Dividing Fractions

Teachers, we’re all familiar with the struggle of teaching abstract math concepts. This is especially true for the process of dividing fractions. The very phrase ‘keep-change-flip’ strikes fear into the hearts of teachers and students alike. “Just flip the second fraction and multiply,” we say over and over to blank-faced middle schoolers. But what if there was an easier way; one that is both easier for us to teach and easier for students to understand? Perhaps the secret lies in a hands-on resource.

The established strategy of teaching dividing fractions in the three stages of ‘keep, change, flip’ actually does not help to build the students’ understanding. To truly grasp the concept, students need to be able to picture it in their minds, model it in their hands, and apply that knowledge to their work.

I only taught 10th grade for one year (normally 6th and 7th grade are my jam), but what shocked me most wasn’t their algebra struggles ... it was how many of those big kids were still trying to “KFC” every time they saw a fraction. Adding, multiplying, simplifying .. it didn’t matter! If there was a fraction, someone was flipping something. There’s got to be a better way!

Concrete-Representational-Abstract: Bruner’s Theory of Cognitive Development

Let’s take a step back and talk about the theory behind why hands-on resources are a math teacher’s best friend and the key to teaching fraction division. Developmental theorist Jerome Bruner (you may know him as the inventor of the Spiral Review) found that children learn best when they are introduced to a new concept in a three-stage framework. This is why you should replace your favourite three-word phrase of ‘keep-change-flip’ with ‘Concrete-Representational-Abstract’ (CRA). Bruner (1982) suggests first allowing children to explore the concept with a ‘concrete resource’ (AKA manipulatives AKA hands-on resources), meaning, a real, physical resource they can see and touch. Only after the children can demonstrate their understanding with the concrete resource should they be introduced to the same concept but in a representational way, for example, by speaking the problem in a sentence. Once those two building blocks are in place, the children can demonstrate their learning in an abstract way, for example, by interpreting and solving a written equation.

Don’t *start* with abstract

Work your way **towards** it.

Activity: Pattern Block Construction Zone

In this activity, students use pattern blocks as a visual, concrete tool to make sense of fraction division. This is the ‘concrete’ part of Bruner’s CRA. In case you don’t have pattern blocks on hand in your classroom, the activity includes a set of printable pattern blocks: the perfect amount for students to complete this activity.

Hands-on math activity showing 1 whole *hexagon), 1/2, 1/3, and 1/6 (trapezoid, rhombus, triangle) with pattern blocks — The activity pack contains a hexagon to model 1 whole, a trapezoid for half (½), a rhombus for thirds (⅓), and a triangle for sixths (⅙).

Students begin with a hexagon. This represents 1 whole shape.

Students place two trapezoids on top of the hexagon and discover that two trapezoids fit inside one hexagon, with one trapezoid occupying half the space inside the hexagon. Trapezoids represent half a shape.
Students repeat this to find that three rhombuses fit inside one hexagon.
Then, they find that six triangles fit inside one hexagon.
Build on this understanding by asking students to experiment with finding out how many triangles fit inside the other shapes.

Now that the students have completed the ‘concrete’ part of CRA, move on to representational and prompt them with the question: ½ ÷ ⅙. Prompt the students to use the trapezoid to represent ½ and triangles to represent ⅙. You can say “This question is really asking, how many triangles fit inside a trapezoid?” The students will find that three triangles fit inside a trapezoid, meaning ½ ÷ ⅙ = 3.

Printable fraction division activity for 6th grade classroom — Students *see* what’s really happening when we divide fractions.

November Content Series: Hands-On Fraction Operations

This activity kicks off my new content series, ‘Hands-On Fraction Operations’. The series will focus on bringing manipulative resources into how we teach fraction operations for middle school students. The activities will be interactive, engaging, and provide all the resources necessary. All you need is a printer and you’re ready to go!