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Writing Sensible Word Problems
Creating meaningful, unambiguous word problems can be
difficult. When word problems involve
fractions and fraction operations, special care must be taken to avoid
strange or ambiguous part/whole relationships. For instance, although the original word
problems in Table 4.2 may sound plausible, they are mathematically
deficient.
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Original Word Problem
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Mathematical Issues
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Improved Word Problem
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Bill has half a pizza.
Jim has a third of a pizza. How
much do they have altogether?
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Deficiency: This problem is ambiguous. The two pizzas might not be the same
size. Until you know that both
fractions refer to the same pizza or to pizzas of the same size, the problem
is not well-defined.
Strategy: Rephrase
the problem so that both parts refer to the same whole.
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Bill and Jim share a pizza. If Bill eats half and Jim eats a third,
how much do they eat together?
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Mr. Smith’s fifth grade class has twenty five
students. If one-third of the students
vote to go to the zoo and two-thirds vote to go to the museum, how many
students voted to go to the zoo?
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Deficiency: This problem is strange. One-third of twenty five equals eight and
one-third. In the context of the
original problem, “eight and one-third students” has no sensible
interpretation.
Strategy: Change contexts so that any fractional part of
the whole has meaning.
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Mr. Smith’s aquarium holds 25 gallons of water. How many gallons has he put into the
aquarium when it is one-third full?
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Table 4.2: Writing
Meaningful Word Problems
Table 4.3 shows two word problems and their inverses. In general, the same attention used to
avoid ambiguity and strangeness when writing word problems should be applied
when writing their inverses. Fortunately,
if a word problem is well written, formulating a satisfactory inverse is
often easier than writing the original problem.
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Word Problem
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Inverse Word Problem
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Bill and Jim share a pizza. If Bill eats half and Jim eats a third, how
much do they eat together?
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Together, Bill and Jim eat 5/6 of a pizza. If Jim ate 1/3 pizza, how much did Bill
eat?
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Mr. Smith’s aquarium holds 25 gallons of water. How many gallons has he put into the
aquarium when it is one-third full?
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Mr. Smith pours eight and one-third gallons of water
into his aquarium. If, at that
point, the aquarium is one-third full, how many gallons of water does it
hold?
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Table 4.3: Word and
Inverse Problems
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