Fractions and Decimals in Binary

I returned from a work trip to London this week and started a new math chapter with my younger son – decimals.

One of the most interesting parts of home schooling over the last four years has been learning how to teach some basic math concepts.  Topics such as fractions and decimals are things that I hadn’t really given any thought to in 25 years, and obviously I had never taught either of these subjects previously.  There were definitely some false starts and mistakes the first time around with my older son, and I’m hoping to do a better job this time around.

Last year we did a project that my younger son really liked – creating a binary adding machine out of Duplo blocks.  On the plane back from London I decided that talking about decimals in binary might be a fun way to get him thinking about decimals in general.   So, for our Family Math project today we spent about an hour reviewing binary numbers and introducing fractions and decimals in binary.

The first step was talking about the usual base 10 representation of numbers and reminding the boys about how to represent numbers in binary.  We hadn’t talked about binary numbers in over a year, but it seemed as though they had some memory of the basic ideas:

The next step was reminding them about the binary adding machine.   Although the motivation for the project today was to write some fractions in binary, a quick reminder of some of the basic arithmetic rules in binary seemed like it would be a good idea.

 

With that review out of the way, we moved on to talking about decimals.  The starting point was base 10 decimals and specifically talking about what the digits after the decimal point represented.   With the base 10 representation written down on the board, it was easy to shift over to talking about decimals in binary.  Talking through the ideas here also turns out to be great review of powers and fractions (who knew . . . ha ha):

Although I promised in the last video that the next step would be looking at 1/3 and 1/5 in binary, that wasn’t actually the next step.   First we needed to do a quick review of how to convert base 10 numbers to binary (and also how to multiply by 2 in binary).  We did this step with our snap cubes:

With all of this review out of the way, we moved on to writing down 1/3 in binary.  Both of the kids thought this was a really fun exercise.  I think they were really surprised to see that the same procedure that worked for integers also worked for fractions:

The last step was converting 1/5 to binary.  The only difference between 1/5 and 1/3 in binary is that the fractions you encounter in the conversion from base 10 are tiny bit more complicated.

After we finished up here, the boys spent a little time figuring out the representation for 1/7.  There’s a neat trick involving multiplying by 8 that helps you see that the representation is right, and they managed to eventually find that trick, too, which was fun.

Definitely a fun morning and hopefully a neat way to introduce decimals.