Download Unit 5 - Long Beach Unified School District

Unit 5 Trigonometry
Algebra 2
Unit Goals – Stage 1
Number of Days: 24 days
3/20/17 – 4/28/17
Unit Description: This year’s last additions to the function families are the trigonometric functions. Students worked with trigonometric ratios and circles
in Geometry and were introduced briefly to radian measure. Developing their understanding of trigonometric functions, students will define radian
measure and the six trigonometric functions in terms of the unit circle. The concept of a periodic function is developed as students graph sine and cosine
by plotting functional values for benchmark angles. The graphs of the remaining four trigonometric functions are deduced from the students’ knowledge
of sine and cosine. Graphs are again transformed beyond the parent functions. Students will complete Unit 5 with an introduction to the trigonometric
identities.
Materials: Graphing calculators, Desmos
Standards for Mathematical
Transfer Goals
Practice
Students will be able to independently use their learning to…
SMP 1
Make sense of problems
• Make sense of never-before-seen problems and persevere in solving them.
and persevere in solving
• Construct viable arguments and critique the reasoning of others.
them.
SMP 2
Reason abstractly and
Making Meaning
quantitatively.
UNDERSTANDINGS
ESSENTIAL QUESTIONS
SMP 3
Construct viable
Students will understand that…
Students will keep considering…
arguments and critique the • The radian measure of an angle is the length of the arc on the unit circle • In what fields, careers or events
reasoning of others.
subtended by the angle.
is the concept of a periodic
SMP 4
Model with mathematics.
function useful?
• The unit circle in the coordinate plane enables the extension of
SMP 5
Use appropriate tools
trigonometric functions to all real numbers, interpreted as radian
• Why is triangle similarity crucial
strategically.
measures of angles traversed counterclockwise around the unit circle.
for defining the trigonometric
SMP 6
Attend to precision.
functions?
• For general angles, the sine and cosine functions can be viewed at the
SMP 7
Look for and make use of
y- and x-coordinates of points on circles.
• What is the effect on the function
structure.
when parameters are included in
• For angles between 0 and 90 degrees, the trigonometric functions are
SMP 8
Look for and express
the function, such as
well defined because the ratios of side lengths are equivalent in similar
regularity in repeated
y = a sin b(x – k) + h?
triangles.
reasoning.
Standards for Mathematical
Acquisition
Content Clusters Addressed
KNOWLEDGE
SKILLS
[s] A-CED.A Create equations that
Students
will
know…
Students will be skilled at and/or be able to…
describe numbers or
•
The
three
trigonometric
ratios
and
their
• Graph trigonometric functions, showing period, midline
relationships.
reciprocals.
and amplitude.
[s] F-IF.C
Analyze functions
•
The
trigonometric
values
for
special
angles.
•
Write a function that describes a relationship between
using different
two quantities.
•
2π
radians
=
360°.
representations.
•
Determine an explicit expression, a recursive process, or
•
The
Unit
Circle
and
the
characteristics
of
its
[m] F-BF.A Build a function that
steps for calculation from a context.
four
quadrants.
models a relationship
•
Identify the effect on the graph when f(x) is replaced by
•
The
characteristics
of
the
graphs
of
the
six
between two
f(x) + k, k f(x), f(kx), or f(x + k) for specific values of k
trigonometric
functions.
quantities.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
1
Posted 3/15/17
Unit 5 Trigonometry
Algebra 2
Unit Goals – Stage 1
[a] F-BF.B
[a] F-TF.A
[a] F-TF.B
[a] F-TF.C
Build new functions
from existing
functions.
Extend the domain of
trigonometric
functions using the
unit circle.
Model periodic
phenomena with
trigonometric
functions.
Prove and apply
trigonometric
identities.
• The effect on the graph of each of the
parameters in the equations:
y = a sin b (x – k) + h, y = a cos b (x – k) + h,
y = a tan b (x – k) + h, y = a csc b (x – k) + h,
y = a sec b (x – k) + h, and y = a cot b(x – k) +
h.
• The Pythagorean Identities:
sin2 θ + cos2 θ = 1, 1 + tan2 θ = sec2 θ, and
1 + cot2 θ = csc2 θ.
• The Cofunction Identities.
• The Negative Angle Identities.
• Vertical asymptotes of the tangent, cotangent,
secant and cosecant functions.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
2
both positive and negative; find the value of k given the
graph.
• Graph all 6 basic trigonometric functions and identify the
key features of the graphs.
• Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and
midline.
• 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.
Posted 3/15/17
Unit 5 Trigonometry
Algebra 2
Assessed Grade Level Standards
Standards for Mathematical Practice
SMP 1
Make sense of problems and persevere in solving them.
