In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jessica Trevizo. Her topic, from Precalculus: graphing the sine and cosine functions.

How could you as a teacher create an activity or project that involves your topic?

For this activity students can either work with a partner or work individually. I enjoyed this activity because students are able to derive the sine and cosine functions on their own using fun materials other than the original paper and pencil. The knowledge that students should gain from this activity is the relationship between the unit circle and the sine/cosine function. Along with this activity, students will be practicing previous concepts learned such as converting degrees to radians, finding the domain/rage, and finding the x-intercepts and y-intercepts. Also, amplitude, period, and wavelength are important vocabulary words that can be introduced and applied to the parent functions. To complete the activity assign the students to write a paragraph comparing and contrasting both functions. In their paragraph make sure students include a discussion of the intercepts, maxima, minimum, and period. It is essential for the students to know how to graph the parent functions of sine and cosine and where they come from before teaching the students about the transformations of the functions.

http://illuminations.nctm.org/Lesson.aspx?id=2870

**A.1 What interesting word problems using this topic can your students do now?**

Real life word problems that involve the sine and cosine function can be used to keep the students engaged in the topic. Both of the functions can used to model situations that occur in real life in a daily basis such as; recording the path of the electric currents, musical tones, radio waves, tides, and weather patterns. Here is an example of a word problem, “Throughout the day, the depth of the water at the end of a dock in Bar Harbor, Maine varies with tides. The table shows the depths (in feet) at various times during the morning.” With the data provided the students are able to do several things such as: be able to use a trigonometric function to model the data and find the depth of the water at any specific time. Also, if a boat needs at least 10 feet of water to moor at the dock, the students should be able to figure out safe dock times for the boat.

**How can technology be used to effectively engage students with this topic?**

Most of the students are familiar with sound waves. As an engage go to www.onlinemictest.com and have the students observe the sound waves that appear on the screen as you speak. Many students will recognize the various sine and cosine functions on the screen. With the online mic test students are also able to make relationships between the sound and the wave. Download several different tones and play them so the sound waves of the tones appear on the screen. Have the students sketch the graph of a soft high note, soft low note, loud high note, and a loud low note. The following graphs should look similar to the figure below. Once all of the students have recorded their own observations have the students work with a partner to compare their graphs. Also give the students a minute or two so they can compare and contrast the 4 different graphs by using the new vocabulary that they learned such as amplitude and period. Students are able to remember the new vocabulary when they have opportunities to have discussions that require them to use them.