Area Doubles, Perimeter Doesn't
Students are investigating how changing the dimensions of a rectangle affects its area and perimeter. The teacher gives a 4 cm by 6 cm rectangle (area 24 cm², perimeter 20 cm) and asks students to double both dimensions. A student concludes, "If you double the sides, the area doubles to 48 and the perimeter doubles to 40." Another student says, "The perimeter should quadruple too, because you doubled two sides." The teacher wants students to reason about why area and perimeter scale differently, and to connect this to the structure of the formulas rather than just recomputing. The class has access to grid paper and can draw scaled rectangles.
1. Determine the correct new area and perimeter, and explain the structural reason the two quantities scale differently.
Read the worked answer
Doubling both dimensions gives an 8 cm by 12 cm rectangle. New area = 8 × 12 = 96 cm², which is 4 times the original 24 cm². New perimeter = 2(8 + 12) = 40 cm, which is 2 times the original 20 cm.
The structural reason: area is a product of two lengths, A = l × w. Doubling each factor multiplies the product by 2 × 2 = 4. Perimeter is a sum of lengths, P = 2l + 2w = 2(l + w). Doubling each length doubles the sum, so the perimeter doubles.
So the first student is half right (perimeter doubles to 40) but wrong about area (it quadruples to 96, not 48). The second student is wrong: doubling two sides does not quadruple the perimeter because perimeter adds lengths; the factor of 2 appears once in the sum, not twice.
2. Describe how grid paper would let students *see* the 4× area relationship, and predict what happens to area and perimeter if only one dimension is doubled.
Read the worked answer
On grid paper, the original 4 × 6 rectangle covers 24 unit squares. Doubling both dimensions produces an 8 × 12 rectangle that can be partitioned into four 4 × 6 blocks (two rows of two), so students can literally count four copies of the original - a visual proof that area scales by 4.
If only one dimension is doubled (say 8 × 6), the area becomes 48 cm², exactly 2 times the original, because only one factor in the product changed. The perimeter becomes 2(8 + 6) = 28 cm, which is not double (it was 20). This is a useful contrast: area responds multiplicatively to each dimension changed, while perimeter responds additively, so a single-dimension change alters perimeter by an amount (2 × the change) that depends on which dimension changed, not by a simple factor. Students should record a table of (dimensions, area, perimeter) to generalize: doubling both dimensions → area × 4, perimeter × 2; doubling one dimension → area × 2, perimeter increases by 2 × (doubled side).