# Lesson 3

Building Quadratic Functions from Geometric Patterns

### Problem 1

1. Sketch or describe the figure in Step 4 and Step 15.

2. How many small squares will there be in each of these steps?
3. Write an equation to represent the relationship between the step number, $$n$$, and the number of small squares, $$y$$, in each step.
4. Explain how your equation relates to the pattern.

### Problem 2

Which expression represents the relationship between the step number $$n$$ and the total number of small squares in the pattern?

A:

$$n^2+1$$

B:

$$n^2-1$$

C:

$$n^2-n$$

D:

$$n^2+n$$

### Problem 3

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

### Problem 4

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

1. Find the product of each pair of numbers. Then, plot some points to show the relationship between the first number and the product.

first
number
second
number
product
1 6
2 7
3 8
5 10
7 12
2. Is the relationship between the first number and the product exponential? Explain how you know.

### Solution

(From Unit 6, Lesson 1.)

### Problem 5

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

1. Complete the table. What do you notice about the areas?

length
(meters)
width
(meters)
area
(square meters)
1 9
3 7
5
7
9
2. 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.
3. On the coordinate plane, plot the points for length and area from your table.

Do the values change in a linear way? Do they change in an exponential way?

### Solution

(From Unit 6, Lesson 1.)

### Problem 6

Here is a pattern of dots.

1. Complete the table.
2. How many dots will there be in Step 10?
3. How many dots will there be in Step $$n$$?
step total number
of dots
0
1
2
3