options: Apply and extend previous understandings of numbers to the system of rational numbers.

Perform arithmetic operations with complex numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Represent complex numbers and their operations on the complex plane.

Reason quantitatively and use units to solve problems.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Use complex numbers in polynomial identities and equations.

Extend the properties of exponents to rational exponents.

Perform operations on matrices and use matrices in applications.

Compute fluently with multi-digit numbers and find common factors and multiples.

Perform operations on vectors.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
problem_activity: 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:
\((10+4i)-(\underline{\phantom{00}} + \underline{\phantom{00}}i) = \text?\)
If the result of this subtraction is a real number, what could the second complex number be?
If the result of this subtraction is an imaginary number, what could the second complex number be?
Student Response
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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.


correct_option_index: 1