Order of Operations: Diagnosing 3 + 4 × 2²
During a fifth-grade lesson on order of operations, Ms. Patel writes 3 + 4 × 2² on the board. She collects several answers, including 28, 19, and 11. Ana explains that she added 3 + 4 to get 7, then multiplied by 2² to get 28. Ben says he computed 4 × 2² = 16 and then added 3 to get 19. Carlo says he multiplied 4 × 2 = 8 and then added 3 to get 11. Ms. Patel wants to use the disagreement to develop meaning for the conventional order rather than simply announce a rule. She has access to two-color counters, paper bags, and a number line. The class has already worked with exponents as repeated multiplication and with multiplication as equal groups.
1. Diagnose the reasoning behind 28 and 11. What does each reveal about the student's understanding of operation order and exponents?
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Ana's 28 comes from (3 + 4) × 2² = 7 × 4 = 28. She evaluated left to right and performed addition before multiplication. She may understand the exponent, since 2² = 4, but she does not treat multiplication as a higher-priority grouping. Her error is about operation precedence.
Carlo's 11 comes from 3 + 4 × 2 = 3 + 8 = 11. He treated 2² as 2, so his error is about the meaning of the exponent, not about multiplication priority. Ben's 19 is correct: 2² = 4, 4 × 4 = 16, and 3 + 16 = 19. Separating Ana's precedence error from Carlo's exponent error is important because the two students need different follow-up instruction.
2. How would you use a context or representation to justify the conventional order rather than just stating a rule?
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Use a grouping context. Interpret 3 + 4 × 2² as 3 plus four groups of 2². Since 2² means a group of 4, four groups contain 16, and adding 3 gives 19. The multiplication describes equal groups, so it must be completed before the addition combines that group total with 3. With counters, make four bags of four counters and place 3 loose counters beside them; the total is 19.
Then contrast the grouping: (3 + 4) × 2² means 7 groups of 4, which is 28. The parentheses change what is grouped. The exponent applies only to the base immediately before it, so 2² is 4, not 2. This representation makes the convention a matter of consistent grouping rather than an arbitrary list of steps.
3. Write a follow-up item that would reveal whether students can transfer the concept, and explain what response would count as evidence of understanding.
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Ask students to compare 2 + 3 × 4² and (2 + 3) × 4². They should compute the first as 2 + 3 × 16 = 2 + 48 = 50 and the second as 5 × 16 = 80, then explain why the values differ. A second item could ask them to insert parentheses in 3 + 4 × 2² to produce 28 and to explain why 19 needs no parentheses.
Evidence of understanding includes identifying that the exponent applies only to 4, that multiplication is completed before addition, and that parentheses change the grouping of the entire sum. A response that says only 'because PEMDAS says so' is procedural recall; a response that explains the equal-groups meaning or the changed grouping is transfer.