Math is everywhere! – fractals on the Franz Josef glacier

One of the stops on our New Zealand adventure was at the Franz Josef glacier on the West coast. We went on the full day hike which gave us plenty of time to explore the various ice formations on the glacier under the careful eye of our guide. Along the way up the glacier, I took the following series of pictures:

All of these were taken on the way up the glacier. Can you tell in what order I took them? If you’re like my students (and a few others I have shown these to), you will likely be incorrect.

I realized as I was walking that this might be because of the idea of self-similarity, a characteristic of fractals in which small parts are similar to the whole. When I showed this set of pictures to my geometry class, I then showed them a great video video zooming in on the Mandelbrot fractal to show them what this meant.

The formations in the ice and the sizes of the rocks broken off my the glacier contributed to the overall effect. Here is another shot looking down the face of the glacier in which you can see four different groups of people for a size comparison:


The cooler thing than seeing this in the first place was discovering that it’s a real phenomenon! There are some papers out there discussing the fact that the grain size distribution of glacial till (the soil, sand, and rocks broken off by the glacier) is consistent throughout a striking range of magnitudes. The following chart is from Principles of Glacier Mechanics by Roger Leb. Hooke:








In case you are interested in exploring these pictures more, here are the full size ones in the same A-B-C-D order from above:

Oh, and in case you are wondering, the correct order is B-C-A-D.

Bugs on your windshield – An introduction to definite integrals

Considering how tired I was this morning on the first day back to school, I could only imagine how the students might be feeling. Today was the first day of our definite integrals unit in Calculus, and I decided to start off class today nice and easy with the following question:

Suppose I pay you to clean the classroom according to the following plan. I’ll give you $400 for the
first hour, $200 for the second, $100 for the third, and so on. If it takes you 6 hours to clean the room,
how much do you make?

They joked about the silliness of the plan and what they would do given this opportunity. Then they got down figuring out the solution. They were a bit rusty and many assumed there was something complicated going on, so some started recalling geometric series and writing functions involving 2^x. These students quickly gave in to peer pressure and just calculated the total directly. It’s always interesting to see how more experienced students decide not to take the simplest route (though in high school, I think I was one of them.)

The other warm-up question I gave for the day was the following:

The images below represent the windshield of the bus during one of Mr. Weinberg’s trips in New Zealand.
The windshield initially had no bugs on it.

The students were a bit annoyed at having to do this, but they got a much needed review of approximating derivatives. Most students used a central difference, with only a couple using just a forward or backward difference. The fact that they did both was really useful during discussions later on about using left, right, and midpoint calculations for integrals.

As tends to occur with my students, especially at this point in the year when they know most of my ideas don’t come out of nowhere, they demanded to see some of the pictures I took. I was, of course, happy to oblige:

I was able to show them a few more actual scenic pictures, which kept things light as they needed to be before diving into the tedium of calculating areas under curves manually.

The rest of the lesson went great and was essentially unchanged from last year, with the exception of using the following data table instead of a table of velocity vs. time:

Originally I was going to start the lesson with this, but added the second warm-up activity when I thought it might seem a bit too contrived to just throw a table like this at them without any feasible way of actually generating it. I also gave the warning that though the values in this table was made up (though some thought that it seemed completely in character for me to actually take the pictures every hour for the purposes of Calculus), it would be possible to generate such a table using the procedure they followed in the warm-up question.

We talked about how we might estimate the total number of bugs during each two hour interval if we knew the rate and assumed that rate was constant. The left hand and right hand sums came straight out of this. A couple students immediately thought about averaging together the two rates to do midpoint, and later on that led very nicely to a visual discovery of the trapezoidal rule. When we looked at what this process then meant graphically, most students seemed to find the overall concept pretty simple.

The mechanics of doing a left/right/midpoint sum with a function initially appeared more complicated, but having them set up the calculation using a table to organize the values (as with the smash rate table) made a big difference.

Overall, I think the students last year got along fine with the more traditional introduction finding displacement from a table of velocity vs. time data. They got the concept fine, as did the students this year when I showed them how it was really the same as what we did. I think it made a difference to be able to introduce the topic in a more quirky way that grabbed their attention slightly more than something that was just plain easy to understand.

Why my trip to New Zealand will make me a better teacher this week….

I just returned today from an amazing three week tour of New Zealand with my wife. My plan is to post photos and captions somewhere in cyberspace, though I haven’t figured out exactly where, and given the start of the new semester this coming week, it may take some time before I am able to do so.

Given that it was the end of the semester before we left, there was no need to even think of bringing work along. Instead, I was able to spend my time focused on the most breathtaking 3,500 kilometers of driving I’ve ever done, giving mountain biking a try (with the scars to show for it), and staring down trails like this:

It amazes me how taking time to completely take my mind off of work and teaching somehow tends to result in doing some of my best brainstorming about work and teaching. Making time for genuine renewal is a real productivity booster. I read The Way We’re Working Isn’t Working by Tony Schwartz a couple years ago towards the end of the school year, an excellent book which explores this idea in depth. I found myself agreeing with all of the concepts then, even though I had done the complete opposite throughout the year. It is counter-intuitive to take a break in the midst of stress – you think about how many little tasks you can get done in the ten minutes you might spend taking a walk, or the thirty minutes you might spend running a few miles, and it becomes too easy to rationalize not  taking a break even though there is plenty of evidence to show that it does good things for you.  It’s the same principle behind the Google twenty percent rule through which employees are given 20% of their work week to work on whatever projects they want to work on.

I made the decision to keep most of my tech toys at home on this trip. I checked email occasionally and looked at tweets, but was otherwise fully immersed in the various adventures we had scheduled for ourselves. It was the right decision, including from a teaching standpoint for this reason: I find myself starting the semester with a big list of ideas for activities and potential projects to engage and involve students through my classroom. I am excited to share my vacation with students on a basic level, but am more excited to show how bug splatters lead to finding definite integrals, or how hiking on a glacier made me think about self similarity. I will share those ideas as I put some structure to them and share them with students over the next week or so.

In the meantime, here is just a taste of another #anyqs that is brewing at the moment:

Finally, a video look at this curious landmark from the North Island: