Three Rules for the Same Tile Pattern
An eighth-grade class works on a tile pattern: Figure 1 has 5 tiles, Figure 2 has 8, Figure 3 has 11, and each new figure adds a row of 3. Three students give rules for the number of tiles t in Figure n. Ana writes t = 3n + 2. Ben writes t = 3(n − 1) + 5. Cara writes t = 3n + 5. Ana and Ben both get 5 when n = 1, but Cara insists hers is right: "You start at 5 and add 3 each time, so 5 + 3n." Several classmates side with Cara, arguing that Ana's rule "doesn't show the starting number." The teacher has 15 minutes left before a quiz that will ask students to write a rule from a table where the input starts at 0.
1. Evaluate each rule: which are equivalent, why Cara's fails, and what is the precise conceptual error?
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Ana and Ben are equivalent. 3(n − 1) + 5 = 3n − 3 + 5 = 3n + 2. Ben's form is the recursive-to-explicit bridge: "5 tiles at Figure 1, plus 3 for each step after the first."
Cara's rule is off by one step. 3n + 5 gives 8 at n = 1 - that is Figure 2's value. Her sentence "start at 5 and add 3 each time" correctly describes the change between consecutive figures, but adding 3n to the start value assumes n counts steps taken from the start. Since n counts the step number, the multiplier applies to (n − 1) steps, or equivalently the constant must be the value at n = 0.
The error is index origin, not arithmetic. The y-intercept is the value when n = 0, which never appears in the table. Cara is treating the first table entry as if it were the intercept. Test at n = 1: 3(1) + 5 = 8 ≠ 5. Her own substitution disproves the rule, which is why the correction must come from computation, not from teacher authority.
2. Design the 15 remaining minutes so that Cara's group resolves the conflict themselves.
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- Extend the table backward (4 min). n: 0, 1, 2, 3 → t: __, 5, 8, 11. Ask: "What would Figure 0 look like?" Students see a figure with just the 2 constant tiles - no row of 3. Fill in 2.
- Test all three rules at n = 0 and n = 2 (5 min). Ana: 2, 8. Ben: 2, 8. Cara: 5, 11. Cara's rule is correct for the shifted sequence; the table exposes it.
- Name the structure (4 min). Common difference 3 → the coefficient of n. Value at n = 0 → the constant. That is the whole idea of slope and intercept in a pattern, and it explains both why Ana's rule "doesn't show 5" (5 = 3(1) + 2, it is built in) and why Ben's form is legal.
- Exit slip (2 min). One table starting at n = 0, one starting at n = 1; write the rule for each.
Do not simply tell Cara she is wrong. The misconception is about where counting starts; only substituting into her own rule and comparing to the figure produces the needed conflict.
3. How should tomorrow's quiz be designed to discriminate real understanding from pattern-matching, and how should Ben's answer be scored?
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Item design:
- One table whose inputs start at 0 → answer read directly as the intercept.
- One table whose inputs start at 1 → requires (n − 1) reasoning or back-solving to n = 0.
- One context where 0 is meaningful ("a candle is 12 cm tall and burns 2 cm per hour; height after h hours") and one where 0 is not (figure numbers). Students should notice the difference in what the intercept means.
- One item giving a rule and asking for Figure 1's value, to check substitution skill separately from rule-writing.
Scoring Ben: full credit. 3(n − 1) + 5 is algebraically identical and shows stronger reasoning about the first step. However, require that he can also produce 3n + 2 on request - standardized and later coursework expect slope-intercept form, and point-slope fluency should not come at the cost of recognizing the standard form. A two-column quiz ("your rule" / "equivalent standard-form rule") makes both visible at once.