Hands-on Critical Thinking: Equilateral Triangles

Equilateral TriangleAbout a month ago, I was working with my Geometry class on equilateral triangles. Using a ruler and a compass, I demonstrated how to construct an equilateral triangle. In fact, there are a number of websites that demonstrate this the same way that I did (see below).

After having the students practice drawing their own equilateral triangles, I introduced them to a two-part activity.

First, I gave them each a blank piece of paper and asked them to construct an equilateral triangle with each side equal to six inches. With a paper that is 8.5 in x 11 in, this was a fairly simple task for the students. Many of them were able to complete in a short amount of time. Those that struggled with it were allowed to ask their neighbor for support.

Second, I gave them a second blank piece of paper and asked them to construct an equilateral triangle with each side equal to twelve inches. At first, they looked at the paper and thought it was impossible. They struggled with it for quite a while on their own, sketching out different designs. Then, they asked if they could talk to one of their neighbors to brainstorm and explore different ideas. Soon, the students ended up in groups of three and four, thinking through all the mathematics they knew, trying to figure out how it could be done. Finally, one group asked if they could construct the triangle in parts, cut it out, and piece it back together. That question changed the whole atmosphere of the class. Quickly, every group saw the solution and were eager to share their approach with the class.

Steps for constructing an equilateral triangle with sides equal to twelve inches.

Step One – Measure six inches from the long side of the paper on both ends and mark the paper.

EquiTri2

Step Two – Draw a line through both marks.

EquiTri4

Step Three – From the bottom right corner of the inner rectangle, measure twelve inches so the twelve-inch mark of the ruler crosses the opposite side (thereby creating a diagonal).

EquiTri5

Step Four – Measure the distance from the left side that the diagonal crosses the top side and mark that measurement below. Draw a line through both marks.

EquiTri7

Step Five – Cut the two adjacent triangles and piece together to form an equilateral triangle.

EquiTri8

The great part about this activity was that it led perfectly into our discussion of 30-60-90 triangles. Before exploring the properties of the 30-60-90 triangles, the students were already able to see their use in constructing (and forming) other geometric figures.

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