Pre-Calculus Unit H: Vectors
Math Florida Standards
Unit Overview
Precalculus students may have previous experience with vectors from Physical Science Honors: understanding what a vector is,
finding components of a vector, and solving problems involving velocity and other quantities that can be represented by vectors. In
this unit students will recognize vector quantities as having both magnitude and direction; represent vector quantities by directed
line segments; use appropriate symbols for vectors and their magnitudes; find the components of a vector by subtracting the
coordinates of an initial point from the coordinates of a terminal point; solve problems involving velocity and other quantities that
can be represented by vectors; add and subtract vectors; given two vectors in magnitude and direction form, determine the
magnitude and direction of their sum; and multiply a vector by a scalar.
When adding vectors students must add vectors end-to-end, component-wise, and by the parallelogram rule. When subtracting
vectors students must represent vector subtraction graphically, by connecting the tips in the appropriate order, and perform vector
subtraction component-wise. When multiplying a vector by a scalar students will represent scalar multiplication graphically by
scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise; compute the magnitude of a
scalar multiple cv using ||cv|| = |c|v.; and 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).
Students must understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes; 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.
Textbook Resources
Glencoe McGraw Hill Precalculus copyright 2011
Connect Ed McGraw Hill
Sections:
8.1, 8.2, 8.3
Mathematics Formative Assessment System Tasks
The system includes tasks or problems that teachers can
implement with their students, and rubrics that help the
teacher interpret students' responses. Teachers using MFAS
ask students to perform mathematical tasks, explain their
reasoning, and justify their solutions. Rubrics for interpreting
and evaluating student responses are included so that
teachers can differentiate instruction based on students'
strategies instead of relying solely on correct or incorrect
answers. The objective is to understand student thinking so
that teaching can be adapted to improve student achievement
of mathematical goals related to the standards. Like all
formative assessment, MFAS is a process rather than a test.
Research suggests that well-designed and implemented
formative assessment is an effective strategy for enhancing
student learning.
Content Standards
Standards for
Mathematical
Practice
MAFS.912.N-VM.1.1
MAFS.K12.MP.1.1
MAFS.912.N-VM.1.2
MAFS.K12.MP.2.1
MAFS.912.N-VM.1.3
MAFS.K12.MP.4.1
MAFS.912.N-VM.2.4
MAFS.K12.MP.6.1
MAFS.912.N-VM.2.5
MAFS.K12.MP.7.1
Other Resources
Vector Introduction (Despicable Me)
Vector Resource
Velocity and Acceleration
Airplane Resources
Dancing Vectors
Vector Lesson Ideas
Khan Academy Precalculus
Mathematics Formative Assessment System Tasks (MFAS)
Unit Scale (Multidimensional) (MDS)
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Pre-Calculus Unit H: Vectors
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these
standards. Examples:

3.0
Prove the rules for adding, subtracting, and multiplying vectors, in component form, to be true algebraically.
The Student will:
 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). (MAFS.912.N-VM.1.1)
 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. (MAFS.912.N-VM.1.2)

Solve problems involving velocity and other quantities that can be represented by vectors. (MAFS.912.N-VM.1.3)

Add and subtract vectors. (MAFS.912.N-VM.2.4)
a) 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.
b) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
c) 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 graphically by connecting the tips in the appropriate order, and perform vector subtraction
component-wise.
Multiply a vector by a scalar. (MAFS.912.N-VM.2.5)
a) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication
component-wise, e.g., as c <vx, vy> = <cvx, cvy>
b) 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).

2.0
The student will recognize or recall specific vocabulary, such as:
 Vector, magnitude, component form, initial point, terminal point, scalar, direction, components
The student will perform basic processes, such as:
 Use a protractor and ruler
 Use trigonometric and inverse trigonometric functions
1.0
With help, partial success at 2.0 content but not at score 3.0 content
This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015
Pre-Calculus Unit H: Vectors
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
Domain: Number & Quantity: Vector & Matrix Quantities
Cluster: Represent and model with vector quantities (Major Cluster)
Standard: Solve problems involving velocity and other quantities that can be represented by vectors. (MAFS.912.N-VM.1.3)
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Certain situations can be represented using vectors.
Know
Declarative knowledge: Facts, vocab., information




Vector arithmetic
Trigonometric functions used to write
vectors
Vector formulas
Write vector in component form
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Analysis
 Solve problems involving velocity and other quantities that can be represented by vectors
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Graphing points, Use a protractor, Right triangle trigonometry, Unit circle
This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015
Pre-Calculus Unit H: Vectors
Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
Common Core State Standard:
Solve problems involving velocity and other quantities that can be represented by vectors. (MAFS.912.N-VM.1.3)
Score
4.0
3.5
Learning Progression
I can…
 Solve problems involving acceleration and velocity represented by
vectors.
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
I can…
 Solve problems involving velocity and other quantities that can
be represented by vectors.
Sample Tasks
A car turns from a road into a parking lot and into an available parking space.
The car’s initial velocity in 4.0 m/s at a bearing of N45˚E. The car’s velocity just
before the driver decreases speed is 4.0 m/s N80˚E. The turn takes 3 seconds.
Calculate the average acceleration of the car during the turn.
(Hint: Acceleration is the change in velocity divided by the time interval.)
A quarterback running forward at 5 meters per second throws a football with a
velocity of 25 meters per second at an angle of 40˚ with the horizontal. What
is the resultant speed and direction of the task?
3.0
2.5
2.0
1.0
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can…
 Find the component form of a vector when given the magnitude and
direction angle.
 Find the direction angle when given a vector in component form.
Find the component form of a vector when given the magnitude and
direction angle.
v =12, q = 60˚
Find the direction angle when given a vector in component form.
<-5, 9>
I need prompting and/or support to complete 2.0 tasks.
Sample High Cognitive Demand Tasks:
This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015
Pre-Calculus Unit H: Vectors
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
MAFS Mathematical Content Standard(s)
Design Question 1; Element 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). (MAFS.912.N-VM.1.1)
MAFS Mathematical Practice(s)
Design Question 1; Element 1
Model with mathematics. (MAFS.K12.MP.4.1)
Marzano’s Taxonomy
Analysis
Teacher Notes
Students will need access to a computer.
Questions to develop mathematical thinking, possible
misconceptions/misunderstandings, how to
differentiate/scaffold instruction, anticipate student
problem solving strategies
1.
2.
3.
1) Using maps.google.ca find a lake for which you will measure the distance across.
2) Take a screen shot of the lake and enough surrounding area to create two connecting vectors to the opposite side of
the lake.
3) Open Paint and paste the screenshot. Print off the picture as well as save it to your H: Drive.
Task
4.
*These tasks can either be teacher created or
modified from a resource to promote higher 5.
order thinking skills. Please cite the source for
any tasks.
6.
7.
4) On the paper copy, draw two vectors for which the resultant vector will be a vector across the lake.
5) Determine the magnitude and direction of the resultant vector. You may not take any measurements across water. In
practicality, your survey equipment would be in one central location, therefore, you may only measure the angles where
the two vectors begin; all other angles must be calculated mathematically.
6) Create a scenario to represent the resultant vector.
7) Using PowerPoint or any other multimedia tool, illustrate your work. Show your screenshot, your vectors used, and all
work associated.
Adapted from Vector Lesson Ideas
This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015