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Pre-Calculus Unit E: Trigonometric Functions
Unit Overview
Precalculus students have previous experience with trigonometric functions from Algebra 2: general understanding of what a
radian is and the unit circle. In this unit students will use special triangles to determine geometrically the values of trigonometric
functions; use the unit circle to explain symmetry and periodicity of trigonometric functions; graph trigonometric functions;
choose trigonometric functions to model periodic phenomena; and use inverse functions to solve trigonometric equations that
arise in modeling contexts.
When using inverse functions to solve trigonometric equations students will need to evaluate the solution(s) by hand and/or
using technology and interpret the solution(s) in terms of the context.
Students must understand what trigonometric functions and their inverses are and how to use them to model and/ or solve
application problems and that restricting a trigonometric function to a domain on which it is always increasing or always
decreasing allows its inverse to be constructed.
Math Florida Standards
Content Standards
Standards for
Mathematical Practice
MAFS.912.F-TF.1.1
MAFS.K12.MP.2.1
MAFS.912.F-TF.1.2
MAFS.K12.MP.4.1
MAFS.912.F-TF.1.3
MAFS.K12.MP.7.1
MAFS.912.F-TF.1.4
MAFS.912.F-TF.2.5
MAFS.912.F-TF.2.6
Textbook Resources
Glencoe McGraw Hill Precalculus copyright 2011
Connect Ed McGraw Hill
Sections:
4.2, 4.3, 4.4, 4.5 (Graphing Tangent Only), 4.6
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.
MAFS.912.F-TF.2.7
Other Resources
Applications of Trigonometric Functions
Deriving the Unit Circle from Special Right
Triangles
Regents Prep
As the Wheel Turns Task
Khan Academy Precalculus
Mathematics Formative Assessment System Tasks (MFAS)
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Pasco County Schools, 2014-2015
Pre-Calculus Unit E: Trigonometric Functions
Unit Scale (Multidimensional) (MDS)
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
Within this unit the standards require in-depth demonstration of understanding and application related to trigonometric functions, their graphs, and
inverses. This being stated, to move to a 4 the teacher may provide more difficult problems related to the later standards in this unit.
The Student will:
 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle; Convert between degrees and radians.
(MAFS.912.F-TF.1.1)

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures
of angles traversed counterclockwise around the unit circle. (MAFS.912.F-TF.1.2)


Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of
sine, cosine, and tangent for x, +x, and 2 x in terms of their values for x, where x is any real number. (MAFS.912.F-TF.1.3)
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. (MAFS.912.F-TF.1.4)

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5)

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be
constructed. (MAFS.912.F-TF.2.6)

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in
terms of the context. (MAFS.912.F-TF.2.7)
2.0
The student will recognize or recall specific vocabulary, such as:
 Trigonometric function, inverse trigonometric functions, unit circle, coterminal angles, degrees, radians, period, amplitude, phase shift, frequency,
midline, odd and even functions
The student will perform basic processes, such as:
 Properties of special right triangles
 Graph an input and output
 Right triangle trigonometry
 Symmetry
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 E: Trigonometric Functions
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: Functions: Trigonometric Functions
Cluster: Model periodic phenomena with trigonometric functions (Major Cluster)
Standard: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5)
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Real world situations can be modeled using trigonometric functions.
Know
Declarative knowledge: Facts, vocab., information


Sine, cosine, tangent, amplitude, frequency,
midline, period, phase shift
Given multiple graphs and/or practical
application problems, determine what
trigonometric function would best fit the
information
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Knowledge Utilization
 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Recognize parent trigonometric function visually, Graph parent trigonometric functions
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 E: Trigonometric Functions
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:
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5)
Score
Learning Progression
I can…
 Write the equation of an appropriate trigonometric function from a
graph and/or data given
Sample Tasks
A pet store clerk noticed that the population in the gerbil habitat varied
sinusoidally with respect to time, in days. He carefully collected data and
graphed his resulting equation. From the graph, determine the amplitude,
period, horizontal shift and vertical shift. Write the equation of the graph.
4.0
3.5
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
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 E: Trigonometric Functions
I can…
 Choose trigonometric functions to model periodic phenomena with
specified amplitude, frequency, and midline
A pet store clerk noticed that the population in the gerbil habitat varied
sinusoidally with respect to time, in days. He carefully collected data and
graphed his resulting equation. From the graph, determine the amplitude,
period, horizontal shift and vertical shift. Which trigonometric function would
best model the graph? Defend your answer.
3.0
2.5
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can…
 State key components of a trigonometric graph
o Amplitude, frequency, midline, phase shift
A pet store clerk noticed that the population in the gerbil habitat varied
sinusoidally with respect to time, in days. He carefully collected data and
graphed his resulting equation. From the graph, determine the amplitude,
period, horizontal shift and vertical shift.
2.0
1.0
I need prompting and/or support to complete 2.0 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 E: Trigonometric Functions
Sample High Cognitive Demand Tasks:
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)
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (MAFS.912.F-TF.2.5)
Design Question 1; Element 1
(MAFS.K12.MP.2.1) Reason abstractly and quantitatively.
MAFS Mathematical Practice(s)
Design Question 1; Element 1
Marzano’s Taxonomy
(MAFS.K12.MP.3.1) Construct viable arguments and critique the reasoning of others.
(MAFS.K12.MP.4.1) Model with mathematics.
(4) Knowledge Utilization
Teacher Notes
Questions to develop mathematical
thinking, possible
misconceptions/misunderstandings
, how to differentiate/scaffold
instruction, anticipate student
problem solving strategies
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 E: Trigonometric Functions
Task
*These tasks can either be teacher
created or modified from a resource
to promote higher order thinking
skills. Please cite the source for any
tasks.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015