Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

7.2.2 Constant and Exponential Change

7.2.2 • Constant and Exponential Change

Pattern 1

Pattern 2

1. Look at Pattern 1.

a. How is the number of dots in each step changing?

Compare your answer:

Each step has 2 rows of dots, with each row having the same number of dots as the step number, plus one extra dot.

b. How many dots will be in Step 5 in the pattern, and how will they be arranged?

Compare your answer:

In Step 5, there will be 2 rows of 5 dots plus one additional dot.

2. Look at Pattern 2.

a. How is the number of dots in each step changing?

Compare your answer:

Each step has as many rows and columns of dots as the step number.

b. How many dots will be in Step 5 in the pattern, and how will they be arranged?

Compare your answer:

Step 5 is a 5-by-5 array of dots.

Consider the following table.

Step	Number of dots in Pattern 1	Number of dots in Pattern 2
0	1	0
1	3	1
2	5	4
3	7	9
4
5
10
12
$n$

3. Complete the table with the number of dots for Pattern 1.

a. In Step 4, what is the number of dots in Pattern 1?

Step 4 has 9 dots in Pattern 1.

b. In Step 5, what is the number of dots in Pattern 1?

Step 5 has 11 dots in Pattern 1.

c. In Step 10, what is the number of dots in Pattern 1?

Step 10 has 21 dots in Pattern 1.

d. In Step 12, what is the number of dots in Pattern 1?

Step 12 has 25 dots in Pattern 1.

e. In Step $n$ , what is the number of dots in Pattern 1?

Compare your answer:

$2 n + 1$

4. Complete the table with the number of dots for Pattern 2.

a. In Step 4, what is the number of dots in Pattern 2?

Step 4 has 16 dots in Pattern 2.

b. In Step 5, what is the number of dots in Pattern 2?

Step 5 has 25 dots in Pattern 2.

c. In Step 10, what is the number of dots in Pattern 2?

Step 10 has 100 dots in Pattern 2.

d. In Step 12, what is the number of dots in Pattern 2?

Step 12 has 144 dots in Pattern 2.

e. In Step $n$ , what is the number of dots in Pattern 2?

Compare your answer:

$n \cdot n$ or $n^{2}$

5. Use the graphing tool or technology outside the course. Plot the number of dots at each step number for Pattern 1 and Pattern 2.

Access multimedia content

a. Pattern 1

Compare your answer:

b. Pattern 2

Compare your answer:

6. Explain why the graphs of the two patterns look the way they do.

Compare your answer:

The points representing the first pattern lie on a line because getting from one step to the next involves adding the same number of dots (2 dots). The points representing the second pattern do not lie on a line. They “curve” upward because the number of dots added at each step increases from step to step.

Self Check

How many squares will be in Step 5?

35
30
25
20

Additional Resources

Writing Expressions for Patterns

Example 1

Find the number of squares in Step 5 -

First, look to see the pattern. Every step, another row and another column are added, plus the two extra squares.

Step 1 - 1 row of $3 + 2$
Step 2 - 2 rows of $4 + 2$
Step 3 - 3 rows of $5 + 2$

So the width is always 1 by $n$ , or $n$ . The length is $n + 2$ , and then another 2 is added.

Then, draw pictures or create a table:

Step	# of squares
1	5
2	10
3	17
4	26
5	37

At Step 5, there are 37 squares.

Example 2

Find the expression for Step $n$ .

Multiply the length and width, and then add the 2 extra squares:

$n (n + 2) + 2 = n^{2} + 2 n + n$

Try it

Writing Expressions for Patterns

Use the figure below to find the number of squares in Step 5 and then Step $n$ .

Here is how to determine the number of squares in each step:

Notice that each figure adds a row and column, and then subtracts one square. The number of rows is always 2 more than the step number.

Begin by making a table:

Step	# of squares
1	8
2	15
3	24
4	35
5	48

There are 48 squares in Step 5.

Step $n$ would be found with $(n + 2) (n + 2) - 1$ or $(n + 2)^{2} - 1$ .