Lesson 1 | Unit Circle and Trigonometric Functions | 11th Grade Mathematics

Objective

Sketch sine graphs to model contextual situations and identify features of sine graphs.

Common Core Standards

Core Standards

The core standards covered in this lesson

F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Interpreting Functions

F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Interpreting Functions

F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.TF.A.4 — Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Trigonometric Functions

F.TF.A.4 — Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
F.TF.B.5 — Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Trigonometric Functions

F.TF.B.5 — Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

G.SRT.C.6

Similarity, Right Triangles, and Trigonometry

G.SRT.C.6 — Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Identify appropriate situations for a sinusoidal model.
Connect sine functions with the height of a rotating object over time.
Identify midline, phase shift, period, and amplitude of sine functions.
Sketch a sine graph to model a contextual situation.
Identify sine functions as an example of periodic behavior.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The basic idea of a periodic function (one that repeats) will be new to students. Start connecting the idea of the fact that it repeats with the idea that the angles around the circle repeat. (F-TF.4)
Check out these videos: "Invention of the Ferris Wheel" on Travel Channel and "Features of Sinusoidal Functions" on Khan Academy.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Show the first video of Ferris Wheel by Dan Meyer.

Graph the height of the red car off of the ground over time. Where will the red car be after three minutes?

Guiding Questions

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References

101Questions Ferris Wheel

Ferris Wheel by Dan Meyer is made available on 101Questions under the CC BY 3.0 license. Accessed Feb. 26, 2018, 10:38 a.m..

Problem 2

What features of the function do you see? What type of Ferris wheel would this correspond to?

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Create a graph of a function that represents the height above the ground of the passenger car for a 225- foot diameter Ferris wheel that completes three turns, where each turn takes 2 minutes. Assume passengers board at the bottom of the wheel, which is 5 feet above the ground, and that the ride begins immediately afterward. Provide appropriate labels on the axes.

Explain how the features of your graph relate to the situation.

References

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 1 — Exit Ticket, Question #1

Algebra II > Module 2 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems connecting graphs and contextual situations.
Include problems finding periods of graphs and extrapolating to emphasize periodic behavior.
Include problems that measure the x-axis both in time and in degrees.

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 1 — Problem Set
Dan Meyer [Makeover] Ferris Wheel — Use questions from this blog post

Lesson 2

Lesson Map

Topic A: Trigonometric Ratios in Application and on the Unit Circle

Sketch sine graphs to model contextual situations and identify features of sine graphs.

F.IF.B.4 F.IF.C.7.E F.TF.A.4 F.TF.B.5

Sketch cosine graphs to model contextual situations and identify features of cosine graphs.

F.IF.B.4 F.IF.C.7.E F.TF.B.5

Find values of specific points on the unit circle using geometry.

F.TF.A.2 F.TF.A.4

Evaluate sines and cosines of points at reference angles on the unit circle.

F.TF.A.2 F.TF.A.4

Describe the relationship between the unit circle and tangent.

F.TF.A.2 F.TF.A.3

Define and evaluate the trigonometric functions of tangent, cosecant, secant, and cotangent.

F.TF.A.2 F.TF.A.3

Convert between degrees and radians and evaluate trigonometric functions written in radians.

F.TF.A.1 F.TF.A.3

Evaluate transformations of sine, cosine, and tangent such as $${2{\pi-x}}$$, $${\pi-x}$$, and $${\pi+x}$$.

F.TF.A.1 F.TF.A.3

Topic B: Graphing Sine, Cosine, and Target

Graph transformations of sine and cosine functions (Part I).

F.IF.C.7.E F.TF.A.1 F.TF.A.3 F.TF.A.4

Graph transformations of sine and cosine functions (Part II).

F.BF.B.3 F.IF.C.7.E

Graph transformations of tangent functions.

F.BF.B.3 F.IF.C.7.E

Identify equations and graphs of all six trigonometric functions.

F.BF.B.3 F.TF.A.3 F.TF.B.5

Model contextual situations using trigonometric functions (Part I).

F.TF.A.3 F.TF.B.5

Model contextual situations using trigonometric functions (Part II).

F.TF.A.3 F.TF.B.5

Unit Circle and Trigonometric Functions

Objective

Common Core Standards

Core Standards

Interpreting Functions

Interpreting Functions

Trigonometric Functions

Trigonometric Functions

Foundational Standards

Similarity, Right Triangles, and Trigonometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Target Task

References

Additional Practice

Lesson Map

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