Lesson 11

The purpose of this Math Talk is to elicit strategies and understandings students have for squaring numbers that involve square roots, including the imaginary number \(\sqrt{\text-1}\) that was introduced in the previous lesson. These understandings help students develop fluency and will be helpful later in this lesson when students will multiply imaginary numbers and decide how to write square roots of negative numbers as real multiples of \(i\).

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

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’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 \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

In the previous lesson, students used the notation \(\sqrt{\text-1}\) to represent a number that was a solution to \(x^2=\text-1\) as an introduction to imaginary and complex numbers. In this activity, students are formally introduced to the symbol \(i\) used to represent the imaginary unit. Students solve quadratic equations of the form \(x^2=k\) for various integers \(k\) 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.

Remind students that in the previous lesson, they wrote \(\sqrt{\text-1}\) to describe an imaginary solution to the equation \(x^2=\text-1\). But just as \(x^2=4\) has two solutions, 2 and -2, \(x^2=\text-1\) also has two solutions. \(\sqrt{\text-1}\) is one of them. Explain to students that the \(\sqrt{}\) symbol is supposed to mean the positive square root of a real number, so mathematicians decided to use a different symbol for the two imaginary solutions to the equation \(x^2=\text-1\). The two square roots of -1 are written \(i\) and \(\text- i\). That means that \(i^2 = \text-1\) and \((\text- i)^2 = \text-1\). Display this graph for all to see:

Similarly, mathematicians try to avoid writing \(\sqrt{\text- 4}\), since the \(\sqrt{}\) symbol is supposed to be reserved for positive square roots of real numbers. Ask students, “How can we describe the square roots of -4 using the number \(i\)?” (If we square \(2i\) and \(\text- 2i\), we get -4).

Then display these equations for all to 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, recording their thinking for all to see. Ask students, “What do you notice about solutions to \(x^2=k\) if \(k\) is positive? What if \(k\) is negative?” (If \(k\) is positive, the two solutions are on the positive and negative real axis. If \(k\) is negative, the solutions are on the positive and negative imaginary axis.)

In the previous activity, students found that negative real numbers have two imaginary square roots. In future lessons, students will use the quadratic formula to find complex solutions, which involves interpreting expressions like \(\sqrt{\text-36}\). In this activity, students learn and apply the convention that for any positive real number \(a\), \(\sqrt{\text-a}=i\sqrt{a}\).

Explain to students that sometimes negative numbers end up inside of the square root symbol as a result of the steps used to solve equations. In the previous activity, we saw that negative numbers have two square roots, one on the positive imaginary axis, and one on the negative imaginary axis. By convention, the \(\sqrt{}\) symbol with a negative number inside refers to the square root on the positive imaginary axis. For example, \(\sqrt{\text-4}=2i\).

Students may need to work through some intermediate steps in order to express the square roots in terms of \(i\). For example, students could break a square root apart like this: \(\sqrt{\text-36} = \sqrt{\text-1}\boldcdot \sqrt{36} = i \boldcdot 6 = 6i\).

Select students to share their responses and encourage students to show that their answers make sense by squaring. Mention that for answers that involve \(i\) multiplied by a radical, it is customary to write the \(i\) first so it doesn’t look like it might be part of the radicand. For example, \(\sqrt{\text-15}\) is usually written as \(i\sqrt{15}\) rather than \(\sqrt{15}i\).

In an earlier lesson, students represented the imaginary number line as an axis perpendicular to the real number line. In this lesson, students used the real and imaginary number lines together to represent purely real and purely imaginary numbers. 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 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.

Display this image for all to see:

Ask students, “What would happen if we tried to add a real number to an imaginary number? How could we represent \(5+i\) visually?” Select 1–2 students to give brief responses before giving students time to work on the task. Follow with a whole-class discussion.

Select 1–2 students to share their thinking for the last question 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 students have seen before have been pairs of real numbers, like \((\text-8, 7)\), 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 you have used before? How is it different?” (Before, we needed two real numbers to make 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 \(\text- 5i+7\)?” (This one has the real part and imaginary part switched from the task, but it should be the same process. Start at 0 and more 7 spaces along the positive real axis, and then move down 5 spaces to represent \(\text- 5i\).)