Unit 4, Lesson 13
Multiplying Complex Numbers
Algebra 2
13.1
Warm-up
Standards Alignment
Building On
HSN-RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Addressing
Building Toward
HSN-CN.A.1
Know there is a complex number such that , and every complex number has the form with and real.
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is to elicit strategies and understandings students have for multiplying imaginary numbers. Later in this lesson, students will multiply complex numbers and write them in the form , so it will be helpful for students to see various strategies for multiplying imaginary numbers.
Launch
NoneActivity
NoneWrite each expression in the form , where and are real numbers.
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. Highlight ways that students regrouped factors of or in order to simplify their answers. If no student mentions the fact that squaring an imaginary number always results in a real number, ask students to discuss this idea.
13.2
Activity
Materials
NoneActivity Narrative
In this partner activity, students take turns matching equivalent expressions using the fact that . They build on a previous activity in which they multiplied imaginary numbers and saw how this changed the numbers’ representation on the complex plane. Fluency with strategically using the fact that will be an important skill in the next activity when students multiply numbers that have both real and imaginary parts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and to critique the reasoning of others (MP3).
13.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is for students to build fluency expressing the product of two complex numbers in the form , where and are real numbers. In order to do this, students must use the fact that .
Look for students who answer the last question as and students who answer 13, and highlight these during discussion.
Lesson Synthesis
The purpose of this discussion is for students to articulate how to multiply complex numbers. Display this equation for all to see, leaving room to record student thinking for all to see during the discussion:
Ask students, “Mentally, without going through all the trouble of writing each side in the form , what about the structure of the expressions on each side of the equal sign tells you that the equation is true or false?” (Looking at the left-hand side, we can make the imaginary part by adding the product with the product , which gives . On the right-hand side, however, the imaginary part is , so the equation is false. The two complex numbers don’t have equal imaginary parts, so they can’t be equal.)
Then, display this equation for all to see:
Again, ask students to use arguments about the structure of the expressions on each side of the equation to mentally determine whether or not it is true. (The imaginary parts match because on the left-hand side, , and on the right-hand side, . The real parts don’t match, however. On the right-hand side, the real part is . On the left-hand side, the real parts of the two factors also multiply to -6, but there are multiples of that are real that haven’t been accounted for.)
If time allows, ask students to verify the arguments for one or both equations by writing each side in the form , where and are real numbers.
Student Lesson Summary
One way to multiply two complex numbers is to use the distributive property.
Remember that , so:
When we add the real parts together and the imaginary parts together, we get: