Mia’s Growing Tile Pattern: Counting Steps
Mia is working on a growing pattern in math centers. Figure 1 has 3 square tiles in a row. Figure 2 has 5 tiles, and Figure 3 has 7 tiles. The teacher asks how many tiles will be in Figure 10. Mia says, "23, because I started with 3 and added 2 ten times, since it’s figure 10." The teacher has a table showing figure numbers 1, 2, 3, and 4 with their tile counts, plus a pile of tiles. The learning goal is for students to describe a rule and use it to predict a later figure, not to memorize a formula. Mia can explain "add 2 each time" but has not connected the starting value to the number of steps between figures.
1. Is Mia’s answer correct? Diagnose her off-by-one error and explain what her method shows about her understanding of the pattern.
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Mia’s answer is not correct; Figure 10 has 21 tiles. Her method shows real understanding of the constant change: she knows each new figure adds 2 tiles. Her error is counting the number of steps. From Figure 1 to Figure 10 there are 9 jumps, not 10, because Figure 1 is the starting figure. So 3 + 2 × 9 = 21. She added 2 ten times, which would take her from Figure 1 to Figure 11. The teacher should recognize that Mia’s additive reasoning is appropriate for grade 2 and that the problem is the starting point, not the "add 2" rule. The judgment call is to correct the step count without replacing her reasoning with a formula she does not understand.
2. What question or table-based move would help Mia see the number of steps without telling her the answer?
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Have Mia extend the table and draw arrows between figure numbers. Ask: "How many jumps from Figure 1 to Figure 2? From Figure 1 to Figure 3? From Figure 1 to Figure 4?" She should see 1 jump, 2 jumps, and 3 jumps. Then ask: "So how many jumps from Figure 1 to Figure 10?" She can count the arrows or notice that the number of jumps is one less than the figure number. If she still says 10, have her count aloud while touching each arrow from 1 to 10. The move keeps the answer hidden and uses the table as evidence. A strong follow-up is to ask her to predict Figure 5 first, where she can check by building, then return to Figure 10.
3. What would you look for in Mia’s next explanation to decide she can use the rule for Figure 20?
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Look for Mia identifying 19 steps of 2 from Figure 1 to Figure 20, then adding those 38 tiles to the starting 3 to get 41. She might say, "19 jumps of 2 is 38, plus 3 is 41," or use a table to extend. Grade-2 additive reasoning is acceptable; she does not need to write 2n + 1. If she says "20 jumps" or "20 plus 20 plus 1" without explaining why, she may still be guessing. The teacher should ask her to show the number of jumps for a smaller case, such as Figure 5, and then transfer. Readiness is shown by explaining the step count, not by producing 41 alone.