There have been a few topics that I have really struggled with knowing whether or not to include in our geometry course this year as we try to embrace the CCSS Geometry standards. One of them was Pythagorean Relationships. Do we need to provide students an opportunity to determine when three given segment lengths will create an acute, right, obtuse triangle – or not a triangle at all? I have always enjoyed using both the Triangle Inequality Theorem and Pythagorean Relationships from Geometry Nspired. And I have read and re-read CCSS 7.G.2:
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
And I must admit that I am worried about this learning objective in grade 7. Will grade 7 teachers realize that this is The Triangle Inequality Theorem, which turned out to be a pretty important theorem in my study of mathematics? How will grade 7 teachers provide students the opportunity to explore when three segment lengths do or do not determine a triangle? Will they provide students the opportunity to explore? Or will they just tell them the necessary conditions for three segments lengths to determine a triangle?
And then more questions – Should we infer from this learning objective that we students should know when a triangle is acute, right, or obtuse? Or is this one of those topics that we might still want to provide an opportunity for our students to learn even though it won’t be CCSS-tested? My dilemma really has to do with time. I could have just told my students the relationship to classify a triangle by its angles using its side lengths, but then again, I really can’t. Since I have been teaching geometry through student exploration and observation, I find that I really can’t ever make generalizations for them. It goes against all that I have come to believe. So I knew that if my students were going to learn about Pythagorean Relationships, it was going to take most of the class period to get them to the point of making those generalizations. Finally I asked my seniors whether I needed to discuss the topic with my geometry students. They have taken several high stakes standardized tests, and they answered with a resounding yes. So we spent a day on Pythagorean Relationships. At least for now.
I enjoyed reading Hannah’s journal on look for regularity in repeated reasoning.
We had a good day in class – students explored and thought. They entered into the practice of make sense of problems and persevere in solving them. It took them a while to determine the relationship for classifying the triangle according to angle measures using the side lengths. But by the end of the class, they were able to articulate that relationship. And the results that I have gotten on subsequent questions regarding whether a triangle with three given side lengths is acute, right, obtuse, or not a triangle provide evidence that students have not only made sense of the mathematics but also remember it.
And so the journey continues ….