Introducing Dividing by Fractions

This week, we introduced multiplying and dividing by fractions. I thought Making Math Meaningful Level 6 plowed through the first page too quickly. We have been spending more time exploring concrete versions of this activity. Yesterday, we focused on scooping measured cups of water into a bowl and scooping the water back with fractional measuring cups (1/2-cup, 1/3-cup, 1/4-cup).

The first couple of clips may seem dull to you, but there is a payoff! If you can believe it, Pamela picked up the pattern for division problems like this (a ÷ 1/D) by the time we reached the second one. If you do not have time for all of these clips, watch the clip for Problem 2 in which I do a double-take when she solves the second problem in her head!

Here is the first problem we explored yesterday:
Problem 1
Put 6 cups of water in a bowl. Scoop out 1/2-cup at a time. How many scoops did you have?
The math equation for this problem is 6 ÷ 1/2 = _______.

The clip below shows me introducing the problem. I wanted to see how well scaffolding works with academics. First, I asked her which measuring cup she should use to make six cups, and she wisely and logically chose the two-cup measurer. She is not very accurate with measuring liquids, so I put masking tape to show the exact line. I was very particular about getting eye-level to measure. I am doing a great deal of thinking out loud, so Pamela sees me solve problems out loud.


In this step, we scoop 6 cups of water, back into the pot, by 1/2 cups to demonstrate dividing by a fraction. She was a little distracted thinking about You-Tube videos but quickly got on track. Pamela has a little trouble verbalizing the count because of the pause in the action when I dump the full cup back into the bowl Either that, or she was getting ahead of herself).


Pamela invented a game called Fly or Drive last week. She tells me two states, and I have to say whether or not I'd fly or drive. In this clip, I redirect her to get back on track with the fraction lesson: figure out how many 1/2-cup scoops equals 6 cups. She easily gets back to business. We finished the first problem.


Problem 2
Put 5 cups of water in a bowl. Scoop out 1/3-cup at a time. How many scoops did you have?
The math equation for this problem is 5 ÷ 1/3 = _______.

Pamela astounded me!!!! On the second problem, Pamela solved 5 divided by 1/3 in her head!!! Before we did any measuring, she knew the answer was 15. She had already spotted the pattern to division problems with this format: a ÷ 1/D. I am pretty well-versed in her abilities, but today she made me do a mega double-take! This clip is the million-dollar, must-see clip if you are pressed for time today!


Here are the rest of the problems. By the time we reached Problem 5, it was obvious that Pamela knew the pattern. So, I let her tell me the answers for the rest and we only needed to do concrete measuring for the first three problems.

Problem 3
Put 2 cups of water in a bowl. Scoop out 1/4-cup at a time. How many scoops did you have?
The math equation for this problem is 2 ÷ 1/4 = _______.

Problem 4
Put 1 cup of water in a bowl. Scoop out 1/4-cup at a time. How many scoops did you have?
The math equation for this problem is 1 ÷ 1/4 = _______.

Problem 5
Put 4 cups of water in a bowl. Scoop out 1/2-cup at a time. How many scoops did you have?
The math equation for this problem is 4 ÷ 1/2 = _______.

Problem 6
Put 4 cups of water in a bowl. Scoop out 1/3-cup at a time. How many scoops did you have?
The math equation for this problem is 4 ÷ 1/3 = _______.

Problem 7
Put 4 cups of water in a bowl. Scoop out 1/4-cup at a time. How many scoops did you have?
The math equation for this problem is 4 ÷ 1/4 = _______.

After three problems, I did a quick review of the first three which we did by hand by scooping water. In this clip, you see us review the answers and then she correctly calculates the answers to Problems 5 through 7 in her head.


On Monday, we will do this with cutting up strips of paper and see if she can come up with a predictable rule written algebraically with letters. In Making Math Meaningful, children work with letters from the earliest books, and Pamela understands that letters can mean number. I am sure, with a tiny bit of guidance she will be able to come up with something like:

a ÷ 1/D = a x D