High School Math 3 Math 3 Unit 5b: Trigonometric Modeling
Approximate Time Frame: 2-3 weeks
Connections to Previous Learning:
Students will use their knowledge of algebra and trigonometric representations and functions to investigate and use trigonometric models. They need to be familiar with the graphs of the trigonometric functions and be confident in solving trigonometric equations.
Focus of this Unit:
Students will use trigonometric functions to model real life phenomena. They should use sin(x) and cos (x) to model “waves” and periodic motion. Examples could include examining the motion of a pendulum, an electrical current, or the relationship of the height of a chair on a ferris wheel to time. Students should also understand transformations and be able to adjust parameters such as amplitude, period, and horizontal and vertical shifts in order to improve these models. Students will work with and modify functions of the form f(x)=Asin[B(x+C)] +D.
Connections to Subsequent Learning:
Modeling with trigonometric functions will not only lead to other mathematics such as polar coordinates but will also connect to many areas of science and engineering.
From the Grade 8, High School, Function Progression document, p.18
Model periodic phenomena with trigonometric functions: Now that students are armed with trigonometric functions, they can model some periodic phenomena that occur in the real world. For students who do not continue into advanced mathematics, this is the culmination of their study of trigonometric functions. The tangent and function is not often useful for modeling periodic phenomena because tan is undefined for , where kis an integer. Because the sine and cosine functions have the same shape, either suffices to model simple periodic phenomena.F-TF.5 Functions are called sinusoidal if they have the same shape as the sine graph. Many real-world phenomena can be approximated by sinusoids, including sound waves, oscillation on a spring, the motion of a pendulum, tides, and phases of the moon. Some students will learn in college that sinusoids are used as building blocks to approximate any periodic waveform.
Desired Outcomes
Standard(s):Create equations that describe numbers or relationships.
- A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential 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.
- N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
- F.IF.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.★
- F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★
- A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
- S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Uses given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
- F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
- Explicit vocabulary instruction with regard to key features of trigonometric and logarithmic functions.
- Guided conversations regarding how trigonometric and logarithmic functions represent contexts.
Understandings: Students will understand that …
- The trigonometric functions sin(x) and cos(x) can be used to model real-life situations that exhibit periodic behavior.
- Changing parameters such as amplitude, period, and midline of a function will alter its graph and that these parameters are related to the context or phenomena being modeled.
Essential Questions:
- What are key characteristics to identify when choosing a function to model a given situation?
- How can the graphs of trigonometric functions be modified to best fit the situations being modeled.
- How do factors such as amplitude, period, midline, and horizontal shift affect these functions and relate to the phenomena being modeled?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students in this unit will encounter many real world problems and will use trigonometric models to solve these problems. Students will engage in problem solving while investigating tasks such as oscillations of a spring, sun spot cycles, or radio waves.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
*4. Model with mathematics. Students will use trigonometric functions to model real world situations. They will then interpret mathematical solutions in terms of the model and be able to describe the significance of particular points or values.
5. Use appropriate tools strategically.
*6. Attend to precision. While working with trigonometric models, students will need to attend to units such as degrees and radians, appropriate function selections, parameters such as amplitude and frequency, and reasonable domains and problem intervals.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Prerequisite Skills/Concepts:
Students should already be able to:
- Graph , evaluate, and solve trigonometric functions.
- Should be able to work in degrees and radians.
Some students may be ready to:
- Use tan(x), cot(x), sec(x), and csc(x) as well as inverse trigonometric functions as models.
- F.TF.3. (+) Use special triangles to determine geometrically the values of sine,
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.4. (+) Use the unit circle to explain symmetry (odd and even) and
Model periodic phenomena with trigonometric functions
- F.TF.6. (+) Understand that restricting a trigonometric function to a domain
to be constructed.
- F.TF.7. (+) Use inverse functions to solve trigonometric equations that arise
interpret them in terms of the context.★
Prove and apply trigonometric identities
- F.TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Knowledge: Students will know…
- That periodic phenomena can be modeled by sinusoidal functions
- Modify f(x)=Asin[B(x+C)] +D to model various real life situations and contexts. They should also be able to explain the contextual meaning of points on these functions and the significance of values and solutions.
- 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.
- Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
- Define appropriate quantities for the purpose of descriptive modeling.
- Create equations in one, two, or more variables.
Academic Vocabulary:
Critical Terms:
Amplitude
Period
Frequency
central axis
midline,
horizontal (phase) shift / Supplemental Terms:
Tangent
Cotangent
Secant
Cosecant
inverse trigonometric functions
2/22/2013 8:10:24 AM Adapted from UbD frameworkPage 1