M3N4 a & b - Meaning of Division

M3N4. Students will understand the meaning of division and develop the ability to apply it in problem solving.
a. Understand the relationship between division and multiplication and between division and subtraction.
b. Recognize that division m ay be two situations: the first is determining how many equal parts of a given size or amount may be taken away from the whole as in repeated subtraction, and the second is determining the size of the parts when the whole is separated into a given number of equal parts as in a sharing model.

If you are asked to write a simple word problem for which 12 ÷ 4 is the appropriate computation, what will you do? Here are two possibilities:

* Cathy has 12 apples and she wants to give them to her 4 friends equally. How many apples will each friend receive?
* James’ mom baked 12 cookies for James and his 3 brothers. If they share the cookies equally, how many will each receive?

In both of these problems, we are sharing the total amount (12) among 4 people. Some people call these division situations “fair sharing” division situation. In a fair sharing situation, you know how many total you have and the number of groups the total is being shared equally.

There are other situations in which division is appropriate. For example,

* Cathy has 12 apples. If she puts 4 apples in a bag, how many bags will she need to put all apples away?
* James’ mom baked 12 cookies. If she puts 3 cookies on a plate, how many plates will she need?

In these situations, you are trying to find out how many groups there are. Thus, you are given the total amount and how many in each group. These situations are often called “measurement” or “repeated subtraction” division situations.

So, where do these differences come? Actually, the difference is closely related to the way we define multiplication. Recall that multiplication is an arithmetic operation that is appropriate when you have equal groups. The number of groups is called multiplier and the number of items in a group is called multiplicand. Division, just like multiplication, is also used in equal groups are involved. The number you are dividing, called dividend, is the total number of items you have. The divisor, however, may be the number of groups as in a fair sharing situation or the number in a group as in a measurement situation. Here is the summary:

multiplicand multiplier product
(Number in a group) x (Number of groups) = (Total)

dividend divisor quotient
Total ÷ (Number of groups) = (Number in a group): Fair sharing division
Total ÷ (Number in a group) = (Number of groups): Measurement division

We often say division is the inverse operation of multiplication, and it is. However, if we pay attention to the multiplier-multiplicand distinction, we get two different situations for division. In one situation, you are trying to figure out the multiplier (number of groups) and in the other, you are trying to determine the multiplicand (number in a group).

Thus, to help students understand these standards, it is very important that they understand the multiplier-multiplicand distinction when they are first introduced to multiplication.