Math // Measurement

Perimeter (age 10+)

Recognise that shapes with the same area can have different perimeters and vice versa; explore this relationship systematically

Ages
10 to 11
Type
Concept
Builds on
1 topic
Leads to
0 topics
Standards
1 code

Quick check

If the learner has 12 square tiles to arrange, can they make different-shaped rectangles and notice that the area stays the same (12 tiles) but the perimeters are all different?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Draw two rectangles with area 24 cm² but different perimeters
  • Find two shapes with perimeter 20 cm but different areas
  • Explain why a long thin rectangle and a square can have the same area but different perimeters

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

1 topic
  • Estimating answers (age 9+)Required

    Measurement // Ages 9 to 10

    Exploring area/perimeter relationship requires area and perimeter formula knowledge

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Perimeter (age 10+)

Ages 10 to 11 // Concept

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Measurement

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Children learn that knowing one of area or perimeter does NOT determine the other: two shapes can share an area yet have different perimeters, and share a perimeter yet have different areas. (They are not completely unrelated, though: for example, a fixed perimeter limits how large the area can be.) The core work is systematic — fix one quantity (e.g. area 24 cm²) and generate shapes that differ in the other, tabulating so the variation is SEEN, not asserted. The target misconception is the intuitive 'Same A – Same B' rule (same area must mean same perimeter, and vice versa). Because a correct answer can come from the faulty intuition, always probe the reasoning.

Common misconceptions

  • 'Same A – Same B': believes shapes with equal area must have equal perimeter (or that changing sides by equal-and-opposite amounts leaves both unchanged).
  • Conflates area and perimeter, so cannot hold one fixed while varying the other.
  • Thinks making a shape 'bigger' must increase area and perimeter together.

Useful representations

  • Squared paper / tile arrangements at fixed area
  • Table of dimensions versus perimeter (or perimeter versus area)
  • Long-thin-versus-square rectangle pair with equal area
  • Fence/string model for perimeter around a fixed area

Teaching practices

  • Fix one quantity and vary the shape systematically (all rectangles of area 24; all shapes of perimeter 20), tabulating the other so children see it change.
  • Confront the 'Same A – Same B' intuition head-on with counterexamples (a 3×4 and a 2×6 rectangle both have area 12 but perimeters 14 and 16).
  • Use squared paper and tiles so children construct and count rather than rely on a remembered rule.
  • Probe reasoning beyond correct answers — a right answer can come from the intuitive rule, so always ask 'why' (Tsamir & Mandel).
  • Link to practical intuition: a long thin field and a square field can hold the same crop area but need very different lengths of fence.

Activities to try

  1. 1

    Lock the area, watch the perimeter change

    Why it works: Keeping area fixed while reshaping shows the perimeter climbing even though the covered surface is unchanged, so same area does NOT force same perimeter.

    1. Start from a 4×6 rectangle on squared paper (or a drag-to-reshape grid) that reads 'Area 24 cm², Perimeter 20 cm'.
    2. Reshape it to 3×8, 2×12, then 1×24, keeping the area pinned at 24 cm² each time.
    3. After each shape, count the squares to confirm the area is still 24, then count the edge units to find the new perimeter (20, 22, 28, 50 cm).
    4. Record each rectangle's dimensions and perimeter in a table so the growing perimeter is visible alongside the unchanged area.
    5. Say aloud what stays the same and what changes.
  2. 2

    Lock the perimeter, watch the area change

    Why it works: With perimeter held constant, the area varies — so same perimeter does NOT force same area either; in both directions, fixing one quantity does not fix the other (although the fixed perimeter does cap the area: here the 5×5 square gives the most).

    1. Fix a perimeter (e.g. 20 cm) and build several rectangles that all have that perimeter: 1×9, 2×8, 3×7, 4×6, 5×5.
    2. For each, count the squares to find the area (9, 16, 21, 24, 25 cm²) and record it in a table.
    3. Note which shape gives the largest area and which the smallest for the same perimeter.
  3. 3

    The string test (predict, then check)

    Why it works: Both rectangles cover the same area (12 cm²) but their perimeters differ (14 cm and 16 cm) — a direct counterexample to 'Same A – Same B'.

    1. Look at a 3×4 rectangle and a 2×6 rectangle on squared paper; confirm both have area 12 cm² by counting squares.
    2. PREDICT whether one loop of string will go exactly around both, and say WHY before testing.
    3. Lay the string around each and measure: the 3×4 needs 14 cm, the 2×6 needs 16 cm.
    4. Draw a THIRD rectangle with area 12 but yet another perimeter, and find the largest perimeter you can make for area 12 (with whole-number sides, the thinnest 1×12 gives 26 cm).

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
Ma6/3.1d

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