This Math Talk focuses on squaring numbers written as the product of an integer and a square root. It encourages students to think about how a square is applied to a product and to rely on what they know about exponents to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students work with imaginary numbers.

In this activity, students have an opportunity to look for repeated reasoning when squaring expressions that involve square roots (MP8).

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?”

In this activity, students are formally introduced to the symbol used to represent the imaginary unit that we had been calling . Students solve quadratic equations of the form for various integers and plot their solutions in the complex plane. The key idea is that, just as positive numbers have two real square roots, negative numbers have two imaginary square roots.

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.)

In this activity, students learn and apply the convention that for any positive real number , . This will be helpful later in the unit when students use the quadratic formula to solve quadratic equations and encounter terms such as .

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 . 

In this activity, students combine real and imaginary numbers with addition in order to plot complex numbers in the complex plane. The crucial difference between the complex plane and coordinate planes that students have used before is that each point in the complex plane represents a single complex number rather than a pair of real numbers as they have seen before.

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 .)