SWBAT use sinusoidal graphs in context to solve problems.

Tides and temperature are periodic, so we can model these scenarios using trigonometric functions.

5 minutes

Since we are going to look at sinusoidal functions in context, I want to get students thinking of different real-life situations where a sine or cosine graph would be applicable. With their groups, ask students to brainstorm different real-world situations in which a sine or cosine graph would be a good model.

Possible Ideas Students will suggest:

- I expect students to say Ferris wheels since they are discussed earlier in this unit. Students will probably suggest other instances where something is moving in a circle and the distance away from the ground is a function of time.
- In my class a students suggested a carousel, but said that the graph would represent the distance from an observer who is not on the ride. It was an interesting conversation about how each horse moves up and down
*and*you are moving in a circle. - Some students may mention a roller coaster, as they associate the shape of the roller coaster with the shape of the graph. If you discuss this interpretation with your class, you could reason on how the time it takes to get to the top of a roller coaster is longer than it takes to go down, so the graph (distance from ground v. time) would not have the symmetry that a sinusoidal graph would.

Let me know if your students come up with other scenarios that demonstrate understanding or a misconception!

20 minutes

Thinking of applications for sinusoidal functions will be a good primer for the work that we are going to do today. We will be looking at two real-life applications of sinusoidal functions. It is important to preface this lesson by letting students know that we are just working with models. Real data is messy and imperfect, but we can use mathematics to make it a little cleaner and to see trends and patterns.

Have students work in their table groups on the tasks on this worksheet. At this point they should be familiar with sketching the graphs of sinusoidal functions, but the temperature graph on the front may be difficult because of the numbers.

Here are some suggestions for some common issues that may arise for the task about the temperature in Fairbanks:

**1. A student who does not know how to find the period of the function:** Ask them what the period would be if there was not a coefficient. Then ask what the period would be if the coefficient were 2. When they say pi, ask them how they got this answer. When the student mentions division, say that we need to use the same operation even though the numbers are not easy.

**2. A student who does not know whether to use radians or degrees to graph:** Ask the student to find the period of the function using both radians and degrees and then ask which period they would rather work with.

**3. A student who does not have their y-axis in the appropriate part of the graph:** For this function, the y-axis should be just to the left of the minimum of the function. When sketching sinusoidal functions, students will often carelessly draw their curve with no thought as to where the maximum or minimum should occur. Ask the student to check their work on a graphing calculator and have them think about what window would make sense in the context of this problem. Another strategy is to draw the sinusoidal function first, and then go back and draw the axis.

For the task about the tides of the river, students can usually sketch the graph from the table but they may have some difficulty with writing the equation. Again, here are some issues that may arise:

**1. A student who cannot find the correct coefficient of x:** Ask how we found the period on the front side of the worksheet. Now we know the period but must find the coefficient. They can set up an equation and solve for the coefficient or just think about what 2pi needs to be divided by to get the period.

**2. A student who cannot use the graph to estimate when travel is not safe:** Have them estimate the points on the graph where the depth is exactly 24 feet. Then ask where the depth would be less than 24 feet. This will get them to see which portions of the function dip below the safe level.

20 minutes

For the Fairbanks Temperature task, choose a student who graphed it correctly and have them explain how they used the equation to sketch the graph. Have them point out the important aspects of the graph and how that relates to the equation that was given.

Poll the class about what they calculated as the warmest and coldest low temperatures of the year. Since it is tedious to find the dates from the graph, there might be a little variation. Ask why students got different answers for these days.

For the last question, students will compare the actual low temperature for today to the predicted value from the function. This should spark a good conversation as students wonder about the validity of the model and why today’s value is very close or not so close. It is a good opportunity to talk about how one year’s data would not make a good mathematical model, but many years would. See if students can think about what the graph would look like if it presents the low temperatures for one year, instead of many. Hopefully students will realize that the more data we have, the smoother the curve of the graph will be.

If you still have extra time, you could also show this graph and talk about why any of this matters and why we would even want a mathematical model. I discuss this in the video below.

For the Thames River task, write down different equations that students came up with and see if students think that all of these equations would produce the correct graph. You can add sine and cosine equations and ones with different phase shifts. In the very unlikely event that your entire class came up with the same function, you could just add them yourself. Also, add some that are incorrect so that you can discuss why they do not work. After deciding on the correct ones, ask students why there are so many different ways that we could have written the functions.