GRADE 11 MATH: TRIGONOMETRIC FUNCTIONS
UNIT OVERVIEW
Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students extend trigonometric functions to all (or most) real numbers. To reinforce their understanding of these
functions, students begin building fluency with the values of sine, cosine, and tangent at π/6, π/4, π/3, π/2, etc. Students make sense of periodic phenomena as they model with trigonometric functions.
TASK DETAILS
Task Name: What’s the Weather?
Grade: 11
Subject: Algebra II/Trigonometry
Depth of Knowledge: 3/4
Task Description: Students apply the Pythagorean Theorem and trigonometric ratios to find the measure of angles on the coordinate plane. They write these measures in degree and radian form. Students also use trigonometric functions to model natural phenomena, reflect on the accuracy of these models, and use them to make predictions about future events.
Standards Assessed:
F-‐TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-‐TF.A.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.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F-‐TF.C.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.
Standards for Mathematical Practice:
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
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TABLE OF CONTENTS
The task and instructional supports in the following pages are designed to help educators understand and implement Common Core–aligned tasks that are embedded in a unit of instruction. We have learned through our pilot work that focusing instruction on units anchored in rigorous Common Core–
aligned assessments drives significant shifts in curriculum and pedagogy.
PERFORMANCE TASK: TRIGOMETRIC FUNCTIONS…….……….………..………...3
RUBRIC AND SCORING GUIDE………7
FORMATIVE ASSESSMENTS……….………...13
INSTRUCTIONAL SUPPORTS………...32
UNIT OUTLINE………..33
EXTENSIONS………..57
GUIDED QUESTIONS……….……….………..58
VISUALS……….………..60
Acknowledgements: This bundle was developed by Alex Sczesnak, Cathleen Cuffie, and Carlos Bermudez from the Metropolitan High School 12X248, in collaboration with the Common Core Fellows.
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GRADE 11 MATH: TRIGONOMETRIC FUNCTIONS PERFORMANCE TASK
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What’s the Weather?
Algebra II – Module 2: Trigonometric Functions – Performance Task
1. Use two different methods to find the measure of the angle below in standard position, showing your work for both. Express your answer in radians.
2. Given sin (θ) = 3√5/11, find cos (θ) and tan (θ). Justify your answer using the Pythagorean identity.
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3. The table below displays the average high temperature, by month, in Central Park.
Average Monthly Temperature (°F)
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
38 42 50 61 71 79 84 83 75 64 54 43
a. Use this data to write a function representing the temperature, f(m), in month m, with January = 0. Graph this function on the axes below, labeling any critical values.
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b. Climate scientists believe that the average high temperature in Central Park for the month of July could increase by as much as 10° by 2050. Assuming that minimum average temperature stays the same, write a second function modeling this change and use it to predict the new average high temperature for the month of October.
c. Although only 12 data points are given in the table above, the function modeled above is continuous. Should average monthly temperature be expressed as a continuous function? Why or why not?
d. The table below displays the average monthly rainfall, by month, in Central Park.
Average Monthly Rainfall (in.)
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
3.65 3.21 2.36 4.50 4.19 6.41 4.60 4.44 4.28 1.40 4.02 5.00
Your friend Sam wants to model this data using a function similar to the one you’ve written. Do you think this is a good idea? Why or why not?
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GRADE 11 MATH: TRIGONOMETRIC FUNCTIONS SCORING GUIDE AND RUBRIC
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What’s the Weather?
