The Growing Tile Pattern
Students build a pattern of square tiles on the floor. Figure 1 uses 4 tiles, Figure 2 uses 7 tiles, and Figure 3 uses 10 tiles, arranged so each new figure adds one row of 3 tiles to the previous figure. Jaden writes the rule 'tiles = 3 × figure number' and predicts Figure 10 will need 30 tiles. Amara writes 'tiles = 3 × figure number + 1' and predicts 31 tiles. The teacher asks them to check Figure 1 and Figure 2 against both rules before deciding. A second group builds a different pattern: Figure 1 uses 2 tiles and each new figure doubles the previous count, giving 2, 4, 8, 16, ...
1. Decide whose rule correctly describes the first tile pattern and justify it using the given figures and a general rule.
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Amara's rule is correct. Test Figure 1: 3 × 1 + 1 = 4 tiles, which matches. Test Figure 2: 3 × 2 + 1 = 7, which matches. Test Figure 3: 3 × 3 + 1 = 10, which matches. Jaden's rule gives 3 × 1 = 3 for Figure 1, but the figure uses 4 tiles, so it fails immediately. The +1 represents the starting tile that is present before any rows of 3 are added. So Figure 10 needs 3 × 10 + 1 = 31 tiles, not 30. A good check is to confirm the rule reproduces every known term, not just the later ones.
2. Compare how the two patterns grow. If each pattern continued, which would have more tiles at Figure 6, and what does that tell you about predicting from a rule versus extending a table?
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Pattern A (linear): tiles = 3n + 1, so Figure 6 = 19 tiles. Pattern B (doubling): 2, 4, 8, 16, 32, 64, so Figure 6 = 64 tiles. Pattern B overtakes Pattern A: at Figure 4, A = 13 while B = 16, and the gap widens because doubling multiplies each term while adding 3 grows steadily. This shows that early terms can mislead you: in the first three figures Pattern A has more tiles (4 vs 2, 7 vs 4, 10 vs 8), yet Pattern B wins by Figure 4. Extending a table is reliable only for the terms you compute; a rule lets you jump ahead, but you must identify whether growth is additive or multiplicative to predict correctly.