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Fraction Classroom · 2.3.0 · Teacher answers & responses

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.

Each small square is 1 square cm. Both rectangles have perimeter 16 cm; the first covers 15 squares and the second 12.5 cm × 3 cm6 cm × 2 cm
Each small square is 1 square cm. Both rectangles have perimeter 16 cm; the first covers 15 squares and the second 12.

Build the idea

model. 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?

  1. Trace the entire outside of the first rectangle. Add all four side lengths, including the opposite sides.
  2. Tile its inside with square-centimetre units. Count five in each of three rows to find 15 square centimetres.
  3. Repeat both measurements for the 6-by-2 rectangle. Its boundary is still 16 cm, but only 12 unit squares fit inside.
  4. State what the example proves: equal perimeters do not force equal areas. Keep cm for boundary length and square cm for covered area.

Worked example: 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.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. Find the perimeter in cm of a rectangle 7 cm long and 2 cm wide.

Answer: 18

Teacher response: Count both long sides and both short sides.

p2. Find the area in square cm of that 7 cm by 2 cm rectangle.

Answer: 14

Teacher response: Two rows each contain seven square-centimetre units.

p3. A 6 cm by 3 cm rectangle also has perimeter 18 cm. Find its area in square cm.

Answer: 18

Teacher response: Six times three gives eighteen square centimetres, which differs from fourteen.

e1. 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: 9

Look for:

  • Counts all outside sides exactly once.
  • Uses an explicit pair of rectangles to disprove the claim.

Teacher response: The areas are 16 and 7 square centimetres; their difference is 9.

Transfer the idea

transfer. Can two rectangles have the same area but different perimeters? Give dimensions and calculate both measurements.

Answer: 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.

Run the 15-minute workshop

teach-1. 3 minutes: Ask learners to predict whether equal edging guarantees equal area and record a reason.

teach-2. 4 minutes: Trace each boundary with a finger, then count the interior squares in a different pass.

teach-3. 5 minutes: Complete the practice and label every result with its measurement unit.

teach-4. 3 minutes: Use the final check and ask learners to revise the starting claim using one counterexample.

Respond to the learner

misconception. The same amount of edging must surround the same amount of space.

Teacher response: Use the two tiled drawings as evidence. Matching a one-dimensional boundary total does not determine how much two-dimensional space lies inside it.

support. Mark opposite sides in matching colours, then separate boundary counting from interior tiling.

extend. Find all whole-number side pairs with perimeter 16 cm and compare their areas; treat rotations as the same rectangle.

Teaching notes