## The Value of Place Value

Katie has been able to count until 100 for a while and now may even be able to get up to 1000.  However, I realized that she doesn’t actually understand the decimal system and place value (i.e. why are two, three, etc. digit numbers written the way they are).  I made some vague attempts with an abacus some time ago, but she was either not ready or not in the mood, and it didn’t get very far.  Today I decided to make a fresh attempt.

The lesson began with a story.  A long time ago, people did not have a way of writing down numbers.  This was very inconvenient, so one day some wise men got together to solve this problem.  The first proposal was to have a separate symbol for every number.  However, after a while it became very tedious to come up with new symbols.  They also realized that people would have trouble remembering so many.  And then the wisest of the wise had an idea.  “We don’t need hundreds of symbols”, he said.  “We just need a few, say 10 (since I have 10 fingers on my hands), and we can write down any number we want.”  He then went on to explain how after reaching 9 you move on to a new column which is called the tens column.  When you place a one in this column it represents a 10, and in general it keeps track of how many 10’s you have.

I tried to show this to the girls by drawing actual columns on a piece of paper and writing two-digit numbers using those columns.  At this point Katie started whining that this was boring so I realized that I needed to either bring out a prop or give them actual problems to solve (or ideally both!).  I had a kit which contained chips with 1’s 10’s and 100’s on them.  As soon as I brought it out, the girls immediately perked up.  In order to have the girls get a feel for the concept, I had them do simple two-digit addition problems (with no carries).  I explained how because of the system one could add 1’s separately from the 10’s.  One of the girls would use the kit to solve the problem and the other one would do it on paper (and then they’d switch).  With some guidance, both were successful.

I think that it will take some time for the girls to internalize the concept of place value, but this felt like a good start.  Eventually, when they have a very good grasp of it, I would like to introduce them to counting in other bases (binary perhaps being the logical next step).  However, this is not a trivial concept and one that is often underappreciated, so I don’t want to rush events too much.