Partial Products and Place Value in 24 × 13
During a fourth-grade unit on multi-digit multiplication, Mr. Alvarez asks students to compute 24 × 13 using any representation. He selects two responses for whole-class discussion. Student A writes: 20 × 10 = 200 and 4 × 3 = 12, so the product is 212. Student B draws a rectangle partitioned into four regions and writes 24 × 10 = 240, 24 × 3 = 72, and 240 + 72 = 312. A third student, C, uses the standard algorithm and gets 312 but cannot explain why the 2 written above the 4 represents 20 rather than 2. Mr. Alvarez wants to use the contrasting work to build place-value understanding rather than simply mark answers right or wrong. He has ten minutes of whole-class time, a document camera, and available base-ten blocks and grid paper.
1. What conceptual error produces Student A's answer, and what specific representation would make the error visible to the class? Explain what students should notice.
Read the worked answer
Student A multiplied only 'matching' places: tens by tens (20 × 10 = 200) and ones by ones (4 × 3 = 12). The cross products 20 × 3 = 60 and 4 × 10 = 40 are missing. This is not a basic-fact error; it is an incomplete application of the distributive property.
A four-region area model makes the omission visible. Draw a 24-by-13 rectangle, partition one side into 20 and 4 and the other into 10 and 3. The four regions are 200, 60, 40, and 12, totaling 312. Student A filled only the two diagonal regions. Asking the class to compare 212 with 312 highlights the missing 100 square units, which is exactly 60 + 40. Students should notice that every part of one factor must multiply every part of the other factor.
2. Student B's decomposition 24 × 10 + 24 × 3 is correct. Explain how it connects to the four-region area model and why both are valid.
Read the worked answer
Student B used the distributive property: 24 × 13 = 24 × (10 + 3) = 240 + 72 = 312. In the area model, the 24-by-10 region splits into 20 × 10 = 200 and 4 × 10 = 40, while the 24-by-3 region splits into 20 × 3 = 60 and 4 × 3 = 12. Combining the two rows gives (200 + 40) + (60 + 12) = 240 + 72.
Both are valid because partitioning a factor does not change the total area. Student B grouped the four regions by the 10-and-3 rows; the area model displays the same four partial products separately. The connection is important: the standard algorithm's 72 is not a mysterious digit string but the combined area of the two regions in the 3-row.
3. Design one follow-up problem and a sentence frame that would distinguish students who understand place value from those who are only mimicking steps.
Read the worked answer
Use 31 × 12 and ask students to predict whether 62 or 372 is reasonable before computing. A student who only multiplies matching places will produce 30 × 10 + 1 × 2 = 302, omitting 30 × 2 and 1 × 10.
Sentence frame: 'I partitioned ___ into ___ and ___, so the four partial products are ___, ___, ___, and ___. Their sum is ___.' A response that includes all four products, especially the two cross products, is evidence of understanding. A response that lists only 300 and 2 is evidence of the same place-value gap. Follow up by asking students to locate the carried 3 in the standard algorithm and explain that it represents 30, not 3.