# Lesson 15

Working Backwards

## 15.1: What's Missing? (10 minutes)

### Warm-up

The purpose of this activity is to focus students' attention on the two parts of a complex number: the real part and the imaginary part. This will be useful in the Information Gap, when students will need to be specific about the information they want and to strategize about how to use it (MP1).

### Launch

Display the following for all to see:

$$(a+bi)-(\underline{\phantom{00}} + \underline{\phantom{00}}i) = \text?$$

Ask, "In order for the result of this subtraction to be a real number, what has to be true about the missing imaginary part of the complex number subtracted from $$a+bi$$?" (It has to match the imaginary part of the other expression so it will sum to 0, which means that $$b$$ must go in the blank.) After quiet think time, invite students to share their thoughts with a partner. Then invite partners to share with the class.

### Student Facing

Here are some complex numbers with an unknown difference: $$(10+4i)-(\underline{\phantom{00}} + \underline{\phantom{00}}i) = \text?$$

1. If the result of this subtraction is a real number, what could the second complex number be?
2. If the result of this subtraction is an imaginary number, what could the second complex number be?

### Activity Synthesis

Ask students to share the expressions they made by filling in the blanks. Highlight what is common about the responses to each question (all responses to the first question have the same imaginary part, and all responses to the second question have the same real part).

Tell students that in the next activity they will continue to reason about the real and imaginary parts of complex numbers based on what they know about how the numbers are combined.

## 15.2: Info Gap: What Was Multiplied? (25 minutes)

### Activity

This info gap activity gives students an opportunity to determine and request the information needed to figure out which complex numbers were multiplied to produce another complex number. The challenge students will face here is that there will be two unknowns in each problem, a real part and an imaginary part, and two equations derived from the information students have about them. Once one of the unknowns is expressed in terms of the other and substituted into the other equation, students will see that they have to solve a quadratic equation. In following lessons, students will return to solving quadratics (with complex solutions), so these problems will be a bridge between complex numbers and quadratics.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

### Launch

Tell students they will continue to investigate what happens when we combine complex numbers. In this activity, they will focus on multiplying. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving the multiplication of complex numbers. Consider providing questions or question starters for students who need a starting point such as: “Can you tell me…?” and “Why do you need to know…?”
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization

### Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
4. Read the problem card, and solve the problem independently.
5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

3. Explain to your partner how you are using the information to solve the problem.
4. When you have enough information, share the problem card with your partner, and solve the problem independently.

### Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What was a strategy you tried when you were working on the problems?”
• “Did it seem at first like there wasn’t enough information to find an answer? If so, how did you figure out what you needed?”

Highlight for students that the real and imaginary parts of complex numbers always combine in the same way when we multiply them, similarly to the way the parts of linear functions combine when we multiply them. This means that we can separate those parts out again if we know enough about the result and what we started with.

## Lesson Synthesis

### Lesson Synthesis

Display the following for all to see (if the third pair of cards was not used, only display the first two):

$$(4-i)(c+di)=34i$$

$$(2+bi)(c+3i)=12$$

$$(a+bi)^2=\text-6+8i$$

Tell students that these are the Info Gap problems with all the given information filled in. Ask, "What was similar about solving these problems? What was different?" Give students 1 minute of quiet think time and 1 minute to share with a partner before inviting students to share similarities and differences. Highlight connections between the type of information that was given and the resulting equations that students had to solve. In particular, the first problem does not require solving a quadratic, but the other two do.

If the third pair of Info Gap cards was not used in the activity, they can be displayed and students can discuss strategies for solving the problem, if desired.

## Student Lesson Summary

### Student Facing

When complex numbers are multiplied, each part of one of the numbers gets distributed to the other one. This means that we'll always see the same pattern:

$$(a+bi)(c+di)=ac+adi+bci+bdi^2$$

We can use the fact that $$i^2=\text-1$$ to rearrange this and make it easier to see the real part and the imaginary part of the result.

$$(a+bi)(c+di)=(ac-bd)+(ad+bc)i$$

Every time we multiply complex numbers, the result is not only a complex number, but it's a specific complex number that comes from combining the parts of the numbers we started with in a specific way. If $$a$$ and $$c$$ are the real parts of the numbers we start with and $$bi$$ and $$di$$ are the imaginary parts, then the result will always have $$ac-bd$$ as a real part and $$(ad+bc)i$$ as an imaginary part.