Understanding By Design Unit Template
Title of Unit
Curriculum Area
Developed By
Introduction to Trigonometry
Algebra 3
Kristina Messina
Grade Level
Time Frame
10-12
2/25/13-3/25/13
Identify Desired Results (Stage 1)
Content Standards
F-TF.1- Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-TF.2- 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.
F-TF.3- (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /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.
F-TF.8- Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
Mathematical Practices
Construct viable arguments and critique the reasoning of others
Model with mathematics
Making sense of problems and persevere in solving them
Understandings
Essential Questions
Overarching Understanding
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Radian measure of angles can be used to model and solve real-life
problems.
Trigonometric functions are used to model real-life problems such as the
movement of an oscillating weight.
Right triangle trigonometry can be used to model and solve real-life
situations like finding the height of a building.
Trigonometric functions are important for modeling periodic behavior
such as business cycles, planetary orbits, and light rays.
Related Misconceptions
Reference angles- students will have trouble at first just using reference
angles to evaluate trig functions with the correct signs of the quadrant
the angle lies in. Students will have a hard time picking out the pattern
within the unit circle and how with each angle of the first quadrant that
are special the only thing that changes throughout each quadrant are the
sign changes.
Right triangles- how to apply it to real-world situations. Students will
struggle with converting the information about a situation into a picture
or a diagram of a right triangle.
Overarching
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How can radian measures
be used in the real-world?
How can trigonometric
functions be used in the
real-world?
How can right triangles
help us solve problems
we face in the real-world?
What are examples of
periodic behavior in the
real-world?
Topical
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What are some special
right triangles that we
know from geometry?
How can we use trig
identities?
How can we solve
trigonometric functions
with a right triangle?
What are the values of
the unit circle?
What are reference
angles? How can you
apply them?
Objectives
Skills
Knowledge
Students will know…
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Students will be able to…
Students will know right triangles and Trigonometric functions can be
used to model and solve real-life situations such as a height of a building,
or the bottom of a pool.
Students will know that reference angles can be used to evaluate and
solve any angle not just acute angles.
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Students will be able to evaluate trig functions using right
triangles and the unit circle.
Students will be able to covert between degrees and radian
measures and convert between Degrees, minutes, and
seconds to degrees.
Students will be able to solve real-world application problems
using trig functions, right angles and reference angles.
Students will be able to use trigonometric identities to verify
trig equations.
Students will be able to find co-terminal angles,
complementary and supplementary angles.
Assessment Evidence (Stage 2)
Performance Task Description
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
Goal
Role
Audience
Situation
Product/Performance
Standards
Other Evidence
Students will be able to evaluate trig functions using right triangles and the unit circle.
Students will be able to covert between degrees and radian measures and convert between Degrees, minutes,
and seconds to degrees.
 Students will be able to solve real-world application problems using trig functions, right angles and reference
angles.
 Students will be able to use trigonometric identities to verify trig equations.
 Students will be able to find co-terminal angles, complementary and supplementary angles.
It is test that covers the basic introduction of trigonometric functions (4.1-4.4)
I will be evaluating the test, looking on the completion, and the work students provide for each question. I will grade
the tests and really pick through students’ work to gauge students’ understanding after taking this summative
assessment to see where students ended after the test.
The summative assessment will be completed in class March 25, 2013.
It will be a completed test that will cover radian measures and angles, unit circle, trigonometric function, right triangle
trigonometry, and finding trig functions of any angle using reference angles and quadrant signs.
F-TF.1- Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-TF.2- 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.
F-TF.3- (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /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.
F-TF.8- Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of
the angle.
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Every lesson I will be asking assessing questions to see where students are in their understanding then use that information to determine if I
need to spend more time of certain concepts before moving on to the next concept.
Entry slips- I am going to use entry slips sometimes to see where my students are at conceptually wise with understanding the previous lesson
content. I will probably have a discussion about the entry slips so that I can get immediate feedback on what they know and build from there.
Exit slips- sometimes I will use exit slips to see what the students learned in the lesson, and use that information to address misconceptions the
next day.
Learning Plan (Stage 3)
Day in Unit
Lesson
Topic
Lesson Learning Objective
Description of how lesson
contributes to unit-level
objectives
Assessment activities
(1) February 25,
2013
4.1 Day 1
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To describe angles
To use radian measure
To use degree measure
To use angles to model and
solve real-life problems
This lesson is the basic foundation of
this unit and of graphing and
analyzing trigonometric functions unit.
It contributes to these objectives:
 Students will know radian
measures and angles
especially linear speed and
angular speed can be used in
real-life situations.