Algebra II – Module 2: Trigonometric Functions – Performance Task Rubric
What’s the Weather? Rubric The elements of performance required by this task are:
• Uses trigonometric ratios and the Pythagorean theorem to find side lengths and angle measures on the coordinate plane
• Writes angle measures in radian and degree form
• Models phenomena with trigonometric functions
Possible Responses Points Section Points
1. a) Uses arctan to find the measure of an angle θ in the fourth quadrant
b) Uses the Pythagorean theorem to solve for the given side length and uses this side length to find either the cosine or sine of an angle θ in the fourth quadrant
c) Writes this angle in radians in standard position Partial Credit
• Draws and labels a right triangle using the given
• Uses only one method to find the measure of the given angle
• Incorrectly identifies the angle θ in the fourth quadrant, leading to an incorrect final answer
2 2
1
(1) (2) (2)
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2. a) Draws and labels a diagram of a right triangle
b) Uses the Pythagorean theorem to find the missing side length c) Uses the missing side length to find the cosine and tangent of the given angle θ
Partial Credit
• Draws and labels a right triangle using the given
• Incorrectly sets up the Pythagorean theorem, leading to an incorrect side length
• Finds only the cosine or tangent of angle θ, but not both
1 2
2
(1)
(1)
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3. a) Writes a function that accurately models the given data set, such as f(m) = –23cos(θπ/6) + 61
b) Draws and labels a graph corresponding to the function
c) Writes a second function reflecting the altered data set, such as f(m) = –28cos(θπ/6) + 66
d) Uses the new function to find f(9) = 66
e) Gives a correct answer “yes” to part (c), using justification such as “because temperature changes gradually over time”
f) Give a correct answer “no” to part (d), using justification such as
“the data in the table does not change periodically and so cannot be graphed using a trigonometric function”
Partial Credit
• Uses trigonometry to model a function in part (a) that is incorrect
• Draws a graph that does not accurately model the given data set
2 2 1 1 1 1
(1)
(1)
Total Points 18
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Performance Level Descriptions and Cut Scores
Performance is reported at four levels: 1 through 4, with 4 as the highest.
Level 1: Demonstrates Minimal Success (0–4 points)
The student’s response shows few of the elements of performance that the task demands as defined by the Common Core standards. The student’s work shows a minimal attempt and lack of coherence. The student fails to use the Pythagorean theorem or trigonometric ratios appropriately and provides little or no justification for his/her answers. The student does not accurately model the given data or
demonstrate understanding in the follow-‐up questions.
Level 2: Performance Below Standard (5–10 points)
The student’s response shows some of the elements of performance that the task demands as defined by the Common Core standards. The student attempts to use some tools, such as the Pythagorean theorem or trigonometric ratios, with limited success. The student might model the given data set with a trigonometric function, but ignore or fail to address some of the constraints of the problem.
Level 3: Performance at Standard (11–15 points)
For most of the task, the student’s response shows the main elements of performance that the task demands as defined by the Common Core standards, with few minor errors or omissions. The student explains the problem and identifies constraints. The student makes sense of quantities and their relationships in the modeling situation. The student uses appropriate tools, such as the Pythagorean theorem or trigonometric ratios. The student might discern patterns or structures and make
connections between representations. The student is able to make sense of the problem and uses a trigonometric function to model the given data set, taking into consideration some or all of the constraints of the problem.
Level 4: Achieves Standard at Highest Level (16–18 points)
The student’s response meets the demands of nearly all of the tasks as defined by the Common Core standards and is organized in a coherent way. The communication is clear and precise. The body of work looks at the overall situation of the problem and process while attending to the details. The student routinely interprets the mathematical results, applies trigonometric concepts in the context of the situation, and reflects on whether the results make sense. The student uses a trigonometric function to model the given data set, taking into consideration all of the constraints of the problem.
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GRADE 11 MATH: TRIGONOMETRIC FUNCTIONS FORMATIVE ASSESSMENTS
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Algebra II – Module 2
Formative Assessment #01 [Initial Assessment]
Standard G-‐SRT.8
1. In ∆XYZ, 𝒎∡𝒀 = 90o, XY = 18 and YZ = 15. Find 𝒎∡𝒁 to the nearest degree and the length of side XZ to the nearest tenth.
2. A tree casts a shadow 20 meters from its base. The height of the tree is 30 meters. Find the angle of elevation formed by the end of the shadow and ground, rounding to the nearest degree. Then find the distance from the top of the tree to the end of the shadow, rounding to the nearest tenth.
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Algebra II – Module 2 Formative Assessment #01 Standard G-‐SRT.8
Formative Assessment #01 Rubric The elements of performance required by this task are:
• Students use the trigonometric ratios and the Pythagorean Theorem to solve right triangles with/without context
Possible Responses Points Section Points
1. Uses trigonometry and/or the Pythagorean Theorem to find 𝑚∡𝑍 to the nearest degree and the length of side XZ
Partial Credit
• Finds either 𝑚∡𝑍 or the length of side XZ but not both
• Does not round one or both answers to the correct place value
2 (1) (1)
2
2. Uses trigonometry and/or the Pythagorean Theorem to find the angle of elevation to the nearest degree and the distance from the top of the tree to the edge of the shadow.
