Naked Problem: Find the missing side…and that is it!
Our Unit on Right Triangles went as follows:
- Pythagorean Theorem
- Pythagorean Triples
- Converse of the Pythagorean Theorem
- 45-45-90 Triangles
- 30-60-90 Triangles
- Trigonometry (basics only ratios of sin, cos, tan)
Each lesson was another isolated idea. It was bad but being new to High school math, my choices were limited to following the course as given.
That was year 1-2
By year 3, I became a rebel without a cause. I attended an AVID Summer Institute and connected with other math teachers all over the country and they were progressively different. I learned how to use questioning, specifically Costas Levels of Questioning.
Feeling empowered, I was able to revamp all lessons to incorporate rigor. My students were now writing questions. They were learning HOW to learn, which was and is crucial to their growth in any area.
I changed the unit entirely to focus on strategies for right triangle applications. Our lesson was all inquiry based where students investigated characteristics of various triangles and developed their own meanings. We used graph paper to cut out squares and rectangles and compared various measurements to prove and disprove their ideas. We even debated the idea of generating pythagorean triples. Our wood shop kids were incorporating right triangles in a project that they were building. They were able to bring their “thinking tricks” to their peers and it was amazing to pull in their expertise. Our learning community was growing stronger!
Applications like geogebra were used to build applets.
Here’s a novel idea…
Instead of using one of the gazillion “solver” apps, have students create their own with Geogebra. For special right triangles, this was especially awesome because students had to consider what shapes, if any, to start with so that the result was always a special right triangle. Hint…Hint…Equilateral triangles, squares, midpoint, segments hidden objects, polygon tool, hide/show. Yes, that is all “geogebra talk” but so worth it! Also, hexagons are fun. Click HERE to see an example…different lesson, same idea.
One of my favorite “aha” moments was asking students to determine which method would be most appropriate and why. They didn’t need to solve…just think and justify. Below is an example from this year from a class in my school. We used padlet to post and communicate.
I had many other lessons such as this but this particular one stuck out because it was my first experience with my students connecting with math in meaningful holistic ways and that was career defining.