Missing Partial Products in a Two-Digit Multiplication
Mr. Alvarez is reviewing a multiplication assessment in his fourth-grade class. Students used an area model to compute 24 × 13. Priya's paper shows (20 × 10) + (4 × 3) = 200 + 12 = 212. She explains, 'I multiplied the tens by the tens and the ones by the ones.' A second student, Devon, wrote 312 using the standard algorithm but cannot explain why the 24 in the second partial product is shifted one place left. Across the class, 14 of 22 students either wrote 212 or made a place-value alignment error in the standard algorithm. Mr. Alvarez has 45 minutes before the next lesson and wants to address the misconception without simply reteaching the procedure. He has base-ten blocks, graph paper, and a document camera available.
1. What does Priya's work reveal about her understanding of multiplication, and why is her rule mathematically incomplete? Use the area model to justify your answer.
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Priya is decomposing both factors by place value but applying the distributive property only to the 'matching' place values: tens × tens and ones × ones. For 24 × 13, the full area model partitions a 24-by-13 rectangle into four regions: 20 × 10 = 200, 20 × 3 = 60, 4 × 10 = 40, and 4 × 3 = 12. Her expression includes only 200 and 12, omitting the two cross-product regions; the correct sum is 312. Her statement that she 'multiplied the tens by the tens and the ones by the ones' shows she is treating digits as separate whole numbers rather than as parts of a decomposition: 24 is 20 + 4 and 13 is 10 + 3, so every part of one factor must multiply every part of the other. The incompleteness is not an arithmetic slip; it is a partial application of the distributive property. The area model makes the omission visible because the missing 60 and 40 regions account for exactly the 100-square difference between 212 and 312.
2. Describe two questions or prompts Mr. Alvarez could use to help Priya see the missing regions, and explain what each is intended to surface.
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First, ask Priya to draw a 24-by-13 rectangle on graph paper, label one side 20 + 4 and the other 10 + 3, and shade the regions that her equation represents. Then ask, 'Where are the 20 × 3 and 4 × 10 regions in your equation?' This prompt is intended to surface the two missing partial products visually, so the incompleteness is not just a correction from the teacher but a gap she can see. Second, ask her to estimate the product: '24 × 13 is close to 25 × 13 = 325, so is 212 a reasonable answer?' This is intended to surface the magnitude error and connect estimation to computation. A follow-up could be to build 24 × 13 with base-ten blocks as 24 groups of 13: ten groups give 240 and three groups give 72, totaling 312. That connects the symbolic work to a grouping interpretation and gives her a second way to check the area model.
3. A colleague suggests telling students, 'Just multiply every digit by every digit and add.' Evaluate this advice using Priya's example and one counterexample.
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The advice is dangerously imprecise because it can reinforce Priya's error rather than correct it. If students literally multiply every digit by every digit as ones, then for 24 × 13 they might compute 2 × 1 + 2 × 3 + 4 × 1 + 4 × 3 = 2 + 6 + 4 + 12 = 24, not 312. The missing idea is place-value scaling: the digit 2 in 24 represents 20, and the 1 in 13 represents 10, so that pair contributes 20 × 10 = 200. A second counterexample is 12 × 11: literal digit-pair addition gives 1 + 1 + 2 + 2 = 6, while the product is 132. The rule works only when each partial product is assigned its place-value weight and all four products are added: (20 + 4)(10 + 3) = 200 + 60 + 40 + 12. A better teacher move is to have students decompose both factors, multiply all part-pairs, and connect each product to a region in the area model. That preserves the colleague's emphasis on all pairs while supplying the place-value meaning that Priya lacks.