options: Use the relation i^(2) = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Know there is a complex number i such that i^(2) = –1, and every complex number has the form a + bi with a and b real.

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
problem_activity: Activity
In the previous activity, students found that negative real numbers have two imaginary square roots. In future lessons, students will use the quadratic formula to find complex solutions, which involves interpreting expressions like
\(\sqrt{\text-36}\)
. In this activity, students learn and apply the convention that for any positive real number
\(a\)
,
\(\sqrt{\text-a}=i\sqrt{a}\)
.
Launch
Explain to students that sometimes negative numbers end up inside of the square root symbol as a result of the steps used to solve equations. In the previous activity, we saw that negative numbers have two square roots, one on the positive imaginary axis, and one on the negative imaginary axis. By convention, the
\(\sqrt{}\)
symbol with a negative number inside refers to the square root on the positive imaginary axis. For example,
\(\sqrt{\text-4}=2i\)
.
Student Facing
Write these imaginary numbers using the number
\(i\)
.
\(\sqrt{\text- 36}\)
\(\sqrt{\text- 10}\)
\(\text- \sqrt{\text- 100}\)
\(\text- \sqrt{\text- 17}\)
Student Response
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Anticipated Misconceptions
Students may need to work through some intermediate steps in order to express the square roots in terms of
\(i\)
. For example, students could break a square root apart like this:
\(\sqrt{\text-36} = \sqrt{\text-1}\boldcdot \sqrt{36} = i \boldcdot 6 = 6i\)
.
Activity Synthesis
Select students to share their responses and encourage students to show that their answers make sense by squaring. Mention that for answers that involve
\(i\)
multiplied by a radical, it is customary to write the
\(i\)
first so it doesn’t look like it might be part of the radicand. For example,
\(\sqrt{\text-15}\)
is usually written as
\(i\sqrt{15}\)
rather than
\(\sqrt{15}i\)
.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills


correct_option_index: 1