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Common Core Standards Template
Domain:
The Complex Number
System
Cluster:
Perform arithmetic operations with
complex numbers.
Class
Standards
Precalculus (Q3)
Notes
N.CN.3 Find the conjugate of a complex number; use conjugates to find moduli
and quotients of complex numbers.
A moduli is “The positive square root of
the sum of the squares of the real and
imaginary parts of a complex number.”
Or
a 2  b2 .
Student Friendly Language: “I can…”
I can determine the absolute value of a complex number.
I can divide complex numbers and simplify the quotient.
I can determine the conjugate of a complex number.
Know
(factual)
Understand
(conceptual)
a + bi is the standard form of a
complex number, where a is the real
part and b describes the imaginary
part.
|a + bi| = a  b .
a + bi and a – bi are conjugates.
2
2
The moduli of a complex number
refers to the distance the value is
away from the origin on the
coordinate plane.
To find the quotient of complex
numbers, we must multiply by the
conjugate of the
denominator/divisor on both the
numerator and denonminator.
Do
(procedural, application,
extended thinking)
Use the distance formula to
determine the moduli of a point.
Instructional Strategies, Applications and Resources:
Connect the complex number system to vectors, magnitude, and finding conjugates of square roots by
comparing and contrasting these topics.
Use a graphic organizer for all of the new vocabulary that will be introduced this chapter. Include definitions,
pictures, page reference numbers, an example, a place for side notes.
Student Misconceptions
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Multiply by the conjugate of the numerator to simplify quotients of complex numbers.
Critical Questions/Problems: Questions that define the standard and are representative of what students will need to
be able to do to show evidence of understanding.
Closed Problems or Questions:
Open Problems or Questions:
Divide, then write your answer in standard form:
3  4i
Write the conjugate:
16
3 1
16
 i b)
a)
16 4
3  4i
3 1
3  4i
 i d) 
c)
16 4
16
6  10i
.
2i
Evaluate | 7 – 3i |
a.
Evaluate: |4 + 3i|
58 b. 4 c. 2 d. -58
Divide, then write your answer in standard form.
4  i
1  4i
8
8
 i b.  i c. i d.
i
a. 
17
17
Divide, then write your answer in standard form.
3  2i
4  3i
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Domain:
The Complex Number
System
Cluster:
Represent complex numbers and
their operations on the complex plane
Class
Precalculus (Q3)
Standards
Notes
N.CN.4 Represent complex numbers on the complex plane in rectangular
There are an infinite number of options
for the polar form of a number given in
rectangular form.
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
Student Friendly Language: “I can…”
I can plot complex numbers in rectangular and polar form.
I can convert a point in rectangular form into point in polar form.
I can convert a point in polar form into a point in rectangular form.
Know
(factual)
Understand
(conceptual)
Complex numbers in rectangular form
are written a + bi, and plotted with the
real part, a, on the x-axis, and the
imaginary part, b, on the y-axis. The
point would have coordinates (a, b).
The polar form of a point is written
 r,   , where r is the
You can have an infinite number of
polar coordinates for a rectangular
point, simply by adding or
subtracting 360 / 2 from
.
Do
(procedural, application,
extended thinking)
To convert from polar to
rectangular complex points, use
x  r cos 
y  r sin 
To convert from rectangular to
polar complex points, use
 moduli of the
r 2  x2  y 2
number, and theta is the angle of the
number.
tan  
y
x
Instructional Strategies, Applications and Resources:
Use a graphing calculator that has apps that can do conversion between the two forms.
Meaningful student connections: explain why they use the variable “r” in polar form.
Teach using “big math ideas.”
Create flow chart for conversion of polar and rectangular points.
Graphic organizer for new vocabulary.
Student Misconceptions
Using the wrong r value, not understanding what a negative r value really means and which
positive or negative r values.

goes with the
They also don’t think of both forms as points rather than vectors, and sometimes draw a line from the origin to
the point.
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Critical Questions/Problems: Questions that define the standard and are representative of what students will need to
be able to do to show evidence of understanding.
Closed Problems or Questions:
In polar coordinates, which of the following is not a
correct representation for the point
 2,   6  b)  2, 11 6 
c)  2,  11  d)  2,  7 
6
6
 2, 5 6  ?
Open Problems or Questions:
Convert from rectangular to polar coordinates:

