I took base-10 for granted during all of my schooling. This makes me sad. It was only when my math ed professor gave a lesson on various bases throughout history that I felt I had grasped a strong sense of what numbers actually are (although this ended when I saw this numberphile video).
But more to the point, when we think it is important to build number sense, it is very useful to see multiple ways of representing one number. Introducing binary, for example, is not only interesting to students but immediately applicable in the digital age.
This week I enjoyed talking about logs, introducing my students to how their parents (although more likely grandparents) did long multiplication problems using log tables, such as the one shown here:
Despite being slightly horrified at how long it took for one problem, the students still were fascinated with how the tables worked. After doing a lesson on exponents and logs, they were even able to get a good sense of how the tables worked, building up their number sense in the process.
When I was taught logs, I was shown a set of rules and informed how to use them. This is missing something of the power that logs have to deal with exponents. Comparing these rules to laws of indices that they already know is essential to get the idea across.
In other news, when I think of exponential growth I start by showing what an exponential graph looks like and ask them to come up with some real world situations that may follow this trend. Investment usually comes up as does population growth. This gives a great opportunity to discuss whether or not a pattern will continue just because that is they way it has been so far. Here are some video clips that go nicely with this topic:
Pumphrey's Math 