# Engaging students: Finding the area of 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 again comes from my former student Deanna Cravens. Her topic, from Geometry: finding the area of a right triangle. How could you as a teacher create an activity or project that involves your topic?
One of the most common questions students ask when working with the area of a triangle is: “Why do I multiply by ½ in the formula?” It is a rather simple explanation for working with right triangles. Students could either do an explore activity where they discover the formula for the area of a right triangle, or a teacher could show this short two minute video in class.

So why do we multiply by ½? If we look at the formula ignoring the ½, you will see that it is the same formula for the area of a rectangle. Each angle in a rectangle forms 90 degrees and if we cut the rectangle along one of the diagonals, we will see that it creates a right triangle. Not only that, but it is exactly one half of the area of the rectangle since it was cut along the diagonal. Another way of showing this is doing the opposite by taking two congruent right triangles and rearranging them to create a rectangle. Either way shows how the ½ in the formula for the area of a right triangle appears and would be a great conceptual explore for students to complete. How can this topic be used in your students’ future courses in mathematics or science?

Students are first introduced to finding the area of right triangle in their sixth grade mathematics class. One way that the topic is advanced in a high school geometry class is by throwing the Pythagorean Theorem into the mix. Students will know that formula for the area of a right triangle is A=½ bh. The way the topic is advanced is by giving the students the length of the hypotenuse and either the length of the base or the height, but not both. Students must use a^2 +b^2=c^2 in order to solve for the missing side length. The side lengths will not always be an integer, so students should be comfortable with working with square roots. Once students utilize the Pythagorean Theorem, they can then continue to solve for the area of the right triangle as they previously learned in sixth grade. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

In this short music video young students at Builth Wells High School did a parody of Meghan Trainor’s “All About that Bass.” They take the chorus and put the lyrics in “multiply the base, by the height, then half it.” This music video can help several different types of learners in the classroom. Some need a visual aid which is done by specific dance movements by the students in the video. Others will remember it by having the catchy chorus stuck in their head. The parody lyrics are also put on the video to help students who might struggle with English, such as ELL students. Plus, it is a good visual cue to have the lyrics on the screen so it makes it easier to learn. No doubt with this catchy song, students will leave the classroom humming the song to themselves and have connected it to finding the area of a triangle.