 Students will be able to covert
between degrees and radian
measures and convert
between Degrees, minutes,
and seconds to degrees.
 Students will be able to find
co-terminal angles,
complementary and
supplementary angles.
Assessing questions throughout the
lesson
Essential question:
 How can radian measures be
used in the real-world?
 How can we convert between
radians and degrees?
 What are co-terminal angles?
(2) February 26,
2013
4.1 Day 2
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To describe angles
To use radian measure
To use degree measure
To use angles to model and
solve real-life problems
This lesson is a continuation of the
foundation concepts of trigonometric
functions. This lesson also contributes
to the same objectives as before.
(3) February 27,
2013
4.2 Day 1
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To identify a unit circle and
its relationship to real
This lesson contributes to these
following objectives and overarching
My Favorite No- a real quick warm-up
On co-terminal angles.
Due to previous lesson will spend
some time talking about the definition
of co-terminal angles and what that
means. Students were very confused
at the end of the pervious lesson and
misconceptions on co-terminals need
to be cleared up before moving on.
Assessing questions
Essential questions:
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(4) February 28,
2013
4.2 Day 2
(5) March 1, 2013
Review
4.1-4.2
(6) March 4, 2013
Review
4.1-4.2
numbers
To evaluate trigonometric
functions using the unit circle
To use the domain and
period to evaluate sine and
cosine function
To use a calculator to
evaluate trigonometric
functions

To identify a unit circle and
its relationship to real
numbers
 To evaluate trigonometric
functions using the unit circle
 To use the domain and
period to evaluate sine and
cosine function
 To use a calculator to
evaluate trigonometric
functions
To review material on radian and
degree measure and the unit circle
sections.
To review material on radian and
degree measure and the unit circle
sections.

understanding:
 Trigonometric functions are
used to model the movement
of an oscillating weight.
 Students will know right
triangles and Trigonometric
functions can be used to
model and solve real-life
situations such as a height of
a building, or the bottom of a
pool.
 Students will be able to
evaluate trig functions using
right triangles and the unit
circle.
This lesson is a continuation of the
previous lesson and contributes to the
same objectives as the above lesson.
Each lesson builds off each other and
deepens their understanding of basic
trigonometric functions.
What are the values of the
unit circle?
Having students reflect on what they
remember about unit circles.
This lesson is to have students assess
what they understand and what they
still need to review before the quiz.
My Favorite No- based off the concept
the majority of the class wants to
review.
Talk as a class, how effort will affect
their achievement in this class, and
what are strategies to improve their
effort and achievement in this class.
How will these strategies shift to
college and the work place?
This lesson is to have students assess
what they understand and what they
still need to review before the quiz.
Unit circle quiz
Bingo game
Assessing questions
Essential questions:
 What are the values of the
unit circle?
Exit slip self-assessment- write down
a concept you still don’t understand,
what you are going to do to
understand the concept, and rate
your effort in this class and explain
your reasoning.
(7) March 6, 2013
Quiz 4.14.2
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To have students evaluate
what material they
understand and what
material they still don’t
understand.
To evaluate what students still don’t
understand.
Quiz 4.1-4.2
Students correct their own quizzes
then for homework go over the
problems they missed to figure out
their mistakes.
(8) March 8, 2013
4.3 Day 1
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To evaluate trigonometric
functions of acute angles
To use the fundamental
trigonometric identities
To use a calculator to
evaluate trigonometric
functions
To use trigonometric
functions to model and solve
real-life problems
This lesson contributes to this
objectives and incorporates geometry
with trigonometry:
 Students will know right
triangles and Trigonometric
functions can be used to
model and solve real-life
situations such as a height of
a building, or the bottom of a
pool.
 Students will be able to solve
real-world application
problems using trig functions,
right angles and reference
angles.
 Students will be able to use
trigonometric identities to
verify trig equations.
 Students will be able to
evaluate trig functions using
right triangles and the unit
circle.
 Right triangle trigonometry
can be used to model and
solve real-life problems like
the height of a building.
This lesson contributes to the same
objectives as above and is a
continuation of the previous lesson.
Assessing questions
Essential questions:
 How can right triangles help
us solve problems we face in
the real-world?
 How can trigonometric
functions be used in the realworld?
 How can we use trig
identities?
 How can we solve
trigonometric functions with a
right triangle?
 What are some special right
triangles that we know from
geometry?
