Name: Neil Moakley Lesson Plan Date:

Oct. 22-24, Oct 31 Subject: Elementary Functions

Class Period 1

UNIT PLAN – Complex Numbers

 This unit will address the following CCSI objectives:

Perform arithmetic operations with complex numbers.

• CCSS.Math.Content.HSN-CN.A.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.

• CCSS.Math.Content.HSN-CN.A.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

• CCSS.Math.Content.HSN-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

The following CCSI standards will also be addressed in part:

Represent complex numbers and their operations on the complex plane.

• CCSS.Math.Content.HSN-CN.B.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

• CCSS.Math.Content.HSN-CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.

• CCSS.Math.Content.HSN-CN.B.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Specifically, we will begin to look at the complex plane and complex numbers in rectangular form. Modulus (absolute value) will be discussed as well.

Name: Neil Moakley Lesson

Plan Date:

October 22 Subject: El Func Class Period 1

SEVEN-STEP LESSON PLAN

VISI

ON

-SET

TIN

G: K

NO

W, S

O, S

HO

W

OBJECTIVE.

KEY POINTS.

SWBAT perform addition, subtraction, and multiplication operations on imaginary numbers

DO NOW (5 + 5 min.) 2 + 3𝑥 + 4 + 5𝑥 = 2 + 3𝑥 ! = 2 + 3𝑥 2 − 3𝑥 = 49 − 9 + 36 =

\ −4 = 3. INTRODUCTION OF NEW MATERIAL (__ min.)

• Review of “𝑖” as imaginary unit, 𝑎𝑖 as pure imaginary number • Introduction to generalized imaginary number 𝑎 + 𝑏𝑖, where 𝑏 ≠ 0 • Introduction of property of uniqueness, i.e. that 𝑎 + 𝑏𝑖 = 𝑐 + 𝑑𝑖 iff 𝑎 = 𝑐 and 𝑏 = 𝑑 • After guided practice example of complex conjugate below, the term is introduced to students and its

“interesting property”, i.e. “a real number product results from the multiplication of these two imaginary numbers) is discussed (anticipating tomorrow’s lesson on division)

2. GUIDED PRACTICE (__ min.)

• Example of property of uniqueness • Do Now questions are reformed as imaginary number questions, students and teacher

solve together: o 2 + 3𝑖 + 4 + 5𝑖 = o 2 + 3𝑖 ! = o 2 + 3𝑖 2 − 3𝑖 = o −49 − −9 + −36 =

1. INDEPENDENT PRACTICE (__ min.)

• Class Exercise #9, p. 28 (property of uniqueness problem) • Written Exercises #1-18 p. 28 (practice with addition, subtraction, multiplication of

imaginary numbers, including of complex conjugates)

5. CLOSING (__ min.)

• Key points from lesson reviewed • Anticipated points from next lesson previewed (we will be covering division of complex

numbers)

HOMEWORK (if appropriate). Finish Written Exercises if incomplete

REI

NFO

RC

EM

EN T

I do not like this lesson plan layout, since I sometimes jump between guided practice and independent practice throughout the lesson. This makes the above lesson unrepresentative of the intended flow

 

Name: Neil Moakley Lesson Plan Date:

October 23 (half-day schedule, 27 minute class)

Subject: El Func Class Period 1

SEVEN-STEP LESSON PLAN

VISI

ON

-SET

TIN

G: K

NO

W, S

O, S

HO

W

OBJECTIVE.

KEY POINTS.

SWBAT divide imaginary numbers DO NOW (3 + 3 min.) 2 + 3𝑖 3 − 4𝑖 = (3 + 4𝑖)(3 − 4𝑖) = 4𝑖𝑖=

𝑖! = 3. INTRODUCTION OF NEW MATERIAL (__ min.) Discussion of division of imaginary numbers

• Cannot easily be done using the same rules as real numbers or polynomials (e.g. synthetic division) • Question to ask is “how do I make my divisor/denominator a REAL number, since I know how to divide

by real numbers” (eliciting answer of “multiply by complex conjugate”) •

2. GUIDED PRACTICE (__ min.)

• Using the “multiply by one” principle, then, what is !!!!!!!!

=

o !!!!!!!!

!!!!!!!!

= !"!!!"

= !"!"+ !

!"𝑖

1. INDEPENDENT PRACTICE (__ min.)

• Written Exercises #19-24 p. 28 (practice with division of imaginary numbers) 5. CLOSING (__ min.)

• Review of common errors observed during circulation. Anticipated: o Mistakes in arithmetic o Multiplying by the denominator-over-denominator, rather than the

CONJUGATE of the denominator-over-conjugate. That is 𝑎 + 𝑏𝑖𝑐 + 𝑑𝑖

𝑐 + 𝑑𝑖𝑐 + 𝑑𝑖

HOMEWORK (if appropriate). Finish Written Exercises if incomplete

REI

NFO

RC

EM

EN T

This lesson needs to be short, to fit into a 27 minute period, and so is a one-part extension of the previous day’s work

 

Name: Neil Moakley Lesson Plan Date:

October 24 Subject: El Func Class Period 1

SEVEN-STEP LESSON PLAN

VISI

ON

-SET

TIN

G: K

NO

W, S

O, S

HO

W

OBJECTIVE.

