The more I create… the more I learn. Here is the 3rd installment of this whole Making Sense Series which has truly forced me to be a better teacher. A more educated teacher. I can’t stress enough how much I’ve learned from diving into the progressions found here. There are many intricate pieces of learning […]
There are certain prerequisites for children to develop a solid understanding of fractions. First, they must understand, through work on additive reasoning, that a whole can be split into parts and that the sum of those parts is the whole. There’s a short step into multiplicative reasoning from here – that a whole can be split into multiple, equal parts and that the whole is the product of the size of each part and the number of parts. Once this is understood, children can begin to think about the whole being worth one and the parts being fractions of one. The ideas that follow are broadly sequential in terms of conceptual development.
Children will need to manipulate various representations of fractions, for example making them with fraction tiles (as both bars and circles); taking strips of paper and ripping them in to equal parts; and drawing bars and circles, dividing them into equal parts. It is worthwhile to get children to do lots of judging by eye and marking equal parts of a whole as well as using squared paper to do so accurately.
Of course, there is a lot of language to work on whilst manipulating these models of fractions. Children need to be shown clearly the link between the total number of parts and the language (but not yet necessarily the written form) of the denominator: two parts – halves; three parts – thirds; four parts – quarters etc.
With a secure start in the basics of splitting a whole into equal parts, children can work on the idea that fractions always refer to something. A third, for example, doesn’t stand alone. It might be a third of an apple or a third of twelve sweets or a third of one whole. Modelling these full sentences and getting children to speak in this way should solidify their understanding of proportion. Through the sharing out of objects, even very young children can work on the concept of fractions of numbers – sharing six sweets between three children means that each child has the same number of sweets and that two sweets is one third of six sweets.
Once children are comfortable with the idea that an object or a set of objects or a number can be split into equal parts, and that each of those equal parts can be described as a fraction of something, that object or that set of objects or that number, they can go on to work at greater depth. By comparing strips of paper or bar models that are the same length yet are split into different fractions, children can look at the relationship between the size of each part and the number of parts. That is, the greater the number of equal parts, the smaller the size of each part. Children should be expected to think about how ¼ is smaller than ½ because ¼ of one whole is one of four equal parts whereas ½ of one whole is only one of two equal parts. Then, questions like this should be relatively straightforward:
The understanding that unit fractions with larger denominators are smaller than unit fractions with smaller denominators will contribute significantly to work in comparing fractions later on.
Children could begin to look at improper fractions and mixed numbers next. Using ¼ fraction tiles, they could make one whole and then see what happens if you add another ¼.
This lends itself to counting in unit fractions but we should exercise caution. Children may be able to chant ‘Three quarters, four quarters, five quarters…’ but early conversion to mixed numbers as well should help to secure their understanding of the relationship between them. Manipulatives like fraction tiles and multi-link cubes are great for representing improper fractions because they can trigger accurate mathematical talk to describe the improper fraction (the total number of cubes as the numerator and how many cubes in each whole as the denominator). The same can be done to describe the mixed number (the number of wholes, then what is left over as a fraction of a whole).
Returning to additive reasoning, children could generate complements to 1 whole and record them as addition and subtraction statements.
A slight change to the representation used can support children to work with complements where denominators are different:
Placing two bar models of equal length one on top of the other is great scaffold for comparing fractions. When the denominators of the fractions are the same, the bars should not even be necessary but when they are different, the image can help to structure thinking.
When dealing with fractions with different denominators, the practice that children had earlier of judging by eye to split a whole into equal parts and marking the divisions themselves becomes crucial, otherwise, things like this could happen:
A standard fraction wall is all that is needed to begin work on equivalence and the first step is of course shading one fraction and looking up or down the fraction wall to find fractions of equal size. When children are comfortable with that, they can begin to look at patterns in the abstract representations, particularly the link between times tables, numerators and denominators.
Using the language of simplifying or cancelling fractions without first talking more generally about the concept is a mistake. If children are well versed in using a fraction wall to find equivalents to a given fraction, it is only a slight tweak to talk about finding the equivalent fraction that has the fewest total parts. It would be tempting to talk about finding the equivalent fraction that is ‘closest to the top’ of the fraction wall but this would be a mistake too. The language of simplifying or cancelling can be used to attach to the concept of finding the equivalent fraction with the fewest total parts to get children thinking conceptually soundly.
One further aspect of thinking of fractions is to consider them as numbers. To do this, plotting fractions on a number line directly beneath the bar model is a good way of linking the two representations.
Representing fractions as a proportion of one, as a part of a quantity and as a position on a number line significantly supports children’s development of proportional reasoning and ensures that future tricky concepts such as calculating with fractions can be built on a secure foundation.