Precalculus, Quarter 3, Unit 3.3
Working With the Complex Numbers System
Overview
Number of instruction days:
8–10
Content to Be Learned
(1 day = 53 minutes)
Mathematical Practices to Be Integrated

Write complex numbers in polar form and
rectangular form.
1 Make sense of problems and persevere in
solving them.

Graph complex numbers in the complex plane
in rectangular and polar form.


Convert polar coordinates to rectangular
coordinates and vice versa.
2 Reason abstractly and quantitatively.

Graph polar functions by plotting points and by
using technology.

Express equations in both rectangular and polar
coordinates.

Find powers and roots of complex numbers in
polar form using De Moivre’s Theorem.

Explain the relationship between rectangular
and polar form.
Develop the habit of creating a coherent
representation between polar and rectangular
coordinates to facilitate quantitative reasoning.
5 Use appropriate tools strategically.

Use a graphing calculator to graph polar
equations.

How do you use De Moivre’s Theorem to find
powers of complex numbers?

What are the similarities and differences
between graphing in the coordinate plane and
graphing in the complex plane?
Essential Questions

Why do the rectangular and polar forms of a
given complex number represent the same
number?

How do you convert points from rectangular to
polar form and vice versa?

What are the four key relationships used to
convert polar equations to rectangular
equations and vice versa?
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Working With the Complex Numbers System (8–10 days)
Standards
Common Core State Standards for Mathematical Content
Number and Quantity
The Complex Number System
N-CN
Represent complex numbers and their operations on the complex plane.
N-CN.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.
ACT’s College Readiness Standards: Mathematics
1.1.a
Define polar coordinates to locate a point on a graph.
1.1.b
Graph polar functions by plotting points and by using technology.
1.1.c
Express two-dimensional points and equations in rectangular and polar coordinates.
1.1.d
Find powers and roots of complex numbers in polar form using De Moivre’s theorem.
Common Core State Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students
check their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of the quantities and their relationships in problem
situations. Students bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily attending
to their referents—and the ability to contextualize, to pause as needed during the manipulation process in
order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating
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Precalculus, Quarter 3, Unit 3.3
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a coherent representation of the problem at hand; considering the units involved; attending to the meaning
of quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
Clarifying the Standards
Prior Learning
As early as kindergarten, students studied patterning to start their conceptual understanding of algebraic
reasoning. Adding, subtracting, multiplying, and dividing rational numbers is the culmination of
numerical work with the four basic operations. Students continued to develop understanding of the
number system in Grade 8; with the introduction of irrational numbers, they expanded their understanding
to real numbers. With the introduction of imaginary numbers in high school, students’ understanding of
the number system expanded to include complex numbers.
Because there are no specific standards for rational number arithmetic in later grades, and because so
much other work in Grade 7 depended on rational numbers, fluency with rational number arithmetic
should have been mastered in Grade 7. Algebra II students have already worked with complex numbers.
They performed arithmetic operations (only addition, subtraction, and multiplication) with complex
numbers and used complex numbers in polynomials (with real coefficients), identities, and equations.
They solved quadratic equations with real coefficients that have complex solutions and extended the
polynomial identities to the complex numbers. They have also shown that the Fundamental Theorem of
Algebra is true for quadratic polynomials.
Current Learning
Precalculus students write and graph complex numbers in polar form and rectangular form. They develop
relationships such as x 2  y 2  r 2 , tan 
y
x
, x  r cos  , and y  r sin in order to facilitate changing polar
equations to rectangular equations and vice versa. They plot simple polar graphs by plotting points.
Students use a graphing calculator to investigate more difficult graphs of polar equations, and they use De
Moivre’s Theorem to find powers and roots of complex numbers in polar form.
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Future Learning
In AP Calculus BC, students will study polar equations. They will graph polar equations by hand, and
they will find the equations of lines tangent to polar curves. They will also find areas of regions created
by polar equations, such as the area of one petal of a rose. Polar equations are used for satellite tracking
and other astronomical applications.
Additional Findings
It is important that students gain experience using multiple representations to deepen their understanding.
“Students should recognize connections among different representations, thus enabling them to use these
representations flexibly.” (Principles and Standards for School Mathematics, p. 309)
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your students’
attainment of the mathematics within the unit.

