More Links: Good article, video, tool, problems, game, and activity

What I've stumbled upon in the past two days:

• Can't keep up with the good articles I'd like to read. The De Morgan Forum pointed me to an article by Liping Ma, whose writing always gets me thinking. 
• Malke's love for the math found in stars primed me to take notice of this video by Toomai on stellations. He notices something intriguing that I'd like to explore when I have more time...
•  I think some day I'll want to explore the tools available at SageMath.

And some older ones (as I begin to slowly clear out my backlog...):

• Megan's thoughts on ranking tasks as a way to get students to grapple with definitions. Hmm, I wonder if I could design a good activity for my calculus or linear algebra courses.
• The link takes you to an illustration of adding cubes, but the page that's on is a huge collection of good problems.
• Kate's logic game, whose name has not been decided, looks great. I'd use it in Discrete Math. Reminds me of an old game called WFF 'n Proof.
• On inventing number systems (with third graders).

Links versus Real Writing

I used to share links more often. I used to write substantive blog posts more often, too. Since I've been writing less, I haven't been comfortable sharing lots of links. Didn't want this blog to descend into just a link-share. But it would be helpful to me to have them here. So maybe I'll start sharing my almost daily finds, even if it's not exciting for y'all.

• What is a radian? Here's Sam's colleague's applet (on geogebratube).
• Trig war for review (thanks again, Sam!)
• Measuring the moon's size through geometry (Anna Weltman)
• With an Eye on the Mathematical Horizon, by Deborah Ball (good article, can only be read online)
• Fermat's Little Theorem (the ad before this video is bizarre - if the ad stays the same...)

These were the tabs I've kept open - some for weeks - hoping to figure out how to use some, how to find time to really process others.

Online Conversations: Math Communication, and Understanding Computer Graphing

I am enjoying two online conversations right now.

Michael Pershan asked:

Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their reasoning. Conflict! Drama!

Why do kids hate writing about math?

I am currently trying to grade my students arguments (as prosecutors) for the murder mystery. Some of them really got into it. Most still didn't explain the math well. My take on this is that students will write (maybe even well) if we give them a good enough context.


In the other conversation, Mr. Honner blogged about what happens when you zoom in super far on Desmos, looking for the hole in a rational function. It gets a bit crazy. The conversation got more interesting for me when Alan Eliasen started explaining "interval arithmetic", which I had never heard of.

Starting Circle Trig in Pre-Calc

I'm teaching four classes this semester, which is a lot for me. That's embarrassing to admit - I know most math bloggers are high school teachers, and teach way more hours a week than I do, with more responsibilities for their students. But for me it's a heavy load. So I'm not prepping as much as usual. I've taught calc and pre-calc dozens of times, so I can usually get by with winging it. And, once in a while, I'm able to conduct a better class by improvising than I ever could have with a tight plan.

That's what happened yesterday in pre-calc. The day before that I had worked hard to get their tests graded, so in the morning I printed out the new unit sheet, and walked into class not particularly sure how I wanted to get us started. I had grabbed a problem from my computer, and asked them to start thinking about it while I handed back tests.


The problem:

Consider three circles, all tangent (externally). Their radii are 4 in, 5in, and 6in. What is the area between them? 

I had asked the students to draw a picture. After they had had plenty of time, I drew my picture on the board. Then I asked them how we might start thinking about the problem. A student suggested finding the area of the triangle formed by connecting the centers. I asked if that triangle's sides actually went through the points of tangency. No one answered. Unlike in a math circle, I rescued them be showing a picture of one circle with a tangent line, and reminding them that they likely proved in geometry that the tangent is perpendicular to the radius (the one that ends at the point of tangency). I don't know what that proof would look like. To me, it seems obvious because of the symmetry. (In the afternoon class, they didn't think it needed proving. It already looked necessary to them.)

To find the area of the triangle, one student suggested drawing in the height. We drew it in, but couldn't yet see how to find its length. One of the students suggested that we could find the measures of the angles. They first suggested using law of sines. That didn't work, so we used law of cosines. Sine of that angle gave the height over a triangle side, so we got the height, which gives us area of the triangle. Then we got the other angles and found the sector areas. The afternoon class did it without the height, so they got to use law of sines.

It was a lot of steps for them, but it was a great review of the triangle trig we'd done earlier in the semester. And maybe they got a small taste of what problem-solving looks like.

When we were done, I had just enough time to explain radians to the morning class. The afternoon class had more time, so we worked out the new circle-based definitions of the trig functions.

What are our intuitions about temperature?

I'm teaching exponential functions and logarithms in pre-calc right now. That means it's time to pull out my murder mystery, in which they will use logarithms to solve an important problem - which of their classmates killed John Doe? Since the murder mystery uses temperature to find the killer, I want to lead in with some thinking about how temperature changes over time.

On Wednesday and Thursday, I told my classes a story, and asked them to draw a graph. I said I was mixing some cake batter up to make a Halloween cake. I asked what temperature it should cook at. We decided to set our oven at 350 degrees. (In one class, I talked about how silly the Fahrenheit temperature scale is, but how, even with Centigrade, zero is just attached to water freezing. It's not the same as zero length, volume, or weight. Temperature is different...)

I also asked what temperature the batter was now. They told me room temperature, and we decided that was about 70 degrees. Then I drew axes on the board, labelled them, and asked the students to graph the temperature of the batter over time. Only one person (out of over 50 in the two classes) came close to the right shape. No one seems to have much intuition about how temperature changes. I did this once before, with the cooling coffee we always think about, and got slightly better results.

Here are my approximations of what students thought:

The green one may have been influenced by our attention in the past week to exponential growth, while the purple one seems to have taken the exponential growth we were studying and limited it by the temperature of the oven. I have often seen students give a linear graph like the blue one, and a logistic-like graph like the orange one. No one wants stuff to heat up fast at first, and then slower.

What makes their intuition bad here? Is there a physical experiment / demonstration we could do to improve their intuition? What would make exponential decay feel like the natural choice to them? Maybe cake is the wrong object to be heating?

Please help me think about this.