Identify when two variables share a periodic relationship and construct their graph.
Describe the key features of a periodic function based on a verbal description or graph.
Quick Lesson Plan
Before getting into any specific trigonometric function, we begin this unit by looking at periodic phenomena. This just means any scenario that exhibits cyclic behavior.
In today’s activity, students will graph and interpret a rough model for the volume of air in one’s lungs during a breathing exercise. Obviously, each body is unique and lung capacity can vary slightly from person to person. However, the idea that the volume of air in one’s lungs goes through a predictable and repeated pattern is the main take-away from today’s lesson.
Start by watching (and completing) this video of a 2 minute breathing exercise. You may wish to share the link with your students as they may need to watch it a few times to collect some of the information needed to answer the questions in the Activity. Note that all times are referring to times since the breathing exercise starts (at 00:03 in the video), not from the time the YouTube video starts. Students may wish to use an additional timer on their phones or to use the video progress bar on YouTube to identify start and stop times for one cycle.
In questions 2 and 3, students use the video and the information in the introduction box to determine the length of each inhale and exhale (each representing half of a breath cycle), and the maximum and minimum values of volume in one’s lungs. In question 4, students will sketch the relationship. Note that students may be using straight lines instead of a wave to connect their maximum and minimum values. This is okay, as long as every cycle is drawn the same way so the graph still shows periodic behavior. The goal of today’s lesson is not to talk only about sinusoidal functions (that’s coming up in Lesson 6.7).
In questions 5 through 7 students interpret the graph and answer key questions related to the scenario, such as the length of one breath cycle (the period), the number of breath cycles over a certain period (a variant of frequency) and the fact that the volume of air in one’s lungs can be found for any t-value simply by knowing the volume of air during the first period.
How much air is in your lungs half-way through your inhale?
How much air is in your lungs half-way through your exhale?
It looks like the volume of the air is the same at t=9 and t=14. Does that mean the cycle starts repeating at t=14?
Do you think everyone’s maximum lung capacity is 6 liters? What part of the graph would change if it wasn’t? Would the pattern still repeat every 12 seconds?
How far into the breathing cycle are you after 93 seconds? How do you know?
Make sure that students understand that the period represents the interval over which one FULL cycle occurs. It is true that this can be found by finding the distance between the maxima or minima, but it is not true that the period is the distance between any two repeating outputs. If you notice students making this assumption, have them look at two t-values with the same output (like t=9 and t=14, roughly) and ask them if the graph between those two endpoints represents a full cycle.
A big idea of this lesson is that knowing the behavior of a function over a single cycle is enough to determine the behavior of the function anywhere. To determine the air in one’s lungs at any time t, you simply have to identify the corresponding t-value in the first cycle. How do we do this? By subtracting lengths of complete cycles until the t-value is in the original interval! This is a great preview of coterminal angles and even uses some modular arithmetic!