# Building Quadratic Functions from Geometric Patterns

ID: mipuv-butin
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Standards: HSF-BF.A.1.aHSA-SSE.A.1HSA-CED.A.2HSF-LE.A.1HSF-IF.A.2

25 questions

# Building Quadratic Functions from Geometric Patterns

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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.

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

1) 4

Show Work

2) ﻿$x$﻿

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3) ﻿$4x$﻿

Show Work

4) ﻿$(x \ + \ 3)$﻿

Show Work

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

5) 4

Show Work

6) ﻿$x$﻿

Show Work

7) ﻿$4x$﻿

Show Work

8) ﻿$(x \ + \ 3)$﻿

Show Work

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

9) 4

Show Work

10) ﻿$x$﻿

Show Work

11) ﻿$4x$﻿

Show Work

12) ﻿$(x \ + \ 3)$﻿

Show Work
##### Expanding Squares

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?

Show Work

16) How many small squares are in step 18?

Show Work

17) Explain how you know.

18) Write an equation to represent the relationship between the step number ﻿$n$﻿ and the number of squares ﻿$y$﻿.

(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 ﻿$y = n^2 - 1$﻿.

##### Growing Steps

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.

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