# Engaging students: Area of a 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 Kelly Bui. Her topic, from Geometry: finding the area of a triangle.

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

As an activity, possibly the “exploration” part of the lesson, students will be paired in partners and the instructor will provide each pair with a different rectangle or square. The goal is to find the area of half of a rectangle.

The condition they must follow is that they cannot “draw” a straight line across the shape, they must “draw” a straight line starting from a corner. At some point, it should be evident that you can only draw a straight line from a corner to another corner. By drawing a diagonal line across the rectangle, they will now have two triangles (if that isn’t clear to them at this point, let them realize it on their own or go over it as a class at the end of the activity). Using rulers or meter sticks, they will have to discover on their own what the area of half of the rectangle is along with what the formula for that looks like. Most students will probably take the area of the entire rectangle and divide by 2. Once they come up with a formula for the area of half of a rectangle, it should look like A=1/2 bh, tell each student to raise half of the rectangle they cut, and announce: “congratulations, you have found the formula for the area of a triangle.”

How can this topic be used in your students’ future courses in mathematics or sciences?

Students begin to see the formula for the area of a triangle in 6th or 7th grade. They know the formula, but often times they don’t understand where it comes from. It can be useful for future homework/test problems that ask for the area of an irregular shape as well as in algebra with unknown lengths. These types of problems require students to think “outside of the box” in order to find the area of an irregular shape. It is not always evident that the irregular shape is simply made out of polygons.

Additionally, this topic will be useful when students are in algebra and they must solve for the area of a polygon that doesn’t have specific dimensions. For example, the trapezoid below has an unknown height as well as an unknown base. It is good for students to know how to apply the area formula of a triangle to solve for the dimensions as well as the area of the entire trapezoid itself. One important thing that should be stressed in the classroom is that formulas are extremely helpful on their own, but they’re even more helpful when they can be applied to different applications.

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

Most students know the formula A=1/2 bh for the area of a triangle, but many students don’t remember the formula used to find the area when all three side lengths are known. Heron’s Formula: √[s(s-a)(s-b)(s-c)] is briefly mentioned in geometry and is often not used in other math courses in high school. Along with his derivation of Heron’s formula, he contributed greatly to ancient society.
Heron of Alexandria was a Greek engineer and mathematician who was known mostly for his work with geometry. He was also a lecturer at the Library/Museum of Alexandria where he would meet with other scholars and discuss work. Additionally, he wrote Metrica, a series of three books which included his work on area and volumes of different types of figures.
It is no secret that Heron had a brilliant mind, and with his engineering and mathematics background, he was actually ahead of the industrial revolution that would take place centuries later. He invented the “Hero Engine, also known as the “aeolipile,” which was powered by steam. Essentially, Heron was the first inventor of the steam engine.
Another one of Heron’s inventions was the “wind wheel,” which is very similar to the modern windmill.

Students will already know that there were many breakthroughs during the industrial revolution, but some of the machines and inventions implemented in the 1800s were actually ideas that were invented centuries before.

Irregular Shape Image: http://www.softschools.com/math/geometry/topics/the_area_of_irregular_figures/

Heron of Alexandria: https://www.britannica.com/biography/Heron-of-Alexandria

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