Equal Perimeter but Different Area in Third-Grade Rectangles
Mrs. Patel's third graders are building rectangles with exactly 14 one-inch segments to form the border. Groups make 1×6, 2×5, and 3×4 rectangles. After sharing, Jamal says, 'All these rectangles have the same perimeter, so they must have the same area. The 3×4 one is just measured differently.' Other students agree because 'perimeter is the outside and area is the inside, and the outside didn't change.' Each group records perimeter and area in a two-column table, but several groups write the same number in both columns because they count border segments as area. Mrs. Patel wants to use this disagreement to teach that perimeter and area are different attributes and that shapes with equal perimeter can have different areas. She has grid paper, 14-inch lengths of yarn, and square tiles.
1. Evaluate Jamal's claim using the three rectangles. What is the most likely conceptual confusion, and how would you represent the difference?
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Jamal's claim is false. All three rectangles have perimeter 14 inches: 2(1 + 6) = 14, 2(2 + 5) = 14, and 2(3 + 4) = 14. Their areas are 6, 10, and 12 square inches, respectively. The most likely confusion is treating perimeter as a global measure of size that determines area, or conflating linear units with square units. Students may also be counting border segments as if they were filling the inside. A representation should make the two attributes separately visible: outline each rectangle with a 14-inch length of yarn to show that the distance around is the same, then fill each outline with square tiles and count the tiles inside. Record perimeter as 14 inches in every row, but record area as 6, 10, and 12 square inches. Ask students to explain why the yarn length is unchanged while the tile count changes. The fixed perimeter is redistributed between length and width; a 1×6 rectangle is long and narrow, while a 3×4 rectangle is more compact and encloses more squares.
2. Describe an instructional sequence using the available materials to help students construct the distinction, including at least one question that presses on units.
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Begin by having each group build one of the three rectangles on grid paper and trace its border with yarn. Compare yarn lengths: all are 14 inches, so the perimeters are equal. Next, fill each rectangle with square tiles and count: 6, 10, and 12. Have groups record both measures in a table with unit labels. Then press on units with the question: 'The yarn is 14 inches long; why can't we report the area as 14?' Students should explain that perimeter counts linear inches around the edge, while area counts square inches covering the inside. A second question is: 'If the outside length is the same, why does the number of squares inside change when we move tiles from a long row to a more square shape?' Have students rearrange the same 14 border segments or compare the 1×6 and 3×4 outlines to see that the same perimeter can enclose different amounts of area. End by asking students to write a sentence distinguishing perimeter and area and to give one example where equal perimeters do not mean equal areas.
3. A student concludes, 'The closer a rectangle is to a square, the larger its area for a fixed perimeter.' Is this a valid generalization? Use the rectangles and one additional example to justify.
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For rectangles with a fixed perimeter, the generalization is valid: area increases as the side lengths become more nearly equal, and the square maximizes area. For perimeter 14, the half-perimeter is 7, and the whole-number rectangles are 1×6 = 6, 2×5 = 10, and 3×4 = 12; the square would be 3.5×3.5 = 12.25, larger than all of them. For perimeter 16, the half-perimeter is 8, and the rectangles 1×7 = 7, 2×6 = 12, 3×5 = 15, and 4×4 = 16 show the same trend: the 4×4 square has the greatest area. The student's phrasing is acceptable for rectangles, but the scope matters. Among all shapes with a fixed perimeter, the circle encloses the greatest area, so 'closer to a square' is not a universal rule for every shape. A precise classroom statement is: 'Among rectangles with the same perimeter, the square has the largest area.' That keeps the claim within the set of examples the class investigated and avoids overgeneralizing beyond rectangles.