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 Brittney McCash. Her topic, from Geometry: defining the terms acute triangle, right triangle, and obtuse triangle.

**D2. How was this topic adopted by the mathematical community?**

** **As the students are walking into class, I will already have a picture of just a standard (acute) triangle on the board (be it Promethean or white). As class begins, I will pose the question of, “Who can tell me what we are looking at?” Of course, the students will tell me a triangle. I will then proceed to show two more triangles, an obtuse and then a right triangle and ask the same question. The answer will be the same for each. After I show all three, I will put a picture of all three together and ask the students what some of the differences are in each. Once we state the obvious (That there are angles of bigger and smaller sizes in each), I will then post a picture of Euclid. I will ask if anyone knows who this is. More than likely no one will. I will then proceed to tell my students that in 300 BC this man, Euclid, wrote a book called *Elements*. In this book. We had 4 sub books that consisted of mainly triangles. When telling this fact I will put emphasis on the word “whole” to show how insane that is. By now, the students should be in awe that someone could write so much about triangles. Then I would state that inside this book, Euclid proved that there were 3 different types of triangles. There is obtuse, right, and acute. We could then discuss as a class what we think each triangle presented at the beginning of class is just by sheer guess, and then see if they were right by then going into the actual discussion of the definition of each term. This is a fun and knowledgeable way to bring in some historical background of what they are learning. This shows that it’s just not going to go away, that it has been around for a while, and is still being thoroughly discussed in classrooms, like ours.

**A1. What interesting word problems using this topic can your students do now?**

** **An interesting topic that is still around today, is the Bermuda Triangle. After a brief introduction of the definition of an obtuse, right, and acute triangle, I will pose this problem: (There will be a picture of the Bermuda triangle with points labeled, (posted below).)

You are captain of the ship *Euclid *and are sailing straight for the Bermuda triangle. Hearing of all the bad things that can happen inside the “triangle,” you want to avoid it as best as you can. Luckily for you, you have a super power. You are able to shift one point of triangle wherever you would like. Using your super power and the knowledge of triangles we discussed previously in class, decide which point you would move, and into what triangle so that you can sail past smoothly. You will need to draw your final result with a justification of why you chose that triangle and point.

This question is not only engaging, but it makes them think abstractly. They have to use their knowledge of triangles and produce a result that fits our discussion. Then not only do they have to draw it, they will need to discuss it as well. Talking about why they chose the method they did, helps students retain and process the information better. Take into account, there are multiple ways to answer this question.

**B1. How can this topic be used in your students’ future courses in mathematics or science?**

** **Triangles are such a widely used topic, that it is almost guaranteed you will see them again. Not only will you use them later on in our course (Geometry), but you will be using them in pre-calculus and so forth. The main triangle you will work with in the future is the right triangle. That is why it is so important for you to learn the difference now between the triangles. Later, you will be discover the different ways you can solve for sides and angles with a right triangle, you will be discovering the different properties that come with each triangle, and how you can draw them using circles. But before you can do any of that, you have to start with the basics, like knowing which triangle is which and their definitions. I would then go in to explain that now only would triangles be used in classes, but in the real-world as well. They are everywhere we look, literally. Every time we look, we are looking at a specific angle. In the video games we play, we are always making decisions based off of the angles we can use, it’s how we build things; it’s everywhere! To have a basic understanding of something so usable in our world, would be essential to success!

**Resources: **