# Monthly Archives: September 2015

## Reasoning with Slopes

My geometry class has started investigating shapes on coordinate grids. Doing this reminded them that they knew how to use the Pythagorean Theorem and how to find the slope of a line segment. Here’s an example of what they worked on:

In looking at the slopes of side AB and side BC, which I wrote on the board as slope of AB = $\frac{+3}{-3}$ and slope of BC = $\frac{+3}{+5}$.

Contemplating the situation, one of my students asked, “If you combine those two slopes, will you get the slope of the other side?” I’m thinking that by “combine” he means “add” and of course you cannot add these two slopes to get the slope of the other side. But instead of saying that I asked him, “What do you mean by ‘combine’?” He responded by explaining that if you add the numerators and denominators separately, it seemed like you would get the slope of the other side. He was thinking about this:

(+3) + (+3) = (+6) and (-3) + (+5) = (+2) which leads to a slope of $\frac{+6}{+2}$. But why would that be true? Vectors. It turns out that this student was learning about vectors in his physics class. I’m not sure that he consciously made the connection, but he did seem to be thinking about vectors. If you travel from A to B and then from B to C, you will have traveled 6 units up and two units to the right. That’s the same result you would get with vector addition.

Imagine what would have happened if I had asked the wrong question, or just replied without asking the question that I did.

Filed under problem solving, teaching

## Another year begins

It seems like school ended just yesterday, or maybe the day before. And yet the new school year is already up and running.

We had two days of orientation last week – organized by students, with adult assistance – and they were really great. The first day was once again at Fort Williams Park and the weather, though quite hot, was sunny. Three years in a row. How can we be so lucky? The difference this year was that 9th and 10th graders were organized by paired advisory groups. Each group had a couple of juniors with them, as mentors, and their adult advisors, of course. The team-building activities were facilitated by juniors with adult assistance. The senior class was off on their own, team building at a ropes course nearby. They joined us at the end of the day, so we had most of the school together, out there under the trees, to debrief the activities of the day.

On Thursday, we were in Portland, at Baxter. Since we have all four grades for the first time this year, we are squished into a building that is too small. So, we needed a satellite campus. This was not a surprise. In fact, several people have been working on this problem for a couple of years. Every time they thought that they had a solution, something got in the way. Late in the summer, the building search team was able to secure a second space in Portland, a short walk from our main building. The space is large enough to house a few (like three) classes, but small enough so that the 9th graders, who will be spending their mornings there, will have a cozy space to make their own.

Once again our faculty has expanded. We’ve added art, engineering, and computer programming teachers, along with math, science, humanities, and special education. We have an outstanding team that shows a great commitment to our mission and purpose, even as we continue to refine what that is.

My year looks to be very interesting and challenging. I have a whole bunch of new courses to create, starting this term with Introduction to Logic and Transformational Geometry. Were those really my ideas? What was I thinking? The first time through is always the most difficult. It will be nice when this year is over. I think that we will have some pretty solid classes.

We’ve emerged from the terrible two’s to begin our third year. We’re toddlers. We’re ready.