DO NOW 11/18/2015 Convert standard forms to vertex forms.

CC 3 Solving Quadratics LT 3C Complex Number System NCTM 1 (learning objective) Pose one question for each LT for students to “pre-view” what is coming and to gather data that will inform lesson design TM: Justin did a really good job with this … when he had the students list all the words they were confused about or haven’t seen before, so they could focus on learning their meaning in that lesson.

“ bi “ is the imaginary part. Complex Number System Complex number is a number that has a real and imaginary part of the form: a + bi “ a “ is the real part. “ bi “ is the imaginary part. The imaginary number i is defined as which normal we say has no real answer.(but there a complex one  ) Both a and b exist in the Real numbers. What are some examples of what a and b could be? Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

I. Complex Number System(Properties) (LT: 3B) Name the parts of the following Complex Numbers Adding Complex Numbers- (-4 + 7i) + (5 – 3i) Imaginary Real -4 + 5 = 1 7i - 3i = 4i 1 + 4i Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

Don’t forget to distribute the negative Name the parts of the following Complex Numbers Subtracting Complex Numbers- (11 - 8i) – (-7 + 15i) Don’t forget to distribute the negative Imaginary Real 11 + 7= 18 -8i -15i = -23i 18 -23i Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

Try it out! a.) (1 + 6i)+(-2 – 3i) b.) (2-3i)+(4 + 2i)- (-6+ 5i) Complete the specified operation. a.) (1 + 6i)+(-2 – 3i) b.) (2-3i)+(4 + 2i)- (-6+ 5i) Move 5: Action Plan / Improve; NCTM 1, 8 5 minutes alone…..make a decision about practice after previewing problems in book, aligned to LTs. Write in margin of notes to compare with teacher choice

DO NOW 11/19/2015 Simplify.

Let’s talk about i. The letter i denotes the imaginary unit, that is… Let’s talk about i. The letter i denotes the imaginary unit, that is….. a. For each integer k from 0 to 8 write ik . b. Describe the pattern you observe, and algebraically prove your observation, In particular simplify i195 Move 2: Investigation Before Explanation / Imagine; MP 1, 6; NCTM 2, 7 Need to have un-stucking questions that guide students in thinking about their approach and/or plans Need advancing questions to guide students in thinking more deeply (especially students who want to focus on execution)

Multiplying Complex Numbers- (-4 + 7i)(5 - 3i) = -20 +37i -21i2 Remember i2= -1 -20 + 12i + 35i -21i2 Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

Try it out!! 1. (2+3i)(3-2i)= 2. (2 + 6i)(2 - 6i)=

If a + bi is the complex number than a - bi is the conjugate Dividing Complex Numbers- Multiply top and bottom by the conjugate of the denominator. Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

The conjugate of the denominator is Now multiply across B3: Dividing Complex Numbers- = The conjugate of the denominator is -2 - 6i Please check slide for mistakes  Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

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D: Purpose We need imaginary numbers for complex situations like sound, electricity and magnetism. Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 Provide reasons why this concept is fundamental in mathematics TM: I have been playing with the idea that we could pose two and then have student reflect and state any where they could see this concept showing up in the real world. Thoughts?

E: Properties of Imaginary Numbers What Pattern do you see? Given complete the following:

CC 5 :Complex Number System LT: 3A I can explain the Complex Number System defining Imaginary, Real, Rational, Irrational, Integers and Natural Numbers and their relationship and define a complex number as a number with the form a+bi where a and b are real numbers and i2= -1. LT:3B I can use the relation of i2=-1 and the commutative, associative and distributive properties to add, subtract and multiply complex numbers. NCTM 1 (learning objective) Pose one question for each LT for students to “pre-view” what is coming and to gather data that will inform lesson design TM: Justin did a really good job with this … when he had the students list all the words they were confused about or haven’t seen before, so they could focus on learning their meaning in that lesson.

I. Complex Number System(Properties) (LT: 3B) A1: Adding and Subtracting Complex Numbers- To add/subt. complex numbers we combine like terms, the Real parts to the Real parts and the imaginary to the imaginary parts. Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

B. Visual Complex Numbers Real Numbers Irrational Numbers Integer Whole Numbers Natural Numbers Imaginary Numbers Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 Graphs or tables or visual relationship TM: What questions will T. pose to strengthen the understanding of the visual? Add definition of Complex number…

B. Visual (other possibility) Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 Graphs or tables or visual relationship TM: What questions will T. pose to strengthen the understanding of the visual? Add definition of Complex number…

C: Process Graph the following Complex Numbers: Imaginary Axis Real Axis Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 Explain the “how” to approach and think through problems and model one DOK 2 or 3 example using the following process: Student tries problem alone, and writes one question (you do) Student pairs with peer to answer questions or come up with a single question (we do) Student watches teacher model and if question is not answered, asks the question (I do) Student tries another similar problem (you do) TM: great to see the Youdo , we do… spelled out.

I. Complex Number System(Properties) (LT: 3B) A2: Multiplying Complex Numbers- To multiply complex numbers we double distribute and combine like terms. Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

I. Complex Number System(Properties) (LT: 3B) If a + bi is the complex number than a - bi is the conjugate A3: Dividing Complex Numbers- Multiply top and bottom by the conjugate of the denominator. Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?

I. Complex Number System(Properties) (LT: 3B) Now multiply across B3: Dividing Complex Numbers- = The conjugate of the denominator is -2 - 6i Please check slide for mistakes  Move 3: Notes / Plan; MP 2, 6; NCTM 3, 6 TM: Do we need to say anything about the definition being a general form and need thought bubbles to break down the difficult concepts ? What questions will you be posing to the students to clarify or recall concepts?