Unit 4, Lesson 15
Working Backward
Algebra 2
15.1
Warm-up
Standards Alignment
Building On
Addressing
HSN-CN.A.2
Use the relation and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Instructional Routines
NoneMaterials
NoneActivity Narrative
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 later in the lesson, when students will need to ask specific questions about information that they need to solve a problem (MP6).
Launch
Arrange students in groups of 2.
Display the complex number for all to see.
Ask students,
- “How could you refer to just the 10 in this complex number?” (It is called the real part of the complex number.)
- “How could you refer to just the 4 in this complex number?” (It is called the imaginary part of the complex number.)
Activity
NoneHere are some complex numbers with an unknown difference:
?
- If is a real number, what real numbers could go in the blanks?
- If is an imaginary number, what real numbers could go in the blanks?
Student Response
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Building on Student Thinking
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
Activity
Instructional Routines
MLR4: Information Gap CardsActivity Narrative
In this activity, students try to find two particular complex numbers but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.
The Information 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).
This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.
Lesson Synthesis
Display these equations for all to see (if the third pair of cards was not used for the Information Gap lesson, display only the first two):
Tell students that these are the Information 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 the 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 Information Gap cards was not used in the activity, that pair can be displayed, and students can discuss strategies for solving the problem, if desired.
Math Community
Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:
- “I picked the norm ‘.’ It really helped me/my group because .”
- “I picked the norm ‘.’ During the activity, I realized that the norm ‘’ would be a better focus because .”
Student Lesson Summary
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:
We can use the fact that to rearrange this and make it easier to see the real part and the imaginary part of the result.
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 and are the real parts of the numbers we start with and and are the imaginary parts, then the result will always have as a real part and as an imaginary part.