Good Bounce.

I finished up my unit on isometries today with my geometry classes and with the few extra minutes that I had, I began showing them how to predict where a ball will bounce after it hits something. At first, they kind of looked at me and wondered why I was explaining this to them, but once they made the connection to reflections and what we had been doing in class, they were hooked. I explained to them that the angle in incidence is equal to the angle of reflection (which was something I assumed they knew) and I actually had a student raise his hand and say "I actually didn't know that before just now and I can use that the next time I play pool." In the past when I taught this, my students blew me off and didn't really care. This year was different: they wanted to learn more. After we went through a few 'nice' examples of bouncing balls off walls that were perfectly straight, someone asked me about what if the wall was curved; could you still make a similar prediction as to where the ball would end up? This goes a little beyond the scope of the course, but I figured if they're asking I'm certainly not going to stop them. I explained to them all about tangent lines and points of tangency and everyone learned something.
Throughout the entire unit, students were asking me, 'Why do I need to know this?' I told them we were getting there and I wouldn't let them down, to which most of them went back to not caring. Perhaps next year I should start with this lesson. Thinking in terms of a 3 Acts-type format, I could show them a video of someone playing pool, bouncing a ball off a wall and stop it before it stops asking, 'Will it go in?' Or I could show a clip of a Dennis Rodman jumping up for a rebound and just before the ball bounces off the rim, 'Is he in the right spot at the right time?' (by the way, my students are always in disbelief when I tell them that Rodman holds the rebound record because he analyzed the angle the ball came in and he figured out where it would bounce to). This might catch their interest, we can dive into the material, and then come back to it in the end to calculate an answer.
Any ideas?

Because It's The Cup

In the spirit of the playoffs beginning today...

What's the first question that comes to mind?

How Awesome.

The above image was a problem that I presented to my class a few weeks ago. We were concluding our unit on area and perimeter, so I decided to kick it up a notch and ask them to find the area of this figure. We had practiced irregular figures such as this one already (actually about 90% of what we did was irregular figures due to the fact that they are literally everywhere in the world), but I don't think anything this abstract or complex. I just made it up on the spot, purposely giving them little information. Below is what occurred when a student volunteered to solve it on their own in front of the class.

Naturally, I did not turn this brave individual down. I encouraged the class to copy down the problem and try it on their own while this brave young soul gave it a shot. The picture above does not due the entire situation justice. As a geometry teacher, I try to teach my students the curriculum (obviously) but more importantly I teach them how to think, a skill that many lack. The conversations that went on throughout this problem (I'd say it took them a good 20-25 minutes until they collaborated, asked questions, fixed mistakes, etc.) were absolutely amazing. At one point, I looked at the clock and realized how long they'd been working; I considered stopping them but I let them go. Why should I stop them from working and thinking original thoughts? I come back to a previous point I made: as a teacher its my job to make sure that they are thinking and problem solving; nothing should stand in the way of that, its in their best interest (but more on that later). Just look at the difference in pictures; I love when students write all over diagrams. It sheds light on their thought process and how they internalize information. As I said, I intentionally didn't draw this to scale and left certain parts unlabeled. I actually tried to give them as little information as possible so they would be forced to think and ask questions. Ultimately, I could care less about their answer, I want to know what's going on in their minds. They had to ask if the 'triangle' on top was isosceles or if the bottom was a rectangle or if both sectors were ninety degrees or if those two vertices were 'midpointish.' And most of the time, I replied with vague answers and forced them to come up with ways to verify their own questions (in the beginning of the year, they hated this, but now they're used to it and rise to the challenge). The majority of the students came to an answer, and most were correct. Those that weren't understood where they went wrong when we discussed it as a class. Its amazing what happens when you just let your students experiment even if it means taking extra time. 

I try to do stuff like this all the time. Some of my colleagues have argued with me because its not real-world, or because it takes 25 minutes for one problem, or because its not entirely written 100% into the curriculum and there is no time. All of these are bogus (while relevancy is an important aspect of teaching, especially math, it is not the only aspect that should be focused on). In the end, if its good for the students, I'm going to do it. I didn't have one student complain about this problem and I had 100% participation without fighting with them. In talking with them, I could tell that many of them learned from it. Students love a challenge when they know they have a chance at success, and that's what I gave them.

I could go on and on about this experience, but ultimately it comes back to a question I pose to my department members on a regular basis: If it's good for the students, why would I not do it?

Curiosity?

Now I'm curious. I wonder what this would like. Would you be able to predict the number of letters in any number (even though it would be easy to count)? What kind of relationship exists? I smell a project....

12 Inches of Math (informer!)

We built our first snowman with our daughter this weekend, and it was a tremendous amount of fun. She didn't do much work, but she gets excited everytime she sees it.

I've been trying to harness my inner-nerd this year; trying to find math in everything I see and do. I figure if I want my students to see math in the world, I need to be able to see it too. I imagine this is how Mystery Guitar Man approaches his videos: searching for music in his everyday surroundings and discovering how he can use those sounds to create something new and original that is both visually and audibly pleasing in order to send the message that music is everywhere (hence my previous post, worlds collide). I feel that I'm getting better at recognizing various situations and their mathematical values, but it's taking some time to develop this mindset.

After building our snowman, I realized that there really wasn't that much snow on the ground to begin with, which you can tell from the dirt and mud in the picture, it was just the perfect wet snow for building. I wondered, how much snow did we use in building that snow man? If we would've used all of the snow in our backyard, how big would our snowman be and would that beat the world record? How much snow actually fell (not just how many inches deep)?

I realize that if I'm following the mathematical storytelling model, you should come up with these questions, but they were just some thoughts. Essential info will be posted later (before the snow melts).

We Built This City



How tall is he? And how much does he weigh? How much does he have to eat in order to live a healthy lifestyle? How does he rock so hard?

Betelgeuse, Betelgeuse, Betelgeuse!!

I was talking with one of my colleagues about how we can improve problem solving skills within our students to encourage higher-order thinking skills. In the past, I've shared with him Dan Meyer's 3 Acts and he was as amazed as I was. Together, we've been working on creating problems for algebra1, 2, and geometry that are set up for this process, and he shared with me a video similar to this one that he showed his students. It's a simple video; there's nothing fancy about the way its made and there is no narration, but the possibilities are endless in the way its interpreted. He used it to illustrate scientific notation. My first thought was to ask the question, 'How many Earth's fit inside Betelguese? How many suns fit inside VY Canis Majoris?' You could relate it to similarity, 3-dimensional space, fractions, or a hundred other topics. Regardless of how I end up using it, I plan on removing the diameters from the video so that my students can do some research to find the information that they think they need to answer their questions.
This video further illustrates to me how many questions can be generated from any picture or video. The phrase 'a picture is worth a thousand words' has become more real. I now try to look at my everyday surroundings from a mathematical perspective. I always tell my students that if math didn't exist, nothing else would either, but they always blow me off. Now I'm beginning to gather solid, concrete examples that they can see. I think the way I'm going to approach it from now on is show my students these examples so we can explore them as a class, and then use them to illustrate that math truly is everywhere. If I can get them to believe this, them I'm one step closer to getting them to appreciate everything that happens around them at any given moment.

So, what are your thoughts? What questions come to mind when you see this video?