Same Perimeter, Different Gardens: A Class 4 Design Challenge

In this guide

Two school-garden plans need the same length of fencing. One looks long and narrow; the other looks almost square. Must they leave the same amount of ground for plants? Invite children to decide with a finger, a strip of paper and a grid before anyone reaches for a formula.

Give the garden team one clear rule

Write this original brief on the board: “Design a rectangular garden on the square grid. Its sides must follow grid lines, its side lengths must be positive whole numbers, and its complete boundary must use exactly 20 unit lengths. There is no gap for a gate. Find two designs that enclose different numbers of squares.” A square counts as a rectangle here. Turning a design around does not make a new garden.

Let one grid edge represent one unit of length. Each complete little square represents one square unit of area. Keep those units visible beside the drawings. A large classroom model can use floor tiles; the numbers describe the model, rather than measurements of an actual school plot.

• 0-5 minutes: trace the edge of a notebook, then cover part of its surface with equal paper squares. Name the two different jobs.
• 5-12 minutes: draw and check the starter garden together. Count every outer edge once.
• 12-24 minutes: groups build at least two valid gardens and complete the recording table.
• 24-34 minutes: compare counterexamples and test whether the list of whole-number rectangles is complete.
• 34-40 minutes: complete an individual claim-and-evidence ticket.

Start with a tempting plan

Draw a rectangle eight squares long and two squares wide. Ask one learner to walk a finger around the boundary while another counts: 8 + 2 + 8 + 2 = 20 units. Next count the inside in two rows of eight: 16 square units. The same drawing has supplied two different measurements.

Now offer a 6-by-5 proposal. It covers 30 squares, so it looks attractive. Does it satisfy the fence rule? Give groups time to check rather than announcing that it is wrong. Their job is to find the exact requirement it fails, not to guess which design the teacher prefers.

Use a small recording sheet

Ask pairs to copy this table into their notebooks. They can draw gardens on a separate grid and label them A, B and C. One learner builds or draws, one traces the boundary, one counts the inside, and one checks the record; in a pair, combine roles and swap after the first design.

Plan	Length and width	Boundary calculation	Inside squares
A	8 by 2	8 + 2 + 8 + 2 = ___	___ square units
B	6 by 4	___	___ square units
C	5 by 5	___	___ square units
Our new plan	___ by ___	___	___ square units

Make the checking physical. Place equal strips along the four sides without overlapping corners. Count rows of squares inside, touching each row only once. If a group uses multiplication, ask what the two factors represent in its drawing. A correct number is more useful when children can point to its meaning.

Compare the claim, not the neatness

Read this fictional claim: “All gardens with a 20-unit fence have the same area.” Ask each group to choose two of its own drawings as evidence. A counterexample needs only two valid gardens with different areas; children do not have to test every possible shape to disprove that claim.

For a further challenge, find all the different whole-number rectangles allowed by the brief. If two neighbouring side lengths together make 10, the complete boundary is 20. Start with width 1, then increase it one unit at a time. Stop when the remaining choices would only rotate rectangles already drawn. Keep the argument about this defined set of rectangles; do not announce a rule about every imaginable garden shape.

Teacher check: dimensions, areas and a complete list

The valid length-width pairs, counting rotations once, are 9 by 1, 8 by 2, 7 by 3, 6 by 4 and 5 by 5. Every boundary totals 20. Their areas are 9, 16, 21, 24 and 25 square units. Plans A, B and C therefore enclose 16, 24 and 25 square units.

The 6-by-5 starter proposal needs 22 boundary units, so it fails the exact fence allowance. Two valid examples such as 8 by 2 and 5 by 5 disprove the equal-area claim. Among the listed whole-number rectangles, 5 by 5 encloses the most squares. That conclusion is limited to the stated task.

Listen for three useful distinctions

If children count only two sides, invite them to trace the whole boundary. If they count grid lines instead of spaces, colour one inside square and establish what one square unit covers. If they write “24 units” for an area, ask whether the answer describes a journey around the edge or a covering of the inside.

Offer raised grid lines, large floor squares, ready-drawn rectangles or a partner who records dictated calculations. Children can explain through pointing and oral language before writing. Keep the mathematical goal on the relationship between boundary and covering; precise ruler work or fast multiplication need not decide who can contribute.

For the exit ticket, ask: “Draw two gardens with equal boundary but different areas. Label the units and finish: I know the fence is equal because…; I know the area is different because…” A secure response contains two valid calculations and points to the corresponding parts of each drawing.

Use the site’s mathematics worksheet collection for further examples, or return to the concrete representation habit in Build 34 Two Ways when learners need help connecting a model and a calculation.

The boundary-and-covering distinction is supported by SCERT Delhi’s Class IV Mental Maths, Fields and Fences, printed page 74. The garden brief, numbers, recording sheet and worked comparison above are original classroom materials.

AI assistance was used to finalize this contribution. The views expressed are personal views of the author; official statements are identified through linked official sources.