Dividing a quantity unevenly is an abstract idea that most children struggle with. As the problems become more complex, people struggle all the more to see what to do, for example:
- Jack and Jill share £28 in the ratio 5:2, how much does Jack receive?
- Jack and Jill share some money in the ratio 5:2. Jack receives £15, how much does Jill receive?
- Jack and Jill share some money in the ratio 7:3. Jack gives £20 of his money to Jill, so now they have the same amount each. How much money do they have altogether?
Even a child who successfully gets their head around the process of ‘adding the numbers, divide by that amount, multiply by each number separately,’ for question 1, is then straight away often stumped by what to do when faced with question 2. The abstract calculation is a nightmare for most.
There are two effective techniques for making ratio problems concrete that somehow seem to slip people for a long time, and those who know about them seem to come across by chance. Here’s the first.
The Box Method
Draw out boxes to represent the ratio.
It quickly makes much more sense to a person now that they will need to share out the £28 from question 1 into 7 boxes.
By this point it’s painfully obvious to anyone that Jack receives £20, and Jill receives £8.
This method copes equally well with question 2.
It is clear that the £15 needs to be shared between 5 boxes.
Which must be the same for Jill.
So Jill must receive £6.
Drawing out boxes would be unfeasible for ratios such as 15:45, but again, the box method copes with this well. Let’s share £480 in the ratio 15:45 between Jack and Jill. Instead of drawing out each box, by now pupils should understand that there needs to be ‘fifteen boxes and forty five boxes,’ so this can be represented as follows:
Questions accompanied by a format of this kind serve as a nice bridging step towards mastery of the full abstract calculation; since the boxes can no longer be counted, a pupil should easily recognise that they need to ‘figure out how many boxes there are’ by adding 15 to 45, then dividing the £480 by 60. The multiplication notation then serves as straight forward instruction as to what to do next to calculate the amount of money each person receives.
Jack receives 15 x £8 = £120, while Jill receives 45 x £8 = £360.
The box method easily generalises to work with three and four part ratios as well.
Question three is from the Singapore Maths curriculum, intended for Year 7s. I gave this question once to a Year 12 A-Level student who had achieved a grade A at GCSE. She couldn’t solve it. Personally I would tend to turn to algebra more often than not to solve any novel problem - something I’ve heard maths graduates refer to as a ‘brute force’ approach; it will almost always work, but isn’t necessarily the most elegant approach. I wrote out the equation that represented the problem, , and after a brief ‘oh!’ she quickly crunched through the algebra to arrive at the solution. Importantly though, she couldn’t build the algebraic model herself.
Although not a replacement for fluent algebraic modelling, pictorial representations like the box method can provide an alternative, often simpler, approach. In this case consider the following:
For them to have same amount after Jack passes over £20, he must give two of his boxes to Jill, so that they each have five boxes.
We’re told that Jack gives £20 to Jill, so each of those boxes must have been worth £10. If there are ten boxes in total, then there must have been £100 altogether. This the same result that you would arrive at by creating the algebraic model above.
This would obviously begin to struggle with odd ratios in a way that the algebraic model wouldn’t, but it can still be simpler for young children to grasp and helps to build conceptual understanding while pupils are still developing their procedural fluency.