Area and Perimeter Confusion in a Garden Design Task
Ms. Chen’s third graders are designing rectangular vegetable gardens. Each group gets 24 feet of string to represent fencing and must find possible dimensions. Maya builds a 6-by-6 square and says it uses 36 feet of fencing because “6 times 6 is 36.” Another group builds a 4-by-8 rectangle and says it uses 32 feet because “4 plus 8 is 12, times 2 is 24,” then reports the area as 24 square feet. During the share-out, several students agree that the 6-by-6 garden uses more fencing because it “looks bigger.” Ms. Chen has square tiles, grid paper, and the string. The upcoming assessment asks students to solve problems involving area and perimeter and to distinguish between them. She has one 45-minute block before the review day.
1. Diagnose the confusion in Maya’s and the other group’s statements. What does each student appear to understand, and what is missing?
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Maya multiplies two side lengths when asked for fencing, which is an area procedure applied to a perimeter question. She may know that 6 × 6 = 36 and that area is measured in square units, but she has not attached perimeter to the attribute “distance around.” The other group computes 4 + 8 = 12 and doubles it to get 24, which is a correct perimeter procedure, but then labels the result as area. That suggests they can follow a perimeter formula but do not distinguish what the number measures. The class’s comment that the 6-by-6 garden “looks bigger” is also telling: a square with side 6 has area 36 square feet, while the 4-by-8 rectangle has area 32 square feet, so the square does have greater area-but both use the same 24 feet of fencing. Students are conflating “more area” with “more perimeter.” The missing idea is that perimeter measures the boundary length in linear units, while area measures the covering in square units. The task should make both attributes visible with the string and tiles, not just supply two formulas.
2. Describe a tile-and-string sequence that would help students distinguish area from perimeter. Include specific questions Ms. Chen should ask.
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First have each group lay the 24-foot string around the outside of a 4-by-8 rectangle on the floor or grid paper. Ask: “What does the string measure? Is it covering the garden or going around it? What unit would we use for this measurement?” Then have them cover the inside with square tiles or grid squares. Ask: “What do the tiles measure? Why are the units square feet rather than feet?” Record both: perimeter = 24 feet, area = 32 square feet. Repeat with the 6-by-6 square: perimeter = 24 feet, area = 36 square feet. Ask: “The fencing is the same. Why is the area different? Which garden gives more growing space? Which uses more fence?” This directly confronts the “looks bigger” comment. For the group that reported area as 24, ask them to point to where 24 appears in the model: “Is 24 the distance around or the number of tiles inside?” Finally, give a quick sort of cards such as “amount of fencing,” “grass seed needed,” “distance walked around the edge,” and “tiles to cover,” asking students to name area or perimeter and the unit.
3. How could Ms. Chen use the fixed 24-foot fencing constraint to build the idea that rectangles can have the same perimeter but different areas-and the reverse?
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With 24 feet fixed, students can build all whole-number rectangles: 1 by 11, 2 by 10, 3 by 9, 4 by 8, 5 by 7, and 6 by 6. Have them record perimeter and area in a table. Ask: “Which rectangle uses the most fencing? Which encloses the most space? Are those the same question?” Students should see that every rectangle has perimeter 24 feet, but the areas increase as the shape becomes squarer: 11, 20, 27, 32, 35, and 36 square feet. Then reverse the constraint: give 24 square tiles and ask for rectangles with area 24 square units. Students can build 1 by 24, 2 by 12, 3 by 8, and 4 by 6, and compute perimeters 50, 28, 22, and 20 feet. Ask: “All of these have area 24. Why do they need different amounts of fence?” This pair of tasks separates the two attributes: fixed perimeter does not fix area, and fixed area does not fix perimeter. The table and the physical models give students evidence to explain why the formulas are different, rather than two rules to memorize.