I thought I knew what place value represented but I was interested to read about our number system's origins in Mathematics Explained for Primary Teachers. A summary of what Place Value represents is that the place in which a digit is written represents the value of a power of ten. So for example, in the number 3487, the 7 represents 7 ones, the 8 represents 8 tens, the 4 represents 4 hundreds and the 3 represents 4 thousands. Zero is a place holder, because it holds the place when there are none of that value there. So for 1001, there are two place holding zeros as there is 1 unit, no tens, no hundreds and 1 thousand.
I like to understand the links between the areas of maths and to do that it is useful to look at place value in terms of 'powers of ten'.
- Ten is 10 to the power of one or '10'
- One hundred is 10 to the power of two or '10x10'
- One thousand is ten to the power of three or '10x10x10'
From that you can see that each of the powers of ten is worth ten times that of the value less than it, so ten tens are one hundred, ten lots of one hundred are 1000 and so on. The reason that this is important to know is that it helps with calculations as once you have nine of a certain power, when you add one more, and you have ten of them, they are exchanged for one in the next value up. Confusingly for children who read from left to right, this is worked from right to left, so with nine units, if one more is added then it is exchanged for one ten, in the place to the left of the units. Ten tens will be exchanged for one hundred, again to the left of the tens column in place value. It is the principal on which carrying one over in addition or doing subtraction by decomposition is based.
Although some of these are higher level than your child will begin with, it is useful to understand it fully as an adult guiding a child through the ideas, so that you can answer any questions your child might have, and so you are fluent in it yourself!
I based the place value exploration on the book 'A Place for Zero, A Math Adventure'. The girls enjoyed the story and I thought that it simply explained the concepts behind Place Value. We based our activities around the Montessori golden bead material that I have. A base ten set is similar, and you can print out sets of number cards from units up to thousands from here. Another effective way of explaining place value with concrete materials is with money - £1 coins represent the hundreds, ten pences represent the tens and one pences represent the units.
I downloaded Place Value worksheets one and two from the teaching ideas website, purely to use as a guide for myself, rather than as a worksheet for the girls. I wrote large numbers out in pen and asked them which number represented hundreds, which was tens, which was units and so on and got them to circle the correct number. I asked them to write down three hundred and eight, or two hundred and ninety seven and so on. The next day we did the same but in thousands.
The way we usually work our living maths is to read the book together, chat about it as we go or afterwards, then do a related activity that hopefully builds on their understanding. They really enjoy doing maths this way and I always get a cheerful 'yeah' when I suggest doing it which is so nice! The third day I introduced the Montessori golden bead material (base ten printables here to use - scroll down to find them, but they aren't as good as the real thing but better than putting it off for ever until you can get the real deal!).
.... then came back and assembled them on the place value mat as shown and put the number cards together correctly before telling me the number that they had.