Partial Credit
• Finds either the angle of elevation or the distance, but not both
• Does not round one or both answers to the correct place value
2 (1)
(1)
2
Total Points 4
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Algebra II – Module 2 Formative Assessment #02 Standard F-‐TF.2
1. Find the exact value of the cosine and sine of –390o.
2. Find the measure of the given angle in standard position.
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Algebra II – Module 2 Formative Assessment #02 Standard FT-‐F.2
Formative Assessment #02 Rubric The elements of performance required by this task are:
• Students find the angle measure corresponding to a coordinate on the unit circle, and use a given angle measure to find a coordinate on the unit circle
Possible Responses Points Section Points
1. Uses knowledge of the unit circle and benchmark values for sine and cosine to find values for the angle –390o
Partial Credit
• Uses a method other than benchmark values (for example, a calculator) to solve
• Finds either cosine or sine of the angle, but not both
2 (1)
(1)
2
2. Uses knowledge of reference angles and benchmark values for sine and cosine to determine the angle measure
Partial Credit
• Uses the wrong reference angle but arrives at logically consistent answer
2
(1) 2
Total Points 4
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Algebra II – Module 2 Formative Assessment #03 Standard F-‐TF.1
1. Convert 𝟏𝟏𝟏𝟐𝝅 to degrees.
2. Convert 480o to radians.
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Algebra II – Module 2 Formative Assessment #03 Standard FT-‐F.1
Formative Assessment #03 Rubric The elements of performance required by this task are:
• Converts between angle measures in radians and degrees
Possible Responses Points Section Points
1. Converts the angle to 165o
1 1
2. Converts the angle to !!!radians
1 1
Total Points 2
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Algebra II – Module 2 Formative Assessment #04 Standard FT-‐F.8
1. Given 𝒔𝒊𝒏 𝜽 = 𝟖𝟔 find 𝒄𝒐𝒔 𝜽 and 𝒕𝒂𝒏 𝜽 in Quadrant I. Use the Pythagorean identity to justify your answer.
2. Given 𝒕𝒂𝒏 𝜽 =𝟑 𝟐𝟓 find 𝒔𝒊𝒏 𝜽 and 𝒄𝒐𝒔(𝜽) in Quadrant III. Use the Pythagorean identity to justify your answer.
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Algebra II – Module 2 Formative Assessment #04 Standard FT-‐F.8
Formative Assessment #04 Rubric The elements of performance required by this task are:
• Use the Pythagorean identity to find cos(Ѳ), sin(Ѳ), or tan(Ѳ) when given cos(Ѳ), sin(Ѳ), or tan(Ѳ)
Possible Responses Points Section Points
1. Uses the Pythagorean identity to find cos(Ѳ) and tan(Ѳ) in Quadrant I
Partial Credit
• Uses a method other than the Pythagorean identity to solve for cos(Ѳ) and tan(Ѳ)
• Correctly uses the Pythagorean identity to find cos(Ѳ) and tan(Ѳ), but fails to account for values in Quadrant I
2 (1)
(1)
2
2. Uses the Pythagorean identity to find cos(Ѳ) and sin(Ѳ) in Quadrant III
Partial Credit
• Uses a method other than the Pythagorean identity to solve for cos(Ѳ) and sin(Ѳ)
• Correctly uses the Pythagorean identity to find cos(Ѳ) and sin(Ѳ), but fails to account for values in Quadrant III
2 (1)
(1)
2
Total Points 4
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Algebra II – Module 2 Formative Assessment #05 Standard FT-‐F.5
1. Describe the difference between the graphs of the trigonometric functions 𝒇 𝜽 = 𝒄𝒐𝒔(𝜽) and
𝒈 𝜽 = 𝟑𝒄𝒐𝒔 𝜽
𝟔 − 𝟓. Make sure to use precise mathematical vocabulary.
