How Do I Love Thee? Patterning

Module 3: Assignment #1

Right triangles are a key topic in Geometry and relate to many other concepts that we cover including the Pythagorean Theorem, volume, area, proofs, and trigonometry.  It is important that students see patterns in figures in order to solve problems and proofs in Geometry.  When I thought of existing patterns with right triangles I immediately thought of special right triangles.  We usually cover this idea after learning the Pythagorean Theorem with some direct instruction on the shortcuts to the figures you see on the left below.  The shortcuts are x – x – x root 2 for 45-45-90 triangles and x – x root 3 – 2x for30-60-90 triangles.

We don’t spend a lot of time going over where the shortcuts came from.  Students are just expected to memorize the shortcuts and be able to use them, which is really hard for some students.  I think it would be even more effective to have students explore the patterns themselves.  In the chart below I displayed these patterns that the shortcuts come from.  For example, since the Pythagorean Theorem states that a squared plus b squared equals c squared we know that in a 45-45-90 if each congruent leg was one unit long, the hypotenuse would be the square root of that sum, 2.  This continues for any 45-45-90 triangle.

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In addition to the exploration, students should come up with their own patterns because “making patterns for oneself is a lot more fun than memorizing” (Root-Bernstein, 1999).  I noticed that the some of the numbers and vocabulary terms with special right triangles rhymed so wrote a little chant or poem.  It has patterns in the way I decided to write it as well as what it represents.

It was very difficult for me to explore and find some new patterns using right triangles and the Pythagorean Theorem.  I had to really push myself to plug in some numbers and play around with the theorem.  Using the chart below I plugged in the same number (1) for the a value in every triangle.  I then took my previous c value as my b value for the next triangle. To be completely honest, I saw there was a pattern, however I didn’t know what it meant.  I went to Google and found out my answer within a few minutes of searching.  There were tons of images, like the shell I drew below depicting the exact pattern I had found.  Amazing!

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This process of patterning is can be difficult, but is important for any learner.  Through this assignment I not only gained experience patterning, but gained a deeper understanding of my content as well.  I think too many times students are just memorizing information to get by and pass the test.  As teachers we need to foster deeper learning and understanding through cognitive tools like patterning.  It is natural to look for patterns all around us, let’s not lose that when teach our students.

Root-Bernstein, R. S., & Root-Berstein, M. (1999). Sparks of Genius: The 13 Thinking Tools of the World’s Most Creative People. Boston, MA: Houghton Mifflin.