We started with this exercise in geometry yesterday for a few reasons:
1. We want students to see that there are multiple approaches (and sometimes, even multiple solutions) to problems.
2. We want students to realize that their answer is no good without an explanation.
We got some great answers. We are using TI-Nspire technology in our classrooms, made complete with the TI-Nspire Navigator System, and so students typed in responses on their handheld. They had lengthy explanations, and so I’ve shown only a few below:
In our discussion, we ended up talking about prime numbers, the Riemann hypothesis, the Fibonacci sequence, square numbers, factors, divisibility, powers of 2, binary numbers, and more. Some students answered with words, and some students used symbols. All students entered into the CCSS Mathematical Practices: Look for regularity in repeated reasoning.Construct viable arguments and critique the reasoning of others.
We also looked at a few “snapshots” from geometry in a TI-Nspire document. We wanted students to learn how to grab and drag a point and to begin to notice what stays the same and what changes when they move an object. We wanted students to begin to make conjectures about what could be happening mathematically. Our snapshots came from Geometry Nspired activities, the one below from Transformations: Translations.
On this page, students noticed that the measure of angle A and the measure of angle A’ were always equal. They noticed that the areas and perimeters of the two triangles were always the same. We asked them whether they could make the red arrow point up, and so they spent some time making the connection between the vertical translation and the distance between A and A’.
Our most important work in class yesterday, though, was that we began to establish our community of learners. We began to learn how to listen to one another, and we began to get brave enough to talk to one another.
I am looking forward to our next class meeting tomorrow (we are on an A/B block schedule), and I am hopeful that at least a few of the students are, too.