Building Quadratic Functions from Geometric Patterns

ID: mipuv-butin
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
Standards: HSF-BF.A.1.aHSA-SSE.A.1HSA-CED.A.2HSF-LE.A.1HSF-IF.A.2
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25 questions

Building Quadratic Functions from Geometric Patterns

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Quadratic Expressions and Area

Figure A is a large square. Figure B is a large square with a smaller square removed. Figure C is composed of two large squares with one smaller square added.

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Write an expression to represent the area of Figure A given the side length is...

1) 4

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2) xx

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3) 4x4x

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4) (x + 3)(x \ + \ 3)

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Write an expression to represent the area of Figure B given the side length is...

5) 4

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6) xx

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7) 4x4x

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8) (x + 3)(x \ + \ 3)

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Write an expression to represent the area of Figure C given the side length is...

9) 4

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10) xx

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11) 4x4x

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12) (x + 3)(x \ + \ 3)

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Expanding Squares


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13) If the pattern continues, what will we see in Step 5?


Sketch or describe the figure.

14) If the pattern continues, what will we see in Step 18?


Sketch or describe the figure.

15) How many small squares are in step 5?

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16) How many small squares are in step 18?

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17) Explain how you know.

18) Write an equation to represent the relationship between the step number nn and the number of squares yy.


(If you get stuck, try making a table.)

19) Explain how each part of your equation relates to the pattern.

20) Sketch the first 3 steps of a pattern that can be represented by the equation y=n21y = n^2 - 1.

Growing Steps


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21) Sketch the next step in the pattern.

22) Kiran says that the pattern is growing linearly because as the step number goes up by 1, the number of rows and the number of columns also increase by 1. Do you agree?

True or false? Write below.

23) Explain your reasoning.

To represent the number of squares after nn steps, Diego and Jada wrote different equations. Diego wrote the equation f(n)=n(n+2)f(n) = n(n + 2). Jada wrote the equation f(n)=n2+2nf(n) = n^2 + 2n.

24) Are either Diego or Jada correct?

25) Explain your reasoning.