Download Lesson Plan - Howard County Public School System

Lesson Title: Complex Numbers
Date: _____________ Teacher(s): ____________________
Course: Common Core Algebra II, Unit 3
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
N.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.
MP2:
MP3:
MP6:
MP7:
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Attend to precision.
Look for and make use of structure.
Common Core Algebra II, Unit 3
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Have students simplify these radicals
√36
√32 √48 √45
Have students formulate answers in pairs and
randomly select pairs to answer. Let others critique
and evaluate answers.
Highlight the separation of the factors and the ability
to take the roots separately.
Point out that even the simple √36 can be
√(9)(4) = (√9) (√4) = (3)(2) = 6
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Have students locate these numbers on the complex
plane:
4+5i
and 3+2i
Then ask them, in their pairs to find the sum of these
numbers on the complex plane.
Display a large number line and have pairs find
approximate locations of these numbers on a number
line of their own construction Ask pairs to show and
justify approximate locations of these four numbers
on the large number line. (Look for evidence of MP2
and MP6.)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Display these problems and ask students to solve them.
x2= 16
x2= 81
x=√16
x= √81
x= √-64)
Ask for answers to the first four. Now ask, “Why are there two answers to the first two problems? Why is
there just one answer to the second two?” Have students formulate answers in pairs and randomly select
pairs to answer. Let others critique and evaluate answers. Have pairs locate these numbers on the number
line. Now ask for the answer to the last problem. (Remind pairs that they are looking for a number that,
when multiplied by itself, equals -64. Allow for no solution at this point, but foreshadow that you will
address this conundrum in today’s lesson). (Look for evidence of MP2 and MP3.)
2. Continue the theme with these problems
x=√(362) x =(√36)2 x = √(72) x = (√7)2
x = √(172)
x = (√17)2
Challenge pairs to explain how the calculations for these are different and yet why the answers are the same.
(Allow students to point how they can find the answers without proceeding through the two steps each time,
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Complex Numbers
Course: Common Core Algebra II, Unit 3
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
but stress that the interim step still exists.) Have pairs locate these numbers on the number line.(Look for
evidence of MP2,MP7)
3. Add these two problems to the set of problems
x= √(-42)
x=( √-4)2
x= (√-1)2 x= (√-1)2
Based on the challenge in part 2, many pairs may find the answers quickly. Have pairs locate the numbers
on the number line. Now have them go back to the calculations involved in part 2. In particular, ask pairs to
compare the calculations. Ask them to compare the interim values. Ask pairs to report differences and
obstacles. Let others critique and evaluate answers. Highlight responses that point to the difficulty of the
second of each pair in finding the interim value of the square root of a negative number. (Look for evidence
of MP2 and MP3.)
4. Highlight this need for numbers to serve as the square roots of negative numbers. We will need to
“imagine” a number which is the square root of a negative number. The most basic is √-1, which we will
called “i” (for imaginary). So finding the last example in part 3 would look like this, showing it’s interim
value (√-1)2 = ( i)2 = -1. Ask pairs to refer to the lesson launch and find √-4 by separating it into factors.
Have students formulate answers in pairs and randomly select pairs to answer. Let others critique and
evaluate answers. Give pairs this set of problems to evaluate. (If necessary point out the similarity to the
lesson launch problems.)
√-36 √-32
√-48
√-45
Have students formulate answers in pairs and randomly select pairs to answer. . (Look for evidence of MP2
and MP7.)
5. Ask pairs to locate the six answers in section 4 on the number line. After some discussion of inadequacy of
the number line, show the need for an imaginary axis to be added vertically to the horizontal real axis. Help
students locate some of the answers in part 4 on the imaginary axis. Have pairs evaluate these numbers: √9
+ √25 √-9 + √-25 √9 + √-25. Point out the special difficulty the third will pose as a combination. Ask
pairs to locate their answer on the number line system. Have students formulate answers in pairs and
randomly select pairs to answer. Let others critique and evaluate answers. Define the combination of real
and imaginary numbers in the third number as a “complex” number. Use the a+bi format.
6. Have students simplify these individually or in pairs:
i2=
(√-1)=
√-9 + √81
(2i)2=
5 + (2i)2=
((√9)i)2=
Have randomly selected students or pairs present their answers and justify them. Let others critique
and evaluate answers.
7. As an extension:
i3=
i4=
Have randomly selected students or pairs present their answers and justify them. Let others critique
and evaluate answers.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any
outside observer) that your students have developed a deepened and conceptual understanding.
Students will be able to simplify common radicals.
Students will identify the need for imaginary numbers and complex numbers when encountering the square root
of a negative number.
Students will be able to simplify radicals of negative numbers
Students will be able to locate real, imaginary, and complex numbers on number line or complex plane.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Complex Numbers
Course: Common Core Algebra II, Unit 3
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
The lesson closure and the end of lesson examples will provide a sense of the students’ ability to connect the
concept of the remainder and the value of a function. Feedback from a homework assignment can also be used
as a formative assessment.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed),
etc.
Key Terms: imaginary numbers, complex numbers
Anticipated Misconceptions: It is sometimes difficult to distinguish the complex number plane from the x-y
plane. The need to have a two number line system for one variable, rather than the luxury of mere real number
lines for two variables, is necessitated by the separate imaginary part of complex numbers. It should be stressed
that we are working with and graphing one number that needs two axes. If we had two complex variables, we
would not have enough dimensions to represent a relation between them graphically.
Resources: What materials or resources are
essential for students to successfully complete the
lesson tasks or activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Number lines
There are numerous skill problems in simplifying
complex radicals in Algebra 2 texts that can be used.
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
How well did I motivate interest in the concept of square roots of negative numbers?
To what level did I allow students to self discover the concepts needed to define imaginary and complex
numbers?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.