Last time I described an idea about how to use matrices to study simple weather models. Really simple weather models; in fact, the model we used was a two-state Markov system. And like all good simple models, it was interesting enough and at the same time inaccurate enough to give us some meat to chew on.

I used it as one session in a teacher institute I just helped present (October 2019), where “matrices” was the topic we were given for the five-day, 40-contact-hour event. Neither my (excellent!) co-presenter Paola Castillo nor I would normally have subjected teachers to that amount of time, and we would never have spent that much time on that topic. But we were at the mercy of people at a higher pay grade, and the teachers, whom we adore, were great and gamely stuck with us.

One purpose I had in doing this session was to show a cool use for matrices that had nothing to do with solving systems of linear equations (which is the main use they have in their textbook).

Some takeaways:

Just running the model and recording data was fun and very important. Teachers were unfamiliar with the underlying idea, and although a few immediately “got it,” others needed time just to experience it.Making the connection between the randomness in the Markov model and thinking about natural frequencies did not appear to cause any problem. I suspect that this was not an indication of understanding, but rather a symptom of their not having had enough time with it to realize that they had a right to be confused. The diagram of the model was confusing.

Let’s take the last bullet first. The model looked like this:

[image error]Our two-state Markov weather model. Use one die to update today’s weather to tomorrow’s.

Of course I also explained it in words: “We will all start on a sunny day. To find the next day’s weather, roll one die; if you get a 1–5, tomorrow will be sunny. If you get a 6, it will rain. If it’s raining, you look over here to find the next day’s weather. Notice how it’s different…” and so forth.

But the numbers on the arrows flummoxed some teachers. One improvement might be to put the numbers closer to the circles that represent the states, like this:

[image error]

Another idea we came up with was to make spinners, one for each state:

[image error]Imagine these are spinners!
Spin “today’s weather” to find tomorrow’s.
This looks almost horoscopical