Last week, in Number Sense 019, we took a look at the area model for multiplication, and demonstrated how the distributive property works. I mentioned that this week, we would look at place value notation, or perhaps expanded notation, which is how we write numbers greater than nine.
Why nine? We have ten fingers, don't we? But somehow, when we get to nine, we start reusing digits instead of having a digit for ten, and starting over at the number one greater than ten. Or, in numbers based on counting on all our fingers... 10. Well, we can count zero as one of our digits to bring the number of digits up to ten, but we didn't always have zero. For a long time in our history, and probably all of our prehistory, zero wasn't part of the mathematical arsenal. Yet we still count by ten. It's a great mystery.
Be that as it may, we still have to solve the problem of dealing with numbers greater than nine. There are some practical solutions. If you are measuring beer, you simply invent larger containers. You don't have 10 pints, you have 1 gallon and 1 quart. Nine gallons? That's a firkin. Eight firkins? Put it in a puncheon. More than a puncheon? Three of them are a tun. That's a lot of beer. What were we talking about?
Oh, yes. Numbers greater than nine. The solution we've come up with is to re-use the nine digits (ten, including zero) and keep track of how many times we've reused them. By convention, we put the number of tens we've already counted on the left side.
So here we've got to nine. Add one more, and we sweep those we've counted into a stack, and start counting again with one.
One stack of ten, and two more. We can number the stacks, to keep track.
Two stacks of ten, and five more. 25.
Ten stacks of ten fit in that little bag, so here we have 123. It is a lot more convenient to write 123 than it is to manipulate that many counters.
As I mentioned last week, 123, or any number greater than nine, is actually shorthand for an addition. 123 can also be written 100 + 20 + 3. This is called expanded notation (123 is standard notation.)
We can combine this with the distributive property model from last week to deconstruct the standard multiplication algorithm. We'll work on 34 x 17.
The undistributed area model looks like this. Not much help there.
But when we write our numbers in expanded notation, instead of one complicated multiplication problem, we have four simple multiplication problems. I say they are simple because they are all, in effect, single digit times single digit multiplications, nothing more than a bit of repeated addition, and we can look up the answers in our multiplication table.
The first multiplication:
This one is very easy, there are three large teal areas. 3 x 1 = 3. Now, how many little unit area squares are in each teal area? If we divided one of the teal squares into unit squares, we'd have ten stacks of unit squares, each stack ten high.
Recall our earlier representation of 123.
The little bag holds ten stacks of ten, the same number as the unit squares in our teal area. So our teal square is 100 unit squares, and three of them is 300.
The second multiplication:
7 x 3 is 21, but we are multiplying by 3 tens, not 3. So our result should be ten times as large: 210
The third multiplication:
4 stacks of 10 is 40
And finally
7 x 4 = 28
So, 34 x 17 is 300 + 210 + 40 + 28
Notice this is not the standard method of multiplication taught in most schools. The standard method incorporates some shortcuts and uses “carrying” if the product of any single digit by single digit multiplication is a two digit number. In this case 4 x 7 and 3 x 7 both would require a number to be “carried” in the standard method, except 3 x 7 would not, because the three is the leftmost digit in the top number in which case carrying isn't required, but you have to remember to add the earlier carried number from 4 x 7, which is two, to the product of 3 x 7, which is 21, so you write down 23 instead.
I suspect most kids, if taught to multiply large numbers using expanded notation, the area model, and partial products (the name of the method I demonstrated above) would discover the shortcuts on their own. And their skill in multiplication wouldn't rely on memorizing procedures, but on simple mathematical principles they understand.
Have fun in the comments.