The Growing Tile Pattern
Students build a pattern with square tiles on the floor. Figure 1 uses 4 tiles, Figure 2 uses 7 tiles, and Figure 3 uses 10 tiles. Each new figure adds the same number of tiles as the one before it. The class records the numbers in a table and looks for a rule that lets them predict any figure without building it. A student claims Figure 10 will use 31 tiles. Another student says the rule is "multiply the figure number by 3, then add 1." The teacher asks the class to test the rule on the figures they already built before trusting it for Figure 10.
1. Find the pattern rule and decide how many tiles Figure 10 will use. Show how you checked the rule against Figures 1, 2, and 3.
Read the worked answer
The tile counts are 4, 7, 10. Each figure adds 3 tiles (7 − 4 = 3, 10 − 7 = 3).
Test the rule "figure number × 3 + 1":
- Figure 1: 1 × 3 + 1 = 4 ✓
- Figure 2: 2 × 3 + 1 = 7 ✓
- Figure 3: 3 × 3 + 1 = 10 ✓
The rule works for all three known figures, so it is a good rule.
For Figure 10: 10 × 3 + 1 = 31 tiles. The student who said 31 is correct.
Another way to see it: start with 4 tiles in Figure 1, then add 3 tiles for each of the 9 extra steps: 4 + 9 × 3 = 4 + 27 = 31.
2. A new student says, "Because each figure adds 3, Figure 100 will use 4 + 100 × 3 = 304 tiles." Is this reasoning correct? Explain the error and give the correct count.
Read the worked answer
The reasoning is not correct. The mistake is in how many times you add 3.
Figure 1 already has 4 tiles. To get from Figure 1 to Figure 100, you take 99 steps, not 100 steps. So you add 3 ninety-nine times.
Correct count: 4 + 99 × 3 = 4 + 297 = 301 tiles.
Check with the rule: 100 × 3 + 1 = 301. Both methods agree.
The new student's answer of 304 is 3 too many because they counted one extra step. A good habit is to test a rule on a small case first: using their method for Figure 2 gives 4 + 2 × 3 = 10, but Figure 2 really has 7 tiles, so the method fails.