Fraction Addition and Subtraction with the Denominator-ator

Creating and critiquing good mathematical tasks with variation theory
26 November 2019
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Adding and subtracting fractions is tricky.

Many of the adults I teach are confused when it comes to fractions. It can be difficult to remember that addition and subtraction of fractions require common denominators, which stay the same when you add and subtract, while multiplication operates on both the numerator and the denominator. I have written about this: The big deal about fractions

Fraction addition confuses. A fraction operation such as 2/3 + 3/4 requires five operations to get the two fractions to a common denominator and then add the numerators. It can be difficult to explain why this is needed. Addition also seems to begin with multiplication, which further confuses. Lack of fluency with multiplication facts can provide a stumbling block, as the working memory is busy multiplying rather than focussing on the process of fraction addition.

Adult students want to go back to a rote, successful method they can almost remember from school. They revert to a method known as cross-multiplication, the butterfly method and “upside down picnic table”.  Nix the Tricks speaks out against this! I have a Pinterest board of Bad Maths teaching resources, full of them.

A problem with a memorised method without understanding is that it is easily forgotten or confused. Neither does a memorised method help develop understanding of the nature of rational numbers (fractions and decimals) and rational number sense.

Early in fraction topics, time is spent finding equivalent fractions. The idea that there are multiple ways of expressing the same number (in fact infinitely many ways) is new compared with whole or natural numbers. For instance ½ can be expressed as 2/4, 3/6, 100/200, 0.5 etc. We seldom express two as anything other than 2, so whole numbers appear to have just one label.

Students are asked to find equivalent fractions with given denominators, which may seem like a mathematical exercise with little purpose. As teachers we know that they will need to change denominators for addition and subtraction, but we don’t really want to do that first. It’s a bit circular.

While pondering this, I invented The Fraction Assistant. I really want to call it the Denominator-ator but it’s a bit of a mouthful for students who are not yet sure what a denominator is. At my Maths Jam meeting, someone said they thought it was for rating denominators. To which I replied, “But why would you rate them when denominators are so common!” (What is it about maths people and puns?)

The Fraction Assistant consists of strips of equivalent fractions, which are slid up and down a frame until the denominators are the same. This provides a powerful image of what happens when converting to common denominators. The plan is not for learners to continue to use the Fraction Assistant, but rather to gain understanding. By using the Fraction Assistant for some addition and subtraction exercises, studying the equivalent fraction strips and making some of their own, students can move on to fraction addition and subtraction without the assistant.

The frame of the Fraction Assistant displays the terms, denominator and numerator, so that learners become familiar with them. In addition, the lower half is shaded for all the fractions and the frame, making it clear what needs to be made the same.

The strength of the Fraction Assistant is that it isolates the concept of finding common denominators from the mechanism of Lowest Common Multiple and conversion. Those come later once it is clear why we need to find common denominators and can build on the imagery of the equivalent fractions strips.

We have made a prototype “Print-and-Learn” that you can download from our website, along with suggestions for teaching.

Get the Print-and-Learn

We would love feedback from teachers and learners, and suggestions for further development.

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