Before reading further, ask your students the following question:
What would you say would be a good story or model for 1 ¾ ÷ ½ ?
If they are typical, they will make up stories about sharing among two people, or about taking half of a given quantity.
In her book Knowing and Teaching Elementary Mathematics, Liping Ma reports that only half of the American elementary teachers she worked with could compute the quotient to the above question correctly, and almost all failed to come up with a correct representation.
How can teachers understand division with fractions beyond “invert and multiply” and help their students build rational number sense?
A good place to start is by extending understanding of whole number division to fractions. Bring in to class a pitcher and an assortment of measuring cups. Fill the pitcher with 6 cups of water and ask students, “How many 3-cup portions can I get from the 6 cups in the pitcher?” When they respond 2, ask how they know. After discussion about why this is a division problem, record 6 ÷ 3 = 2 on the board and discuss that the 6 cups are being divided into 3-cup portions, and the quotient tells us how many portions, or groups of 3, can “fit” into 6 cups. Pour the 6-cups into 3-cup portions to check the students’ result.
Next, ask how many 2-cup and 1-cup portions can be made from the original 6 cups. Discuss and record 6 ÷ 2 = 3 and 6 ÷ 1 = 6, and check by pouring. Notice that as the portion size decreases, the number of portions increases. So how many ½-cup portions can we get from the 6 cups? Have students think this through and justify their answers to each other. Ask, “Is this still a division problem?” and discuss. When the class can agree that the problem is still the same, all that has changed is the size of the portion, pour the 6 cups into ½-cup portions and count. Record 6 ÷ ½ = 12 after confirming the quotient. Move on to 6 ÷ 1/3 = 18, 6 ÷ ¼ = 24, and see if students begin to notice a pattern, that 6 is being multiplied by the denominator of the fraction. Reason about why this should be so.
Next use 2/3-cup and ¾-cup and see what happens. There are half as many portions of the 2/3-cup as there are 1/3-cup, and one third as many ¾-cup portions as there are ¼-cup portions. Discuss why this is so. It seems that, not only are we multiplying by the denominator, we are also dividing by the numerator. Finally, connect what you have already learned about multiplying fractions to conjecture that dividing by a fraction is the same as multiplying by the fraction with its numerator and denominator switched. At this point you can introduce the vocabulary term reciprocal, one of whose definitions is “interchanged”, as in interchanging the numerator and denominator.
Extend to dividing by a mixed number by considering 1 ½ -cup portions, etc.
Finally, ask students the question at the top of the page again. See if their stories now involve how many ½’s can fit into 1 ¾ , a proper interpretation of division.