# Engaging students: Defining sine, cosine and tangent in a right triangle

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 Williams. Her topic, from Precalculus: defining sine, cosine and tangent in a right triangle.

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

I know of a good project/activity for the students to do that will be extremely engaging. You could either do this for an elaborate activity for your students or maybe an opening activity for day 2 of a lesson. For my class, I would get a square cookie cake, and have the slices cut into right triangles. I would allow each student to have a piece (but not eat it just yet). The students will be provided with rulers and a protractor. The students will each measure the hypotenuse of their cookie cake and the degree of whichever angle you would like them to measure, however each student should be measuring the same parts so do this unanimously). As a class, decide on an average for the measurements for everyone to use so that the data is not off. Then take the supplies away from the students and ask the students to find the rest of the missing sides and angles of their piece of cookie cake. They will also be provided with a worksheet to go along with this activity. This is a good review activity or al elaborate activity to allow further practice of real world application of right triangle trigonometry. Then go over as a class step by step how they solved for their missing angles and side lengths and make each group be accountable for sharing one of them. This allows the students to all be actively participating. Through out the lesson, make sure to tell the kids as long as they are all participating they will get to eat their slice when the lesson is done. Lastly, allow the students to eat their slice of cookie cake.

How does this topic extend what your students should have learned in previous courses?

Prior to learning about right triangle trigonometry the students will know how to use the Phythagorean Theorem to find how long the missing side length is of a right triangle. The students know basic triangle information such as, the sum of the angles in a triangle is 180 degrees. The students already know the difference between the hypotenuse and the other two legs. The students know that hypotenuse will be the longest leg and the leg across from the 90 degree angle. The students will also know the meaning of a fraction or ratio. The students may need some refreshing of memory on some parts of prior knowledge, but as teachers we know this is an extremely important part of a lesson plan. Even as teacher we tend to forget things and require a jog of memory. A simple activity such as headbands or a kahoot with vocabulary would be an excellent idea for accessing the students prior knowledge. This allows the students to formally assess themselves and where they stand with the knowledge. Also, it allows the teacher to formally assess the students and see what they remember or parts they are struggling on. This allows the teacher to know what things to spend more time on.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Technology is always an amazing aspect of the classroom. Like stated above, a vocab review using headbands or kahoot would be a good idea for this type of lesson that DEFINITELY needs prior knowledge to be applied in order to succeed. Also, showing the students how to plug in sine, cosine, and tangent is crucial. They have seen these buttons on the calculator but they do not know what they mean or how to use them. Using an online TI on display for the class is great. I had to do this with my 10th grade students to make sure they understood how to use the 3 buttons. Also, when using arcsin, arccos, and arctan it can be confusing. Using technology to show the class as a whole is the best route to go. Also, technology can used as review for a homework assignment or even extra credit for the students. It benefits them by getting extra review and extra credit points. I found a website called http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/object_interactives/trigonometry/use_it.html , which is a golf game that requires review of triangles and trigonometry. It allows the students to practice the ratios of SOH-CAH-TOA using a given triangle.

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