Constant and Exponential Change

7.2.2 Constant and Exponential Change

7.2.2 • Constant and Exponential Change

Pattern 1

A pattern of dots. Step 0 has 1 dot. Step 1 has a row of 2 dots on the bottom and 1 dot on the row above. Step 3 has 3 dots on the bottom row and two dots on the row above. Step 3 has a bottom row with 4 dots and the row above had 3 dots.

Pattern 2

A pattern of dots. Step 0 has no dots. Step 1 has 1 dot. Step 2 has two rows with 2 dots in each row. Step 3 has three rows with 3 dots in each row.

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:

Scatterplot of points on a coordinate grid. The x-axis represents the step number from Pattern 1. The y-axis represents the number of dots.

b. Pattern 2

Compare your answer:

Scatterplot of points on a coordinate grid. The x-axis represents the step number from Pattern 2. The y-axis represents the number of dots.

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?

Additional Resources

Writing Expressions for Patterns

Example 1

A pattern of squares. Step 1 has 3 rows of squares. The top row has 1 square, the middle row has 3 squares, the bottom row has 1 square. They are shaped like a plus sign. Step 2 has 4 rows of squares. The top row has 1 square, the second row has 4 squares, the third row has 4 squares, the last row has 1 square. The squares in the top and bottom row are aligned above and below the second squares in rows 2 and 3, respectively. Step 3 has 5 rows. The top row has 1 square, row 2 has 5 squares, row 3 has 5 squares, row 4 has 5 squares, and the bottom row has 1 square. The squares in the top and bottom row are aligned above and below the second squares in rows 2 and 4, respectively.

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 $n$ .

A pattern of squares. Step 1 has 3 rows of squares. The bottom row has two squares and the top two rows have 3 squares. Step 2 has 4 rows of squares. The bottom row consists of 3 squares and the top three rows above it contain 4 squares. Step 3 has 5 rows of squares. The bottom row has 4 squares in it while the 4 rows above it contain 5 squares each.

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 $(n + 2)^{2} - 1$ .