Counting in Different Bases

                                                 

A suggestion that came up when I spoke to my friend D----- today was that she introduce her 5 year old to the idea of counting in different bases.  Her son likes to count and add and subtract, and she was looking for something else he might enjoy.  So I remembered when my little Mojojo was about that age, and she loved learning to count in different bases.  There's lots I like about this activity, but one thing is that it's a perfect example of a low floor, high ceiling activity.  Kids don't need much to start working with different bases, but they can use it to tie together lots of math later on.

Essentially, I explained it to Mojojo like this:

We say we count in Base 10, because there are 10 different numbers that we use: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.  When we count, after we reach 9, we have to start re-using numbers: 1-0, 1-1, 1-2, etc.  But what if we only had 4 different numbers? Then we would count 0, 1, 2 3 and run out of numbers!  So we'd have to count 10 next!  When Mojojo was 4-5, she would laugh hysterically when I skipped from 3 to 10, and then we'd count together: 11, 12, 13, 20!  And she would burst out in giggles again.  By the time we went from 33 to 100, she could barely speak from laughing at all the numbers we were "skipping".

Doing this with young children makes it easy for them to understand place value when it comes up at school.  Putting a 1 in the tens place means you've run through all ten numbers once -- or whatever your base is.  If you're counting in base 4, instead of a tens and hundreds place, you'd have a fours and sixteens place.  This is pretty easy to prove to kids because you can count with them and they'll find that 10 is the fourth number they reach and 100 is the 16th.  Later, when they learn more about exponents, they can tie this understanding to squares and cubes.  (Another cool activity for older kids is to see whether they can express fractions and decimals in different bases -- full disclosure, this bends my brain around a little.  To write 1/4 in base four is 0.1, and 1/16 is 0.01.) 

Operations in bases are fun, and they can be a big help to a deeper understanding of algebra later on.  But getting kids started with them can happen at 4 or 5!

One last thought on these -- if you have a kid who wants to know if we ever use this, YES.  I don't necessarily think practical applications make something more fun to learn, but computers work with binary, which is base 2.  And if you have a cable box that you reset, you can watch it count in hexadecimal (base 16!) which includes the letters A, B, C, D, E & F.  Mojojo points out that colors on the computer have hex values which are written in hexadecimal, which is a different way of writing a specific color than the RGB values.