KEY POINTS.

SWBAT quickly calculate integer powers of 𝑖, and calculate the modulus of imaginary numbers

DO NOW (5 + 5 min.)

• !!! !!!! !

=

• What is the reciprocal of !!!!!

? • What is 𝑖! + 3𝑖! + 2𝑖!? • What is 𝑖!"? HINT: Try to use rules of exponents to save some work • REVIEW: Include note of observed student mistakes/misconceptions in review of problems, esp #1

3. INTRODUCTION OF NEW MATERIAL (__ min.)

• Segue from final Do Now question review into a discussion of the pattern of exponential powers of 𝑖, and how they can be quickly calculated. Use both a pattern grid/remainder approach and an “exponent rules” approach

• Discussion of geometric look at absolute value • Brief review of number line extensions from natural -> rational, irrational, zero, negatives, etc • Plot a few real numbers, with student input • Imaginary numbers as an extension to this number line. Introduce complex plane. • Plot a pure imaginary number • Discussion of geometric look at absolute value. What are the absolute values of these samples? What

does absolute value mean geometrically? (First with real, then extend to pure imaginary) • Ask student volunteers to come to the board and plot two (non-pure) imaginary numbers, one that has

at least one negative component • How might we calculate the abs value of these, if seen as “distance from zero”? (Try to elicit

Pythagorean Theorem) • Give name as “modulus” • IF TIME: Introduce idea of “argument”, though students will not be able to calculate for non “special”

angles. 2. GUIDED PRACTICE (__ min.)

• Integrated into above lesson. 1. INDEPENDENT PRACTICE (__ min.)

• Written Exercises #25-32 p. 29 (practice with exponential powers of 𝑖) • Look back on WE 1-24…assign sections of the class to calculate the absolute value of

certain of these results

5. CLOSING (__ min.)

• Note to anticipate a quiz at the beginning of next week, and that I will be assigning small groups for a minor project/quiz grade in the material

• Anticipation of next section: solving quadratics (including complex roots) • Test prep specific reminders

HOMEWORK (if appropriate). Finish Written Exercises if incomplete

REI

NFO

RC

EM

EN T

This is the last lesson of the week; Friday is a practice SAT administration.

Name: Neil Moakley Lesson Plan Date:

October 28 Subject: El Func Class Period 1

SEVEN-STEP LESSON PLAN

VISI

ON

-SET

TIN

G: K

NO

W, S

O, S

HO

W

OBJECTIVE.

KEY POINTS.

SWBAT show certain general properties of complex numbers

DO NOW (5 + 5 min.)

• !!"!!"!

!!

• 12 + 5𝑖 • (12 + 5𝑖)(12 − 5𝑖) • What relationship do you see between the answers to problems 2 & 3?

3. INTRODUCTION OF NEW MATERIAL (__ min.)

• Segue from final Do Now question review into a discussion of the relationship 𝑧 ! = 𝑧𝑧 • PREPARED: “Show that” proof for above: o 𝑧 ! = 𝑎! + 𝑏!

!= 𝑎! + 𝑏!

o 𝑧𝑧 = 𝑎 + 𝑏𝑖 𝑎 − 𝑏𝑖 = 𝑎! + 𝑏! 2. GUIDED PRACTICE (__ min.)

• PREPARD: Show that 𝑤 + 𝑧 = 𝑤 + 𝑧 1. INDEPENDENT PRACTICE (__ min.)

• PREPARED: Groupwork on “show that” proofs: WE #42-44 p. 28, and 𝑤𝑧 = 𝑤 |𝑧| 5. CLOSING (__ min.)

• PREPARED: Individual presentation of results

HOMEWORK (if appropriate). • Prepare for quiz • Take the absolute value of three imaginary numbers, using either the complex number

property discussed today or the geometric approach from last week.

REI

NFO

RC

EM

EN T

Students were unable to complete the “Do Now” exercises, and this lesson was changed to a reteach of these concepts.

Name: Neil Moakley Lesson Plan Date:

October 29 Subject: El Func Class Period 1

SEVEN-STEP LESSON PLAN

VISI

ON

-SET

TIN

G: K

NO

W, S

O, S

HO

W

OBJECTIVE.

KEY POINTS.

Students will review the week’s material and take a quiz on complex numbers

DO NOW (5 + 5 min.)

• !!!!!

• 5 + 7𝑖 ! • Absolute value of both

3. INTRODUCTION OF NEW MATERIAL (__ min.)

• No new material; students are assessed with a 55 point quiz, including two extra credit questions (see attached)

2. GUIDED PRACTICE (__ min.)

• NA 1. INDEPENDENT PRACTICE (__ min.)

• NA 5. CLOSING (__ min.)

• NA

HOMEWORK (if appropriate). NA

REI

NFO

RC

EM

EN T

This is the last lesson of the week; Wednesday through Friday are test prep review.