The Fraction Fair-Share Problem
A Grade 5 class is splitting 5 identical rectangular pizzas among 6 students so that everyone gets the same amount. Maya says each student should get 5/6 of a pizza. Leo says that is impossible because you cannot cut one pizza into sixths evenly, so each student should just get 1/2 of a pizza and the leftovers stay in the box. Their teacher asks the class to test both claims. The class also has a second task: 3 identical granola bars shared equally among 4 students. Priya notices that in both problems the number being shared is smaller than the number of people, and wonders whether the answer is always less than 1 whole.
1. Evaluate Maya's and Leo's claims about the pizza. Which reasoning is mathematically sound, and how would you justify it with a model or number sentence?
Read the worked answer
Maya is correct. Sharing 5 pizzas among 6 students means each student gets 5 ÷ 6 = 5/6 of a pizza. One way to see it: give each student 1/6 from every pizza, so each student collects five 1/6 pieces, which is 5/6. Leo's error is assuming a single pizza must be divided among all 6 students at once; instead the 5 wholes are distributed, and each whole contributes a sixth to every student. Leo's plan (1/2 each) leaves 5 − 3 = 2 pizzas undistributed, so it is not an equal fair share of the whole amount. Check: 6 × 5/6 = 30/6 = 5 pizzas, so nothing is left over.
2. Priya claims that when you share fewer items than people, the share is always less than 1 whole. Test her claim using the granola bar problem and explain whether it always holds.
Read the worked answer
For the granola bars, 3 ÷ 4 = 3/4 of a bar per student, which is less than 1, so Priya's example supports her claim. The claim is true whenever the number of items is less than the number of sharers: if a < b, then a ÷ b = a/b < 1 because the numerator is smaller than the denominator. A fraction less than 1 means each share is part of one whole. If the items equal the people (4 ÷ 4), each share is exactly 1. Priya should state her rule with the condition 'fewer items than people,' because sharing more items than people (for example 7 ÷ 4) gives a share greater than 1.