The Fraction Pizza Problem
Four students share two identical pizzas. Each pizza is cut into 8 equal slices. Ana eats 3/8 of a pizza, Ben eats 1/4 of a pizza, Cara eats 1/2 of a pizza, and Dan eats 1/8 of a pizza. A student says, "Together they ate 6/8 of one pizza, so there is more than one whole pizza left." Another says, "They ate more than one whole pizza." The teacher asks the class to add the fractions, compare to the two wholes, and decide who is right.
1. Add the four amounts and determine how much pizza was eaten in total. Express the answer as a fraction and as slices.
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Convert to eighths: Ana 3/8, Ben 1/4 = 2/8, Cara 1/2 = 4/8, Dan 1/8. Sum = 3/8 + 2/8 + 4/8 + 1/8 = 10/8 = 1 2/8 = 1 1/4 pizzas. In slices: 10 slices eaten. Since each pizza has 8 slices, 10 slices = one whole pizza (8 slices) plus 2 slices. So they ate more than one whole pizza, supporting the second student.
2. How much pizza is left out of the two pizzas? Show your reasoning with a subtraction and check with slices.
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Two pizzas = 16/8 (16 slices). Eaten = 10/8 (10 slices). Left = 16/8 − 10/8 = 6/8 = 3/4 pizza, which is 6 slices. Check: 10 + 6 = 16 slices. The first student confused the leftover 6/8 with the amount eaten; 6/8 is less than one whole, so there is not more than one pizza left. The correct statement is that 3/4 of a pizza remains.
3. Explain the error in the statement "they ate 6/8 of one pizza." What misconception does it reveal?
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The statement likely comes from adding numerators only (3+1+1+1 = 6) and keeping 8, ignoring that the fractions have different denominators. Before adding, all amounts must be expressed in a common unit (eighths): 1/4 = 2/8 and 1/2 = 4/8. The correct sum is 10/8, not 6/8. The misconception is treating fractions with unlike denominators as if the denominators were the same. A second issue is unit confusion: 6/8 refers to part of one pizza, but the total spans two pizzas, so comparing to one whole misstates the situation.