SMP 2
Reason abstractly and quantitatively.
SMP 3
Construct viable arguments and critique the reasoning of others.
SMP 4
Model with mathematics.
SMP 5
Use appropriate tools strategically.
SMP 6
Attend to precision.
SMP 7
Look for and make use of structure.
SMP 8
Look for and express regularity in repeated reasoning.
Standards for Mathematical Content
[s] A-CED.A Create equations that describe numbers or relationships [Equations using all available types of expressions, including simple
root functions].
A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels
and scales.
[s] F-IF.C
Analyze functions using different representations.
F-IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period,
midline, and amplitude.
[m] F-BF.A
Build a function that models a relationship between two quantities.
F-BF.1
Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
[a] F-BF.B
Build new functions from existing functions.
F-BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative);
find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Include recognizing even and odd functions from their graphs and algebraic expressions for them.
[a] F-TF.A
Extend the domain of trigonometric functions using the unit circle.
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.2.1 Graph all 6 basic trigonometric functions. CA
[a] F-TF.B
Model periodic phenomena with trigonometric functions.
F-TF.5
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
[a] F-TF.C
Prove and apply trigonometric identities.
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.
Key:
[m] = major clusters; [s] = supporting clusters, [a] = additional clusters

Indicates a modeling standard linking mathematics to everyday life, work, and decision-making
CA Indicates a California-only standard
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
3
Posted 3/15/17
Unit 5 Trigonometry
Algebra 2
Evidence of Learning – Stage 2
Assessment Evidence
Unit Assessment
Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.
Concepts and skills that may be assessed in Claim1:
[s]
•
•
[s]
•
[m]
•
•
[a]
•
•
[a]
•
•
•
[a]
•
[a]
•
A-CED.A
Students will create equations in two or more variables to represent relationships between quantities.
Students will graph equations on coordinate axes with labels and scales.
F-IF.C
Students will graph trigonometric functions, showing period, midline and amplitude.
F-BF.A
Students will write a function that describes a relationship between two quantities.
Students will determine an explicit expression or steps for calculation from a context.
F-BF.B
Students will identify the effect on the graph when f(x) is replaced by f(x) + k, k f(x), f(kx), or f(x + k) for specific values of k both positive and
negative;
Students will find the value of k when given the graph of a trigonometric function.
F-TF.A
Students will understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Students will 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.
Students will graph all 6 basic trigonometric functions.
F-TF.B
Students will choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F-TF.C
Students will 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.
Claim 2: Students can solve a range of wellposed problems in pure and applied
mathematics, making productive use of
knowledge and problem-solving strategies.
Standard clusters that may be assessed in
Claim 2:
•
A-CED.A
•
F-IF.C
•
F-BF.A
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Claim 3: The student can clearly and precisely
construct viable arguments to support their own
reasoning and critique the reasoning of others.
Standard clusters that may be assessed in
Claim 3:
•
F-IF.C
•
F-BF.B
•
F-TF.A
•
F-TF.C
4
Claim 4: The student can analyze complex,
real-world scenarios and can construct and use
mathematical models to interpret and solve
problems.
Standard clusters that may be assessed in
Claim 4:
•
A-CED.A
•
F-IF.C
•
F-BF.A
•
F-TF.B
Posted 3/15/17
Unit 5 Trigonometry
Algebra 2
Evidence of Learning – Stage 2
Other Evidence
Formative Assessment Opportunities
• Opening Tasks
• Informal teacher observations
• Checking for understanding using active participation strategies
• Exit slips/Summaries
• Modeling Lessons (SMP 4)
•
•
•
•
•
Tasks
Formative Assessment Lessons (FAL)
Quizzes/Chapter Tests
Big Ideas Math Performance Tasks
SBAC Interim Assessment Blocks
Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
5
Posted 3/15/17
Unit 5 Trigonometry
Algebra 2
Learning Plan – Stage 3
Days
1-2
days
Learning
Target
I will use my
knowledge of
triangles to
investigate the
trigonometric
ratios in the
Opening Task.
I will explore the
characteristics
of the six
trigonometric
functions by…
6-7
days
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Algebra 2
Expectations
(Activities and
Lessons)
OPENING TASK – Are Relationships Predictable?
In this opening task, students will explore the predictable nature of
the trigonometric ratios. Students should see that, no matter the
size of the triangle, the sine ratio for a 60° angle remains constant,
within reason given the measuring inaccuracies on the part of the
students.
A discussion of similar triangles should follow the Opening Task to
further student understanding of the power of trigonometry.
• Reviewing the sine, cosine and tangent ratios.
• Defining the secant, cosecant and cotangent ratios.
