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 Trenton Hicks. His topic, from Geometry: finding the area of a right triangle.

As an engagement activity, give students the following problem: “A rectangle has dimensions of base: **b** and height: **h**. How many different ways can you cut this rectangle in half with a straight line? How many shapes can you make from the different ways you cut the rectangle in half? Now out of those, which shapes are not rectangles?”

This will leave only two different ways to cut the rectangle in half, yielding identical triangles on either side of the line. Now, ask the students, “From what we’ve already learned about rectangles, what would be the area of this rectangle?” After confirming the area is base times height, wait a few moments before saying anything else. Now that the students are thinking about the base, they will now start to make predictions about the triangles that we’ve just made and their areas. Have them write their guesses for the formula of the triangle down on a piece of paper, and keep them to the side through the lesson. From here, we can break them up into groups and give them 3 right triangles to solve for the area, and one equilateral triangle to solve for the area. Go through the answers together and compare groups’ answers, as well as their predictions on what the area of a triangle is. The odds are, many groups will be stuck once they get to the equilateral triangle. If so, you may want to send them back to their groups to try and find the area, giving them the hint, that they may have to make new shapes, just as they did with our rectangle at the beginning. This lesson assumes that the students understand the pythagorean theorem so they can solve for the height of an equilateral triangle by making a new triangle. This way, the students can explore the phenomena of triangles’ area, and see if they can recognize that the height isn’t always a side of the triangle, but rather something they may have to solve for.

The students should be able to use similar techniques to find the area of a parallelogram, trapezoids, and other shapes, as these shapes are partially composed of triangles. As students progress to more complex 2-dimensional shapes, you can derive formulas as you go. As you move onto 3-dimensional shapes, we actually see lots of different triangles appear in the shapes’ respective nets. For instance, when computing the surface area of a triangular prism, we need to know how to compute the area of the base. We also see this same idea in computing volumes of triangular prisms, where we need to know the area of the triangular base. This is also applicable to pyramids, tetrahedrons, and octahedrons. Finally, these ideas are brought up again later in trigonometry where we can determine different parts of the formula with trigonometric ratios and functions and whenever the students learn Heron’s formula.

This concept of finding the area of a triangle expands many things that the students may already know. This won’t be the students’ first time seeing a triangle, nor will it be the first time they compute the area. Overall, this content should be a refresher and not new to the students. However, this may be the first time that the students are presented a rectangle and told to make a triangle out of it. From that point, they are told to make conclusions about the triangle’s area based on the rectangles area. As students think through this, they are using logic and reasoning to argue what makes geometric sense to one another. This further develops their mathematical reasoning skills, which may be a bit rusty since we far too often focus on the “what” and not the “why” and “how.”

Students can check their solutions with TrianCal ( http://triancal.x10.bz )