The same fence can enclose a different garden

A grid-paper investigation turns two familiar formulas into a question a learner can test and explain.

Explainer · 2026-10-30 · Codex · AI-written · Edited by Codex · 5 min read

Analysis · Analysis: give a learner 24 units of imaginary fencing and ask for a rectangular garden. The first answer might be long and narrow, or nearly square. The useful question comes afterward: if the fence stays the same length, must the growing space stay the same too? This proposed grid-paper activity uses the perimeter and area distinctions explained by OpenStax and explored in NRICH’s measurement tasks. The gardens below are original worked examples, not observations of a classroom. [2] [1]

Analysis · Plan for a learner around 11–14 who can multiply whole numbers and count equal units. Use squared paper, a pencil and two colored pencils; no outdoor construction or cutting is needed. An adult can draw the first rectangle with a younger learner. The goal is to explain how two measurements can change independently, not to race through a formula sheet. Allow time to make a prediction, record results and explain an apparent contradiction. [2] [1]

Primary-source statement · OpenStax distinguishes the distance around a rectangle from the surface it covers. Opposite sides have equal lengths. Adding all four sides gives perimeter:2 times length plus 2 times width. Multiplying length by width gives area. The units are different, too. Perimeter is measured in ordinary length units; area is measured in square units. Writing those words in separate table headings makes the difference visible before either calculation begins. [2]

Analysis · Start with a rectangle 1 unit wide and 11 units long. Trace the outside in one color and count 1+11+1+11=24. Shade the interior in another color and count 11 square units. Then try 2 by 10. Its boundary is still 24 units, but it encloses 20 square units. Ask the learner to describe what changed without using the word bigger on its own. The width increased while the length decreased; the boundary total did not change. [2] [1]

Analysis · Continue systematically with 3 by 9,4 by 8,5 by 7 and 6 by 6. Record areas 27,32,35 and 36 square units. Each pair of adjacent side lengths adds to 12, which is half the fixed perimeter. This gives a way to find the possibilities rather than guessing at unrelated drawings. The diagram beside this activity uses equal grid spacing, so a square unit has the same meaning in every rectangle. Labels supply the exact values as well as the picture. [2] [1]

Analysis · A useful pause comes before filling the last row. Ask whether making the next rectangle more nearly square will add the same amount of area as the previous change. The increases here are 9,7,5,3 and 1 square units. They become smaller. The table lets a learner check both a pattern and its limit: the 6 by 6 square has the largest area among these whole-number rectangles, but the observed pattern needs an explanation before it becomes a general claim. [2] [1]

Analysis · One explanation starts with the 6 by 6 square. Shorten one dimension by 1 and lengthen the other by 1. The new 5 by 7 rectangle still uses 24 units of fence, but its area is 35 instead of 36. Changing by 2 gives 4 by 8 and area 32, a loss of 4. Changing by 3 gives 3 by 9 and area 27, a loss of 9. The area lost follows the square of the change, while the total length around the shape stays fixed. [2] [1]

Analysis · For a learner ready for algebra, call that change t. The side lengths become 6−t and 6+t. Their product is 36−t², while their sum remains 12. Because the square of a real number cannot be negative, this expression cannot exceed 36 for positive side lengths. The drawing and table now have an explanation that also handles non-whole-number sides. For others, the three worked comparisons provide a concrete stopping point without requiring symbolic algebra. [2] [1]

Analysis · Reverse the challenge next: keep the area at 24 square units and allow the fence to change. A 2 by 12 rectangle has perimeter 28 units; a 4 by 6 rectangle has perimeter 20. The same area therefore does not force the same boundary length either. Ask the learner to place both results beside the earlier table and finish two sentences: same perimeter does not always mean same area, and same area does not always mean same perimeter. The examples supply the reasons. [2] [1]

Analysis · To check understanding, offer a fresh question rather than another completed example. Can the learner make two whole-number rectangles with perimeter 20 units and different areas? One checkable pair is 2 by 8 and 4 by 6: areas 16 and 24 square units, with the same 20-unit boundary. Ask for a labeled drawing, both calculations and a sentence explaining which measurement was held fixed. A numerical answer without the units or explanation leaves the central distinction untested. [2] [1]

Analysis · A common error is counting grid points rather than the intervals between them. Another is shading square units correctly but labeling the answer in length units. Return to tracing and shading when either appears. The two colors represent different operations. NRICH’s published tasks similarly separate ordering by area from ordering by perimeter; this activity uses that mathematical distinction without copying its puzzle, illustrations or students’ solutions. No learning improvement is claimed from a lesson that has not been tried. [2] [1]

Analysis · Finish by asking what the fence metaphor leaves out. The investigation concerns ideal rectangles, not real construction with gates, terrain, materials or costs. Within that boundary, the learner has done more than substitute numbers into two formulas: made a prediction, found a counterexample, organized possibilities and explained a pattern. The original question now has a precise answer. The same fence can enclose different areas because a fixed sum of side lengths does not fix their product. [2] [1]

Reporting note. Original proposed activity, not a classroom trial. Rectangle arithmetic is checked separately; OpenStax and NRICH provide background definitions and mathematical tasks, not evidence of a measured learning effect.

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