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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.

Figure 4.6.4.1. Growing Tile Patterns

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.

Table 4.6.4.2.
Step Number Number of Light
Colored Tiles
Number of Dark
Colored Tiles
Total Number
of Tiles
1      
2      
3      
4      
5      
\(x\)      

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?