Problem | Teachers' Notes | Hint | Solution | Printable page |

We're going to start with the same animation we looked at in Twizzles Venture Forth . If you haven't done so yet, it would be worth getting in some twizzle arithmetic practice using the notes for that problem as a guide.

In this animation, you can change the value of the blue twizzle z .

The red twizzle takes the value (z-i) .

The green twizzle takes the value (z+i) .

The grey twizzle takes the value (z-i)(z+i) .

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Switch on 'Show Tracings' and draw a small loop around each of these solutions with the blue z twizzle. Observe carefully what happens to the paths traced by the red, green, and grey twizzles and how the paths relate to each twizzle's zero spot.

Now click on 'Show Tracings' twice - once to clear the paths, and once to switch tracing back on, and draw a small loop around the other solution. Again, observe carefully the paths of the other twizzles.

Click 'Show Tracings' twice again to clear the paths.
This time, draw a loop that goes around both solutions (so both the solutions are inside your loop). Again, observe what paths the other twizzles trace out. How many times to they turn around their spots?

Here's the question!

When you trace out a loop with the blue z twizzle, what determines how many times the grey (z-i)(z+i) twizzle winds around its zero spot?

Generalise!

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Sine. Rotations. Representing numbers. Roots of polynomial equations. Interactivities. Long problems. Argand diagram. Complex numbers. Learning through exploration. Quadratic equations.