Building Quadratic Functions from Geometric Patterns

ID: rapam-dijiz
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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42 questions

Building Quadratic Functions from Geometric Patterns

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Problem 1


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1) Sketch or describe the figure in Step 4.

2) Sketch or describe the figure in Step 15.

3) How many small squares will there be in Step 4?

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4) How many small squares will there be in Step 15?

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5) Write an equation to represent the relationship between the step number, nn, and the number of small squares, yy, in each step.

6) Explain how your equation relates to the pattern.

Problem 2

7) Which expression represents the relationship between the step number nn and the total number of small squares in the pattern?

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a)

n2n^{2}+1

b)

n2n^{2}-1

c)

n2n^{2}-nn

d)

n2n^{2}+nn

Problem 3

Each figure is composed of large squares and small squares. The side length of the large square is xx. Write an expression for the area of the shaded part of each figure.

8)

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9)

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Problem 4

Here are a few pairs of positive numbers whose difference is 5.

Find the product of each pair of numbers.

10) 1 and 6

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11) 2 and 7

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12) 3 and 8

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13) 5 and 10

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14) 7 and 12

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15) Plot some points to show the relationship between the first number and the product.

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16) Is the relationship between the first number and the product exponential?

True or false? Write below.

17) Explain how you know.

Problem 5

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

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

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

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20) If the length is 5 meters, what is the width?

21) What is the area?

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22) If the length is 7 meters, what is the width?

23) What is the area?

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24) If the length is 9 meters, what is the width?

25) What is the area?

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26) What do you notice about the areas?

27) 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.

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

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29) Do the values change in a linear way?

True or false? Write below.

30) Do they change in an exponential way?

True or false? Write below.
Problem 6

Here is a pattern of dots.

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31) What is the total number of dots in Step 0?

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32) What is the total number of dots in Step 1?

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33) What is the total number of dots in Step 2?

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34) What is the total number of dots in Step 3?

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35) How many dots will there be in Step 10?

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36) How many dots will there be in Step nn?

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Problem 7

Mai has a jar of quarters and dimes. She takes at least 10 coins out of the jar and has less than $2.00.

37) Write a system of inequalities that represents the number of quarters, qq, and the number of dimes, dd, that Mai could have.

Is it possible that Mai has each of the following combinations of coins? If so, explain or show how you know. If not, state which constraint—the amount of money or the number of coins—it does not meet.

38) 3 quarters and 12 dimes

39) 4 quarters and 10 dimes

40) 2 quarters and 5 dimes

Problem 8

A stadium can seat 63,026 people. For each game, the amount of money that the organization brings in through ticket sales is a function of the number of people, nn, in attendance.

41) If each ticket costs $30.00, find the domain of this function.

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42) If each ticket costs $30.00, find the range of this function.

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