In this month's maths article from Caroline Clissold, we'll be looking at fractions and in particular approaching equivalence.

We will be covering:

How to approach equivalence in Year 1

What children need to be able to understand about equivalence in years 2 and 3

Activities to help children understand the concept of equivalence

How Fluency with Fractions can help

Equivalence is a very important aspect of mathematics. It’s basically about finding relationships or aspects that are the same. In measurement we often ask the children to find equivalent units of measure, for example 12kg 500g, 12.5kg and 12 500g, one mile is the same as 1.61km (rounded to nearest two decimal places). In number we ask the children to make equivalent missing number statements, 15 x 3 = 60 - ? (15). Finding solutions to calculations is also all about equivalence. When they see the symbol =, children often see that symbol as an indication that the answer comes next. They need to be shown that equals means equivalent.

### Equivalence in Year 1

In fractions, we often ask the children to find equivalences, for example 1/2 = 2/4, 8/10 = 4/5. Finding equivalences begins in Year 1 when children need to recognise, find and name a half as one of two equal parts of an object, shape or quantity and recognise, find and name a quarter as one of four equal parts of an object, shape or quantity. When they explore this, they will notice that two halves and four quarters are equivalent to one whole and two quarters are equivalent to one half. It is important to give the opportunity for the children to find this out for themselves. A simple way to do this is to give them strips of paper and ask them to keep one whole and then fold a second in half and a third into quarters:

If they do this they can clearly see that 2/4 = 1/2 and the 4/4 and 2/2 are equivalent to one whole. They can also clearly see that 1/4 is smaller than 1/2.

Try introducing the word equivalent to children in Year 1 and expect them to use this word when describing these types of factions.

### Equivalence in Year 2 and 3

In Year 2 children need to recognise, find, name and write 1/3, 1/4, 2/4 and 3/4 of a length, shape, set of objects or quantity, write simple fractions for example, 1/2 of 6 = 3 and recognise the equivalence of 2/3 and 1/2.

In Year 3 the children begin adding and subtracting fractions with the same denominator and in Year 4 they do this for fractions with common equivalent fractions. In Years 5 and 6 they begin, first multiplying, and then dividing fractions. These operations all require an understanding of equivalence.

The NCETM has a great video of children in Year 2 adding and subtracting halves and quarters which demonstrates just what they are capable of if they are given appropriate visual representations. These children are able to use equivalence to do this. It’s worth watching!

All the way through, children need to be given opportunities to compare and order fractions. This often requires them to find equivalent fractions in order to do this accurately.

To develop a conceptual understanding, the children first need to be aware that a fraction is an area and not a shape. Many develop the misconception that if, for example, fractions of shapes look the same they are equivalent but if they are different shapes then they are not. Thus misconception often develops because of the way fractions are presented in textbooks or on worksheets. Here is a typical example:

### Activities to help children understand the concept of equivalence

There are lots of nice activities to help the children understand the concept of equivalence.

Try presenting fractions so that they are not identical. For example, you could ask them to find out whether these are quarters:

By cutting the pieces out and folding them in half, they can see that each piece is one eighth. They should know that 2/8 is equivalent to 1/4, so each part is 1/4 and each quarter is equivalent.

You could give them two pieces of A4 paper. They tear one in half. Ask them to tell you what fraction the smaller piece is of the larger. Most children will tell you half. Ask them to stick the half in the middle of the large piece. Ask them what fraction the piece in the middle is and then ask them what fraction the part around the outside is of the whole. Many will give random guesses as to the latter because they may think that the parts should look the same and so that part can’t be the other half.

You could then ask the children to cut around the smaller piece so that there is one long piece of paper. Ask them what fraction this is of the whole again and they may well give you other random fractions that are different from their first guesses. You could then ask them to cut this long pieces into parts and stick them onto the smaller piece so proving that they are both halves.

You could write 1/4 on the board and ask the children to write a variety of ways to show this, drawing on their experiences of equivalence. For example:

Tangrams are also a great way to explore equivalence:

You could ask the children to explore what fraction each part is of the whole and then other fractions and equivalences. There are lots of ideas that you can adapt to help your children get to grips with the concept of equivalence.

### How can Fluency with Fractions help?

The Fluency with Fractions series from Rising Stars provides great support for using a variety of visual representations of fractions to support understanding. For example, in each year level fraction bar images are used to introduce the concept of fractions as equal parts of a whole, equivalence, counting (linked to the number line) and calculating. Finding and identifying equivalent fractions paves the way for later understanding of equivalent ratios, so the Fluency with Fractions series moves from work on equivalent fractions through decimals and percentages to incorporate ratio and proportion in Year 6. Developing understanding of equivalence early on is essential to becoming confident with these concepts.