I was talking with a group of 5^{th} grade students today…

**Me:** You know that next year you’ll need to be able to divide a fraction by a fraction and not just a fraction by a whole number? (MCC6.NS.1)

**Student:** I can do that right now Mr. Fletch! I don’t need to wait until next year!

**Me:** Oh yeah? How is that?

**Student:** Well all division is, is repeated subtraction right?

I returned 5 minutes later with the following problem and asked the students to solve it using repeated subtraction. *Sam had 1 ½ cups of flour in the cupboard. He needed 2/5 of a cup to make 1 batch of cookies. How many batches of cookies can Sam make?*

I didn’t solve the problem in advance and just whipped together some numbers (silly me). Having worked exclusively with models to this point and never attempting repeated subtraction myself for fractions, I thought it would make sense. My understanding was that repeated subtraction and division went hand-in-hand although it’s not the most efficient method…but an accurate one. The student’s solution.

Everything looked great! I quickly changed the mixed number into an improper fraction, inverted then multiplied. Yes I know it’s super procedural but I wanted to quickly check the students solution *(disclaimer: I understand why invert & multiply works so I’m okay to use it).*

**Me speaking to myself:*** And the answer is… *3 and 3/4. *WHAT THE?!?!?!* How is that?

I repeatedly looked at the student’s work and double checked it to make sure he had made no errors. Everything made sense which completely baffled me and then there it was!

**Me:** Draw a model to represent what you just did.

**Student:** But I have 2 different answers and I know my drawing is right.

**Me**: How many tenths make a batch and how many tenths are “leftover”?

**Student:** Well I need 4/10 and I only have 3/10 and (the light-bulb explodes)……..Oh I get it! It takes 4/10 of a cup to make a batch and the 3/10 in the repeated subtraction is 3/10 of the 4/10 needed. So I have 3/4 of the next batch. So the answer’s really 3 and 3/4 of a batch!

**Rule:** if you can’t make sense of your thinking through a model, you probably don’t understand!

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## About gfletchy

K-8 math consumer trying to listen and learn each day. Stay thirsty my friends!

Well done, g and 5th grade partner in math sense-making.

Beautiful, elegant, inspiring. Just like you!

This student was definitely rocking the SMPs! What a beautiful example of the necessity of context! Awesome job gfletchy!

Turtle and Katie thanks for the encouragement!

What I found most awesome about this student was that in order for him to use repeated subtraction he had to identify a common denominator. I really like the idea of this because the subtraction/common denominator piece lends itself to a tidy model.

It would have been really interesting to see how he would have approached the modeling of division if he didn’t think of repeated subtraction.