Diego’s 613: Diagnosing Two-Digit Addition
Diego, a second grader, solves 46 + 27 on a whiteboard. He writes: "6 + 7 = 13, so I write 13. 4 + 2 = 6, so I write 6. My answer is 613." He shows no regrouping and seems pleased with his work. The teacher has base-ten blocks, a hundreds chart, and 20 minutes before recess. Diego can count by tens and ones and knows that 46 means 4 tens and 6 ones. He sometimes reverses digits when writing. The class is about to begin a unit on two-digit addition with regrouping, and the teacher must decide whether to focus on place value, regrouping, or both, and must avoid simply saying "line up the numbers."
1. Diagnose Diego’s error. What does his work show that he understands and misunderstands about place value? Use an estimate to show why 613 is not reasonable.
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Diego understands that two-digit numbers can be split into tens and ones, and he correctly adds the ones (6 + 7 = 13) and the tens (4 + 2 = 6). His misunderstanding is that he writes the 13 ones as a two-digit chunk and then writes the 6 tens beside it, producing 613. He is not treating 13 ones as 1 ten and 3 ones, so he does not regroup. An estimate shows the answer is unreasonable: 46 is about 50 and 27 is about 30, so the sum should be about 80, not 613. The teacher should use the estimate to help Diego build number sense, not just point out the wrong digit. The key diagnosis is that Diego’s separate addition is sound, but his place-value recording is not.
2. Describe a sequence of two questions or moves with base-ten blocks that would let Diego discover the regrouping, not just hear the rule.
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First, have Diego build 46 as 4 tens and 6 ones, and 27 as 2 tens and 7 ones. Ask: "How many ones altogether?" He combines 6 and 7 to get 13 ones. Second, ask: "Can we trade some ones for a ten? Show me." Diego should group 10 ones into a ten and see that 13 ones become 1 ten and 3 ones. Then ask: "Now how many tens are there altogether?" He should count the 4 tens, the 2 tens, and the new ten to get 7 tens, so the answer is 73. The teacher should let Diego do the trade and say the value, rather than saying "carry the 1." This sequence targets the exact gap: 13 ones is not written as 13 in the ones place; it is 1 ten and 3 ones.
3. How would you decide whether Diego is ready to try another problem independently? Give one specific evidence you would look for.
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Look for Diego explaining a trade without prompts. For example, after building 38 + 24, he should say something like "8 + 4 is 12; that’s one ten and 2 ones" and then count the tens to get 62. If he can make the trade, record the answer correctly, and explain why the 1 ten belongs with the tens, he is ready for a similar problem. If he gets 62 but cannot explain the trade, he may be copying a procedure, so the teacher should ask one more "why" question before moving on. The judgment call is to separate a correct answer from understood place value; readiness is shown by explanation and by handling a new problem, not by repeating the same 46 + 27 example.