Fraction Division

I tried some new things with fraction division this year.  I want to document for it myself as well as those who ask me about it. 

I started this unit by making claims.  I wrote about that in this blog post. 
For the fraction division I used Cuisenaire rods:

Each student has a set of each of the 10 different rods.  Students were instructed not to shout out answers, but to set the “answer” rods aside and keep quiet so everyone has think time.  Allow time for students to figure out each task.  Ask for students to answer and ask several students how they know.

• Orange:  “This is 1. Find ½.”  
• Dark Green:  “This is ⅔. Find 1.”  
• Purple:  “This is ⅖. Find ⅗.”
• Brown: “This is ⅘. Find ⅗.”  

To explore reciprocal relationships that are needed for division:

Held up and showed students the red rod.  

• “Now think back to the red. Let’s pretend it’s a granola bar. This is ⅓ of a granola bar. Find 1 whole.”  “How many times does ⅓ of a granola bar fit into 1 granola bar?”  “What equation can you write to describe this situation?”  Students may come up with several that are correct but ask for a division equation if they are not getting it. 
• Purple: “This is ⅖ of a granola bar.” Find 1 whole.” How many times does ⅖ of a bar fit into the whole bar?”  

When we divided with the Cuisenaire rods it looked something like this:

I have tried to divide using number lines before and it was a hot mess.  Using the rods we were able to understand division as division (as opposed to multiplying by the reciprocal) and used a measurement model for fractions.  

I also pulled some questions out of my Developing Fraction Knowledge Book.  I love this book and highly recommend it.  

Here is an example:

• A landscape architect bought 5 bags of soil, knowing that each flower bed in the garden she was designing required 3/5 of a bag of soil.  How many flowerbeds can she fill with soil?

Students made different sketches to figure out the answer, but all of them focused on dividing.  Some drew flowerbeds and had 3/5 in each.  Others drew 5 bags and divided them into fifths and colored 3 at a time for each flowerbed.  

After a few questions like this we revisited this question:

• This time determine the number of flowerbeds the architect can make with 1 bag of soil.  Show your reasoning.

This brought us back to the idea of the reciprocal relationship and viewing it as a unit rate.  Once we knew the unit rate we could just multiply by 5 because we had 5 bag.  

What surprised me about the reciprocal relationship is that I thought it was just what you multiply by to get one whole.  In working through this activity we are able to see it is also a unit rate.  We can see the relationship between multiplication and division.  I do not feel like understand all of this enough to get all students understanding it yet.  It did help in understanding why we multiply by the reciprocal when we connect it to the unit rate. 

We also explored division with a common denominator.  Fawn Nguyen has a great blog post about this.  She used graph paper and rectangles.  Like the rods, it helps students understand fraction division as it relates to whole number division.  These 2 strategies helped deepen my students' fraction sense around division.  They were now beginning to understand why the number would get smaller or larger than the starting number.  

Christopher Danielson also has some great posts about fraction division:

Questions from middle school teachers: Division of fractions

More on fraction division (you know you love it!)

Partitive fraction division - I used some of the visuals from this blog and had my students see if they could explain the "students" thinking.  

Common numerator fraction division  - I challenged my students to see if a common numerator strategy was possible.  Most of them came back to say that they were able to figure out a strategy for common numerators but were not sure why it worked.  It was a great one to explore together.