If pupils can grasp the big of idea of equivalence, they can go on to develop flexible and playful ways of working that will make them better mathematicians, says Mike Askew...
Mathematically, equivalence is hugely important because representations that look different can all be linked to the same underlying idea. For example, 1/2, 50% and 0.5 are all equivalent representations of theideaofahalf.The representation we choose is usually linked to the context in which it’s used: fractions are oftenusedinthesolutionto division problems when the answer is not a whole number; percentages are used to express probabilities; and decimals are favoured in the context of measures. But the underlying mathematics does not essentially change.
Conceptually, equivalence is a big idea for several reasons. First because it means that numbers and measures can be expressed inaninfinitenumberof equivalent ways by different partitionings and factorisings. For example, 348 is equivalent to 340plus8,or34tensand8ones, or 3 hundreds and 48 ones, or 35 tens minus 2 ones, and so on.
Different representations of the same quantity highlight different properties. Take, for example, the number 64. We can express this as:
164 − 100
Looking at 2 x 2 x 2 x 2 x 2 x 2, I know that 3 cannot divide into 64 exactly (since multiplying any number of 2s together can never result in a multiple of 3). Then 8 x 8 means 64 is a perfect square, and 4³ reveals it is a perfect cube. Listing equivalent expressions for 63 and 65 can illustrate how different these numbers are from 64, making them all more interesting than simply being three numbers in the 60s.
A second reason why equivalence is conceptually important is that, when calculating, it is often easier to answer an equivalent calculation than to answer the one actually given. For example, in mentally figuring out 49 x 4 you may think along the lines of, ‘40 times 4 is 160, 9 times 4 is 36, so the answer is 160 plus 36 – that’s 196’. We don’t directly calculate 49 x 4, but work out the equivalent calculation of (40 x 4) + (9 x 4).
But you might have said, ‘Well, 50 times 4 is 200, so the answer must be 4 less – that’s 196.’ As equivalent equations, the reasoning goes:
49 x 4 = (50 - 1) x 4 = (50 x 4) -
(1 x 4) = 200 − 4 = 196
i.e. 49 can be expressed as 50 - 1, and (50 - 1) x 4 expressed as (50 x 4) - (1 x 4). The advantage of this example is that it can be applied to larger numbers: ‘299 times 3? That’s 300 times 3, 900, minus 3, 897.’ Quicker than paper and pencil or reaching for a calculator.
We work with equivalences all the time in mathematics, often without an explicit awareness that this is what we are doing. Take a simple problem like having a crate of 201 apples and selling 199. How many apples are left? Coming up with an answer of ‘2’ does not challenge most adults, but I’ve seen children write down
And laboriously carry out the subtraction algorithm. (One child I met made 201 tallies and crossed out 199 of them!) As adults we intuitively turn the calculation 201 − 199 into the equivalent one of 199 + [ ] = 201 in the knowledge that counting up from 199 to 201 is easier than, literally, taking 199 away from 201.
As teachers, one of the most important things we can do to help develop pupils’ understanding of equivalence is to talk about and emphasise the correct use of the equals sign.
Children often think ‘=’ means ‘makes’. In other words, the answer has to go after the equals sign. Imagine presenting the following equation:
If asked to fill in the missing number, children will write 17 in the box (it is what 12 + 5 ‘makes’) or even 23 (by adding all the numbers: 12, 5 and 6). From introducing the equals sign we need to emphasis the idea of ‘balance’ and how everything on each side of the equals sign has to match up in some way. Altering the position of the unknown in number sentences helps here:
[ ] + 5 = 12
[ ] = 3 x 5
21 = 3 x [ ]
The other thing to take care over is not to use the equals sign to ‘string together’ a series of calculations that are not equivalent. Take, for example, a child explaining how he mentally worked out 73 − 27. He may say something like, ‘I started with 73 and took away 20, that’s 53. I took away 3, that’s 50, and then took away the 4, that’s 46’. As he explains, record his working out:
73 - 20 = 53 - 3 = 50 - 4 = 46.
The trouble is, nothing here is equivalent: 73 - 20 does not equal 53 - 3, which in turn does not equal 50 - 4 and so on. (Note: above, the equations for 49 x 4 are OK strung together as each is equivalent to 196).
Although it may seem laboured, work with learners to set such chains of calculations out as separate, correct, equations:
73 - 20 = 53
53 - 3 = 50
50 - 4 = 46
So 73 - 27 = 46
Early years: slice it
Provide children with several copies of an array of five by seven dots:
Talk about the array and how to describe it. Encourage children to talk about it having five rows with seven in each row, or seven columns with five in each row. Be consistent that, in this orientation, it is a five by seven array.
Working in pairs, children investigate ‘slicing’ the array into two smaller arrays – either by drawing a straight line across the array or perhaps cutting it in two. Make sure the children are clear that the slicing has to produce two rectangular arrays.
Ask the children to record each array they have made, the number of dots in each array and the total number of dots. Does it still always total to 35? Bring the class together to share their different slicings and show how these can be recorded as equivalent sums of products:
5 x 7 = 5 x 2 + 5 x 5
5 x 7 = 3 x 7 + 2 x 7
Children can make an array of their choice and investigate the different ways of slicing it.
Middle years: balance it
Put up an equation like:
4 x 5 = [ ] x 2
Ask the children to write down what they think should go in the box. Some are likely to write 20 and some may think the answer is 40 (finding 4 x 5 and then doing x 2). Establish the idea that the equals sign means the totals on each side have to be the same – so what number needs to be multiplied by 2 to get 20? Put the following equation up on the board.
10 x [ ] = 20 x [ ]
Ask the children if they can find at least two pairs of numbers to make this equation true.
List suggested pairs on the board:
What pattern do the children notice? Can anyone explain why this pattern works? Arrays can help here. If you draw a 10 by 10 array, what has to be done to it to turn it into a 20 by five array?
Set the children off to investigate other missing box calculations such as:
10 x [ ] = 30 x [ ]
100 ÷ [ ] = 200 ÷ [ ]
Upper years: product posters
Put up the calculation 99 x 75.
Can the children draw at least three different arrays that could be used to find the product of 99 and 75?
It is likely that most of them will choose to draw arrays that are 99 by 75 and to partition these in different ways.
Ask them if it would be possible to start with an array that was bigger than 99 by 75 and to slice bits off. Children should have little difficulty in starting with a 100 by 75 array and slicing off a strip that is one by 75. What if they started with a 99 by 100 array? Get the children to make posters showing three different equivalent calculations to 99 x 75.
Work with the class to record the equivalent equations, for example:
99 x 75 = (100 x 75) - (1 x 75) (Strictly speaking, the brackets are not needed here as, conventionally, multiplication take priority over addition, so this could be written 100 x 75 - 1 x 75)