Practice 1
Find the perimeter in cm of a rectangle 7 cm long and 2 cm wide.
Answer and teaching response
18 · Count both long sides and both short sides.
Keep the boundary length and change the area
Compare two rectangles with the same perimeter and use unit squares to show why their areas can differ.
Get the student and teacher materialsPrepare the group
Learning goal: A learner can distinguish measuring a boundary from measuring its inside and provide a counterexample to equal perimeter implying equal area.
You will need: Squared paper and a pencil. Use drawn side lengths in centimetres; the diagram is a mathematical model rather than a full-size measurement template.
Start with this question
A 5 cm by 3 cm rectangle and a 6 cm by 2 cm rectangle both need 16 cm of edging. Do they cover equal areas?
No. Their perimeters are 5 + 3 + 5 + 3 = 16 cm and 6 + 2 + 6 + 2 = 16 cm. Their areas are 15 and 12 square centimetres.
- Trace the entire outside of the first rectangle. Add all four side lengths, including the opposite sides.
- Tile its inside with square-centimetre units. Count five in each of three rows to find 15 square centimetres.
- Repeat both measurements for the 6-by-2 rectangle. Its boundary is still 16 cm, but only 12 unit squares fit inside.
- State what the example proves: equal perimeters do not force equal areas. Keep cm for boundary length and square cm for covered area.
Run the 15-minute workshop
- 3 minutes. Ask learners to predict whether equal edging guarantees equal area and record a reason.
- 4 minutes. Trace each boundary with a finger, then count the interior squares in a different pass.
- 5 minutes. Complete the practice and label every result with its measurement unit.
- 3 minutes. Use the final check and ask learners to revise the starting claim using one counterexample.
A useful error to discuss
A learner might say: The same amount of edging must surround the same amount of space.
Use the two tiled drawings as evidence. Matching a one-dimensional boundary total does not determine how much two-dimensional space lies inside it.
Practise, then check independently
Use the first three tasks for guided practice. Try the final task without help. Record a drawing or equation as well as the answer.
Practice 2
Find the area in square cm of that 7 cm by 2 cm rectangle.
Answer and teaching response
14 · Two rows each contain seven square-centimetre units.
Practice 3
A 6 cm by 3 cm rectangle also has perimeter 18 cm. Find its area in square cm.
Answer and teaching response
18 · Six times three gives eighteen square centimetres, which differs from fourteen.
Independent check
A 4 cm by 4 cm square and a 7 cm by 1 cm rectangle both have perimeter 16 cm. How many more square cm does the square cover?
Answer and teaching response
9 · The areas are 16 and 7 square centimetres; their difference is 9.
Transfer the idea
Can two rectangles have the same area but different perimeters? Give dimensions and calculate both measurements.
Discuss a possible response
For example, 6 cm by 2 cm and 4 cm by 3 cm both have area 12 square cm. Their perimeters are 16 cm and 14 cm, respectively.
Choose the next teaching step
Look for:
- Counts all outside sides exactly once.
- Uses an explicit pair of rectangles to disprove the claim.
If the idea is not secure: Mark opposite sides in matching colours, then separate boundary counting from interior tiling.
If the learner is ready: Find all whole-number side pairs with perimeter 16 cm and compare their areas; treat rotations as the same rectangle.
Materials for your group
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.