Growing Patterns: Where Does the Extra 2 Come From?
Ms. Brooks's third graders build a tile pattern: Figure 1 uses 5 tiles, Figure 2 uses 8, and Figure 3 uses 11. She asks how many tiles Figure 10 will use. Andre says 30, Bianca says 35, Caleb says 32, and Dana says "you can't tell without building it." Ms. Brooks notices that all four students can state the recursive rule - "add 3 each time" - and all four can correctly build Figure 4 with 14 tiles. She has color tiles, chart paper, and one class period. She wants a task that makes the starting value visible, rather than reteaching a recursion the students already know.
1. Reconstruct the reasoning likely behind each of the four answers and evaluate each one.
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- Andre, 30: computed 3 × 10, applying the growth rate directly to the figure number and dropping the 5 starting tiles. Correct in noticing the rate of 3, but the count at Figure 1 is 5, not 3.
- Bianca, 35: computed 5 + 3 × 10, adding the growth ten times to the Figure 1 value instead of nine. She has both ingredients (start of 5, rate of 3) but an off-by-one error in how many growth steps occur between Figure 1 and Figure 10.
- Caleb, 32: correct. 5 + 3 × 9 = 32, equivalently 3 × 10 + 2. Nine jumps from Figure 1 to Figure 10.
- Dana, "you can't tell": her iterative rule is valid but she does not yet trust that a pattern can be extended without building it. She is not wrong about the recursion; she has no way to jump ahead.
Notice that the three numeric answers cluster around 32 ± 3, which is exactly the size of one growth step - a strong signal that the difficulty is locating the starting value, not the rate.
2. Write an explicit rule for the pattern and explain what each part of it refers to in the tiles. Then describe a "Figure 0" prompt that makes the constant term visible.
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Explicit rule: tiles = 3n + 2, where n is the figure number. Check: n = 1 gives 5, n = 2 gives 8, n = 3 gives 11, n = 10 gives 32.
Meaning of the parts: the 3 is the number of tiles added with each new figure; the 2 is the constant core that never changes (for example, two tiles that form the base every figure shares).
Figure 0 prompt: "Build Figure 1, then take away the 3 tiles that Figure 1 has but Figure 0 would not. How many tiles does Figure 0 have?" Students remove 3 from 5 and get 2. Then build a table: n = 0, 1, 2, 3 → tiles 2, 5, 8, 11. Now the +2 is the value at n = 0, and the rate of 3 is the step between consecutive rows. Recording the table with a Figure 0 row is what converts "add 3" into "3n + 2" - the constant is visible as the entry in the n = 0 row rather than hidden inside the Figure 1 value.
3. A student says "3n + 2 and 'add 3 each time' are the same rule, so it doesn't matter which one I use." Explain the difference, and give a question where the difference changes what a student can do.
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They describe the same pattern but do different work. "Add 3 each time" is recursive: it tells you how to get the next figure from the one before, so it requires knowing where you are. "3n + 2" is explicit: it produces any figure directly from the figure number with no intermediate steps.
They agree on every term - Figure 4 is 14 either way - which is exactly why the student thinks they are interchangeable. They differ in cost and in reach.
Question where it matters: "Which figure number has exactly 47 tiles?" With the explicit rule, 3n + 2 = 47 gives 3n = 45, so n = 15. With only "add 3," the student must build or list upward and has no way to predict how many steps are needed or to be sure the answer is exact. A second good question: "How many tiles would Figure 100 have?" - the explicit rule answers it in one step (302), while the recursive rule demands 99 iterations and invites exactly the off-by-one error Bianca made in the original task.