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Algebra 1

7.2.2 Constant and Exponential Change

Algebra 17.2.2 Constant and Exponential Change

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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?

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

2. Look at Pattern 2.

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

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

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
nn

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?

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

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

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

e. In Step nn, what is the number of dots in Pattern 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?

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

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

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

e. In Step nn, what is the number of dots in Pattern 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.

a. Pattern 1

b. Pattern 2

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

Self Check

How many squares will be in Step 5?

  1. 35
  2. 30
  3. 25
  4. 20

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+23+2
  • Step 2 - 2 rows of 4+24+2
  • Step 3 - 3 rows of 5+25+2

So the width is always 1 by nn, or nn. The length is n+2n+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 nn.

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

n(n+2)+2=n2+2n+nn(n+2)+2=n2+2n+n

Try it

Try It: Writing Expressions for Patterns

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

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.

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