2. Describe the difference between the graphs of the trigonometric functions 𝒇 𝜽 = 𝒔𝒊𝒏(𝜽) and
𝒈 𝜽 = −𝟏
𝟑𝒔𝒊𝒏 𝟑𝜽 + 𝟏𝟕. Make sure to use precise mathematical vocabulary.
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Algebra II – Module 2 Formative Assessment #05 Standard FT-‐F.5
Formative Assessment #05 Rubric The elements of performance required by this task are:
• Understanding the functions f(Ѳ) = acos(b Ѳ) + k and f(Ѳ) = asin(b Ѳ) + k in terms of amplitude, frequency, and midline
Possible Responses Points Section Points
1. Correctly relates coefficients a, b, and k to the modified amplitude, frequency, and midline of the trigonometric function
Partial Credit
• Correctly relates two of the coefficients to the changes in the amplitude, frequency, or midline of the trig function
2 (1)
2
2. Correctly relates coefficients a, b, and k to the modified amplitude, frequency, and midline of the trigonometric function
Partial Credit
• Correctly relates two of the coefficients to the changes in the amplitude, frequency, or midline of the trig function
2
(1) 2
Total Points 4
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Algebra II – Module 2 Formative Assessment #06 Standard FT-‐F.5
Robert and his friend Hilda are riding in a Ferris wheel at the state fair.
Their elevation over the first 10 minutes of their ride is recorded in the table below:
1. Write a periodic function, f(m), relating their elevation to the time m in minutes that they spend in the Ferris wheel.
2. Predict Robert and Hilda’s elevation at time m = 15 minutes.
3. After 15 minutes, the two friends decide they want to get off the Ferris wheel. How much longer will they have to wait before they can exit the ride?
Height Above the Ground TIME
(min) 0 1 2 3 4 5 6 7 8 9 10 ELEVATION
(feet) 0 7 24 40 48 40 24 7 0 7 24
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Algebra II – Module 2 Formative Assessment #06 Standard FT-‐F.5
Formative Assessment #06 Rubric The elements of performance required by this task are:
• Model with trigonometric functions by choosing the appropriate function and adjusting the amplitude, frequency, and midline
Possible Responses Points Section Points
1. Writes a correct function f(m) modeling the given data Partial Credit
• Writes a function that is partially correct but contains one conceptual error
2
(1) 2
2. Correctly predicts the Ferris wheel height at m = 15 to be approx.
seven feet Partial Credit
• Substitutes m = 15 into the function, but makes one calculation error
2
(1) 2
3. Uses algebra to solve for m when f(m) = 0 and states that the friends can exit at m = 16 minutes
Partial Credit
• Uses a method other than algebraic to predict when the friends can exit the Ferris wheel
2
(1) 2
Total Points 6
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GRADE 11 MATH: TRIGONOMETRIC FUNCTIONS INSTRUCTIONAL SUPPORTS
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Unit Outline
I
NTRODUCTION: Building on their previous work with functions, and on their work with trigonometric ratios and circles in geometry, students extend trigonometric functions to all (or most) real numbers. To reinforce their understanding of these functions, students begin building fluency with the values of sine, cosine, and tangent at π/6, π/4, π/3, π/2, etc. Students make sense of periodic phenomena as they model with
trigonometric functions.
Algebra II: Trigonometric Functions
U
NITT
OPIC ANDL
ENGTH:
Ø Module 2: Trigonometric Functions (20 days) C
OMMONC
ORES
TANDARDS:
Ø F-‐TF.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle*
Ø 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
Ø F-‐TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline
Ø F-‐TF.8: Prove the Pythagorean identity sin
2(Ѳ) + cos
2(Ѳ) = 1 and use it to find sin(Ѳ), cos(Ѳ), or tan(Ѳ), given sin(Ѳ), cos(Ѳ), or tan(Ѳ) and the quadrant of the angle*
*TF.1 and TF.8 are partially assessed in the formative and performance tasks but are covered completely in the learning activities.
B
IGI
DEAS/E
NDURINGU
NDERSTANDINGS:
Ø Converting radians to degrees and degrees to radians
Ø Valuing trigonometric functions for any degree measure, even those greater than 2π or 360
°Ø Modeling periodic phenomena with trigonometric functions by manipulating amplitude, frequency, and midline
Ø Using the Pythagorean identity to rewrite expressions and equations involving trigonometry
E
SSENTIALQ
UESTIONS:
Ø How is radian angle measure defined?
