Simplifying equivalent fractions
A) To begin with I would make sure that pupils have a thorough understanding of the concept of fraction. I usually do this simply by ripping up pieces of paper. I ask pupils to fold the paper then lay their ruler on the fold and pull the paper against the ruler; this works better than scissors.
Scrap A4 is fine for this.
With differently coloured pieces of paper, you can look at different fractions. If you have some plastic pockets with the fraction on the front, the pieces of paper can be used in subsequent lessons to investigate the four rules - add (+), subtract (-), multiply ( x ), divide ( V ) and to help understand conversions between mixed numbers and top-heavy fractions.
I begin very simply, tearing the paper into two equal parts and getting pupils to label each half " 12", reminding them that we have cut one whole into two equal parts and each one of the pieces is "one of two parts" , reinforcing that "two out of two parts" makes a whole one (I write this on the board). I repeat this for several fractions. I can remember doing this with a parent maths club and one of the parents asked: "Is that all a fraction is? Why didn't someone get me to rip up paper when I was at school."
Having done this I then put a rectangle on the board and divide it into twelfths. An interactive whiteboard is great as you can easily colour over the diagram and repeat it loads of times. I shade it in as shown and ask the class:"What fraction have I shaded?" The first answer is usually "12".
I write the answer on the board and then say that this isn't the answer that I wanted, not saying if it is correct or not. When I did this recently with an adult group, this then encouraged the quieter ones to respond. The following fractions will be suggested: 6Z12, 3Z6. Let them know that you did want all these as answers. If any of the responses are incorrect they should be discussed - ask why they cannot be correct.
With an interactive whiteboard you can show that there are denominators other than the most obvious (2 and 12), for example 6. Make a rectangle that is the size of two of the small rectangles, in a different colour.
This rectangle can be dragged around the board (or copied) to cover the six dark-blue rectangles three times and the six light-blue rectangles three times.
The challenge is: can they think of any more? Talk about all of these fractions being the same fraction of the original shape, but written as a different number of parts. Write 12 = 3Z6 = 6Z12 - Jshowing how these are equivalent helps them to find other fractions that are equivalent.
While teaching a group of adults the other day we had used this approach and I moved on to "cancelling" fractions. I thought the fact that I had shown them multiplying or dividing top and bottom by the same number would leave them happy to cancel, but this wasn't the case. I had to take the ideas above one stage further.
The fraction I worked with was 16Z24. We began by dividing top and bottom by two, representing this as a product of the factors of 16 and of 24. This reinforced the fact that 2Z2 = 1.
I then wrote the same fraction out, with both numerator and denominator written as a product of their prime factors. We discussed the fact that this was a really long way to cancel fractions, and that being fluent in multiplication tables meant that they would then be able to recognise multiples from the same table - Jin this case cancelling by eight.
Again, I demonstrated this as shown. They were much happier as they had a picture of the process. My guess is that when we get to simplifying algebraic fractions they will find it a lot easier.