Invert-and-Multiply...But Why?

If you teach division of fractions (the other "F" word) using the standard algorithm of invert and multiply, then you may have wondered, as I have, why it works. As a self proclaimed "math nerd," I have been thinking about this off and on for about a year now. As I started to work deeply with the AZCCRS (that's "Arizona College and Career Ready Standards" for those of you who do not live in AZ), I was really impressed at how the standards require the use of drawing and modeling with concrete objects to help students understand and make connections to procedures. I believe this is a key piece math education has been missing for far too long.

However, there was one common procedure (algorithm) that I could never connect to a model or drawing for myself - that is the standard algorithm for dividing by a fractions. The "Invert-and-Multiply" procedure is a mysterious procedure to most (if not all) students and I would guess to most adults, including teachers, as well. No matter how I tried, I was not able to connect a drawing to the standard algorithm of flipping and multiplying that you are probably so familiar with. Three education specialists (two math and one STEM) in my office have tried for months to find a way to demonstrate the connection for students. We were able to use drawings to answer division of fraction questions, but the elusive connection was always absent.

That all changed recently. As part of an effort to roll-out a math program to schools in Pinal County I have been participating in training offered by the Rodel Foundation of Arizona. The training is targeted at 3rd grade teachers. As part of the training we were asked to read selected chapters from "Teaching Student-Centered Mathematics - Developmentally Appropriate Instruction for Grades 3-5" by John A. Van de Walle, Karen S. Karp, LouAnn H. Lovin, and Jennifer M. Bay-Williams. It is a wonderful resource and provides insight into helping students make connections so that they understand mathematics at the deep level required by the AZCCRS. 

I started reading the chapter on fraction operations as required. I expected to read another technical proof on why the invert and multiply routine works. I had seen multiple explanation of this and understand and agree with them. However they always fall short of making the connection using a drawing or model. As I read the section I was encouraged and finally it happened. As I worked through the suggested questions to ask of students, I began to use drawings to help me make sense of the problems I suddenly realized each time I was dividing my drawing by the denominator and then multiplying by the numerator. Although I know I was doing this with every problem, the secret for me to unlocking the connection between the drawing and the algorithm was the context that was given to the problem. It finally made sense! And I understood it! 

I worked several more problems just to make sure I understood it. Then the other two specialists and I practiced at the circular table in the middle of our office space. This is where a lot of great discoveries take place (or a lot of junk piles up - depending on the work load of any given week). We tried what ifs and asked each other to explain it again. After an hour or so, feeling accomplished, we went back to our individual tasks for the day.

A few weeks later, while I was working with a group of middle school teachers, who are working on studying the standards and writing effective lesson objectives in preparation for writing curriculum for their district, the question arose. How can we help students understand the process of dividing fractions. I pumped up my chest, and smiled and stated, "I can show you." Feeling pretty smart, I walked to the white board and asked for them to give me a question. They asked the question (of course it was a word problem) and I started. I drew some rectangles, divided them accordingly and then completely forgot everything that I thought I knew about making connections. (This ever happen to anyone besides me?) I tried another approach. Maybe a different context. Cookies! Yeah, there was something about cookies that helped me understand it last time.

I drew some cookies. No luck.
I drew some ribbon - you know for making bows. No soup.
Rates! - That was it. If it takes 2 and 1/2 hours to travel 3 and 1/8 miles how far can you travel in one hour. Nope - still couldn't put it together.

Every adult (7 middle school and high school math teachers and two math specialists) tried for 45 minutes to make the connections. Finally our time ran out. I left for 10 days of vacation and the teachers moved on to greener pastures. The first thing I did when I got back to work, was grab "Teaching Student-Centered Mathematics" and make the connection for myself again.

So now, after learning and forgetting and relearning, I decided it would be a good idea to have it where I can always access it. Isn't the Internet great! I can post it here, share it, get feedback, and find it when I need it.

First - start by asking students questions where the divisor is a unit fraction (i.e. 3 divided by 1/2, 5 divided by 1/4, 3 and 3/4 divided by 1/8). Also, to help students visualize, give them a context. (How many servings of 1/2 are in 3 containers?) This will help them create pictures to help them with the math.

This is how I approached the question "How many servings of 1/8 are in 3 3/4 containers?"

I first drew a picture of 3 3/4. I divided it into 1/4s. I kind of did this intuitively. I'm not sure if students would work with way at this point, but as they work and get stuck I may direct them in this direction later.

I then needed to decide how many 1/8 fit into the 15/4. I see that if I divide each section into 2 parts I will then have 1/8s and I can simply count.

Simply by counting I see that 15/4 divided by 1/8 is 30. Thus, there are 30 1/8 servings in 3 3/4. The standard algorithm would result in my multiplying 15/4 times 8/1 or (15*8)/(4*1). At this point I was still not able to make the connection. But I could see that in effect when I counted 2 1/8s for each quarter which means I multiplied by 2. 15 * 2 is 30. I can also see that if I had first reduced the fractions when multiplying by the inverse I would have multiplied 15 times 2 since the 8 would reduce to 2 and 4 would reduce to 1.

The next question really got me on the way. "You have 1 1/2 oranges, which is 3/5 of an adult serving. How many oranges (and parts of oranges) make up 1 adult serving?"

More drawing. I started by drawing a representation for the oranges. 

I also knew the drawing would represent 3/5 of an adult serving. I thought to myself, if I could figure out what 1/5 of an adult serving is, then I could simply multiply by 5 so that I would have 5/5 or 1 adult serving. (Here's where I started to put this together.) If 3/2 of oranges is 3/5 of a serving then I divide the 3/2 into 3 equal parts (dividing by 3 - which is the numerator of the divisor) I will know 1/5 of an adult serving. 

Back to the drawing.

I know this mean 1/2 of an orange is 1/5 of an adult serving. Therefor multiplying 1/2 times 5 gives me 5/2 oranges or 2 1/2 oranges. I can now see with the picture that I divided by 3 and although I didn't multiply by 5 using the picture I can see the connection to the standard algorithm. I can see the light!

Now to try something a little more difficult, where the fractions aren't as friendly. "Aidan found out that is she walks quickly during her morning exercise, she can cover 2 3/5 miles in 4/7 or an hour. How fast is she walking in miles per hour?

I start by drawing 2 3/5 miles. I am also thinking that this represents 4/7 of an hour. Again, if I can determine how far in 1/7 of an hour, I will be able to determine how far you can travel in an hour, or mph.

To do this I need to split the 1/5 sections into 4 equal groups, or divide by 4. Since 13 doesn't divide into 4 even groups I will need to rename the sections. I know that if I split each 1/5 section into 4 equal parts, then I will be able to split the 52 sections in 4 even groups.

This means that she travels 13/20 of a mile in 1/7 of an hour. To determine how far she can travel in 1 hour, I multiply 13/20 times 7. This means that she can travel 91/20 miles in one hour, or approximately 4 1/2 mph or exactly 4 11/20 mph. So in summary, I divided 13/5 by 4. In order to do so I changed 13/5 into 52/20. When I divided I dot 13/20 miles for 1/7 hours and then multiplied by 7 to get 91/20 or 4 11/20 mph. I divided by the numerator and then multiplied by the denominator. I have made the connection using the algorithm to what I have done with the drawing.

I still think I can make this connection more explicitly, but I am much closer than I was and think I could help students connect these ideas. In some instances I think that students find this easier they have no knowledge of the standard algorithm and do not enter the conversation with misconceptions that we as adults sometimes bring to the situation.

Let me know if this makes sense to you and/or if you have another way to demonstrate this connection to help students make sense of and understand this often poorly understood algorithm.