Last week, in Number Sense 018, we wanted to begin talking about another model for multiplication, the area model, but we made an excursion into geometry, to talk about area, and got sidetracked by some tessellating goats. So, having defined area last week, we are ready to talk about the area model for multiplication this week.
We measure area with a unit called the square unit. It's value is 1 and it is a square that is one length unit wide and one length unit high. It looks like this:
We also define multiplication as repeated addition. Let's do some repeated addition using our counter and table model, with square units as the counter.
We begin with an empty table (that's the zero) and add three.
We do it again.
Two more times. Now we have 3 added 4 times. In multiplication talk, that's 4 times 3. If we count the squares, we find out there are twelve of them. If we look up 4 x 3 in our multiplication table, we see that 4 x 3 is 12.
To turn our counter and table model for addition into an area model for multiplication, all we need to do is push our counters (unit area squares) together, without any gaps or overlaps.
This shows the multiplication fact 4 x 3 = 12. If we turn the rectangle of area squares on its side...
We have the multiplication fact 3 x 4 = 12. This demonstrates the multiplication is commutative, that is, just like addition, it doesn't matter which number comes first in a multiplication problem, the answer is the same. The area of the rectangle does not change simply because we turn it.
Another thing we can show with the area model is the distributive property of multiplication over addition. The distributive property says that if we multiply a sum of two numbers (an addition) by another number, we get the same answer if we multiply each number in the sum separately, then add the products. Let's break that down:
We multiply the sum of two numbers
( 2 + 3) (the sum of two numbers)
by another number
x 4 (multiplied by another number)
= (we get the same answer if we)
(2 x 4) (multiply one of the numbers in the sum: 2 to get one product)
+ (add the products)
(3 x 4) (the other number in the sum: 3, multiplied separately to get the other product)
on one line: (2 + 3) x 4 = (2 x 4) + (3 x 4)
It is complicated, and can be confusing. The area model can clear up a lot of the confusion, because it works for both addition and multiplication.
When we apply the distributive property twice, both numbers that are multiplied can be sums, not just one of them.
It looks complicated when written in math notation, but it's pretty straightforward when looking at unit squares pushed together into rectangles.
While we don't often think about it in this way, numbers larger than nine are actually addition problems condensed into a shorthand we call place value notation.
What I mean is: 123 is really 100 + 20 + 3. We will take a look at this next week.
Have fun in the comments.