A Quick Geometry Snippet

So, we were reviewing this morning in Geometry getting ready to finish up our circle unit. I was reminding the kiddos of the angle and arc relationships we have been discussing. I had been writing things like \$latex x=frac{1}{2}left ( a+b right )\$ where x is an angle formed by the intersection of two chords and a and b are the measures of the intercepted arcs. However, today as I drew my diagram for it one of my students suggested I mark is as seen below.

I think I am delighted by this and will write it this way from now on. It feels to me that it is a more natural way to think about this relationship instead of having a coefficient of 2 or a coefficient of 1/2 that might not seem at all intuitive. I then drew the following

I’d love to hear from some other teachers about whether this seems at all like an improvement over the more standard way of writing these equations.

That’s all for now, just needed to get that off of my chest!

Highlights of a Stressful Week

So, there have been many scattered thoughts on my mind in the past week but there are also three things that happened that are just completely awesome.

1. My pal John Golden (@mathhombre on twitter) steered a number of his teacher training students over to a post on my blog and twelve of them chimed in with comments. Totally cool! One of them decided to follow my blog and I took the time to respond to each of them. LOVE the idea of new teachers in training dipping their toes into this rich world of teachers blogging and sharing. I am also flattered that John thought my virtual home here was worth a visit.
2. I woke up Wednesday morning with a message on twitter from a teacher in Louisiana who asked if he could use my Geometry book at his school next year. I am so excited by the idea that this work might be used at another school.
3. In my Geometry class this week we are talking about angle and arc relationships. One of my students stayed after class one day this week and she had this to say. “You know, I was thinking, when will this be important? I mean, when will I need to find an arc length like that? Then I realized that the work we are doing to find that length is what is important. Pretty cool.” Wow.

Some Fun Geometry Action

On the heels of learning some right triangle trig I am really trying to develop more proportional logic with my students. Just this week we had a really productive conversation about the following problem.

Being a bit of a bull in a china shop sometimes, I proposed that we should find the height of each triangle, find each chord length and find the height of the trapezoid by finding the difference of the heights. Not elegant, I know. I was trying to make sure that we remembered some right triangle trig. that we remembered our area formula for a trapezoids, and that we try to develop some patience in solving multi-step problems. That was my plan, but as with many school plans, it did not quite unfold that way. One of my students who is a bright and quick problem-solver pointed out that simply finding each triangle area would be enough. I understand that his solution is pretty much the same as mine, but it certainly sounds more efficient. But as soon as I acknowledged that his idea was more efficient than mine another student trumped each of us. She pointed out that I had already asked them to consider the ratio of areas between the two triangles. So, if we know one area, we can automatically know the other area. If we know both areas, we find their difference as suggested by the first student who chimed in. I was so happy that she took my clue from within the problem and that she was clever enough to really save time and energy this way. I made sure to compliment her in class and I bragged about her work to two of my colleagues yesterday. Oddly, this morning when we were reviewing before today’s quiz I reminded her – and the rest of the class – of her clever idea. She had no memory of this conversation. Sigh…

I’m trying to process this and figure out what it might mean for my classroom practice. I understand that I should be more excited by my students’ ideas than they often are. I understand that I will remember context of conversations more easily than they will because I am not dealing with the cognitive load of trying to learn/understand the conversation. I am simply coming at it from such a different place. What I don’t understand is how a student can be so in command of an idea but then not remember the creative process that made her arrive at this clever conclusion. I discussed this in the faculty lounge right after Geometry today and one of her other teachers intimated that this might simply be modesty on her part. I am not sure how much faith I put in that reading.

So, while I am a bit frustrated and confused, I am choosing to focus instead on the positive energy of yesterday’s conversation, on the clever ideas that my students brought to the table, and on the fact that my students did a really nice job on their quiz today.