There are certain topics in mathematics education not appropriate for polite discussion. Number bases other than 10 fit this category well, perhaps because of their association with the maligned “new math” of the 1960s. That’s a shame because there is a lot to learn from them, especially when presented as interactive puzzles.

Below (and here) are eight dials, each with ten evenly spaced tick marks. Press Start/Stop and asking students to observe what happens. How many times does the red counter on the far-right dial move before the counter on the dial to its left moves once? How many times does the counter on the far-right dial move before the third red counter from the right moves once? Can students predict the answers to these questions for the remaining counters on dials four through eight?

Continue by asking students to describe the connection between these dials and our base 10 number system. To make the connection concrete, press Show Numbers to view the count as the red counters move around their dials.

Now comes the real fun: Press the arrow in the bottom-right corner to go to the second dials model. Notice that there are just three tick marks per dial instead of 10. As before, press Start/Stop and ask students to observe the movement of the red counters. How many times does the red counter on the far-right dial move before the counter on the dial to its left moves once? How many times does the counter on the far-right dial move before the third red counter from the right moves once? Can students predict the answers to these questions for the remaining counters on dials four through eight?

All of these questions introduce students to base 3 counting in a natural way, aided by the dynamism of the red counters moving around their respective dials.

It’s instructive (and mesmerizing) for students to watch the counters move for five minutes or more and begin to understand and be able to predict  the pattern in their movements. Students can, and undoubtedly will, hasten the counters’ action by dragging the point labeled speed.

After the counters have moved for a while, tell students to press Start/Stop to freeze their current location. Ask them to determine how many times the counter on the far-right dial has moved over the course of the entire animation. Answering this question is equivalent to converting a number from base 3 back to base 10 (Students can check their answer by pressing Show Total Moves.)

To explore all of these questions in bases other than 10 or 3, simply change the number of tick marks per dial.

I’ll leave you with a challenge from my colleague, Scott: Can you get the number dials to show 33333333? Try it for different bases!

An annotated list of all our elementary-themed blog posts is here.