# UTPA STEM/CBI Courses/Calculus/Modeling Periodic Behavior

Course Title: Precalculus
Lecture Topic: Modeling Periodic Behavior
Instructor: Virgil U. Pierce
Institution: University of Texas -- Pan American

## Backwards Design

Course Objectives
* Primary Objectives- After this module students will be able to:
** Find sinusoidal functions which model periodic data collected from an experiment.
** Determine a graph from a sinusoidal function.
** Understand some basic concepts of 'Best Fit' of a function to slightly a-periodic data.
* Sub Objectives- The objectives will require that students be able to:
** Graph sine and cosine functions.
** Identify the amplitude, average value, period and phase shift of periodic data or a periodic function.
* Difficulties-
** Particular care needs to be taken, and this module is designed particularly to address, the determining of a phase shift of a standard sine or cosine function to match periodic data.
** Students are often confused about the relationships between the wave number, period and frequency
* Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
** Determining functions for making predictions from data collected in: biology, medicine, physics, astronomy, sociology, mechanics and other periodic or almost periodic behaviors.
Model of Knowledge
* Concept Map
** Amplitude and average value of a sinusoidal function.
** Maximum and minimum of a sinusoidal function.
** Period, frequency and wave number of a sinusoidal function.
** Phase shift of a sinusoidal function
** More generally transformations of graphs such as translation, dilation and reflection.
* Content Priorities
** Enduring Understanding
*** Determine the parameters of a sinusoidal function.
*** Use simple algebra to solve for the wave number from the period or frequency and vice versa.
** Important to Do and Know
*** Draw a scatter plot, possibly using technology such as a Speadsheet or Mathematica type application.
** Worth Being Familiar with
*** Translation of graphs of equations
*** Dilation of graphs of equations
*** Reflection of graphs of equations
Assessment of Learning
* Formative Assessment
** In Class (groups)
*** Plotting Sine and Cosine Functions and manipulating the parameters (experiments) -- best done with the help of some technology, possibly in a computer lab with Mathematica.
*** Drawing a scatter plot of the data.
*** Determining the parameters of our plotted data.
*** Group agreement on the correct function to match the parameters.
*** Use of a computer program such as Mathematica to manipulate the phase parameter.
*** Determining what is meant by best fit for an example with a-periodic data.
** In Class (clickers)
*** Matching graphs to functions and vice versa.
** Homework (individual)
*** Matching graphs to functions and vice versa.
*** Repeating the procedure with different experimental data.
* Summative Assessment
** Test or Quiz question on matching a sine function to given data.
*** It could be just matching to a given Maximum and Minimum.
*** It could be a more generally small version of the challenge problem.

## Legacy Cycle

OBJECTIVE
By the next class period, students will be able to:
* Construct a periodic function matching data,
* Identify important parameters (period, amplitude) in the data,
* Discuss the closeness of a periodic model of data to real data.
The objectives will require that students be able to:
* Identify an appropriate variable and time scale,
* Find the amplitude of periodic data,
* Find the period of periodic data,
* Find the vertical shift of the data,
* Find the phase shift to match a function to the data.
THE CHALLENGE
Predator-Prey interactions often exhibit a periodic behavior in the populations of the predators and prey. You are a hungry fox who subsists largely on rabbits. The following data was collected in 2010-2011 from a field at the far edge of your range. When is the best time of year to visit the field?
The following data was collected in 2010-2011:
| Month | January '10 | February '10 | March '10 | April '10 | May '10 | June '10 | July '10 | August '10 | September '10 | October '10 | November '10 | December '10 |
| Population of Rabbits | 11 | 11 | 14 | 21 | 28 | 33 | 38 | 40 | 37 | 28 | 21 | 15 |
| Month | January '11 | February '11 | March '11 | April '11 | May '11 | June '11 | July '11 | August '11 | September '11 | October '11 | November '11 | December '11 |
| Population of Rabbits | 13 | 9 | 12 | 17 | 24 | 32 | 35 | 39 | 40 | 32 | 20 | 17 |
GENERATE IDEAS
* Students should identify the data as likely being periodic, best method is to sketch a graph of the data points.
* Students should then identify the key parameters needed to specify a periodic function.
** A sub lesson should be spent on asking students to graph various trigonometric functions to identify where the parameters are in an expression.
* Students should get used to constructing a graph of the data in order to easily identify the key elements.
MULTIPLE PERSPECTIVES
* The most challenging part of the problem is incorporating the phase shift in the data. Students should discuss how this can be done.
* There are many different solutions.
* Students should discuss why.
* The periodic models will not fit the data perfectly. Students should discuss why.
RESEARCH & REVISE
Students should now:
* Identify the Maximum and Minimum of the data.
* Identify an amplitude, vertical shift and period of the data,
* Identify a candidate function that incorporates all of the parameters except for the phase shift.
* Identify the appropriate phase shift.
* Test their answer by graphing it.
Interested students could be asked to study why predator-prey interactions can lead to periodic behavior.
TEST YOUR METTLE and Go Public
Modify the data above. In practice each group of students should get different data, the groups can then exchange their data and compare their answers.

## Pre-Lesson Quiz

* Sketch a graph of f(t) = sin (3t)
* Sketch a graph of f(t) = sin (6t)
* Sketch a graph of f(t) = sin (2πt/2)
* Sketch a graph of f(t) = 2 sin (t)
* Sketch a graph of f(t) = 1/2 sin (t)
* Sketch a graph of f(t) = sin (t) +5
* Sketch a graph of f(t) = sin (t-π/4)
* Sketch a graph of f(t) = sin (2π/2 (t-1/4))

## Test Your Mettle Quiz

* Find a trigonometric function with amplitude 20, maximum value 50, and period 100.
* Find a trigonometric function with amplitude 20, maximum value 50, and period 100; where the maximum occurs at t = 50.
* Find a trigonometric function with amplitude 20, maximum value 50, and period 100; where the maximum occurs at t = 15.
* Give a trigonometric function which models the population of rabbits in the field,
* Identify the best time of year to visit the field as the fox.