I’ve recently been working to develop, among other things, my approach to introducing addition and subtraction of fractions with different denominators. In what follows, I shall outline my current thinking based on work I have done with two different secondary school classes and with a student teacher, Luke Ito, at certain points during the past year or so. The motivation to set my ideas down in this post came after a brief exchange with Sharon Malley (@mathsmumof2) during the Twitter #mathscpdchat event on March 10^{th} 2020. This is a weekly happening on Twitter on Tuesdays during English term-time from 7 till 8pm UK time, organised by the National Centre for Excellence in the Teaching of Mathematics. Summaries of previous discussions, ably provided by Mary Pardoe (@PardoeMary), are available on the NCETM website.

First, a brief background sketch. With both classes, pupils’ prior learning had provided them with a reasonable conceptual grounding in fractions. Of particular relevance to this account is the fact that they were fairly proficient at identifying whether or not a given fraction was in simplest form and in simplifying any fraction that wasn’t. They were also able to convert improper fractions to mixed numbers and vice versa.

My starting point, unsurprisingly, was to introduce addition and subtraction of fractions with the same denominator, judiciously using a variety of pictorial aids as scaffolding where required. I’m keen for my pupils to become able to work fluently with purely numerical representations and spent two or three periods building up their skills with small sets of questions sequenced as follows:

- Add and subtract fractions where the answer is a proper fraction already in simplest form,

e.g. _{ }^{2}/_{7} + ^{3}/_{7}

- Add and subtract fractions where the answer is a proper fraction not in simplest form,

e.g. _{ } _{ }^{8}/_{9} – ^{2}/_{9}

- Add and subtract fractions where the answer is an improper fraction already in simplest form ,

e.g. _{ }^{4}/_{7} + ^{5}/_{7}

- Add and subtract fractions where the answer is an improper fraction not in simplest form,

e.g. _{ }^{5}/_{9} + ^{7}/_{9}

- Add and subtract mixed numbers where the answer is already in simplest form,

e.g. _{ }2^{4}/_{5} – 1^{3}/_{5}

- Add and subtract mixed numbers where the answer is not in simplest form,

e.g. 2^{5}/_{6} – 1^{1}/_{6}

- Perform more challenging calculations with mixed numbers,

e.g. 2^{4}/_{5} + 1^{3}/_{5}

or 3^{1}/_{6} – 1^{5}/_{6}

The categories I have described here were not shared with the class; instead, I included prompts like “What’s different here?” at the end of each question set.

Pupils were, of course, required to simplify all answers where possible. (We have a mantra that a calculation is either correct and complete or correct but not complete.) We also discussed a range of strategies for tackling the questions involving mixed numbers.

Now for the tricky bit: what if the fractions to be operated on don’t have a common denominator? The insight I had here was to turn this question on its head:

What if I set some questions involving two fractions that *do* have a common denominator but aren’t both in simplest form?

Something like this:

Work out the answer to

^{3}/_{6} + ^{2}/_{6}

Now simplify any fractions you can in the calculation you have just performed.

What is the answer to

^{1}/_{2} + ^{1}/_{3}

and how do you know?

Write down some subtraction facts that must be true based upon the above calculations.

Work out the answer to

^{1}/_{2} – ^{1}/_{3}

Write down some addition facts that must be true based upon the subtraction task you have just completed.

It was my intention to keep this blog as short as possible, so I shall wind it up by reminding you that my aim here was to *introduce *pupils to adding and subtracting fractions with different denominators, building on a certain set of prior knowledge. I invite those readers who are teachers to reflect on whether and how these ideas could be used to support pupil learning in your own setting. In particular, the door is open to proceed as you see fit, for example, using either direct instruction or continuing with the more inquiry based approach I have sought to adopt here.

Thanks for joining me in learning, teaching and doing some mathematics. I hope your time spent proves worthwhile.