Last week in Fundamentals of Mathematics XXXVI we talked about division as another way of expressing the mathematical relationship between three numbers that form the measurement of the sides and area of a rectangle. If students are familiar with geometry and know how to calculate area by multiplying, this definition should make sense. If they are not familiar with geometry, there is another concept, taught along with addition and subtraction, that might prove exemplary: number families.
Number families begin with addition. With single digit addition number facts, such as 1 + 2 = 3. The commutative property gives us the second member of the family: 2 + 1 = 3. Same addends, same sum, different order of the addends. That the order of adding two numbers doesn't affect the sum is easily demonstrated with manipulatives. It could also be pointed out that, by our definition of addition as iterative counting, with the objects lined up in a row, it doesn't matter whether you count from left to right, or from right to left, the total amount doesn't change.
The second two members of the 1,2,3 number family for addition are two subtraction facts: 3 - 2 = 1 and 3 - 1 = 2. There are two ways to approach the subtraction members of the family. The first is to define subtraction as the inverse of addition: a + b = c means c - b = a and b + a = c means c - a = b.
With this definition, we can simply write the two subtraction family members by substituting our 1,2 and 3 into the addition fact to find that a is 1, b is 2 and c is 3, so 3 - 2 = 1 and 3 - 1 = 2.
Another way is to define subtraction as moving leftward on a number line. It's easy to convince oneself that there is no way to move leftward from a small number to arrive at a larger number, so the first number in the subtraction fact must be the largest of the three. This approach has the disadvantage of planting the idea "you can't subtract a larger number from a smaller number" which leads to difficulties later on, so it's not recommended.
So, here is an addition fact family, relating the numbers 1, 2 and 3:
Here's another, relating numbers 2, 3 and 5.
And so on.
Fact families are also used to teach multiplication and division facts.
2 x 3 = 6, so these three numbers are related in a multiplication family, consisting of two multiplication facts: 2 x 3 = 6 and 3 x 2 = 6, as well as two division facts, discovered by defining division as the inverse of multiplication:
Along with the commutative property of multiplication, this gives us the two division members of the 2,3,6 family:
These facts can be illustrated with a rectangular arrangement of unit area squares
two (length of left or right side) times three (length of top or bottom) is six (area of rectangle)
and
six (area of rectangle) divided by two (divided into two groups) is three (each group has three)
or, same rectangle, rotated 90 degrees
three times two is six and six divided by three is two.
This brings us back to the fact that division isn't a closed operation under whole numbers. As can be seen, when multiplying by, say, three, we only get multiples of three: 3, 6, 9, 12, and so on. There is no fact family for 2, 3, 7. Two times three is not seven, and neither is three times any other whole number. We can't make a rectangle with seven squares.
So what do we do with seven squares? Or any other number of squares that doesn't make a neat rectangle?
We simply make the largest rectangle we can, then add in the leftover square.
If we want to write this using the division sign, we first must isolate the multiplication on one side of the equal sign, and we do this by subtracting one from both sides
We can treat 7 - 1 as if it were a single number, six, and write the division equations in the number family 2, 3, (7-1)
Have fun in the comments. :)