Inverse operations and functions are wonderful things. Without them, solving equations would be much more challenging. Yet inverse operations can also be odd beasts. My previous “inverses” post pondered addition and subtraction, which led us (as young students) to expand our initial universe of counting numbers into the integers. Addition and subtraction are operations that only make sense when acting on “like” quantities, or within a single spacial dimension.
Way back when, you may have been introduced to multiplication as scaling (stretching by a factor), as repeated addition, or perhaps just as “multiplication”… something you were going to have to learn how to do.
Multiplication seems to have two fundamental forms:
1) Multiplying a scalar (a unit-less or dimensionless quantity, such as “three” or “one half”) by a dimensional quantity (such as “three inches”) stretches, shrinks, and/or rotates (if multiplying by a negative) the original quantity within its dimension(s). This can also be thought of as taking “groups of” the original quantity: 3 times 5.2 can be thought of as meaning “take three groups of 5.2”. The units (dimensions) of the result will match the units of the initial quantity (if any), and the size of the result will be the product of the two numbers: 15.6.
2) Multiplying a dimensional quantity by another extends the original quantity into the dimensions of the second quantity. As an example, visualize a line segment that is 3 inches long. Multiplying it by 5.2 inches causes the line segment to be extended 5.2 inches into a new dimension perpendicular to the original line, creating a rectangle that measures 3″ x 5.2″ and whose surface area is the product of the two numbers: 15.6 square inches.
When multiplying dimensional factors, the units/dimensions of the result will be the combination of the units/dimensions of the two factors, and have a total number of dimensions equal to the sum of dimensions of the two factors. Inches times Square Inches produces Cubic Inches. Newtons times Meters produces Newton-Meters.
Properties Of Multiplication
Picking up where my musings about Addition and Subtraction left off (working with the set of integers as our numerical universe), and thinking for a minute only of the magnitude of the result, the set of integers is closed under multiplication. This means that the product of two integers will always be an integer. However, when multiplying two dimensional quantities, the product will have different units than either of the original quantities.
Multiplication, like addition, is commutative:
However, this property only applies to the magnitude of the result. When multiplication is being used to describe physical situations, the situations themselves may not be commutative. For example, “two groups of five apples” is potentially a very different arrangement than “five groups of two apples”, even though they both involve ten apples. Yet, a two inch vertical line multiplied by a five inch horizontal line will produce the same figure and surface area as a five inch horizontal line multiplied by a two inch vertical line.
Multiplication is also associative:
Once again, this property applies to the magnitude of the result, as the physical manifestations represented by each association could appear different.
Multiplication also has one new property above and beyond those of addition: the distributive property of multiplication over addition or subtraction. An example of the distributive property at work is:
The distributive property became useful almost immediately to help calculate products you could not remember. If the result of (7)(8) did not come to mind, you could rewrite it as
then use the distributive property to turn one challenging problem into two manageable ones (a key concept in problem-solving).
The distributive property is almost magical. Not only does it help break down a challenging problem into smaller and simpler steps, it also changes the problem I am working with from a product (of a number and sum) into a sum (of two products). That’s pretty neat when you stop to think about it. Unfortunately, errors in applying the distributive property have also lead to many points being lost on problems in math class.
With multiplication mastered (or at least nearly so), we were introduced to division. It “undoes” multiplication, divides a quantity into parts, and “converts” fractions into decimals.
Long division was probably the second major procedure we were taught while being asked to disengage our intuitions. How many students really understand how and why the long division process works? I suspect most just follow the rules.
Properties Of Division
Division, like subtraction, is not commutative:
or in textbook notation:
Nor is division associative:
or in textbook notation:
The lack of these two properties makes division inflexible as an operator when compared to multiplication. Furthermore, if we divide two integers:
we can get an answer (six tenths) that is not an integer. Division forced us to expand our mental universe of numbers to include decimals and rational numbers (numbers that can be expressed as the ratio of two integers).
The greater complexity/sophistication introduced by the rational numbers has a bright side though. It turned out that dividing by an integer had exactly the same result as multiplying by the reciprocal of that integer:
This turns a division problem into a multiplication problem, allowing the commutative and associative properties to be used once again to re-arrange the problem for our convenience:
And one last property for division: like multiplication, it distributes over addition and subtraction:
or in textbook notation:
And then there is the notion of “factoring”, or un-distributing a common factor:
Factoring is the inverse of distributing. It is easy to explain that way (“if all terms have a common factor, un-distribute the factor”), but I don’t know if such an explanation is more or less helpful to those grappling with factoring for the first time. Factoring applies to both multiplication and division, just as the distributive property does. I mention it here for the first time because most people see factoring as requiring familiarity with division: you divide each term by the common factor, before writing the common factor outside of the parentheses. Although I could have written: you multiply each term by the reciprocal of the common factor…”. Hmmm, with all this multiplying by a reciprocal, do I even need division? The same argument could be made for adding the negative of a number vs subtraction.
But how do people conceive of division once they have learned about it: what does (5/3) mean to them?
– five thirds (a fraction)
– five for every three (a ratio or proportion problem)
– five divided by three (a division problem)
– the number that equals five when multiplied by three (inverse perspective)
– five reduced by a scaling factor of three (anti-multiplication)
– one third of five (avoidance = multiplying by reciprocal)
The experience of attempting to divide food or candy equally among a group of people as a child probably provides a nice experiential introduction to the concept division for most people. However, how division by non-integers or negative numbers should be carried out is not intuitive for most.
The inverse perspective requires the most words to describe, just as it did with subtraction, and also produces the most convoluted sentence and thought process.
I have the impression that many students perceive fractions as one topic, proportions as a second (and perhaps related) topic, and division as a third topic. After all, that’s the way they were introduced all those years ago. If students can be taught to see both fractions and proportions as examples of division at work, and to be equally comfortable treating (5/3) from any of the above perspectives – depending on which makes it easiest for them to tackle a problem, then algebra problems become easier for them.
It is interesting that our language has words for simple fractional quantities (a half, a third, a tenth, etc.), and that such proportions seem intuitive to many people, yet their time spent studying “fractions” in math class is something that most people have repressed. This may be due to not having developed both a conceptual and a concrete understanding of fractional notation, or even fractions in general. Sure, the idea of “half a cookie” makes sense, but most people do not seem to have ever really thought through how they would physically determine what proportion of a whole cookie is represented by half a cookie and a third of a cookie. And I suspect this unease with fractions is often transferred to division.
So, in pondering the first two inverse operations introduced to most students, my generalizations from addition and subtraction still apply, with one new generalization. Inverses can:
– not share some convenient properties with the “forward” version of the function, like being commutative or associative
– share some properties with the “forward” version of the function, like being able to be distributed over addition or subtraction
– have ranges that force us to expand the universe of numbers that we had been habitually using before being introduced to the inverse function
– be more challenging to grasp and describe verbally than their “forward” sibling
And one last point to ponder: while addition and subtraction are dimensionally-closed operations, multiplication can increase the number of dimensions and division can reduce the number of dimensions. Using a scalar with multiplication or division will leave the number of dimensions unchanged.