Unit 6, Lesson 11
More Arithmetic with Complex Numbers (Optional)
Integrated Math 2
11.1
Warm-up
5 mins
Which Three Go Together: Complex Expressions
Standards Alignment
Building On
Addressing
HSN-CN.A.2
Use the relation
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items, in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three expressions that go together and can explain why. Next, tell students to share their response with their group and then to work together to find as many sets of three as they can.
Activity
NoneStudent Task Statement
Which three go together? Why do they go together?
A
B
C
D
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any complex number terminology they use, such as “real part,” “complex plane,” or “complex numbers,” and to clarify their reasoning as needed. Consider asking:
- “How do you know_____?”
- “What do you mean by _____?”
- “Can you say that in another way?”
11.2
Activity
15 mins
Powers of
Instructional Routines
5 Practices11.3
Activity
15 mins
Add 'Em Up (or Subtract or Multiply)
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
The structure of a row game gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3). In a row game, pairs of students do different problems that have the same answer. If there are discrepancies in their answers, students must communicate with each other to resolve those discrepancies.
Look for groups who disagree and then work to reach agreement, to share during the Activity Synthesis. Also look for any groups who initially agree, but upon deeper inspection, were both incorrect.
Launch
Arrange students in groups of 2, assigning one student as partner A and the other as partner B. Explain to students that there are two columns of problems and that they are to do the problems in their column only. Students are to complete the problems, and then compare answers with their partner. If they do not get the same answer, they should work together to find the error.
MLR8 Discussion Supports. Display sentence frames to support partner discussions. Examples:
- “First, I
because .“ - “I tried
, and what happened was .” - “I agree/disagree because
.”
Lesson Synthesis
The purpose of the discussion is for students to articulate how they think about adding and multiplying complex numbers. Here are some questions for discussion:
- “How is arithmetic with complex numbers the same as with real numbers? How is it different?” (The arithmetic is the same as with real numbers because it still involves multiplying, exponentiating, adding, and subtracting. It is different because we didn’t divide complex numbers, and also there is an extra consideration that
- “What strategy do you prefer for multiplying complex numbers?” (I like to begin by using the distributive property, then replace any
with -1, then combine the real parts together, and finally combine the imaginary parts.)
Student Lesson Summary
Suppose we want to write the product
Keeping track of the negative signs is especially important since it is easy to mix up the fact that
Next, suppose we want to write the difference
Again, it is important to be precise with negative signs. It is a common mistake to subtract