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
(9) March 11,
2013
4.3 Day 2
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To evaluate trigonometric
functions of acute angles
To use the fundamental
trigonometric identities
To use a calculator to
evaluate trigonometric
functions
To use trigonometric
Assessing questions
Essential questions:
 How can right triangles help
us solve problems we face in
the real-world?
 How can trigonometric
functions be used in the realworld?

functions to model and solve
real-life problems
(10) March 12,
2013
4.3 Day 3
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
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To evaluate trigonometric
functions of acute angles
To use the fundamental
trigonometric identities
To use a calculator to
evaluate trigonometric
functions
To use trigonometric
functions to model and solve
real-life problems
How can we use trig
identities?
 How can we solve
trigonometric functions with a
right triangle?
 What are some special right
triangles that we know from
geometry?
Entrance slip- verifying trigonometric
equation using trigonometric identities
This lesson contributes to the same
objectives as above and is a
continuation of the previous lesson.
Assessing questions
Essential questions:
 How can right triangles help
us solve problems we face in
the real-world?
 How can trigonometric
functions be used in the realworld?
 How can we use trig
identities?
 How can we solve
trigonometric functions with a
right triangle?
 What are some special right
triangles that we know from
geometry?
My favorite No- right triangle
application problem
(11) March 13,
2013
4.4 Day 1
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To evaluate trigonometric
functions of any angle
To use reference angles to
evaluate trigonometric
functions
To evaluate trigonometric
functions of real numbers
This lesson contributes to the
following objectives and concludes
the last section of the unit:
 Trigonometric functions are
important for modeling
periodic behavior such as
business cycles, planetary
orbits, and light rays.
 Students will be able to solve
real-world application
Activity where students find the
reference angle
Assessing questions
Essential questions:
 What is periodic behavior in
the real-world?
 What are reference angles?
How can you apply them?
(12) March 14,
2013
4.4 Day 2
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(13) March 15,
2013
4.4 Day 3
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To evaluate trigonometric
functions of any angle
To use reference angles to
evaluate trigonometric
functions
To evaluate trigonometric
functions of real numbers
To evaluate trigonometric
functions of any angle
To use reference angles to
evaluate trigonometric
functions
To evaluate trigonometric
functions of real numbers
problems using trig functions,
right angles and reference
angles.
 Students will know that
reference angles can be used
to evaluate and solve any
angle not just acute angles.
This lesson contributes to the same
objectives as above and is a
continuation of the previous lesson.
This lesson contributes to the same
objectives as above and is a
continuation of the previous lesson.
This concludes all the material
needed to master to cover all the
objectives of the unit.
Assessing questions
Essential questions:
 What is periodic behavior in
the real-world?
 What are reference angles?
How can you apply them?
Exit slip- What concepts do you still
don’t understand?
Assessing questions
Essential questions:
 What is periodic behavior in
the real-world?
 What are reference angles?
How can you apply them?
Connecting 4.1-4.4 togetherClaim-Support-Question
Prompt: We have been studying
trigonometry in a variety of
contexts: radian and degree
measures, Trigonometric functions
and the Unit Circle, Trigonometric
Functions and Right Angles, and
Trigonometric functions of any
angle. How are these ideas
connected?
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Claim- What connections can
you make among the
trigonometry topics we have
been studying?
Support-What is your
evidence to support these
claims? Can you give
examples?
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(14) March 18,
2013
Review 4.34.4
(15) March 19,
2013
Quiz 4.34.4
(16) March 20,
2013
(17) March 21,
2013
Question- What questions do
you still have regarding this
connection?
To review the material on right
triangle trig and trig functions of any
angle sections.
To have students evaluate what
material they understand and what
material they still don’t understand.
This lesson is to have students assess
what they understand and what they
still need to review before the quiz.
To evaluate what students still don’t
understand.
Pairs- students worked on verifying
trigonometric equations.
Assessing questions
Quiz 4.3-4.4
Half-day
No algebra 3 classes that day
N/A
N/A
Review 4.14.4
To review the material of the unit
This lesson is to have students assess
what they understand and what they
still need to review before the test.
(18) March 22,
2013
Review 4.14.4
To review the material of the unit
(19) March 25,
2013
Test 4.14.4
To test on the material of the unit
This lesson is to have students assess
what they understand and what they
still need to review before the test.
This is to test if students have
achieved the unit objectives.
Pairs-students worked on verifying
trigonometric equations with
trigonometric identities, arc length,
converting between radians and
degrees, and converting between
degrees minutes seconds and degree
decimals.
Worksheet
Assessing questions
Students providing their solutions to
the review worksheet
Graffiti review
4.1-4.4 Test