This month, for Thursday Tool School, I am featuring tools to help students develop an understanding of fraction operations. Last week, I shared two videos to support students with fraction multiplication. Today, I want to focus on dividing with fractions. Of all of the fraction skills, dividing fractions seems to be the easiest– not because it is easier to understand, but because it usually gets whittled down to “yours is not to reason why, just invert and multiply.” This one has always baffled me, why shouldn’t our students know why? It’s actually just as easy for me to help my students develop the concept as it is to teach the algorithm.

By far, the best dividing fractions lesson I have ever seen was one from a math conference here in Texas. The lesson was based on the idea of popcorn. I wish I had a demonstration video to show you (I’m working on it; creating instructional videos for teachers is one of my goals for 2017!). The set-up begins with six “cups” of popcorn and a plastic scooper cup (the same cup you used to measure out the six cups). Ideally, there is an activity sheet to go along with the activity. The one from the conference session (which I have misplaced over the years) shows six blank cups (the same size as the scooper cup) for each situation. Essentially, you can use any “cup” for the activity as long as it is the measuring cup and the dividing cup.

A. The first task asks students to determine how many cups can be made from the bucket of popcorn. Well, when students use the scooper cup to pour the popcorn into the dividing cups, they get six full-cup portions. So, 6 cups divided into 1 “cup” scoops yields 6 portions. This question is a baseline so that students understand the process and can compare to this first task when needed.

B. The second task asks students to determine how many half-cup portions can be made from the bucket of popcorn. Students use a dry-erase marker to mark a line at the half-cup mark on the scooper cup and then use the scooper cup to fill to the half-way mark and pour the popcorn into the dividing cups until all of the popcorn is gone from the bucket. After each pour, have students mark where the popcorn filled to and write the group number on the cup. (Note: There should be two scoops in each cup.) In this situation, they get twelve half-cup portions. So, 6 cups divided into 1/2 “cup” scoops yields 12 portions. Students would then divide and label the cups with the group number on the activity sheet. Be sure to have them record a number sentence to match.

C. The third task asks students to determine how many fourth-cup portions can be made from the bucket of popcorn. Students use a dry-erase marker to mark a line at the fourth-cup mark on the scooper cup and then use the scooper cup to fill to the fourth-way mark and pour the popcorn into the dividing cups until all of the popcorn is gone from the bucket. After each pour, have students mark where the popcorn filled to and write the group number on the cup. (Note: There should be four scoops in each cup.) In this situation, they get twenty-four fourth-cup portions. So, 6 cups divided into 1/4 “cup” scoops yields 24 portions. Students would then divide and label the cups with the group number on the activity sheet. Be sure to have them record a number sentence to match.

At this point, students begin to notice a pattern, because it appears that the answer is 6 times whatever the denominator is. Allow students to verbalize and flaunt this understanding but keep going. Ask the students to continue to test their conjecture.

D. The fourth task asks students to determine how many third-cup portions can be made from the bucket of popcorn. Students use a dry-erase marker to mark a line at the third-cup mark on the scooper cup and then use the scooper cup to fill to the third-way mark and pour the popcorn into the dividing cups until all of the popcorn is gone from the bucket. After each pour, have students mark where the popcorn filled to and write the group number on the cup. (Note: There should be three scoops in each cup.) In this situation, they get eighteen third-cup portions. So, 6 cups divided into 1/3 “cup” scoops yields 18 portions. Students would then divide and label the cups with the group number on the activity sheet. Be sure to have them record a number sentence to match.

Now the students know that they have it. They’ve got this skill down. Continue on.

E. The fifth task asks students to determine how many two-third-cup portions can be made from the bucket of popcorn. Students use a dry-erase marker to mark a line at the two-third-cup mark on the scooper cup and then use the scooper cup to fill to the two-third-way mark and pour the popcorn into the dividing cups until all of the popcorn is gone from the bucket. After each pour, have students mark where the popcorn filled to and write the group number on the cup. (Note: This one is tricky because it doesn’t “appear” to follow the pattern. There should be one and a half scoops in each cup. Therefore there may be one group number on two cups. For example, the first cup will say 1 for the first two-thirds and two for the leftover third. The second cup will say 2 for the bottom third and 3 for the top two-thirds). In this situation, they get nine two-third-cup portions. So, 6 cups divided into 2/3 “cup” scoops yields 9 portions. Students would then divide and label the cups with the group number on the activity sheet. Be sure to have them record a number sentence to match.

Now they’re confused and some students will look both puzzled and defeated, but then one bright person will say, “Wait a minute. If you multiply 6 times 3 and divide it by 2 you get 9.” Try to contain your excitement because this student has just developed the invert and multiply algorithm. This is because when we divided the cups into third-cup portions, we had eighteen portions. However, to get two-third cup portions, we simply group two of the thirds together. So, 18 divided by 2 is 9.

Is your brain on fire? I encourage you to play with this one. Get some beans or some rice and mimic the I tasks as I described them above. It’s a phenomenal activity! Props to the CAMT teacher team who presented it!

I’m sure you’re wondering how to continue. You continue the same process with other fractions. For example, try three-fourths. Or, try one and one-third. That one’s great because students will get to see how remainders when dividing with fractions work. Dividing the popcorn into one and one-third, or four-thirds, cup portions follows the same process. When we divided the popcorn into third-cup portions, we got 18 portions. If we divide those into groups of 4 (for the four-thirds), we get 4 full portions and 2 leftover thirds. So, how do I record that? Well, its 4 wholes and 2 thirds out of 4 thirds needed to make a full group (or 1/2 of a portion). 6 divided by 1 and 1/3 (or 4/3) is 4 and 1/2.

This last task really helps students develop the idea of the remainder. They’re so tempted to write 4 and 2/3 because the denominator of one and one-third is three. This helps them understand why. The one and one-third cup portion is effectively the “whole,” so we compare the leftover two-thirds to the whole that is four-thirds.

I hope you enjoyed the activity and will give it a try (before working with the students :o)). There are lots of ways to build from this into other fraction division concepts. Happy popping!

P.S. I know this would be better as a video, but I don’t have one to share just yet. If you know of a quick start guide to creating videos, please let me know. Thank you!

**Sound Off!** How do you use models and manipulatives to teach fraction skills?

## Leave a Reply