# Variety adds spice to equations

10th March 2006 at 00:00
Q. I feel it is important to have some different ways to look at topics.

Algebra is something that my students find difficult. Can you suggest anything that might open some doors?

A. An approach that I have found really useful is to use coloured envelopes with matchsticks. The colour of the envelope becomes the variable, so blue is "b", and the contents - matchsticks - Jare the solution. Equations can be quickly created and their solution discussed. This idea is developed using a balance with the envelopes and matchsticks to show balance in equations.

An interactive version of the idea is now available at www.mathagonyaunt.co.uk under resources, thanks to Genee World: www.geneeworld.com

The envelopes and matchsticks deal with items and it is difficult to partition a matchstick. What about when the problem is about length? So I started to think about the number line. I wanted to balance the fact that what is done on one bar has to be done on the other, a replication of the balance idea.

Begin with a simple case, such as the one in the diagram. Ask pupils: "What information does this picture give?" In the diagram, the orange bar represents 2a and the blue bar suggests that 2a is the same length as 16.

Write the equation below the number bar.

Ask them what you would have to do to find the value of a. The solution is found by dividing the bar by two. Show this in writing beneath the bar.

You could play with a few examples, maybe including problems where the variable isn't a whole-number answer. You can also include problems such as 5a = - 10; -4a = - 8; and (an interesting one) - 4a = 8; lots of discussion to be had. This reinforces work on integers really nicely.

The natural progression is to questions that involve fractions of the variable. Usually I follow this with the addition or subtraction of a constant value, but I think it would be useful to look at the inverse operation of division. Here is an opportunity to revise what is meant by a fraction - how many thirds do you need to make a whole one - in this case a whole of the variable a. This requires multiplication by 3. Demonstrate this by writing the algebra below.

Invite pupils to the board to write the algebra for various problems. An extension would be to look at mixed number problems, eg 112a as the bar.

This combines revision of fractions with algebra in a graphical way.

Ask them what this picture tells them and what they might do to find the value of a. One pupil might suggest that you divide the line in three. They can do this, though it is messy, as the seven will also have to be divided by three.

The solution of taking away seven and then dividing by three follows.

These examples can also be extended to look at integers, eg 2a + 6 = - 14 and so on, but with the variable on one side only.

The next stage is to look at problems which have an added constant value. I found the pages below when I was looking at work involving modular arithmetic. They include some great artwork, full of ideas. Perhaps you could suggest a joint project with art - it would be exciting and fun.britton.disted.camosun.bc.camodartjbmodart2.htm

I will be demonstrating some interactive activities at the Education Show at the NEC in Birmingham March 9-11 on the Interactive Education Stand at E4, including a new voting system that can put fun into maths. I will also be at The TES stand. Hope to see you there!

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