The goal of this discussion is to review students’ strategies for squaring expressions that include a square root.

To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

The purpose of the discussion is to highlight that just as positive real numbers have two square roots, one positive and one negative, it is also true that negative real numbers have two square roots, one on the positive imaginary axis and one on the negative imaginary axis. Display the complex plane from the task, for all to see and select students to share how they reasoned about the solutions to the equations and plotted their solutions. Record their thinking for all to see. Ask students, “What do you notice about solutions to if is positive? What if is negative?” (If is positive, the two solutions are the same distance from 0 in the positive and negative direction on the real number line. If is negative, the solutions are the same distance from 0 in the positive and negative directions on the imaginary number line.)

If students do not yet correctly express the square roots in terms of , consider saying:

• “Can you explain how you wrote using .”

• “How could rewriting as help you to express the square roots in terms of ?”

Select students to share their responses, and encourage students to show that their answers make sense by squaring. Mention that it is convention to write after a rational number such as or instead of or . When writing an expression that includes multiplied by a radical, such as , be careful not to extend the radical symbol over the . 

Select 1–2 students to share their thinking about how to plot complex numbers, and display their responses for all to see.

Tell students that together, the real number line and the imaginary number line form a coordinate system, and this complex number plane helps to visualize complex numbers. People call the real number line the real axis and the imaginary number line the imaginary axis. One important distinction to make is that points in coordinate planes that students have seen before have been pairs of real numbers, like , but in the complex plane, each point represents a single complex number.

Consider asking students:

• “How is the complex plane the same as coordinate planes that you have used before? How is it different?” (Two real numbers are used to represent a point in the coordinate plane, but in the complex plane, each point represents a single complex number that results from adding a real number to an imaginary number. They are the same because they have two axes, they have positive and negative directions on those axes, and it’s possible to make gridlines that show different coordinates.)
• “In your own words, how would you plot ?” (This one has the real part and imaginary part in a different order from the task, but it should be the same process. Start at 0 and move 7 spaces along the positive real axis, and then move down 5 spaces to represent .)