Lesson 12

The purpose of this Math Talk is to elicit strategies and understandings students have for adding and subtracting integers. These understandings help students develop fluency and will be helpful later in this lesson when students combine like terms to express the sums and products of complex numbers in the form \(a+bi\), where \(a\) and \(b\) are real numbers.

In this activity, students have an opportunity to notice and make use of structure (MP7) because each sum includes several pairs of opposites that sum to 0. Being able to compute efficiently by choosing which parts of an expression to evaluate first will be helpful as students develop fluency with complex number arithmetic.

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?”

The purpose of this activity is to introduce the idea that sums of complex numbers can be rewritten as a single complex number in the form \(a+bi\), where \(a\) and \(b\) are real numbers. In previous lessons, students have used the complex plane to visualize complex numbers, and now they continue that work using the complex plane to visualize why the sum of two complex numbers is another complex number.

Display the image from the task statement for all to see.

Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a brief whole-class discussion. Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image.

Arrange students in groups of 2 and give them a few minutes of work time for the activity, encouraging students to discuss each problem and, if there is disagreement, to work to reach agreement. Follow up with a whole-class discussion.

Select students to share their arithmetic results and connect them to arrows in the complex plane. It is important to discuss that doing operations with complex numbers results in another complex number. In order to see this, it is helpful to write the result in the form \(a+bi\), where \(a\) and \(b\) are real numbers. When a complex number is written this way, \(a\) is called the real part of the number, and \(bi\) is called the imaginary part.

If time allows, ask students, “Is \((2 + 3i) + (\text- 8 - 8i)\) the same as \((\text- 8 - 8i) + (2 + 3i)\)?” (Yes. The diagrams will look different, but, for example, going right 2 then left 8 is the same as going left 8 and right 2.)

In this activity, students compare what happens on the real number line and on the complex plane when we multiply numbers. This reinforces the idea that the complex plane is a way of representing complex numbers, analogous to the real number line, and not like the coordinate plane which represents pairs of numbers.

Students may need to be careful with parentheses when raising \(2i\) to different powers. It may also be helpful for students to write out all the \(2i\) factors, and then collect the \(i\) factors and pair them up.

Display a blank complex plane for all to see. Invite students to share the points they plotted to represent \(2i\), \((2i)^2\), \((2i)^3\), and \((2i)^4\). Draw each point on the plane, and invite students to agree or disagree with each placement.

Once it is agreed where each point should go, display the following for all to see: “How is multiplication on the complex plane similar to multiplication on the real number line? How are they different?” Give students 1 minute of quiet think time to consider their answers to these questions, then invite them to share with their partner. Follow with a whole-class discussion.