CCSS-M G-SRT.B.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
We’ve been teaching our CCSS Geometry course for three years now, and this is the first year that we have been able to spend more than a little class time on proofs of the Pythagorean Theorem. (Our students are coming to us knowing more mathematics than three years ago. Our students are coming to us more willing to take risks and use the Standards for Mathematical Practice than three years ago. We are making progress just in time for our legislators to decide that collaborating with other states to write standards and assessments was a bad idea.)
We started with the Mathematics Assessment Project formative assessment lesson (FAL) on Proofs of the Pythagorean Theorem. This FAL is one that includes student work. Students focus on SMP3: construct a viable argument and critique the reasoning of others.
As students practice look for and make use of structure, I asked them to share what they noticed and wondered.
Then we looked specifically at a diagram drawn to scale, and students noted what they knew to be true (and why).
As we started to examine the student work proofs so that students could critique the proofs, SC asked to go back to the previous page. I wonder what will happen if we reflect the outer right triangles about their hypotenuses into the center square.
What do you think will happen?
The triangles will make a square.
I think I’ve said before that technology slows me down in the classroom. Students notice and wonder more than they did before, and the technology gives us the chance to see what happens so that we can make sense of why it happens mathematically. I am not the only expert in the room. The student who gets mathematics without technology is not the only expert in the room. Our use of technology increases our confidence and lifts all in the room to experts. And so the journey continues …