We have all known our multiplication tables for years, and have successfully answered questions like “what is 6 times 7?”, but do we really understand what multiplication represents?
One interpretation of multiplication, which only works when multiplying by an integer, is “repeated addition”. From this perspective, “6 times 7” is a compact way to represent the sum of six sevens:
This model is handy for filling in gaps in memorized multiplication tables: if we forget a specific product, such as 6 times 7, but remember the product of something close to it, such as 5 times 7, or 7 times 7, we can use addition or the addition of a negative (subtraction) to bridge the gap:
5 x 7 = 35 , plus one seven is 42… or
7 x 7 = 49, less one seven is 42
The repeated addition model also corresponds nicely to our use of the phrase “groups of” in the English language. The phrase “seating is arranged in three groups of six” tells us that chairs are found groups of six. If there are three such groups, their sum can be represented more compactly as 3 x 6, or 18 in total.
A second, more powerful, and always valid, interpretation of multiplication is as “scaling”. When we scale something, we are growing or shrinking it by some ratio or proportion. There are a number of words in English that we use often to describe scaling, and our intuitive sense of what happens when someone asks us to “triple” a quantity defines what is it to “scale a quantity by a factor of 3″. We are stretching it until it is three times its former size.
To illustrate one fundamental conceptual difference between addition and multiplication, consider an opaque bag that is partially filled with marbles. If you are asked to add two marbles to the bag, can you do that without knowing how many marbles are in the bag?
Yes… take two marbles and drop them in the bag. Done. You do not need to know anything about the contents of the bag in order to be able to successfully add two marbles to it.
What if you are asked to triple the number of marbles in the bag? Can you do that without knowing how many marbles are in the bag?
No, you cannot. To triple the number of marbles in the bag (scale their number by a factor of 3), you have to know exactly how many marbles are in the bag at the moment. Multiplication affects everything in the bag – we need to know what is in the bag in order to scale it. Multiplication is more complex than addition.
Same Versus Different Units
Another significant difference between multiplication and addition is that addition and subtraction only make sense when used with quantities that are measured in the same units, or no units. It usually does not make sense to try to join three rocks and two shoes… 2+3=5, but five what? Addition only makes sense when the units of the quantities to be added are the same.
Furthermore, when we add or subtract two quantities that have the same units, the answer to the question will have the exact same units as both of the quantities that were added or subtracted. Two rocks plus three rocks is five rocks.
Unlike addition and subtraction, multiplication easily handles factors with different units: “three workers” times “two hours” equals six worker-hours; “three feet” times “two pounds” equals six foot-pounds. Notice that the units of the answer are different than the units of each of the two factors… the units of the answer are the product of the units of the two factors.
The scaling model of multiplication can be interpreted as “stretching” the original dimension(s) when one factor in the product is a scalar: a quantity with no units, no dimension. In this case, the result will have the same units as the factor being scaled:
2 x 3 square inches = 6 square inches
However, when both factors in a product have units, the scaling model leads us to drag the first quantity into a new dimension. A one-dimensional quantity (such as three inches) times another one-dimensional quantity (say, two inches), produces a two-dimensional quantity (six square inches, or six inches squared). If we use a line to represent “three inches”, it is dragged in a direction perpendicular to its length by the second factor (two inches), producing a rectangular surface that measures three inches by two inches, and whose area is “six square inches”.
If a “six square inch” rectangle is multiplied by a four inch line, it is dragged in a direction perpendicular to both of its dimensions through a distance of four inches, creating a rectangular prism (a box) with a volume of 24 cubic inches (6 square inches by four inches). Multiplication takes us into new dimensions!
When multiplication was first introduced to us as children, our teachers probably put an “x” between two numbers that were to be multiplied. But as we progressed through the grades, we gradually stopped seeing that notation. Once variables are introduced, and everyone’s favorite variable seems to become “x”, using “x” to indicate multiplication as well becomes very confusing. The “x” symbol for multiplication is therefore retired from use, and replaced with several notations, all of which represent multiplication:
It is very important that you recognize all of the above notations as multiplication, and as meaning exactly the same thing as one another. If you find one of them confusing, then rewrite it in your preferred notation… just don’t overlook the fact that the powerful process of multiplication is called for! As expressions become more complex, the operation involved is still always multiplication:
Earlier posts in this series:
Algebra Intro 1: Numbers and Variables
Algebra Intro 2: Addition
Algebra Intro 3: Subtraction
Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Algebra Intro 5: Addition, Subtraction, and Terms
Posts that continue this series:
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Algebra Intro 10: Fractions and Multiplication
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions