Perimeter of Regular and Irregular Shapes
Warm-up
On a geoboard, make a shape and count the elastic band length along each side. Add them up. That is the perimeter. Make two shapes with the same perimeter but different shapes. Which is easier to compute perimeter for? (Regular polygons: just multiply one side by the number of sides.)
Explore
Perimeter investigation: using centimetre grid paper, draw 5 polygons with different perimeters. For each: measure all sides, calculate perimeter, record. Challenge: draw a rectangle with perimeter exactly 24 cm. How many different rectangles are possible? (1x11, 2x10, 3x9, 4x8, 5x7, 6x6.)
Formalize
Missing side problem: a triangle has perimeter 27 cm. Two sides are 9 cm and 11 cm. Write the equation: 9 + 11 + n = 27. Solve: 20 + n = 27, n = 7 cm. This is an algebraic equation embedded in a geometric context.
Perimeter of Regular and Irregular Shapes
Fencing scenario: a farmer wants to fence a rectangular field. The length is 45 m and the width is 28 m. How much fencing? P = 2x(45+28) = 2x73 = 146 m. If fencing costs 12 = $1,752. Multi-step problem connecting perimeter to multiplication and financial literacy.
Practice
Students calculate perimeters of 6 shapes, solve 3 missing-side problems, and solve 2 real-world perimeter problems. Exit ticket: a square has perimeter 36 cm. What is the side length?
Exit ticket
Students calculate perimeters of 6 shapes, solve 3 missing-side problems, and solve 2 real-world perimeter problems. Exit ticket: a square has perimeter 36 cm. What is the side length?
The figure: an L-shape with only four sides labelled: across the top 9 m, down the right 3 m, the lower inner width 4 m, the bottom depth 4 m. Two sides are unlabelled.
Step 1: State the L-shape's secret: opposite distances must MATCH — the two horizontal runs at the bottom must add to the top's 9; the two vertical drops on the left must add to the total right-side height.
Step 2: Find the missing horizontal: bottom pieces are 4 and ? , together spanning 9 → ? = 5.
Step 3: Find the missing vertical: right side drops 3, then the inner step drops 4 more?? — no: total height on the left is 3 + 4 = 7.
Step 4: March the whole boundary, adding as you walk (start top-left, clockwise): 9 + 3 + 4 + 4 + 5 + 7 = 32 m of baseboard.
Step 5: The audit: count the turns — an L has 6 sides; did the sum use 6 numbers? ✓. Missing-side reasoning BEFORE any adding is the entire skill; the addition is dessert.
Step 1: The task: with exactly 12 square patio tiles, design every possible rectangle. Build them: 1×12, 2×6, 3×4.
Step 2: Measure each perimeter by walking the edge: 1×12 → 26 units. 2×6 → 16. 3×4 → 14.
Step 3: Absorb the result: SAME 12 tiles of area — perimeters from 14 to 26. Area does not determine perimeter (the mirror of Grade 3's fixed-fence discovery). The skinny patio spends its edge extravagantly; the chunky 3×4 is thrifty.
Step 4: Apply it as a decision: edging costs 28; the 1×12 costs $52 for the SAME floor space. Shape is money.
Step 5: Push past rectangles: keep 12 tiles but allow L-shapes and zigzags — perimeters climb even higher (a 12-tile staircase can pass 26). Conjecture to leave simmering: for fixed tiles, compact shapes minimize edge; straggly shapes maximize it. (Nature agrees — soap bubbles are round for this reason.)
Step 1: Estimate first, bodies only: one giant step ≈ 1 metre. Pace the room's length (say ~9 steps) and width (~7). Estimated perimeter: 9 + 7 + 9 + 7 = 32 m. Every student records their own paced estimate.
Step 2: Measure properly with the trundle wheel or metre sticks: length 8.4 m, width 6.8 m.
Step 3: Compute the true perimeter — and pick your structure: (8.4 + 6.8) × 2 = 15.2 × 2 = 30.4 m. (Or double each and add: 16.8 + 13.6 = 30.4 ✓ — same by either route, which is itself worth noticing.)
Step 4: Compare estimate to measurement: 32 vs 30.4 — pacing ran ~5% hot. Calibrate: "my step is a touch over a metre." Estimation isn't guessing; it's measurement with cheaper units, corrected over time.
Step 5: The transfer question: "the hallway is about 25 of MY steps long — how many metres, given my calibration?" Owning a personal, calibrated unit is the exit skill — it never runs out of batteries.