We tend to act as if multiplying is repeated addition. This misses a key, crucial difference between the two operations:

WE CANNOT ADD UNLIKE THINGS.

We can absolutely multiply unlike things. Sometimes, the result doesn’t make any real world sense, but we can do it. This is because multiplication doesn’t care about units, and addition does.

Let’s talk about addition first. In order to add, we need to have the same units. We can’t add 3 feet and 6 inches directly: We need to convert one unit to the other. We can convert 6 inches to 0.5 feet (to give use 3.5 feet), or 3 feet to 36 inches (to give us 42 inches), but there’s no sense that 3 feet and 6 inches makes 9 of something.

When it comes to fractions, I think people get confused because we treat the denominators like numbers sometimes and units other times. In 2/3, it’s fair to ask: Is the denominator 3, or is it 1/3?

In mathematics lingo, it’s 3. If you look at the etymology of “denominator,” though, it’s 1/3. 1/3 is the unit we care about. 2/3 is “two things” where “thing” is 1/3 of some whole object.

So when we add 2/3 and 1/3, what we’re doing is adding two “things” to one “thing”, where “thing” is 1/3. In this sense, 2/3 is 2 /3s. 2 apples and 1 apple makes 3 apples, and 2 /3s and 1 /3 is 3 /3s.

This is why we can’t add 2/3 and 1/4 directly: What we’re trying to do is add 2 /3s (one type of thing) to 1 /4 (another type of thing). One reason why 2/3 + 1/4 doesn’t bother students as much as 2 apples + 1 orange is because /3s and /4s both involve numbers, and indeed even numbers that could be added, to give us /7s.

Indeed, many students will tell me that 2/3 + 1/4 = 3/7. And… sit down for the next part, fellow mathematics teachers… there’s a context in which that makes sense.

Imagine: There are two groups of students in a lunchroom. The first group, three students, has two boys. The second group, four students, has one boy. What is the fraction of boys in the entire group?

This is not how we add fractions, but this is a useful thing to do in some contexts. I used to work in market research: There were plenty of times where we would ask, “53% of the first group likes this, but only 42% of the second group does. What percentage of the entire group does?” Being able to realize that simple fractions aren’t useful for comparisons if the group sizes are different is an important mathematical skill. By not recognizing this and simply shaming students for 2/3 + 1/4 = 3/7 misses out on this.

There was a meme that made the rounds a while back. The question was something like: “Joe had half a pizza. Steve had a third of a pizza. But Steve had more pizza than Joe. How is this possible?”

The student had given the obvious and presumably correct answer, and the teacher had marked it wrong and written that 1/3 is always less than 1/2.

I’m pretty sure that if 1/2 of Monaco’s population and 1/3 of France’s population supported something, more people would support it in France than in Monaco. And I’ll go ahead and state that without bothering to look up each country’s population.

(I admit: After the last paragraph, I did look it up. It’s not even close.)

In order to say that 1/2 of a pizza is larger than 1/3 of a pizza, the pizzas have to be the same size. If we don’t have the size of the units for comparison, we can’t claim that.

So… going back to 2/3 + 1/4, we need matching units. 2 /3s gives us the same amount as 8 /12s of the same big (“whole”) unit, and 1 /4 gives us the same as 3 /12s, so 2/3 + 1/4 = 8/12 + 3/12 = 11/12.

What I want to point out at this juncture: There’s a lot of high-level thinking going on here that we tend to just brush aside. We might dwell on some of this, but the part I tend to see getting the shortest shrift is this notion that we need the same units to add.

It’s not that we need “matching denominators” to add. We need the same units… which could be denominators, could be measurement units, could be real world objects, could be radicals, could be just about anything. They can even be a combination of things: \(3i\sqrt{2}/4 + 5i\sqrt{2}/4 = 8i\sqrt{2}/4 = 2i\sqrt{2}\)

Note what we’ve then done in the last step: After working with adding “things” that have the unit \(i\sqrt{2}/4\), we then decide to treat part (and only part!) of that unit as a number again.

How often do we comment to our students that we’re playing these magic tricks? DO we comment to our students that we’re playing these magic tricks? Or do we just let the students figure it out, or not, as they will?

The upshoot of all of this is: Addition is **hard.** Not the operation itself, the operation itself is easy. It’s all this stuff about making sure units match, understanding that units can look just like numbers, understanding that specific parts of our terms can be values in one moment and units in another…. That’s dizzying, especially if the adult in the room is acting like it’s obvious.

Now I’ll talk about multiplication.

Multiplication is conceptually more difficult than addition, when we set units aside. This is probably why we teach addition first, and we do this bit about “10 * 4” being “ten groups of four,” which encourages students to figure out 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4, even though 10 + 10 + 10 + 10 is much easier in this case, and even though 10 * 4 has the same numeric value as those addition expression but it is not the same thing.

But one way in which multiplication is ultimately easier than addition is when we bring units back in. Multiplication multiplies EVERYTHING. It multiplies numeric values, it multiples units, everything.

Indeed, while it is crucial to understand the difference at any given moment between values and units when adding, multiplying simply does not care.

2/3 * 1/5 = 2/15. Done.

3 feet * 5 feet = 15 square feet. Done.

5 pounds * 3 = 15 pounds. Done.

Because multiplication doesn’t care about the distinction, it doesn’t matter whether we interpret 2/3 * 1/5 as “2 divided by 3 mutiplied by 1 divided by 5” or as “2 /3s times 1 /5.” In the former case, it’s just a bunch of numbers to push around; in the latter, its 2 * 1 and /3s * /5s, which gives us 2 and /15s, Either way, we get the same result.

Consider a rectangle that’s 3 meters by 5 meters. Are we taking line segments of 3 m and 5 m and finding the area that they bound (15 m^2), or are we taking 5 groups of 3 square meter units?

Multiplication is commutative over units, so it doesn’t care!

3 * meters * 5 * meters = 3 * 5 * meters * meters = 5 * (3 * meters * meters). It’s all the same to multiplication.

We also say that addition is commutative, but that’s only true of the numeric values. The units, whatever portion of each term we decide is “the unit,” have to be the same.

Now, in practice, in order to keep things tidy and in order to simplify things, we do tend to identify three sorts of things when we multiply: Numeric values that increase magnitude (which we put in the numerator), numeric values that decrease magnitude (which we put in the denominator), and non-numeric units. And we have conventions for how we like things to look, especially when addition comes back into play.

But the operation of multiplication doesn’t care about any of that. 3 * feet * /4 * √2 is fine with multiplication. It just looks awful to a mathematician, who would generally prefer 3√2/4 feet.

I understand why we teach what we teach at early grades, about multiplication modeling repeated addition, and I don’t know when we need to shift away from that thought (I would think earlier rather than later, but elementary education is not my strength). But I am concerned that we wait far too long to do that shift.