Problem 6

The value of a phone when it was purchased was $500. It loses 15\frac{1}{5} of its value a year.

30) What is the value of the phone after 1 year?

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31) What is the value of the phone after 2 years?

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32) What is the value of the phone after 3 years?

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33) Tyler says that the value of the phone decreases by $100 each year since 15\frac{1}{5} of 500 is 100. Do you agree with Tyler?

a)

Yes

b)

No

c)

I need more information.

34) Explain your reasoning.

Problem 8

35) Give a value for rr that indicates that a line of best fit has a negative slope and models the data well.