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What Makes A Good Resource - Representing complex numbers in modulus-argument form

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Created on 16 July 2010 by ncetm_administrator
Updated on 27 August 2010 by ncetm_administrator

Representing complex numbers in modulus-argument form

Resource description:
The resource consists of a sheet of complex numbers (which you can download here). These can be with i or with j, with arguments measured in radians or in degrees. Students also need a large piece of paper to make a poster and pens, scissors and glue.

Teacher comment:

I created this resource to revise the idea of modulus-argument form or polar form for 2nd year Further Maths students. I had found that students appeared to understand the form when it was first described and deduced from right angled triangles, but that they tended to be very sloppy in their use of it. I’ve seen students saying that the argument of

$r(sin\theta -jcos\theta)$

has argument $\theta$. I have since used it with 1st year Further Maths students when introducing the idea.

What I did:

I gave out the sheet to pairs of students and asked them to cut out the numbers and sort them into 3 groups: numbers in Cartesian form, numbers in modulus-argument form, and numbers in neither form.

After an interesting space of time I told them that there were 8 in each of the first 2 categories. Once the numbers were correctly sorted each pair were given a large piece of paper and asked to draw axes for an Argand diagram and to stick the Cartesian numbers on it. They then had to stick the modulus-argument numbers below the Cartesian, without using a calculator they had done

$\sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$

in A2 Maths.

Those who finished this then went through the other numbers, working out which of the 8 numbers stuck on the poster they were and piled them in the appropriate places.

Reflection:

This took a lot longer than I had expected it to. I thought it was going to be a quick check up that the form was understood. It showed that their understanding was very fuzzy. There was a lot of arguing and neither stage (sorting or placing) were as straight forward as I had expected them to be. There were quite a few “NOW I understand” noises. The posters we had at the end were put up on the wall and I found myself referring to them quite often when we did other work on complex numbers, for example multiplying and dividing numbers in modulus-argument form.

Submitted by Claire Willman, Exeter College