Wentzville School District
Curriculum Development Template
Stage 1 – Desired Results
Unit 4 - Polar Coordinates
Unit Title: Polar Coordinates
Course: PreCalculus
Brief Summary of Unit: Students will learn to graph points and equations on the polar plane. Students will then learn to
identify and graph classical curves in mathematics. Also, students will be able to convert coordinates and equations
from rectangular form to polar form and vice versa. In addition, students will be able to perform operations on complex
numbers written in polar form.
Textbook Correlation: Glencoe PreCalculus Chapter 9 (Sections 1, 2, 3, 5)
Timeframe: 3 weeks
WSD Overarching Essential Question
Students will consider…
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How do I use the language of math (i.e. symbols,
words) to make sense of/solve a problem?
How does the math I am learning in the classroom
relate to the real-world?
What does a good problem solver do?
What should I do if I get stuck solving a problem?
How do I effectively communicate about math
with others in verbal form? In written form?
How do I explain my thinking to others, in written
form? In verbal form?
How do I construct an effective (mathematical)
argument?
How reliable are predictions?
Why are patterns important to discover, use, and
generalize in math?
How do I create a mathematical model?
How do I decide which is the best mathematical
tool to use to solve a problem?
How do I effectively represent quantities and
WSD Overarching Enduring Understandings
Students will understand that…
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Mathematical skills and understandings are used
to solve real-world problems.
Problem solvers examine and critique arguments
of others to determine validity.
Mathematical models can be used to interpret and
predict the behavior of real world phenomena.
Recognizing the predictable patterns in
mathematics allows the creation of functional
relationships.
Varieties of mathematical tools are used to
analyze and solve problems and explore concepts.
Estimating the answer to a problem helps predict
and evaluate the reasonableness of a solution.
Clear and precise notation and mathematical
vocabulary enables effective communication and
comprehension.
Level of accuracy is determined based on the
context/situation.
Using prior knowledge of mathematical ideas can
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relationships through mathematical notation?
How accurate do I need to be?
When is estimating the best solution to a
problem?
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help discover more efficient problem solving
strategies.
Concrete understandings in math lead to more
abstract understanding of math.
Transfer
Students will be able to independently use their learning to…
know that there are many different types of coordinate systems, and that each type of coordinate system is best suited
for modeling different real-world contexts.
Meaning
Essential Questions
Understandings
Students will consider…
Students will understand that…
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When is a polar system more useful than
rectangular?
What algebraic operations are easier in polar
form?
Why is the polar system necessary for finding all
solutions for an equation?
What sorts of real-world contexts can be
represented using a polar coordinate system?
What kinds of real-world contexts can be
represented using complex numbers?
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There are situations in nature that can be better
represented using the polar coordinate system
than the rectangular coordinate system.
Polar form makes it possible to find solutions to
equations that couldn’t be found in rectangular
form.
There is a relationship between polar form of a
number, rectangular form of a number, and
complex numbers.
Acquisition
Key Knowledge
Key Skills
Students will know…
Students will be able to….
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pole
polar axis
complex plane
imaginary axis
absolute value of a complex number
r cis
modulus
argument
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Graph points with polar coordinates
Graph polar equations
Find the distance between points plotted on a
polar plane
Identify and graph classical curves (limacon,
cardioid, rose, lemniscate, spiral of Archimedes)
Convert between polar and rectangular
coordinates
Convert between polar and rectangular equations
Convert complex numbers from rectangular to
polar form and vice versa
Find products, quotients, powers, and roots of
complex numbers in polar form
Solve real-world problems that can be modeled
using polar coordinates.
Solve real-world problems that can be modeled
using complex numbers.
Standards Alignment
MISSOURI LEARNING STANDARDS
Perform arithmetic operations with complex numbers.
N-CN-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.
N-CN-2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply
complex numbers.
N-CN-3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
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.
N-CN-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°.
N-CN-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.
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Show Me-Standards
Goal 1: 1, 4, 5, 6, 7, 8
Goal 2: 2, 3, 7
Goal 3: 1, 2, 3, 4, 5, 6, 7, 8
Goal 4: 1, 4, 5, 6
Mathematics: 1, 4, 5