6, 2

a)
Convert from polar to rectangular coordinates:
8, 7 6 
Plot -3 + 4i in the complex plane.
 4 3,  4 b)  4, 4 3 
c)  4, 4 3  d)  4,  4 3 
a)
Calculate | 5 + 12i |.
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Domain:
Vector Quantities and
Matrices
Cluster:
Represent and model with vector
quantities.
Standards
Class
Precalculus (Q3)
Notes
N.VM.1 Recognize vector quantities as having both magnitude and
direction. Represent vector quantities by directed line segments, and
use appropriate symbols for vectors and their magnitudes (e.g., v, |v|,
||v||, v).
Student Friendly Language: “I can…”
I can find the magnitude of a vector.
I can find the direction of a vector.
I can draw a vector on the coordinate plane.
Know
(factual)
Magnitude (
Understand
(conceptual)
v ) of a vector, a, b , is
the length of the vector. The formula
for finding magnitude is
v  a 2  b2 .
Direction of a vector is found using the
formula tan  
Vectors are equivalent if their
direction and magnitude are equal,
regardless of their position on the
coordinate plane.
There are 2 parts to a vector:
direction and magnitude.
Do
(procedural, application,
extended thinking)
Determine the magnitude of a
vector.
Use a variety ways to represent a
function.
Compare two vectors to determine
if they are equivalent.
y
.
x
Instructional Strategies, Applications and Resources:
Student Misconceptions
Two vectors are equivalent if they share the same direction or magnitude, not both.
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Critical Questions/Problems: Questions that define the standard and are representative of what students will need to
be able to do to show evidence of understanding.
Closed Problems or Questions:
Find the direction of v : v = 7i – 2j
a. 344o b. 16o c. 164o d. 196o
Open Problems or Questions:
Find the direction of v: v =
1,  4 .
Convert the vector v into component form:
v = 4i + 2j
a.
2, 4 b. 4, 0 c. 0, 2
d.
4, 2
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Domain:
Vector Quantities and
Matrices
Cluster:
Represent and model with vector
quantities
Standards
Class
Precalculus (Q3)
Notes
N.VM.2. Find the components of a vector by subtracting the coordinates of
Head-minus tails rule.
an initial point from the coordinates of a terminal point.
Student Friendly Language: “I can…”
I can identify the head of a vector.
I can identify the tail of a vector.
I can use the head-minus-tail rule to find a vector in component form.
Know
(factual)
If the initial point of vector v
is
x1 , y1 and the terminal point
of v is
Understand
(conceptual)
The head of the vector is the end
with the arrowhead on it. The tail
does not have any marking on it.
Do
(procedural, application,
extended thinking)
Be able to determine if two vectors are
equivalent vectors by performing the
head-minus-tail rule.
x2 , y2 , then the
component form of
v  x2  x1 , y2  y1 .
Instructional Strategies, Applications and Resources:
2-Column Notes used to show equivalent vectors.
Student Misconceptions
Two vectors aren’t the equivalent unless they have the same initial and terminal point.
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Critical Questions/Problems: Questions that define the standard and are representative of what students will need to
be able to do to show evidence of understanding.
Closed Problems or Questions:
A vectorv has initial point (3, 7) and terminal point
(3, -2). Find its component form.
a.
0, 9 b. 9, 0
c.
0,  9
d.
Open Problems or Questions:
Find the component form of the vector below.
9, 0
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Domain:
Vector Quantities and
Matrices
Cluster:
Represent and model with vector
quantites.
Standards
Class
Precalculus
Notes
N.VM.3. Solve problems involving velocity and other quantities that can be
represented by vectors.
Student Friendly Language: “I can…”
I can determine the actual velocity of an airplane when factoring in the velocity of the wind.
I can find a resultant vector.
I can determine the speed and direction of any vector.
Know
(factual)
Understand
(conceptual)
The sum of 2 vectors is called the
resultant.
A vector is composed of speed and
direction. Speed is the magnitude
of the vector, and direction is the
direction angle of the vector when
in standard position.
The sum of two vectors represents
the effect of 2 or more forces acting
on an object.
The graph of the sum of 2 or more
vectors would represent the single
vector that starts at the origin and
ends at the head of the second
vector.
Do
(procedural, application,
extended thinking)
Determine the speed and compass
heading a pilot should set an
airplane at in order to arrive at the
correct destination when figuring in
the force of wind.
Determine the resultant force on an
object if 2 or more forces are
pushing or pulling on that object.
Instructional Strategies, Applications and Resources:
Use sequencing for multi-step problems.
KWL process.
Student Misconceptions
Compass headings and direction angles are not the same. The calculator gives the direction angle, and must be
converted into a compass heading.
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Standards Template
Critical Questions/Problems: Questions that define the standard and are representative of what students will need to
be able to do to show evidence of understanding.
Closed Problems or Questions:
Two forces, one of 120 pounds and the other 200
pounds, act on the same object at angles 30o and -30o
respectively, with the positive x-axis. Find the
direction of the resultant of these two forces.
a) 14o b) -8.2o c) 0o d) 18o
Given v =
5,  2 and w = 6,1 , find the angle
between v and w.
a) 58.7o b) 148.7o c) 31.3o d) 121.3o
Open Problems or Questions:
What force is required to keep a 2000-pound vehicle
from rolling down a ramp inclined at 30o from the
horizontal?
A storm front is moving east at 30.0 mph and north at
18.5 mph. Find the resultant velocity of the front.
An airliner’s navigator determines that the jet is flying
475 mph with a heading of 42.5o north of west, but the
jet is actually moving 465 mph in a direction 37.7o north
of west. What is the velocity of the wind?
Two forces, one of 150 pounds and the other of 200
pounds, act on the same object at angles of 20o and
-30o, respectively, with the positive x-axis. Find the
magnitude and direction of the resultant force.
Standards of Mathematical Practices:
1.
2.
3.
4.
Make sense of problems & persevere in solving them.
Reason abstractly & quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.