Addition and Multiplication: Units

It is the habit among mathematics teachers, particularly at the elementary level, to present multiplication as repeated addition. The inimitable Keith Devlin, among others, has ranted about this, but it’s easy enough to see the temptation. When dealing with integers, multiplication and iterated addition will return the same numbers. Historically, it may be the case that multiplication was developed as a shortcut for addition, although the so-called Russian peasant method of multiplication is related to an Egyptian method that’s some 3650 years old, so the awareness that effective multiplication requires methods beyond simple addition is very old.

Devlin’s rant is based in significant part on the conceptual problem of multiplying non-integers. If you’re multiplying \(5.4 \times 4.1\), what does that even mean? And if the dodge response to this is that \(5.4 \times 4.1 = 54 \times 0.41\) (that is, 0.41 repeated 54 times), which it does, then what does \(\sqrt{2} \pi\) mean? In what sense can we “repeatedly add” an irrational number an irrational number of times?

That’s a fair point. The rebuttal is that the point of introducing multiplication as “repeated addition” is to get children used to the concept of multiplication before getting into heady mathematical theory. Devlin’s essay pre-empts this point as well, but I want to address a separate difference based on my natterings on the abstract unit.

So, what is a number, anyway?

At least until we get into college, mathematics consists largely of performing operations on numbers. At the elementary level, those operators are mostly restricted to addition, subtraction, multiplication, and division. Of these, addition and multiplication are the most important: If we wanted, subtraction could be fully abandoned once we introduce negatives, and division involves undoing multiplication (and could, theoretically, also be abandoned in favor of inverse multiplication, but I think that would lead to a bunch of needless complications).

What are we doing when we add two numbers?

  • Bobby has two apples and Sally has three apples: Together, they have five apples.
  • There are seven birds in the tree. Two more land on the branches. Now there are nine birds.
  • \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\)
  • \(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\)
  • \(4x + 12x = 16x\)

Until we get to fractions, radicals, and algebra, the matter of addition (and the issue of “what is a number?”) is straightforward: We have two different groups of the same sort of thing, and when we combine those groups, we add the numeric value and leave the units untouched. We can teach students that every “number unit thing” (whatever we want to call it) consists of a count and a unit. The part that looks like a “number” is the number and the part that looks like words or stuff or groups or things or measuring unit is a unit.

As I’ve discussed before, because the unit part is boring to math, we ignore it. It dies from loneliness and neglect, and by third grade or so, it’s only referred to (if at all) in the most casual of ways.

And therein lies madness, because we resurrect it for fractions, but students have long forgotten about it. By the time we get to “combining like terms” in algebra, many students are lost.

Here’s the hook in fractions: For the first time, we have things we’re adding where the “units” include things that look decidedly like numbers. Clever students might have noticed that five nickels and eight nickels make thirteen nickels, which can then be converted using \(13¢ \times 5¢ = 65¢ = $0.65\). But “nickel” doesn’t look like a number: It looks like a unit. The conversion between nickels, cents, and dollars is part of the same concept as the conversion between inches and feet, which also don’t look like numbers.

Meanwhile, the bottom parts of \(\frac{1}{4}, \frac{2}{7},\) and \(\frac{3}{44}\) are unambiguously numbers.

So, to summarize and expand:

  1. A numeric term can be separated into a counter and a unit.
  2. In order to add two numeric terms, the units must be the same.
  3. A unit can be explicit or abstract (in which case, it need not be expressed).
  4. A unit can be basic (foot, dollar, apple, thing) or complex (five feet, two dollars, a seventh of a thing).
  5. To add, combine the counters and leave the unit unchanged.

The fourth point is a key concept to addition. It makes possible things like adding fractions with unlike denominators and combining like terms. It also makes addition itself a significantly harder process. At the “apples and butterflies” stage of addition, addition consists of taking the number part, combining it, and leaving the unit alone. Here’s the process of addition when fractions with unlike denominators are involved:

  1. For each fraction, find an equivalent fraction so that the units are the same. Do this by multiplying the fraction by \(\frac{x}{x}\), where \(x\) is a factor of the other fraction’s denominator. Note that it’s simpler at this step to multiply each fraction by the other fraction’s full denominator, i.e., \(\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d}{b\cdot d} + \frac{c\cdot b}{b\cdot d} = \frac{a\cdot d + c\cdot b}{b\cdot d}\), but this is often sneered at in favor of the more sophisticated method of finding the least common denominator because it’s more likely to require simplification at the end.
  2. Separate each fraction into a counter and a complex unit, and add the counters.
  3. Examine the result to see if there’s an equivalent fraction with a larger unit; if so, use that equivalent value.

