This grid activity helps learners see that area and perimeter answer different questions. Area counts the square units covering a shape; perimeter measures the distance around its boundary. Learners draw rectangles, count squares, compare examples with the same area and explain why the boundary lengths can differ. Use centimetre grid paper or a drawn grid; no special materials are required.
Materials and setup
Give each learner a sheet with 1 cm squares, a pencil and a ruler. Draw a rectangle that is 3 squares long and 4 squares wide. Label the side lengths and ask learners to predict the number of unit squares inside before counting. Keep the same unit throughout the activity. A square grid makes each unit square visible, while the labels remind learners that length and area use different units.
Build the first comparison
Ask learners to draw a 2 by 6 rectangle. Count its rows and columns: 2 rows of 6 squares give an area of 12 square units. Add the four side lengths: 2 + 6 + 2 + 6 = 16 units around the edge. Now compare it with the 3 by 4 rectangle. It also covers 12 squares, but its perimeter is 3 + 4 + 3 + 4 = 14 units. The equal areas do not make the perimeters equal.
Find a family of equal perimeters
Set a perimeter target of 14 units. Learners can test 1 by 6, 2 by 5 and 3 by 4 rectangles. Each has perimeter twice the sum of its side lengths: 2(1+6), 2(2+5) and 2(3+4), which all equal 14. Their areas are 6, 10 and 12 square units. Ask what changes and what stays fixed. The side-length pairs change, the perimeter stays at 14 and the area increases across these examples.
Student recording table
| Length × width | Area (square units) | Perimeter (units) |
|---|---|---|
| 1 × 6 | 6 | 14 |
| 2 × 5 | 10 | 14 |
| 3 × 4 | 12 | 14 |
| 2 × 6 | 12 | 16 |
Prompts that uncover reasoning
- How did you count the squares inside?
- Which numbers did you add to find the distance around?
- Can two rectangles have the same area but different perimeters? Show an example.
- Can two rectangles have the same perimeter but different areas? Use the table.
Listen for a clear distinction between covering the inside and tracing the boundary. If a learner multiplies length by width for perimeter, invite them to trace each outer side with a finger and add the four lengths. If they count every boundary grid segment twice, mark the outside edge once and count only that path. Praise the explanation and the diagram together, not just a memorized formula.
Answer key and extension
The 3 by 4 and 2 by 6 rectangles both have area 12 square units. Their perimeters are 14 and 16 units respectively. The rectangles 1 by 6, 2 by 5 and 3 by 4 all have perimeter 14, while their areas are 6, 10 and 12. Extend the activity by asking learners to make a rectangle with area 18 and list possible whole-number side pairs, then calculate each perimeter. Finish with an exit ticket: draw two rectangles with the same area and explain how their perimeters compare.
