Wednesday 23 January 2013
There are plusses and minuses of living in any particular location – and one of the plusses of teaching in the north of Scotland is that during the winter months we sometimes get enough snow that school is cancelled. Not everyone is going to consider that a plus (and depends on what particular day it occurs) but when you live near a nice hill for sledding (or "sledging", if you’re British) – as I and my family do – then a day cancelled due to snow means some fun sliding down a hill on some piece of equipment that usually offers little in terms of directional control (but a very low coefficient of friction). Today our school was closed due to snow, so me and my 13-year old daughter spent some time on our local sledding hill. My inability to control the direction (and spin) of my sled is obvious in the video below. I was not thinking about school - just enjoying the snow - but once I got home I did have some thoughts about the association between snow and mathematics, and even an idea for a good student exploration that would fit into the new internal assessment program. See below the video.
Snow and Mathematics
After returning home from the hill and warming up with some hot chocolate– and knowing that snow would be a topic of discussion at school – I thought of opportunities for mathematical exploration in connection to snow. It just so happens that I came across a wonderfully written short piece on associations between winter and mathematics in a recent edition of The New Yorker magazine submitted by Gregory Buck, a professor of mathematics at Saint Anselm College in Manchester, New Hampshire USA and called The Wondrous Mathematics of Winter. I may not have my Math HL students read it but I will definitely share it with my Theory of Knowledge classes. Reading it again got me to thinking of some exploration ideas that would investigate how to do some mathematical modelling (or "modeling", if you're American) connected to snow.
We're told that each snowflake displays some innate symmetry and that each snowflake is unique. Apparently this is due to the conditions in which a snowflake is created. A snowflake is first 'born' when water vapor in a cloud freezes around a dust particle or a bit of pollen. As water molecules are added the snowflake 'grows' but the way in which this growth occurs is affected by temperature and humidity which will vary depending on the path the growing snowflake takes to descend from the cloud to the ground. Until recently this process has proven very difficult to model using mathematics. Nearly a year ago, a team of mathematicians based at universities in Germany and the UK succeeded in modelling snowflake growth on a computer using laws of physics and mathematics. See the article Snowflake Growth Successfully Modeled from Physical Laws in the March 16, 2012 issue of Scientific American. The mathematics used in the computer model involved partial differential equations. So this is a topic that is beyond Math HL, even the Calculus HL Option, so it is not a suitable topic for students to research with a possible exploration in mind to satisfy their internal assessment requirement. But I would suggest having students read about this example of mathematical modelling (e.g. the Scientific American article) to illustrate a good real-life example of using mathematics to model a natural process. I do have an idea that involves mathematical modelling and snow that would be suitable for a student exploration - probably best suited for a student in HL.
Modelling Melting Snow
Anyone who lives in a location that gets enough snow to cancel school from time to time will have seen (or been involved in) the building of a snowman. Inevitably the snowman melts away - but at what rate does the snow melt? Of course, it depends on various factors - the most important of which is temperature. If it is known that a particular portion (e.g. half) of a snowman melts away during a certain amount of time (e.g. 12 hours), then can we determine how long it will take for the rest of the snowman to melt? Questions about the rate at which a snowman, or any particular formation of snow, melts is a ripe area for applying some mathematical modeling. And I think that it can serve as a good example of illustrating to students the characteristics of a mathematical model and some insights into how one goes about trying to construct a mathematical model - especially of a natural process, such as snow melting.
I found a nice Java applet that demonstrates dynamically how the differential calculus topic of related rates (only in HL syllabus) can provide a useful way to mathematically analyse how a snowball melts. See Related Rates - Melting a Snowball.
If you have the free Wolfram (Mathematica) CDF Player installed on your computer, you can also interact with a dynamic illustration (shown below) of solving the following related rates problem: If a snowball melts so that its surface area decreases at a rate of 1 find the rate at which the diameter decreases when the diameter is 10 cm.