How would you explain the mathematical expression 12 ÷ 3 = 4 to your child?
The concept of division usually brings to mind images of sharing sweets among a group of friends. This is a simple enough way of giving context to expressions like 12 ÷ 3 = 4 (and let’s be honest, any excuse to get the sweets out is to be welcomed). The more hands-on children can get with mathematical objects, the better equipped they’ll be to make sense of all those symbols.
But sharing is just one way to think about division. When teaching division to your child, it’s also helpful to have other representations at hand.
How to teach division as sharing and grouping
In school, children are usually taught division in terms of sharing and grouping. They’ll be asked to ‘share an amount equally between’ or to ‘group an amount into equal sets’. There’s a subtle difference.
Let’s go back to 12 ÷ 3 = 4.
Pete has 12 grapes.
He shares them between 3 of his friends. How many grapes does each friend get? After sharing out the grapes one at a time into three piles, Pete will end up with this:
We say 12 grapes shared between 3 people to give 4 grapes each. The answer, in this case, is the value of each equal share (portion size). Let’s spell this out a bit more in terms of the terminology around division:
The dividend is the thing to be divided (12 grapes), the divisor is the number of groups (3 friends). The quotient is the number of items in each group (portion size), which is 4.
Now let’s look at the same calculation in terms of grouping.
Pete has 12 grapes.
He wants to put them into bags of 3. How many bags will he have? Pete fills up the bags one at a time, ending up with:
This time, we say that when 12 grapes are put into bags (groups), we end up with 4 bags, each containing 3 grapes. Note that the answer here is not the number of grapes in a group, but the number of bags.
The dividend is the same as before (12 grapes) but now the divisor is the number of items in each group, 3, and the quotient is the number of groups/bags, 4.
The grouping method of division shows clearly that division is the inverse (opposite) of multiplication. The expression 12 ÷ 3 = 4 holds true because 4 x 3 = 12 (and vice versa). We can literally see 4 bags each containing 3 grapes when grouping. Put another way, when we solve the bags of grapes problem, we are asking how many 3s are there in 12? This is the same as asking what do I multiply 3 by to get 12?
How to teach division (and multiplication) with arrays
Arrays offer a powerful way of visualising the link between multiplication and division. They make no distinction between grouping and sharing. An array simply arranges objects in columns and rows and by doing so, they offer at least 6 interpretations of multiplication and division:
Saying ‘3 groups of 4’ is represented by the calculation 4 x 3 = 12
Saying ‘4 groups of 3’ is represented by the calculation 3 x 4 = 12
Saying ’12 splits into 3 equal shares of 4′ is represented by the calculation 12 ÷ 3 = 4
Saying ’12 splits into 4 equal shares of 3′ is represented by the calculation 12 ÷ 4 = 3
Saying ’12 splits into 4 groups of 3′ is represented by the calculation 12 ÷ 3 = 4
Saying ’12 splits into 3 groups of 4′ is represented by the calculation 12 ÷ 4 = 3
Why does all this matter?
A cynic might wonder why we bother with all these representations, when surely the only thing that matters is getting the answer.
To help children develop as mathematical thinkers we need to dig into the structures that underlie all the calculations they are expected to do. Rather than just rattling off 12 ÷ 3 = 4 from memory, it helps to attach such calculations to concrete representations because that’s how these calculations will be encountered in the real world. Grouping and sharing are both valid ways of thinking about the same calculation. Arrays extend our thinking even further. As we’ve seen, by studying those representations we bring out important connections – like the link between division and multiplication. Those symbols make a lot more sense when studied in context.
Language also matters: it’s easy to mix up sharing with grouping, but we’ve seen that the answer (or quotient) represents a different quantity in each case. Mathematical calculations are very precise, so our choice of words must be too.
With a rich repository of representations and precise language at hand, your child will have the foundations to engage in mathematical dialogue and find patterns and connections among all those calculations.