• Defining the independent variable in the trigonometric functions
as the acute angle of a right triangle and the dependent
variable being the trigonometric value of that angle.
• Finding unknown side length and angle measures of right
triangles.
• Finding coterminal angles.
• Converting between degree and radian measure.
• Using radian measure to solve for arc length and sector area.
• Using reference angles to evaluate a trigonometric function.
• Using trigonometric functions to solve real-life problems.
• Answering questions such as:
o What is the difference between exact and approximate
values?
4
mean?
7
o
What does sin θ =
o
How do you know which side is the “opposite” side and
which is the “adjacent” side?
How can you convert 30° to radians without using a
formula?
Do sine, cosine and tangent of an angle always have the
same sign?
Given tan θ = 3 , does θ = 60°?
o
o
o
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
6
• Section 9.1
• RealLife STEM
Video:
Parasailing to
Great Heights
• Section 9.2
• Section 9.3
Curriculum Intranet
•
Are Relationships
Predictable?
Conceptual Understanding:
• Right Triangle Trig Ratios:
George W. Ferris’ Day Off
• Radian Measure: Diggin’ It
• Side Lengths: Water
Wheels and the Unit
Circle
• Getting Familiar with the Unit
Circle
Procedural Skills and Fluency:
• Exploring Trig Ratios –
Use it
• MathOpenRef: Sine Function
• MathOpenRef: Coterminal
Angles
• MathOpenRef: Radian
• Illuminations: Pi Fight Game
• Finding the Value of a
Relationship: Inverse
Functions and Angles of
Elevation and Depression
• Blank Unit Circle
• MathematicsVisionProject:
Relationships with Meaning
Posted 3/15/17
Unit 5 Trigonometry
Algebra 2
Learning Plan – Stage 3
Days
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Algebra 2
Expectations
(Activities and
Lessons)
Learning
Target
Curriculum Intranet
•
MathematicsVisionProject:
Finding the Value of a
Relationship
Applications:
• STEM Video Performance
Task: Parasailing to Great
Heights
I will use
trigonometric
functions to
model real-world
situations by…
•
•
•
•
•
•
4-5
days
I will work with
trigonometric
identities by…
3-4
days
•
•
•
•
Exploring the characteristics of the sine and cosine functions.
Transforming sine and cosine functions.
Using the graphs of the sine and cosine functions to generate
the graphs of tangent, cosecant, secant, and cotangent.
Using and interpreting frequency.
Using trigonometric functions to solve real-life problems.
Answering questions such as:
o What are the characteristics of the six trigonometric
functions?
o How are the transformations of the trigonometric functions
like the transformations of previous functions?
o How do parameters affect the function
y = a sin b(x – k) + h?
o Why don’t the graphs of the tangent, cotangent, secant and
cosecant functions have amplitude?
o What are the characteristics of real-life problems that can
be modeled by trigonometric functions?
o Sinusoidal graphs can be modeled with either sine or
cosine functions. How is that possible?
• Section 9.4
• Section 9.5
• Section 9.6
Deriving trigonometric identities from definitions of the
trigonometric functions.
Simplifying trigonometric identities.
Verifying trigonometric identities.
Answering questions such as:
o Is there a process to verify a trigonometric identity?
o What is the difference between an equation and an
identity?
• Section 9.7
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
7
Conceptual Understanding:
• Unit Circle Graphs of Sine
and Cosine Applet
• Geogebra: Explore the
Trigonometric Functions and
Their Graphs
• Which One Doesn’t Belong:
Trigonometric Graphs
Procedural Skills and Fluency:
• Desmos – Cosine Curve:
Period and Amplitude
• Desmos - Cosine Curve:
Phase Shifts
Applications:
• Modeling: High Tide
• MathematicsVisionProject:
Solving Right Triangles Using
Trigonometric Relationships
Procedural Skills and Fluency:
• Quizlet Dynamic Matching
Game
Posted 3/15/17
Unit 5 Trigonometry
Algebra 2
Learning Plan – Stage 3
Days
1-2
days
1-2
days
Learning
Target
I will check my
understanding of
trigonometric
functions by
participating in
the FAL.
I will prepare for
the unit
assessment on
trigonometry
by...
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Algebra 2
Expectations
(Activities and
Lessons)
FORMATIVE ASSESSMENT LESSON –
Representing Trigonometric Functions
Incorporating the Standards for Mathematical Practice (SMPs)
along with the content standards to review the unit.
1-2
days
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Curriculum Intranet
•
FAL: Representing
Trigonometric Functions
Procedural Skills and Fluency:
• Review: Relationships with
Meaning
Unit Assessment (LBUSD Math Intranet, Assessment)
8
Posted 3/15/17