Why do this problem?

Working on this problem gives students an insight into the geometric effects of squaring complex numbers while offering an opportunity to practise manipulating complex numbers algebraically.
 

Possible approach

Set students the first challenge: "I want you to find some complex numbers whose squares are real. Once you have found some, plot them on an Argand diagram. Be ready to explain what you notice."

Once students have tackled this, bring the class together to explain what they found, then set the second challenge: "Now look for complex numbers whose squares are imaginary. Again, plot them on an Argand diagram and be ready to explain what you notice."

Bring the class back together to discuss their findings, and then set the main task:
"Now explore the squares of other complex numbers. You could start by looking for the numbers that square to give the answers you found to the first two questions, and then explore complex squares more generally. If you notice anything interesting, see if you can make a conjecture. Can you prove what you find? Don't forget to plot the numbers and their squares on an Argand diagram to help you to visualise what is happening. Be ready to report back on what you found."

Students may find it useful to use GeoGebra or other software to explore.

At the end of the lesson, bring the class together to share what they have found. Focus particularly on the geometrical results they have noticed.
 

Key questions

What do you get if you square $a+ib$?

Working backwards:

if $(a+ib)^2$ is real, what relationships must $a$ and $b$ satisfy?
if $(a+ib)^2$ is imaginary, what relationships must $a$ and $b$ satisfy?

What does this look like on an Argand diagram?
 

Possible extension

Students could have a go at Complex Puzzle

Possible support

Give students plenty of time to explore the GeoGebra worksheet in A Brief Introduction to the Argand Diagram, and perhaps invite them to create their own worksheet that squares complex numbers.