Saturday, June 2, 2007

M2N3a - Meaning of Multiplication

M2N3. Students will understand multiplication, multiply numbers, and verify results.
a. Understand multiplication as repeated addition.

In my last post, I discussed a particular property of division. I am going backward this time to talk a bit more about multiplication.

So, let’s start from the very beginning. What is multiplication? M2N3 (a) states that children are to understand multiplication as repeated addition, but is it? If multiplication is just another form of addition, why do we need a separate operation?

Multiplication is an operation that can be used when you have equal size groups to determine the total number of objects in those groups. For example, if you have 4 plates and on each plate there are 6 strawberries, we can express the situation with an equation, 6 x 4 = 24 (or 24 = 6 x 4). On the other hand, if one of the plates has 5 strawberries and another has 7, then, we can no longer use a multiplication sentence to express this relationship.

What is important to remember here is that mathematical operations and mathematical sentences are used to represent relationships among quantities in different situations. Clearly, the example above could have be written as 6+6+6+6=24 as well. However, repeated addition may be a way to calculate the product (the answer for a multiplication problem), but not necessarily the same thing as multiplication as an operation. This becomes a problem later when students start studying multiplication of fractions and decimals.

How else is multiplication different from addition? Well, you are probably familiar with the cliché, “you can’t add apples and oranges.” Addition (and subtraction) requires two numbers that are referring to the same thing (4 apples and 5 apples, 4 oranges and 5 oranges, but not 4 apples and 5 oranges – unless we change our referents to 4 fruits and 5 fruits). On the other hand, multiplication can be performed between two numbers that are referring to two different things – 6 strawberries and 4 plates.

So, if we can find the total amount by repeated addition, what is the advantage of using multiplication? One advantage is that when you write 6 x 4, it is much easier to tell that there are 4 plates. If you write 6+6+6+6, you have to count the number of 6’s to get that information. Clearly if there are only four 6’s, it may not be too difficult but imagine if you had the situation that can be written as 24 x 31.

Finally, because the two numbers in a multiplication sentence tells us specific information about the situation, we should be consistent in the way we write multiplication sentences. Let’s go back to the original example of plates and strawberries. Could we have written 4 x 6 = 24 as well as 6 x 4 = 24? The answers are clearly the same, but do they mean the same thing? Well, that depends if we have any agreement on how we are to write multiplication sentences. If we agree that a multiplication sentence is to be written
{number in a group } x {number of groups} = total number of objects
then, the situation must be written as 6 x 4 = 24 as we don’t have 4 strawberries each on 6 plates.

Some people may argue that since the answers are the same, then it really doesn’t matter which way you write. One thing to keep in mind is that 6 groups of 4 things and 4 groups of 6 things are two different situations. If we want multiplication sentences to be useful in communication, we need to feel confident that when we write “6 x 4,” it can be interpreted only one way. Otherwise, the sentence becomes a useless as a tool for communication.

Mathematically speaking, the first number is the number in a group (called “multiplicand”) and the second is the number of groups (called “multiplier”). The sentence “6 x 4 = 24” should be read as “6 multiplied by 4 equals 24.” However, because the language of “times” (which is not really a mathematical term) is much more common, some may prefer to say the first number is the number of groups. After all, if you read “6 times 4,” it feels much more natural to thing of “6 times of 4.” My strong preference is to write the multiplicand first, but the important thing in classrooms is that teachers (hopefully within the same system – across grades) stay consistent one way or the other.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.