Fractions, Decimals and Percentages Ideas
When asked about which parts of maths pupils (and adults) find tricky, it would be rare for fractions, decimals and percentages not to appear - often accompanied by a grimace. Depending on how they have been introduced, it is also common for pupils to think of these three entities as completely separate topics, which is why we start with links.
The first link to be made when working on fractions - even with young pupils - is to division. It is worth asking older pupils what a fraction is; I have not yet had a pupil suggest that it has anything to do with division. Ask pupils to look at the division symbol (obelus) and a fraction written down - do they see any connections? Encourage pupils to write division calculations as fractions when appropriate - this can often simplify the calculation when the larger numbers in the numerator and denominator have common factors.
Work on fractions, decimals and percentages as a unit with older pupils so that pupils can see the connections between the different representations.
There is a natural progression from percentages to decimals when they are introduced as fractions with a denominator of 100 since this allows a natural link to decimals within the place value system.
Apart from fraction circles, squares and towers, using squared paper for highlighting parts of rectangular shapes and blank bar models split into halves up to tenths, provides visual images for pupils to work with. Plain paper is also useful for folding into various fractions.
Place value charts are helpful for pupils to see the value of the columns. Using numeral headings rather than words (such as thousandths) helps pupils to see that the numbers are ten times smaller. Place value sliders help illustrate that the the decimal point stays put as the digits move when a number is multiplied or divided by a multiple of ten.
Very long number lines (created by adding paper together) which represent the part of the number line between 0 and 1 are useful for placing various proper fractions and promoting discussion about how to compare fractions. Asking pupils to place fraction pieces on the number line can help pupils to see that unit fractions do not increase in size - but rather decrease - as the denominator increases, which can be a common error made by pupils.
Base 10 materials are great for helping pupils to see a whole block split into one hundred squares which is very useful for understanding percentages as fractions with a denominator of 100; this then links nicely with their representation as decimals using place value charts. Use the reverse of the base 10 block so that the squares are clearly seen and encourage pupils to see the block as representing one whole before flipping it over to see the 100 squares more clearly.
Allow younger pupils to pursue their own learning opportunities: painting half a picture to create a symmetrical image; sharing items between two pupils so that they have half the amount each; building towers that are then split into two, which invites a discussion regarding why we cannot do this with all numbers and which numbers don't work. We can therefore connect odd and even numbers with multiples of two, division by two and halves. These connections may seem obvious to us but pupils need frequent opportunities to explore these links.
Pupils need to see different representations of fractions as: part of a unit, as a number on a number line, the division of two numbers, as an operator and as a ratio. Asking groups of pupils to create a resource such as a video, poster, cartoon or drama sketch to explain the different representations of fractions at some point during a unit of learning can be very effective. When working on fractions with older pupils, it can be illuminating to give pupils proper fractions (on sticky notes) to be placed on to the class number line. Even some of those who appear to have grasped fractions concepts will attempt to place 3/4 between the numerals 3 and 4 rather than 0 and 1.
We can teach pupils to follow a process or a rule but if they don't understand why it works it can be rapidly forgotten when we move on to another topic.
One such rule is to, 'multiply the top and the bottom by the same number' to produce equivalent fractions. I have often asked pupils who recited this rule to explain why it works and several pupils have not seen the equivalent fractions as different representations of the same entity at all, despite the name. Discovering that we are actually multiplying by a/a, and that this is the same as multiplying by one, can be a revelation for pupils.
If pupils are given fraction squares, fraction circles or fraction towers, they will see which fractions are the same size or height and with time, will often come to the conclusion of how to quickly create an equivalent fraction once they have discovered many equivalent fractions that they have made themselves and start to see the connections between them.