The Area Model That Gives Two Answers
A 4th-grade teacher poses: "A rectangle is 3½ meters by 2⅔ meters. What is its area?" Students work in pairs. Pair 1 multiplies 3×2=6 and ½×⅔=⅓ and writes 6⅓ m². Pair 2 converts to improper fractions (7/2 × 8/3 = 56/6 = 9⅓ m²) and writes 9⅓ m². Pair 1's answer is wrong but their method "looks like" distributing. The teacher must diagnose the error, choose a representation to repair it, and decide what to do with Pair 2, who are correct but fast.
1. Diagnose Pair 1's error precisely and explain why an area model is the right repair tool.
Read the worked answer
Pair 1 applied the distributive property incompletely. They multiplied the whole-number parts and the fraction parts separately but omitted the cross terms:
(3 + ½)(2 + ⅔) = 3·2 + 3·⅔ + ½·2 + ½·⅔ = 6 + 2 + 1 + ⅓ = 9⅓
Pair 1 computed only 3·2 and ½·⅔, dropping 3·⅔ (the strip along the side) and ½·2 (the strip along the bottom).
Why the area model: it makes the missing pieces visible. Draw the rectangle split into four regions-3×2, 3×⅔, ½×2, ½×⅔-and have students label each region's area and sum them. The two "missing" strips appear as real regions, so the omission becomes concrete rather than a rule to memorize. This also connects to the standard algorithm for multiplying mixed numbers.
2. What should the teacher do with Pair 2, who finished correctly and quickly?
Read the worked answer
Use them as a resource, not as idle finishers-but verify understanding first. Ask Pair 2 to explain why 7/2 × 8/3 works and to connect it to the area model (e.g., "show where 56/6 comes from in the picture"). Fast correct answers can mask procedural-only understanding.
Then:
- Have Pair 2 model their improper-fraction method to the class alongside the area model, so students see the two methods agree.
- Extend them with a task that requires reasoning, e.g., "Estimate first: is 9⅓ reasonable? Why is 6⅓ too small?" or a new problem with a variable side.
This keeps advanced students challenged, gives the class a second valid strategy, and reinforces that both methods rest on the same area reasoning-the teacher's real target.