Area and Perimeter in a Garden Design
Mr. Patel gives pairs a design task: draw a rectangle on grid paper with an area of 24 square feet and a perimeter of 20 feet, where each grid square is 1 square foot. Rosa draws a 6-by-4 rectangle: 24 squares inside and 20 unit segments around. Eli draws a 24-by-1 rectangle and says, 'Area is 24, so it works,' but its perimeter is 50. Tia draws a 5-by-4 rectangle, gets an area of 20 and a perimeter of 18, and says, 'That's close enough.' During the share-out, a student asks whether a square counts as a rectangle, and another says Rosa's rectangle has 'more area than perimeter' so it must be wrong. Mr. Patel has 15 minutes for the share-out and wants the discussion to settle the task and the definition question without turning into a vocabulary drill.
1. Explain the difference between Eli's and Rosa's understanding of area and perimeter. Why do several rectangles share an area of 24, but only one shape (up to rotation) meets both constraints?
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Eli treats area as the only attribute of the shape and checks one constraint. Area counts the square units covering the rectangle; perimeter measures the distance around it, so both must be checked. Factor pairs of 24 give the possible dimensions: 1×24, 2×12, 3×8, and 4×6. Their perimeters are 50, 28, 22, and 20. Only 4×6 (or 6×4) satisfies both constraints, because perimeter depends on the sum of the side lengths while area depends on the product; many pairs multiply to 24, but only one of those pairs adds to 10. Rosa's work checks both constraints independently, which is exactly the reasoning Eli is missing.
2. How would you respond to Tia's 'close enough' and to the student who says Rosa's rectangle has 'more area than perimeter'?
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For Tia, return to the grid and count: 20 squares inside does not equal 24, and 18 unit segments around does not equal 20; the task's numbers are exact constraints, so 'close' fails both. Ask her to list factor pairs of 24 and test perimeters, which turns the guess into a search. For the area-versus-perimeter comment, the two quantities have different units-square feet versus feet-so they cannot be compared as numbers. Rosa's 24 and 20 are both correct but answer different questions. A quick demonstration with a 6-by-4 grid labeled '24 square feet inside' and '20 feet around' separates the attributes without a definitions lecture.
3. A student says a square is not a rectangle. Give a definition-based answer and a task that would build the idea.
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A rectangle is a quadrilateral with four right angles; that definition does not require unequal side lengths. A square has four right angles and four equal sides, so it satisfies the definition and is a rectangle with the extra property of equal sides. A 4-by-4 square has area 16 and perimeter 16, and it is a special case of the same family that includes 4-by-6. Task: give students a set of quadrilaterals with right angles in different orientations and side lengths, including squares, and ask them to sort first by 'four right angles' and then by 'all sides equal.' The sort should show squares inside the right-angle group, not beside it. Ask students to name a property every rectangle has and one that only some rectangles have.