### Student Page 4.6.4 Growing Tile Patterns

There are 10 growing tile patterns to investigate in Figure 4.6.4.1.

Patterns A through D are in Rows A through D respectively, with Step 1 in Column 1 and Step 4 in Column 4.

Patterns 1 through 4 are in Columns 1 through 4 respectively, with Step 1 in Row A and Step 4 in Row D.

Diagonal 1 begins in the upper left corner at Step 1 and moves diagonally to the lower right corner at Step 4.

Diagonal 2 begins in the upper right corner at Step 1 and moves diagonally to the lower left corner at Step 4.

Choose one tile pattern to investigate. Solve each problem for the pattern you choose.

#### 1.

Study the pattern.

##### (a)

Describe how the number of white tiles is changing as the pattern progresses from Step 1 to Step 4.

##### (b)

Describe how the number of shaded tiles is changing as the pattern progresses from Step 1 to Step 4.

##### (c)

Draw Step 5. Explain how you know it follows the same pattern.

##### (d)

Is Step 5 possible for all 10 patterns? Why or why not?

#### 2.

Complete Table 4.6.4.2 showing the number of white, shaded, and total tiles for each step.

Step Number | Number of Light Colored Tiles |
Number of Dark Colored Tiles |
Total Number of Tiles |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

\(x\) |

#### 3.

Find patterns in Table 4.6.4.2. How are the numbers changing in each column?

#### 4.

Find equations to fit white, shaded, and total tiles, respectively, assuming the pattern continues predictably. Enter the equations in the last row of the table.

#### 5.

How are the equations you found related to how each pattern of tiles is growing?

#### 6.

How are the equations related to each other?

#### 7.

Suppose you have 200 white tiles and 200 shaded tiles.

##### (a)

What is the largest step number of your chosen tile pattern you can create? How do you know?

##### (b)

If the tiles are 1-inch squares, how large will the project be?

#### 8.

Patterns E and F are shown in Figure 4.6.4.3. Solve Student Page Exercise 4.6.4.1–4.6.4.7 for one of these patterns.