Way back in 2014, I wrote a blog post titled Covering Your Bases that offered an interactive Web Sketchpad experience with various number bases. As I noted then, “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.”

My thoughts on this subject have not changed, and indeed I now have another interactive number base model to share.

Page 1 of the websketch below (and here) begins in familiar territory with our base 10 counting system. Pick a circular counter and drag it across the vertical divider. You’ll see that “Sum of Counters” is updated and the table in the lower-right corner indicates which counter you selected. As you drag more counters across the divider, both the sum of the counters and the table will update accordingly. Notice that if the top row of the table reads 1 3 5 2, you can understand this both as a way to keep track of counters (You moved one 1000 counter, three 100 counters, five 10 counters, and two 1 counters) and as the base 10 representation of the sum—one thousand, three hundred, fifty two.

Tap in the lower-right corner of the websketch to navigate to page 2. There you’ll find another counter model, but now the counters are labeled 1, 2, 4, 8, 16, 32, 64, and 128. Here are some ideas for questions you might pose:

• What do you notice about the numbers on the counters?
• Drag some of the counters across the vertical divider. How are they recorded in the table? How is this similar to page 1?
• What sums can you make with the counters? Since there are only eight counters, do you think the possible sums are limited?
• If you’re given a sum, can you describe a general method for picking counters with that sum?
• Suppose the top row of the table reads 10110101. What sum corresponds to this number?
• How does this model relate to base 2?

Subsequent pages of the Web Sketchpad model display base 3 through base 9 models of counting.