Area Measurement of Squares and Rectangles
Warm-up
Two rectangles on grid paper: 3 x 8 and 4 x 6. Count tiles for each. (Both have 24 tiles = area 24 sq units.) Calculate perimeters. (3x8: 2x(3+8)=22. 4x6: 2x(4+6)=20.) Same area, different perimeter. Interesting: same area does not mean same perimeter.
Explore
Fixed perimeter investigation: using 24 square tiles arranged as a rectangle, find all rectangles with perimeter 24. (1x11, 2x10, 3x9, 4x8, 5x7, 6x6.) Calculate area for each. Graph: perimeter (constant) on x-axis, area on y-axis. Which rectangle has the greatest area? The square.
Formalize
Traditional dwelling connection: a longhouse floor is 30 m long and 8 m wide. Floor area? 30x8=240 m squared. Perimeter (the walls)? 2x(30+8)=76 m. If the community wanted the same floor area but a more square shape, what dimensions? About 15x16: area=240, perimeter=62 m. Less wall material, same living space.
Area Measurement of Squares and Rectangles
Missing dimension: a rectangular garden has area 72 m squared and one side is 9 m. What is the other side? 72/9=8 m. This is division applied to area: A = l x w, so w = A/l. It is also an equation: 9w = 72.
Practice
Students calculate area for 6 rectangles/squares, solve 3 missing-dimension problems, and complete the fixed-perimeter investigation. Exit ticket: a rectangle has perimeter 36 cm and length 12 cm. Find the width and area.
Exit ticket
Students calculate area for 6 rectangles/squares, solve 3 missing-dimension problems, and complete the fixed-perimeter investigation. Exit ticket: a rectangle has perimeter 36 cm and length 12 cm. Find the width and area.
Step 1: Tile the 7 m × 4 m bed with square-metre tiles (grid paper model). Count by rows: 4 rows of 7 → 7 + 7 + 7 + 7 = 28 square metres.
Step 2: Compress the count: 4 rows of 7 is 4 × 7 — the area formula A = length × width is just row-counting written small. It's the multiplication array wearing garden clothes.
Step 3: Units discipline: 28 SQUARE metres (m²) — tiles, not fence-lengths. Draw one m² tile in the corner as the unit's portrait.
Step 4: Run it backwards (the real test): "a rectangular pen has area 54 m² and one side 6 m — find the other." 54 ÷ 6 = 9 m. Area formulas run both directions; division is the reverse gear.
Step 5: Estimate a real irregular thing with the same tool: trace your hand on cm grid paper — count full squares, pair partial squares into wholes. ~90 cm²? The tile-counting idea handles ragged shapes; formulas only handle tidy ones. Both are AREA.
Step 1: The challenge: on cm grid paper, draw five different rectangles with area exactly 24 cm². Collect: 1×24, 2×12, 3×8, 4×6 … and the fifth? Allow half-units: 1.5 × 16 ✓. (Or an L-shape totalling 24 — if the class is ready to accept non-rectangles into the club.)
Step 2: For each, record the perimeter alongside: 50, 28, 22, 20, 35 cm. Same area, wildly different fencing — reinforcing the independence discovered in earlier grades, now with student-chosen dimensions.
Step 3: The factor-pair harvest: whole-number rectangles of area 24 correspond exactly to factor pairs of 24. Geometry and multiplication facts are the same knowledge drawn differently.
Step 4: The composite move (new skill): an L-shape of area 24 — split it into two rectangles (say 4×3 and 4×3), sum the parts. Decompose → compute → recombine is THE area strategy for every complicated shape from now through high school.
Exit: find the area of a 10×8 rectangle with a 3×2 bite removed from a corner. (80 − 6 = 74 cm² — subtraction is decomposition's twin.)
The trap, sprung deliberately: "1 m = 100 cm, so 1 m² = 100 cm²." Sounds airtight. Let's audit it.
Step 1: Draw 1 m² as a square, 1 m on each side. Convert the SIDES: 100 cm by 100 cm.
Step 2: Tile it with cm²: each row holds 100 tiles, and there are 100 rows → 100 × 100 = 10,000 cm². One square metre is TEN THOUSAND square centimetres — a hundred times more than the trap claimed.
Step 3: Locate the error precisely: lengths scale by 100, but area is length × length, so it scales by 100 × 100. Squared units scale by SQUARED factors. (The picture makes it undeniable: 100 rows of 100.)
Step 4: Test the repair on a new pair: 1 m² in mm²? Sides: 1,000 mm each → 1,000,000 mm². Confidence check: does a million tiny squares in a metre-square feel right? Each mm² is a pen-tip dot — yes.
Step 5: Where this bites in real life: flooring quoted per m² vs a room measured in cm — misconvert and the quote is off by ×100. The squared-scaling idea returns in Grade 9 (similar figures) and never stops mattering.