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Advanced Topics Unit B: Trig Functions
Unit Overview
Math Florida Standards
Content Standards
Students will understand the construction of the unit circle, work with trigonometric functions graphically and algebraically
and apply trigonometric identities and properties.
Textbook Resources
Prentice Hall Geometry
Pearson Success Net
Section: 8-2
Glencoe McGraw-Hill Pre-calculus textbook
McGraw-Hill
Sections:
4-1, 4-2, 4-3, 4-4, 4-6, 5-1, 5-2, 5-3, 5-4, 5-5
Additional Lessons:
Deriving the Unit Circle from special right triangles
(F-TF 1.3)
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.
MAFS.912.F-TF.1.3
MAFS.912.F-TF.1.4
MAFS.912.F-TF.2.5
MAFS.912.F-TF.2.6
MAFS.912.F-TF-2.7
MAFS.912.F-TF.3.8
MAFS.912.F-TF.3.9
MAFS.912.F-IF.3.7
Standards for
Mathematical Practice
MAFS.K12.MP.2.1
MAFS.K12.MP.7.1
MAFS.K12.MP.8.1
Other Resources
As the Wheel Turns Task
Applications of Trigonometric Functions
Regents Prep
Mathematics Formative Assessment System Tasks
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Advanced Topics Unit B: Trig 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
2.0
Explore composition of trigonometric functions with polynomial, rational, exponential, and/or logarithmic functions.
The Student will:
 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 π–x, π+x, and 2π–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)
 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios. (MAFS.912.F-TF.3.8)
 Prove the addition and subtraction, half-angle, and double-angle formulas for sine, cosine, and tangent and use these formulas to solve problems.
(MAFS.912.F-TF.3.9)
 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
(MAFS.912.F-IF.3.7)
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and
amplitude, and using phase shift.
The student will recognize or recall specific vocabulary, such as:
 Unit circle, trigonometric function, sine, cosine, tangent, symmetry, midline, period, frequency, phase shift, end behavior, x- and y-intercepts, amplitude
The student will perform basic processes, such as:
 Identify key information on the unit circle
 Identify some features of trigonometric functions
 Use trigonometric properties and identities
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
Advanced Topics Unit B: Trig 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
Advanced Topics Unit B: Trig 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.
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
Advanced Topics Unit B: Trig 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
Advanced Topics Unit B: Trig 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.
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
Advanced Topics Unit B: Trig 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