Mathematics K–6 · Years 5–6

Same fence, different paddocks: fixed perimeter, changing area

Measurement and space

Practical, model not builtLow risk

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The idea

Rectangles with the same perimeter can have very different areas, and the area is largest when the sides are closest to equal.

What you need

a loop of string exactly 50 cm around (knotted and checked with a ruler)
1 cm grid paper and four pins or drawing pins on a corkboard
a recording table with columns length, width, perimeter, area
50 cm of thin dowel cut into 1 cm pieces (optional, to model fence panels)

How to do it

Pin the 50 cm loop into a 1 cm by 24 cm rectangle on the grid. Record the sides, confirm the perimeter is 50 cm, and count the squares inside.
Move the pins to 2 by 23, 5 by 20, 10 by 15, 12 by 13 and 12.5 by 12.5. Record the area each time.
Plot area against length on grid paper and describe the shape of the graph.
Predict which whole-centimetre rectangle has the largest area, then check by trying the neighbours of your answer.
Reverse the question: pin rectangles of area 24 square centimetres (1 by 24, 2 by 12, 3 by 8, 4 by 6) on the grid and find each perimeter by counting the grid edges around it. Only the 1 by 24 rectangle has a perimeter that the 50 cm loop fits.

What you should see

With perimeter 50 cm the areas are 24, 46, 100, 150, 156 and 156.25 square centimetres for the six rectangles, so the square has the largest area and 12 by 13 is the largest with whole-centimetre sides. The graph rises steeply, then flattens near the top at 12.5 cm and falls again beyond it by symmetry. For a fixed area of 24 the perimeters are 50, 28, 22 and 20 cm, so the rectangle nearest a square needs the least fence.

What changes

What you change: the length of the rectangle
What you measure: the area
What you keep the same: the perimeter fixed at 50 cm by the loop
right-angled corners on the grid
sides measured to the nearest centimetre

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

Rectangles with the same perimeter have the same area.
A longer rectangle always has more area.

Safety card

Low riskLearners carry it out

Hazards

drawing pins

Controls

pins stay in the corkboard; count them out and in

Note

No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics K-10 Syllabus (2022), NSW Education Standards AuthorityMA3-GM-02 MA3-2DS-02 MAO-WM-01
Australian Curriculum v9AC9M5M02

Sources

The pages the author read to write this activity.

All Concept Studio activities