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 Andrew Cory. His topic, from Geometry: finding the area of a right triangle.

How can this topic be used in your studentsâ€™ future courses in mathematics or science?

Finding the area of a right triangle opens up the door to all sorts of applications in the future. The next step is the Pythagorean theorem which is used constantly throughout many math courses. The study of right triangles also opens up the world of trigonometry with students will be using in nearly every math course they go on to take. Once knowledge is learned of right triangles, other triangles can be manipulated to look like right triangles, or to create right triangles within normal triangles. Triangles are even utilized when determining things about other shapes as well, such as dividing rectangles into 2 triangles and other manipulations. If they go on to pursue geometry further, the Pythagorean theorem is one of the first couple of theorems proved and used in book 1 of Euclid.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Pythagoras was a Greek philosopher that contributed to right triangles. He is credited with discovering possibly one of the most important right triangle properties. A legend says that after he discovered the Pythagorean theorem, he sacrificed an ox, or possibly an entire hecatomb, or 100 cattle, to the gods. The legitimacy of this legend is questioned because there is a widely held belief that he was against blood sacrifices. The Pythagorean theorem was known and used by Babylonians and Indians centuries before Pythagoras, but it is believed he was the first to introduce it to the Greeks. Some suggest that he was also the first to introduce a mathematical proof, however, some say this is implausible since he was never credited with proving any theorem in antiquity.

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

Applications such as Geogebra can be used for any type of geometry activity. It is a great way to help kids visualize what is happening with shapes in geometry, something that is usually a struggle for students. For helping students understand how to find the area of a right triangle, it can easily be shown that if you take a rectangle, or a square, and cut it in half diagonally, you get two right triangles. And since the area of a right triangle is half of the area of a rectangle or square. The various ways that shapes can be manipulated virtually can be a big help for students that learn in different ways. Being able to view shapes in different ways opens doors for students who traditionally struggle seeing a shape in their head, and using it to solve their problems.

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