Paving-Stone Patio Fence
A third-grade class has 24 one-foot-square paving stones to make a rectangular patio against the school wall. The principal says the patio must use all 24 stones, with whole-number side lengths, and no stones cut. The class wants the least amount of fencing for the three sides not against the wall. One student suggests a 1-foot by 24-foot strip because it uses all the stones. Another suggests a 4-foot by 6-foot rectangle because “6 and 4 are friendlier.” The class must list possible whole-number dimensions organized in a table, calculate the perimeter of each, and decide which design uses the least fencing. They also must explain why two rectangles can have the same area but different perimeters, using the 24-stone constraint.
1. List all whole-number dimension pairs for a rectangle with area 24 square feet. Then give the perimeter of each rectangle.
Read the worked answer
The factor pairs of 24 are 1 × 24, 2 × 12, 3 × 8, and 4 × 6. Perimeter is 2 × (length + width): 1 × 24 has perimeter 2 × 25 = 50 feet; 2 × 12 has perimeter 2 × 14 = 28 feet; 3 × 8 has perimeter 2 × 11 = 22 feet; 4 × 6 has perimeter 2 × 10 = 20 feet. Each rectangle uses 24 square feet because the side lengths multiply to 24.
2. Which design uses the least fencing for the three sides not against the wall? Remember one side is the school wall.
Read the worked answer
For each rectangle, the wall replaces one side, so fence only three sides. If the 4 × 6 rectangle puts the 6-foot side on the wall, fencing is 4 + 6 + 4 = 14 feet. If the 3 × 8 rectangle puts the 8-foot side on the wall, fencing is 3 + 8 + 3 = 14 feet. Other choices use more: the 2 × 12 with the 12-foot side on the wall uses 2 + 12 + 2 = 16 feet, and the 1 × 24 with the 24-foot side on the wall uses 1 + 24 + 1 = 26 feet. So the least fencing is 14 feet, achieved by either 4 × 6 with the 6-foot side on the wall or 3 × 8 with the 8-foot side on the wall.
3. Explain why the 1 × 24 and 4 × 6 rectangles have the same area but different perimeters. Use multiplication in your explanation.
Read the worked answer
Both areas are 24 square feet: 1 × 24 = 24 and 4 × 6 = 24. Perimeter depends on adding the side lengths, not just multiplying them. The 1 × 24 rectangle has a long side and a short side, so 1 + 24 = 25 and the perimeter is 2 × 25 = 50 feet. The 4 × 6 rectangle has side lengths closer together, so 4 + 6 = 10 and the perimeter is 2 × 10 = 20 feet. Same area does not force the same perimeter because different pairs of factors can multiply to 24.