Why is this skill important?
Fractions are always a bit of a bug-bear in mathematics. Many people remember them with loathing and fear! Adding, subtracting, multiplying and dividing fractions never seemed to be something most people could remember how to do with any ease or, even if they did, could recall with any sort of understanding.
The key to adding and subtracting fractions, and also to using fractions in most everyday life situations is to understand the concept of equivalent fractions. As is explained below, these are pairs of fractions which indicate the same portion of the cake! This idea – namely that fractions, which are written differently, can actually be altered to be identical, is crucial in calculations involving fractions.
Fractions are complicated in mathematical terms, because they refer to both a mathematical concept and also a mathematical operation. This means that ¼ has to be understood as referring to one of four equal slices of a cake. But it also has to refer to a division calculation when we divide a quantity or amount by four. Thus ¼ of 32 is the division 32 ÷ 4.
It is this dual role that gives fractions such an important place in mathematics but that also makes them hard to get our heads around!
What is this skill?
A fraction may be defined as an equal part of a whole. Thus a quarter (¼) is one of four equal parts, a third (⅓) is one of three equal parts and so on, Clearly, we can have more than just one part, so ⅔ indicates that the cake is divided into three slices and that we have two of them!
The bottom number – mathematically known as the ‘denominator’ – indicates the number of slices in this particular cake, e.g. if we are talking about ¾ we know the cake is cut into 4 equal slices or quarters.
The top number – mathematically known as the ‘numerator’ – indicates the number of slices that we have managed to get our hands on, e.g. in ¾ we have three of the four slices.
It follows that we can read a fraction as a division. Three quarters is also three divided by four. It is good to remember this as it indicates that a fraction is always less than one.
We can, however, have mixed numbers – where the number consists of a fractional part and also a whole part, e.g. 2¾ or 1½. Mixed numbers are numbers like any other, and we can count in fractions just as we can count in steps of other numbers such as 10s or 100s, e.g. ½, 1, 1½, 2, 2½, 3, 3½, 4… etc. Counting along a line of fractions helps children to understand that fractions are numbers on a line, like other numbers.
A pair of equivalent fractions is a pair of fractions which look different but which in fact indicate the same part of a cake. The most common example is that one half is also two quarters: ½ = 2/4. But there are plenty of other examples of useful pairs of equivalent fractions:
⅔ = 3/6
¼ = 2/8
2/10 = 1/5
5/10 = ½
50/100 = ½
25/100 = ¼
75/100 = ¾
So how is this skill taught?
Since fractions are used to refer to a mathematical concept and also to a mathematical operation, namely division, the teaching needs to focus separately on these two aspects.
Fractions as parts of a whole
Getting the idea that the smaller the number at the bottom of the fraction, the larger the slice and the larger the number at the bottom, the smaller the slice is the key to understanding fractions. This is easiest to see in pictorial terms – by seeing fractions as equal slices of a cake or equal sections on a strip. Each section here is 1/8
Fold the strip in half to see that 4/8 = ½.
Fold the strip into quarters to see that 2/8 = ¼ and 6/8 = ¾
Fraction strips can be created for any denominator (bottom number of the fraction). So that we can make a tenths fraction strip or a thirds fraction strip to show these fractions. This will clearly demonstrate to children that the larger the number on the bottom, the more spaces on the strip there are, and the smaller each one is.
Fractions as divisions
The use of a fraction strip is equally useful in demonstrating to children how to use fractions as divisions. For example, if we want to find one quarter of 12, we can show 12 stickers on a fraction strip for quarters.
One quarter of 12 is 3, as we can see on the strip. Three quarters of 12 is 9 as we can again see on the strip.
In this way, we can demonstrate that ¼ of 12 = 3 is another way of writing the division 12 ÷ 4 = 3. We have divided the strip into 4 pieces. There are 3 stickers on each quarter.
Children need to understand that to find a given fraction of an amount we need to divide by the bottom number in the fraction (the denominator). If we want several parts, we have to multiply the answer by the number we want (the top number of the fraction).
- So if we want to find 1/8 of 32, we divide 32 by 8. This gives us 4.
- If we want to find 3/8, we multiply 1/8 (4) by 3, getting 3 x 4 = 12.
- 12 is 3/8 of 32.
Practise Together: These activities are intended to be shared. Read the Explanation of the skill being practised and then play the game or share the task. Watch out for the points highlighted in the Explanation and if necessary, help your child, following the advice in ‘How this skill is taught’ section. Shared activities are not only more fun – they enable you to actively support your child’s learning.
Explanation & Worksheets: Having practised a skill together using the shared activities, children can then rehearse the skill using the ‘Child alone’ sheets. These are presented in order of difficulty 1-5 and should only be given to the child AFTER the Practise Together activities. In this way you can be sure that the child has acquired this skill first. We cannot rehearse what something have not yet learned!
Test: Take a test, questions from this area
Counting in sequence
Read & write numbers
Counting in tens & hundreds
Counting in sequence: Count any sequence of numbers from 1 to 10,000 forward or back with confidence
Read & write numbers: Read and write the numbers 0-10,000
Place value: Understand that 4392 is made up of 4000 + 300 + 90 + 2 and that 4092 has no hundreds
Money: Begin to understand that £6.54 is six pounds and 54 pence and that £6.04 is six pounds and 4p while £6.40 is six pounds and 40p
Counting in tens & hundreds: Count in tens or hundreds forward and back from any number, e.g. 284, 294, 304, 314, etc. understanding how to cross a multiple of 10, 100 or 1000
Count multiples: Count in (add or subtract) multiples of 10, 100 or 1000 (800+300)
Writing fractions: Understand how fractions are written, e.g. ½ and ¾ and begin to realise that ½ is the same as 2/4 or 3/6 etc.
Number Concepts: Count in different ways, understand how numbers work, become fluent in the ways of numbers
Adding and Subtracting: Mentally add or subtract numbers with confidence and develop written ways of adding and subtracting larger numbers or more of them!
Multiplying and Dividing: Know the times tables and use these to perform mental multiplication and divisions; develop written methods for multiplication and division.