This is an account of a particularly successful topic from Y5 on multiplying fractions and integers. It’s fast becoming one of my favourite concepts to teach.
I showed multiple representations of a calculation, showing how to make it with cubes:
In this example, it was important to make four lots of two thirds first before rearranging the coloured cubes: later when they draw it on a number line, they’ll need to be thinking of four jumps of two thirds. The language of number of parts, size of each part and whole was supported by the bar model and came in handy for the empty box problems later.
Children needed to make other examples of calculations, talking about it with a partner:
Most children were fine with this and ready to move on to pictorial representation on a number line. Some needed a bit more consolidation with how the cubes represent a calculation. This is of a set of scaffolded questions where children made the calculation with cubes, then drew it:
For those ready for a number line, I explained a worked example before modelling how to do one from scratch:
The explanation had to be clear on how to construct the number line accurately, which involved making decisions about how many squares to leave between each whole number (using the denominator of the ‘size of each part’), and how many squares each jump needed to be (using the numerator of the ‘size of each part’). The final decision to make was how many jumps to draw, using the number of equal parts. The whole, and the answer to this particular calculation, was where the last jump ended up in the number line.
When children practised doing this, they chose from using a scaffold of minimally different questions or tackling varied questions:
Some children needed a few lessons to keep practising this and each time we returned to it, I started by showing multiple representations of the same calculation:
Some children grasped this concept quickly and needed to work on deeper tasks. I made some subtle changes to the language I used to describe what was happening (bottom right corner of the above image) and used the same bar model to introduce inverse operations with aim of learning to solve empty box multiplication of fractions:
Bar models helped to explain what was known and what was unknown, using the language of whole, number of parts and size of each part. I showed how it looked on a number line too:
Some managed to get on to dividing fractions and whole numbers, using bar models (which they were familiar with for integer multiplication and division) and number lines to show what was happening:
All the while, more children began to move from basic multiplication of fractions to some deeper thinking about multiplicative reasoning and fractions.
A few weeks later, children worked out the area of rectangles involving fractions and had internalised much of the language to be able to represent questions like this: