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.

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Step Two – Draw a line through both marks.

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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).

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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.

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

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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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Applying Bloom’s Revised Taxonomy to Mathematics

A few months ago, I designed a training on applying Bloom’s Taxonomy to create higher order thinking questions in mathematics.

I showed two figures: (1) A right triangle and (2) an equilateral triangle.

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Right Triangle

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Equilateral Triangle

The following were questions that I developed to help the instructors think through the various levels of Bloom’s Revised Taxonomy in regards to mathematics.

  • Knowledge – What can you tell me about the first triangle?
    • The students provide any information they know about the mathematical concept.
  • Comprehension – What makes the first triangle a right triangle?
    • The students use the information they already know about triangles to rightly identify a specific triangle.
  • Application – Based on what you know about right triangles, why is the second triangle not a right triangle?
    • The students apply the information they already know about triangle to differentiating one triangle from another based on their characteristics.
  • Analysis – How is the first triangle similar to a rectangle?
    • The students compare the characteristics of a right triangle with those of a rectangle.
  • Evaluation – How would you prove that all right triangles fit in a circle, with each vertex (or corner) of the triangle touching the circle?
    • The students extend their understanding of triangles by proving a well-known theorem of geometry (see Thale’s Theorem)
  • Synthesis – How could you use the right triangle to design our next engineering project?
    • The students integrate information they know and understand about the right triangle into designing a new project.

Any comments/suggestions?

The Möbius Strip and the Klein Bottle

I have always been interested in figures like the Möbius Strip and the Klein Bottle. The Mobius strip is a is a one-sided nonorientable surface and the Klein Bottle is a closed nonorientable surface. Both figures have a Euler characteristic of 0. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle can be embedded in R4. As an aspiring geometer, I always found it intriguing to visualize both of these figures. Since the Möbius strip can be embedded in three-dimensional Euclidean space R3, it was much simpler to visualize than the Klein Bottle. Still, it was figures like the Möbius Strip and the Klein Bottle that inspired me to pursue geometry beyond the foundations of Euclidean geometry, namely hyperbolic and elliptic geometry. Interestingly, the angles of a triangle in hyperbolic geometry add up to less than 180o.

 

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Möbius Strip

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 Klein Bottle