As I was organizing my student work pictures this morning, I realized I had tweeted out this awesome work, but never blogged about it.

My students are very comfortable with putting fractions in the numerator. They use them all of time when decomposing, adding and comparing fractions like in these two examples…

One day two of my students asked if there could be a fraction as the denominator. I asked them to try it out and see what they thought.

They wrote their question and then started playing around with some fractions in the denominator. I think they were trying to find one that jumped out and made sense to them. They drew some pictures to see if they could illustrate what it would look like. The 1/1.5 was cut into thirds with 1.5 shaded but when I asked what they would name what they just drew, they said 1.5/3. Hmmmm. They tried talking about 1/ 2/5, didn’t like that one, and moved to 1 / 2/8. After shading, one student wrote the 1,2,3,4 over top of each 2/8. I asked why and he wrote the 4 next to 2/8 and said that there were four of the 2/8’s in his picture, so it is “kinda like four.”

I asked him what he saw as the whole in the picture and he said 2/8, and wrote 2/8 / 2/8=1. I asked them to now think about what the 1/ 2/8 would be, walked away, and came back to this amazing work…

When I came back they said they had realized that 1/ 2/8 was really an improper fraction since 2/8 / 2/8 was =1. I especially love the 2/8 x 4 = 1 and 1 x 4 = 4…did they just divide fractions by multiplying by the reciprocal without even knowing it?! When I asked what that meant, they said since 2/8 was really 1 in their picture, it took four of them to make four wholes.

Another student started playing around with it and came up with this explanation. Even though it is not correct, I especially like the “1 group of 2/8” because he is thinking about “1 piece the size of 2/8” which we talk about a lot, but having trouble thinking about the entire fraction in the denominator. It makes sense because when we talk about 2/3, we could say it is 2 pieces the size of 1/3 or 2 x 1/3.

This student was dealing with the same dilemma, but I love how he attempted to put it into a context…

One student listened to all of their conversations and simply wrote this as his thought about the fraction as the denominator. I have to revisit this one with the class, so interesting!

-Kristin

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Paula Beardell KriegThis is so interesting! Hopefully they will remember this exploration fondly when they get to calculus, where they will be seeing fractions in the denominator all the time.

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mathconfidenceA great fraction as a denominator is 12 divided by 1/2. This example can also help students know why Keep Change Flip works and perhaps will help them remember it as well.

When asking students if they have a favorite division with fractions problem, i tell them i have one 12 divided by 1/2 and they can adopt it 🙂

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