Accelerated GSE Pre-Calculus
Unit Eight Information
Curriculum Map: Vectors
Concept 1: Perform arithmetic operations with complex numbers
Concept 2: Represent and model with vector quantities
Concept 3: Perform operations on vectors
Content from Frameworks: Vectors
Unit Length: Approximately 15 days
20152016
TCSS – Accelerated GSE Pre-Calculus – Unit 8
Curriculum Map
Big Idea / Unit
Students will use geometric interpretations to model problems.
Unit Essential Questions:
How are vectors used to
model problems?
Prerequisites: As identified by the GSE Frameworks
Length of Unit
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Use the Pythagorean Theorem.
Solve right triangles using trigonometry and inverse trigonometry.
Use reference angles to find the measure of an angle in standard position.
Apply transformations (translations, rotations, reflections) to objects on a coordinate grid.
Use the triangle inequality: The sum of the lengths of any two sides of a triangle is greater than the length
of the third side.
Use a ruler and protractor to measure lengths and angles.
Multiply matrices.
Perform operations on complex numbers—add, subtract, multiply, divide, and conjugate.
Recognize multiplication as repeated addition and exponentiation as repeated multiplication.
Use fractional exponents to write radicals.
Find distance and midpoints in the plane.
15 Days
Concept 1
Concept 2
Concept 3
Perform arithmetic operations with
complex numbers
Represent and model with vector
quantities
Perform operations on vectors
GSE Standards
MGSE9-12.N.CN.3
Find the conjugate of a complex number; use
conjugates to find moduli and quotients of
complex numbers.
MGSE9-12.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.
MGSE9-12.N.CN.5
Represent addition, subtraction,
multiplication, and conjugation of complex
numbers geometrically on the complex plane;
use properties of this representation for
TCSS
GSE Standards
MGSE9-12.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).
MGSE9-12.N.VM.2
Find the components of a vector by subtracting the
coordinates of an initial point from the coordinates
of a terminal point.
MGSE9-12.N.VM.3
Solve problems involving velocity and other
quantities that can be represented by vectors.
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MGSE9-12.N.VM.4
Add and subtract vectors.
MGSE9-12.N.VM.4a
Add vectors end-to-end, component-wise, and by
the parallelogram rule. Understand that the
magnitude of a sum of two vectors is typically
not the sum of the magnitudes.
MGSE9-12.N.VM.4b
Given two vectors in magnitude and direction
form, determine the magnitude and direction of
their sum.
MGSE9-12.N.VM.4c
Understand vector subtraction v – w as v + (–w),
where (–w) is the additive inverse of w, with the
same magnitude as w and pointing in the
opposite direction. Represent vector subtraction
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TCSS – Accelerated GSE Pre-Calculus – Unit 8
computation.
MGSE9-12.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.
graphically by connecting the tips in the
appropriate order, and perform vector subtraction
component-wise.
MGSE9-12.N.VM.5
Multiply a vector by a scalar.
MGSE9-12.N.VM.5a
Represent scalar multiplication graphically by
scaling vectors and possibly reversing their
direction; perform scalar multiplication
component-wise, e.g., as c(vx, vy) = (c⋅vx, c⋅vy).
MGSE9-12.N.VM.5b
Compute the magnitude of a scalar multiple cv
using ||cv|| = |c|v . Compute the direction of cv
knowing that when |c|v ≠ 0, the direction of cv is
either along v (for c > 0) or against v (for c < 0).
MGSE9-12.N.VM.11
Multiply a vector (regarded as a matrix with one
column) by a matrix of suitable dimensions to
produce another vector. Work with matrices as
transformations of vectors.
Lesson Essential Question
 How can I represent complex numbers
graphically?
• How does the complex plane show
addition, subtraction, multiplication, and
conjugation of complex numbers?
• What are two ways to represent a
complex number, and what are the
advantages of each form?
• How are operations on real numbers
represented in the complex plane?
• When given two points on the complex
plane, what does it mean to find the
distance between them and the midpoint of
the segment connecting them?
TCSS
Lesson Essential Question
 How are vectors and scalars similar and
different?
• How can I use vector operations to model,
solve, and interpret real-world problems?
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• How can I represent addition, subtraction,
and scalar multiplication of vectors
geometrically?
• How do geometric interpretations of addition,
subtraction, and scalar multiplication of
vectors help me perform computations
efficiently?
• What are some different ways to add two
vectors, and how are these representations
related?
• In what ways can matrices transform vectors?
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TCSS – Accelerated GSE Pre-Calculus – Unit 8
Vocabulary
Complex number
Complex plane
Rectangular form of a complex number
Cis Ѳ
Polar form of a complex number
Complex conjugate of z
Modulus of a complex number
Argument of z
Re(z)
Im(z)
Resources – Concept 1
 Complex numbers – refresher
(power point)
Vocabulary
Vocabulary
Vector
Scalar
Initial point
Terminal point
Magnitude of a vector
Components of a vector
Parallel vectors
Equivalent vectors
Zero vector
Resultant vector
Tail to Head representation
Parallelogram representation/rule
Velocity
Resources – Concept 2
 Vectors in the City
These tasks were taken from the
GSE Frameworks.
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 Complex Operations
 How far and Halfway (graphing)
Differentiated Activities
Concept 1
Differentiated Activities
Concept 2
 Graphic Organizer (It’s Not that
Complex)
 Walking and Flying around Hogsmeade
TCSS
Resources – Concept 3
Teacher
7/30/2015
These tasks were taken from the
GSE Frameworks.
A Delicate Operation
Putting Vectors to Use
Differentiated Activities
Concept 3
 Graphic Organizer Vectors
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TCSS – Accelerated GSE Pre-Calculus – Unit 8
Unit 7 Checklist – Vectors
Good luck to ________________________Date_______ Period___
Keep this list handy and refer to it periodically to see how you are doing. If you know how to each of these you should do well on
an exam.
Unit 7 – Vectors
In this unit I :
sort of
really
can convert vectors between component form and magnitude-direction form.
can convert complex numbers between rectangular and polar coordinates.
can use vectors to model and solve problems involving quantities that have both
magnitude and direction.
can convert between algebraic expressions and verbal descriptions / geometric
representations of operations on vectors (add, subtract, multiply by a scalar, multiply
by a transformation matrix) and algebraic equations or expressions.
can convert between algebraic expressions and verbal descriptions / geometric
representations of operations on complex numbers (add, subtract, multiply, divide,
conjugate, average).
can explain the effect of various operations on the magnitude / direction of vectors
and on the modulus / argument of complex numbers.
can use a variety of methods (tail-to-end, parallelogram, component-wise) to perform
operations on vectors and explain the relationships among these methods.
TCSS
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