Monday, 30 September 2013

Factors, multiples, primes and Venn diagrams, all in one lesson!


Venn diagrams
We have all had the problem as teachers of thinking a class has understood a lesson we have taught only for the next time we see the class to despair they haven’t remembered  thing. 

It’s even worse for the poor learner, they may remember the aim of your lesson but as for the content it’s gone; what a loss of confidence in their own abilities. 

In an effort to help pupils remember I always try to use colour and imagery along with the other types of standard teaching techniques that all good teachers employ. These techniques are simple and work.

These two good friends of mine aid pupils’ retention of facts, techniques and concepts.

Prime factors

Finding prime factors, once learnt is easy. I find the pupils don’t recognise the question in exams, so I spend  long time before exams just going over the wording of the questions. 

Once they have made the connection between the words and the activity life becomes easier for everyone. I always use the tree diagram method describing it to them as the numbers family tree and we are tracing back to the origins of the number given. 

Once a prime number has been found we colour it in and continue with the other branches of the family. So the number 180 we look like the diagram below.

Prime factors of 180
So using all the numbers that we have coloured in 180 = 2 x 2 x 5 x 3 x 3. Notice how using colour makes the primes stand out therefore it is far easier for the pupils to record the numbers.

LCM, HCF and confusion …

Once the above is mastered the and they have decoded the wording of the questions you might think finding the Lowest Common Multiple (LCM) should be easy. However anyone who has tried to take the next step with a class knows the utter confusion that is caused.

Lets say we have been asked too find the LCM of 180 above and 225. The pupils will happily produce a tree diagram of 225 which will look like the one below.

So 225 = 5 x 5 x 3 x 3 or 32 x 52. Now trying to explain how we find the Lowest common multiple using the primes can be tricky to teach and even worse to learn and remember. 

Have a look in a few text books, imagine yourself to be learner trying understand how it works for the first time.

Venn diagrams

Having struggled to teach this for a number of years I then hit upon the idea of using Venn diagrams. Below I have put the prime factors into Venn diagram.
The factors of 225 are in the left had set, 180 in the right and any common factors are in the overlap of the two sets. Colour in everything and multiply together,

LCM = 5 x 3 x 5 x 3 x 2 x 3 x 2 =  2700

After some practise the students really like this method. I then introduce finding the Highest Common Factor (HCF) using Venn diagrams. 

Using the same two numbers 180 and 225 another Venn diagram is drawn, this time only the intersection of the two sets is coloured. (I always make a point of using a different colour from the LCM). My diagram now looks like this.
The Highest Common Factor is found by multiplying 3 x 5 = 15. Repeat this exercise several times, just colouring the intersection.

Final touch

I have used colours and images to  identify the LC and HCF, the final touch is to try and remember which is which. I now use what appears to them as a paradox HIGHEST/LOWEST. 

By this I mean to find the HIGHEST common factor I use the lowest amount of numbers (5 x 3). To find the LOWEST common multiple I use the HIGHEST amount of numbers in the Venn diagram (5 x 3 x 5 x 3 x 2 x 3 x 2). 

Using everything above you will have success when teaching this topic and the pupils will remember far more than if you approached it via the more conventional route.

Book recommendation: A source of different ideas and inspiration for all Maths teachers is 100+ Ideas for Teaching Mathematics (Continuum One Hundreds)

As one reviewer on Amazon said
'Mike Ollerton is a real inspiration for any maths teacher who is interested in making their pupils think as well as showing us just how enjoyable it can be (for the teacher and pupil alike).' Well worth investigating and reasonably priced.

Why not follow me on Twitter @croftsr1.

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