When I did my Masters of Education, I wrote a paper on fractions and explored the research surrounding the challenges that students have with fractions. Since that time, I have grown in my own understanding of how best to guide students in their understanding of fractions.
To start, I am going to focus on division of fractions simply because I have a slight hatred of the phrase ‘flip and multiply’. To be honest, I cringe when I hear it.
I am a big fan of Fawn Nguyen and know that she has written a post on division of fractions. My ideas are similar, the conceptual understanding behind teaching fractions doesn’t change, but perhaps my approach is slightly different. Her post motivated me to share my ideas mostly because I know many teachers struggle with teaching fractions conceptually and I figured another post on the concept might contribute a few others ideas for fellow educators to use.
Dividing a whole number by a whole number
Introducing a new idea requires students to access and make connections with the mathematical concepts that they already understand. In the case of division of fractions, we first need them to think about division of whole numbers:
My experience is that most students are quickly able to share:
My goal is to use this knowledge of whole numbers and guide students in applying it to the division of fractions. I like to start with a problem that is very visual and for which students can easily ‘make sense’. I am sure that I have borrowed this problem from someone, but have been using it so long that I no longer remember where I got it from. The problem is a bit contrived (perhaps very much so), but I love seeing the solutions that my students come up with as it always comes through in allowing students to show excellent understanding of a whole number being divided by a fraction. Often without recognizing the leap they have made.
Dividing a whole number by a fraction
Notice that I provide my students with rectangles to act as the brownie – perhaps I shouldn’t as without the picture, it would leave the problem solving more open. However, the difference I see in the willingness of all students to engage in the problem when the visual is provided is completely different from when I just provide the problem. I see the visual as scaffolding for students who read the problem and struggle on where to start. Reflecting on it, I should probably provide the question, give thinking time, and then provide the picture for those who require it.
My students’ solutions usually look like one of the following:
My students explore a whole number being divided by a fraction for quite a bit. We really focus on the visual and do not yet approach representing the mathematics using abstract notation.
Eventually we move on to division of a fraction by a fraction. We spend a long time discussing how we can compare fractions using Which One Doesn’t Belong? and Convince Me That. Which brings us to the following:
Dividing a fraction by a fraction
All students begin by representing the fractions using a rectangle as that is the norm in our classroom. Some student prefer to use folded paper to represent the fractions instead of drawing rectangles. (I am against using circles to represent fractions, but that is for another post).
Many students see the relationship between comparing two-thirds and one-sixth immediately while other prefer to have equal parts in both fractions and redraw the visual representations:
Students quickly see that four green rectangles are needed. They are applying their knowledge of whole number division to a fraction divided by a fraction.
We then approach more challenging problems, but use the same ideas as previously.
For this problem, students recognize the need to have equal parts in both fractions and redraw the visual representations in order to better compare the fractions.
We then have the following problem:
Students recognize that we need one whole set of red squares plus one part of the red (which is made of three parts). Written mathematically, this means:
After time exploring different problems of fractions being divided by a fraction using visual representations, we are ready to ‘mathematize‘ our understanding by applying our understanding and use of equivalent fractions by finding common denominators.
We spend a lot of time discussing and sharing with each other what it means that the denominator of the fraction now equals one. What does it mean to divide by one? Why is dividing by one an important mathematical concept?
Using our understanding, we look at other ways we can have a denominator equal to one. Through exploration, this leads us to using the reciprocal:
Which ultimately leads to (cringe) “Flip and Multiply”. However, this time it is with reasoning and understanding and not simply a rule thrown out for students to blindly follow.