Represent complex numbers on the complex plane in rectangular and polar form.

Know and explain why both rectangular and polar forms represent the same complex number.

Transform complex numbers in a complex plane from rectangular to polar form and vice versa.

Use De Moivre’s Theorem to find powers of complex numbers.

Write and graph the polar form of a linear equation.

Convert equations from polar form to rectangular form and vice versa.
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Instruction
Learning Objectives
Students will be able to:

Use polar coordinates to represent points on a plane.

Convert points between rectangular and polar coordinates.

Convert equations from polar to rectangular form, and vice versa.

Graph complex numbers in the complex plane.

Find powers and roots of complex numbers in polar form using De Moivre’s Theorem.

Review and demonstrate knowledge of important concepts and procedures related to the complex
number system.
Resources
Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 2006, Teacher Edition
and Student Edition

Section 9-1 (pp. 553-560)

Section 9-3 (pp. 568-573)

Section 9-4 (pp. 574 – 579)

Section 9-6 (pp. 586 - 591)

Section 9-8 (pp. 599 – 606)

TeacherWorks All-In-One Planner and Resource Center CD-ROM
Exam View Assessment Suite
TI-Nspire Teacher Software
Polar Coordinates activity can be found at education.ti.com. See the Supplementary Materials section of
this binder for the student and teacher notes for this activity.
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations.
Materials
TI-Nspire graphing calculators
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Instructional Considerations
Key Vocabulary
rectangular form of a complex number
polar coordinates
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary taught in previous grades or units: amplitude, complex numbers, complex plane
(Argand plane), definition of i, and imaginary number.
Students in Precalculus will be continuing the study of complex numbers. This will be the development
stage of complex numbers on the polar plane. Students have not converted points from the rectangular to
polar coordinate system; therefore, they will need to study those conversions before using complex
numbers in the two planes. Students will need to use their knowledge of the trigonometric functions in
converting complex numbers from rectangular to polar coordinates.
Students will need a reminder of the powers of i. Have students make a graphic organizer to review their
knowledge. Students can use an identifying similarities and differences strategy to represent the cyclical
nature of imaginary numbers. They can make a comparison matrix to show how multiplying 1 by
itself x amount of times gives a pattern:
i0
1
i
1
i
i
2
–1
i3
–i
i
4
1
i
5
i
Have students show what kind of parabolic graphs are associated with complex numbers. Also, have
students make a quadratic function with complex roots, solve using conjugates, and share out their
findings with the whole class.
Draw a coordinate plane on the board. The x-axis will represent all real numbers and the y-axis will
represent imaginary numbers. Student will get a card with different real/imaginary numbers written on it.
The students will use sticky notes to locate the position of the numbers on the graph. (The lines and sticky
notes will remain here until the closing when the students will be asked to come back up to the board and
reevaluate their positions.) Students should be able to answer the essential questions after completing this
assignment.
Have students in groups of three apply their knowledge by identifying similarities and differences
between rectangular and complex coordinates, making a graph of each kind on poster board, and then
comparing the coordinates. Each group should share out their findings and connect their graph to a realworld situation. Extend this activity by including the polar plane. Students may make a graphic organizer
to show how the points will look on each of these graphs.
Students must have a conceptual understanding of how to use the calculator to graph polar functions.
Students should work in cooperative groups and use technology to graph polar functions. Be sure to pay
attention to the calculator Teaching Tips provided in each section. Additionally, the Graphing Calculator
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Exploration on p. 602 in the textbook provides an opportunity for students to quickly explore roots of a
complex number using technology. The Polar Coordinates activity on education.ti.com is an additional
option aligned with this unit. In this activity, students have brief introduction to the polar coordinate
system. They develop a basic understanding of the polar coordinate system and locate points given in
polar form. Students also convert between polar and rectangular coordinates and sketch graphs of polar
functions. TI-Nspire resources can be found using the TI-Nspire Teacher Software on your school
computer. The content tab of the TI-Nspire desktop software contains links to Advanced Mathematical
Concepts, Glencoe McGraw-Hill, 2006 textbook. These resources are accessible by chapter and section.
Notes
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