Ambiguous Wording in a Garden Fence Problem
A sixth-grade math item reads: 'A rectangular garden is three times as long as it is wide. The perimeter is 48 meters. What are the dimensions?' One student writes width = 6 m and length = 18 m, using length = 3 × width. Another student writes width = 4.8 m and length = 19.2 m, arguing that 'three times as long as it is wide' means the length is the width plus three more widths, or length = 4 × width. The teacher says the first interpretation is standard but the wording is ambiguous. The class is asked to revise the problem so that only the first interpretation is possible, and to explain why the second interpretation, while nonstandard, is linguistically defensible. The students must also show the equations for both readings and check each against the 48-meter perimeter.
1. Solve the problem under both interpretations. Show the equation for each and check the perimeter.
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First interpretation: length = 3 × width, so L = 3W. The perimeter is 2(L + W) = 48, so L + W = 24. Substitute: 3W + W = 24, so 4W = 24 and W = 6. Then L = 18. Check: 2(18 + 6) = 2(24) = 48. Second interpretation: length = width plus three more widths, so L = W + 3W = 4W. Then 4W + W = 24, so 5W = 24 and W = 4.8. Then L = 19.2. Check: 2(19.2 + 4.8) = 2(24) = 48. Both pairs satisfy the perimeter; the wording is what changes the equation.
2. Revise the sentence 'The garden is three times as long as it is wide' so that only the first interpretation is possible. Explain why your revision is clearer.
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A precise revision is: 'The length of the garden is three times its width' or, even more explicitly, 'Let the width be w meters. Then the length is 3w meters.' The first version removes the comparative phrase 'as long as,' which some readers can interpret as an additive comparison, and states a direct multiplicative relationship. The second version is clearest because it gives the equation itself: L = 3W. It leaves no room for the reading 'length = width plus three more widths.' In a math item, the equation is often better than a verbal phrase because it defines the relationship exactly and lets students check the perimeter without guessing the intended comparison.
3. Which interpretation is standard, and why is the other interpretation linguistically possible even though it is nonstandard in math class?
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The first interpretation is standard: 'three times as long as' normally means length = 3 × width, a multiplicative comparison. The second interpretation comes from everyday phrasing such as 'three times longer than,' which some speakers read as 'the original length plus three more times the width,' or 4 × width. That reading treats 'longer than' as an additive increase rather than a direct ratio. In careful math wording, 'three times as long as' and 'three times longer than' should not be treated as identical, because the second phrase is more ambiguous. The item should use an equation or the phrase 'three times the width' to avoid the additive reading.