Naturally, this is not exactly how the process is presented to children. We speak of “common denominators” instead of identical units (although Common Core does speak of \(\frac{a}{b} = a \times \frac{1}{b}\), as a Third Grade concept).

This is one reason why I feel that moving away from concrete units as the norm in early years does a disservice to students: Success in fractions, and later in algebra, involves being able to convert between complex units that include a numeric portion. Procedurally, there’s no difference between adding apples, fourths, feet, or millions.

What about Multiplication?

Introducing multiplication as repeated addition implies something obvious to any random First Grader: Multiplication is harder than addition. After all, if it involves addition, and then something else, it must be harder than addition, right?

Wrong. This is the real danger of relying on “multiplication is repeated addition”: Once we step away from decimals, multiplication is easier than addition.

Most teachers who teach fractions are already aware of this. How do we multiply fractions?

  1. Multiply the tops.
  2. Multiply the bottoms.
  3. Simplify if needed.

The third step is the same as addition: “Examine the result to see if there’s an equivalent fraction with a larger unit; if so, use that equivalent value.” I’ve written it in fewer words, the trick that teachers use to make the business of fractions seem simpler than it is. Otherwise, though, multiplication is easier than addition when it comes to fractions: There’s no need to convert to equivalent fractions. The denominators don’t need to be the same. Just go for it.

And it’s not that we don’t care about units when it comes to multiplication. We do, very much. It’s a common mistake for students faced with a multiplication problem such as “Find the area of a room that is 10 meters by 12 feet” to say, “120”. Forget the units, they think: There are some numbers, and it says “area”, and we know the formula for area. Multiply the numbers. Done!

ACT Study Guides and suchlike will explain the error in the thinking: “We can’t multiply numbers with different units.”

And I am here to tell you, in as strong and bold a voice as I can: “YES, WE CAN.”

The error in thinking of 120 is not that we can’t multiply numbers with different units, it’s that what doing so in this case creates is a weird unit that nobody would ever use in the real world. A valid answer to the question “What is the area of a room that is 10 meters by 12 feet?” is “120 foot-meters”. It’s just that a foot-meter is a silly unit.

When we add, we decide on what the complex unit is, set it aside, add the counters, then combine it back with the complex unit. We may then need to reassess the numeric value, setting the base unit aside, and decide if we can simplify it at all. That’s addition.

Here’s multiplication: Combine all the things! That’s it. Multiply the counters. Multiply the units. Multiply everything! If we need to simplify, we’ll divide, which involves breaking some stuff up into multiplicands and then deciding if any of the same multiplicands are on the top and the bottom. So even that involves multiplication.

Indeed, the simplification process involved in adding fractions? That involves multiplication.

In other words, successful addition involves carefully attending to like-units, and then doing math on the counters, setting the units aside. Successful multiplication involves just mashing everything up and then sorting it out.

They are, in short, different operations. Addition cares about the distinction between “numeric value” and “units”; multiplication doesn’t.

Fractions: Adding requires the same denominator. Multiplication doesn’t care.

Algebra:

Adding requires combining like terms: \(3x + 4y + 5x = 8x + 4y\).

Multiplication doesn’t care: \(3x \times 4y \times 5x = 3 \times 4 \times 5 \times x \times x \times y = 60x^2y\).

Radicals:

Adding requires the same radicand: \(3\sqrt{2} + 5\sqrt{8} + \sqrt{7} = 3\sqrt{2} + 10\sqrt{2} + \sqrt{7} = 13\sqrt{2} + \sqrt{7}\).

Multiplication doesn’t care: \(3\sqrt{2} \times 5\sqrt{8} \times \sqrt{7} = 3 \times 5 \times \sqrt{2 \times 8 \times 7} = 15 \sqrt {2 \times 2 \times 2 \times 2 \times 7} = 60\sqrt{7}\)

Distance v Area:

Adding requires the same measurement unit: 10 meters + 12 feet requires converting one distance to the other’s unit.

Multiplication doesn’t care: A room with sides of 10 meters and 12 feet has an area of 120 meter-feet. What’s a meter-foot? I don’t know. Let the engineers figure out a use for it. The point is, we can do it; we can’t add with unlike units.

And so…

I’m still on the fence about whether it’s useful to originally present multiplication as repeated addition. I think it might help students understand why we’d want to multiply in the first place, and the language (particularly “times”) is certainly suggestive of the relationship. But I think it’s a relationship that ought to be abandoned as soon as students understand basics of multiplication, and I also question the habit of curricula to continue to imply that addition comes before multiplication throughout fractions, algebra, radicals, and on. At some point, I think the curricula should shift the precedence in light of the fact that, procedurally at least, multiplication is easier than addition.

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