Ø What is the unit circle and how does it define the value of any trigonometric function at a given angle?
Ø How can we manipulate the amplitude, frequency, and midline of a
trigonometric function?
Ø How can we use trigonometric functions to model periodic data?
Ø What is the Pythagorean identity and how can we use it to rewrite expressions and equations?
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V
OCABULARY/K
EYT
ERMS: Ø T
OPIC1:
R
EVIEWo
TRIGONOMETRY,
FUNCTION,
SINE,
COSINE,
TANGENT,
RIGHT TRIANGLE,
OPPPOSITE/
ADJACENT SIDES,
HYPOTENUSE,
SUPPLEMENTARY,
COMPLEMENTARY,
INVERSE(
OR ARC)
Ø T
OPIC2:
U
NITC
IRCLEo
UNIT CIRCLE,
DEGREES,
STANDARD POSITION,
REFERENCE ANGLE,
INITIAL SIDE,
TERMINAL SIDE,
QUADRANTØ T
OPIC3:
R
ADIANS ANDD
EGREESo
RADIANS,
PI,
ARC,
ARC LENGTHØ T
OPIC4:
P
YTHAGOREANI
DENTITYo N
O NEW VOCABULARY INTRODUCEDØ T
OPICS5
AND6:
M
ANIPULATING/M
ODELING WITHT
RIGF
UNCTIONSo
AMPLITUDE,
FREQUENCY,
MIDLINE,
PERIOD/
CYCLEA
SSESSMENT EVIDENCE AND ACTIVITIES:
I
NITIALA
SSESSMENT:
SEE
FORMATIVE
ASSESSMENT
#01
FOR
INITIAL
ASSESSMENT F
ORMATIVEA
SSESSMENT:
SEE
FORMATIVE
ASSESSMENTS
#02-‐06
F
INALP
ERFORMANCET
ASK:
SEE
“WHAT’S
THE
WEATHER?”
PERFORMANCE
TASK L
EARNINGP
LAN ANDA
CTIVITIES:
ØØ Day 1–3: Diagnostic/Review
Objectives:
• Students complete the formative assessment #01 [initial assessment].
• Students use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems.
UDL Supports/Suggestions:
• Teacher can scan over students’ initial assessments to come up with small learning groups for students with incorrect responses.
• Students that got both assessment items correct can be paired with students who got only one question correct.
ØØ Day 4: Radians
Objectives:
• Students understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
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• Students complete radian activity 1 from Project Maths.
(http://www.projectmaths.ie/documents/radian_activity_1.pdf)
UDL Supports/Suggestions:
• Provide students with worked examples for guidance.
ØØ Day 5: Radians
Objectives:
• Students understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
• Students complete formative assessment #03.
UDL Supports/Suggestions:
• Teacher can scan over students’ formative assessments to come up with small learning groups for students with incorrect responses.
ØØ Day 6: Unit Circle
Objectives:
• Students understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
• Students complete NCTM “paper plate” activity.
(http://www.loganschools.org/curriculum/mathematics/Paper_plate_unit_circle.pdf)
UDL Supports/Suggestions:
• Pair students with strong literacy skills with students with weak literacy skills.
ØØ Day 7: Unit Circle Extensions
Objectives:
• Students are able to explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measure of angles traversed counterclockwise around the unit circle.
• Students complete Shmoop trigonometric functions worksheet.
(http://www.shmoop.com/common-‐core-‐standards/handouts/f-‐tf-‐worksheet_2.pdf)
UDL Supports/Suggestions:
• Provide worked examples for students.
ØØ Day 8: Unit Circle Extensions
Objectives:
• Students are able to explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measure of angles traversed counterclockwise around the unit circle.
• Students complete the illustrative mathematics activity “Trigonometric functions for arbitrary angles.” (http://www.illustrativemathematics.org/illustrations/1692)
UDL Supports/Suggestions:
• Provide a worked example for a similar problem.
ØØ Day 9: Pythagorean Identity
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