Exit 245. Intentionally written to be confusing.

Every few months, a meme circulates asking people to solve this math problem (or one like it):

6 ÷ 2(1 + 2)

People begin arguing over what the answer actually is, with most people saying (for this problem) either 1 or 9. They insult people with the other answer, because apparently that’s what people do on the Internet. And several of my friends tag me every time this goes around, either to resolve an argument or just to read what people are saying.

The correct answer is 9, but I’ll come back to that later. The issue is that this problem is intentionally written to be confusing.

I find the whole thing annoying. For one thing, this is something that we were supposed to learn somewhere around sixth grade. Apparently, we as teachers and educators aren’t doing our job if adults’ mathematics skills have atrophied to the point that no one can do a simple yet fundamental problem from sixth grade. So, naturally, everyone tags their friend with the college degree in mathematics (me) to resolve a sixth grade problem. I wonder, do these same people tag their friends with degrees in English when they forget to spell a word? Do they tag their friends with degrees in music theory when they don’t remember the name of a song? I’m writing this so that when this meme goes around again, I won’t have to type a long response; I can just share this link.

Anyway, Order of Operations says this (paraphrased, my own words): First, resolve expressions in parentheses and other grouping symbols. Then resolve exponents. Then evaluate multiplication and division from left to right. Then evaluate addition and subtraction from left to right. This was somewhat of an arbitrary distinction, but mathematicians and others in related fields have done it this way for centuries, and an organized set of rules needs to exist in order for most problems to have a well-defined solution (for example, whether you follow these rules or not determines whether the answer to the above problem is 9 or 1 or something else entirely). Math books and teachers often abbreviate this rule with the acronym “PEMDAS,” which stands for Parentheses, Exponents, Multiply and Divide, Add and Subtract.

So, the correct answer as written is 9, and this is why:

6 ÷ 2(1 + 2)
Do what is in parentheses first: 6 ÷ 2(3)
Multiply and divide from left to right. As you read from left to right, 6 divided by 2 comes first, and that equals 3, so now the problem is 3(3).
This equals 9.

There are several common incorrect interpretations of these rules which lead to the arguments. Before I continue, it should be pointed out that the disagreement has nothing to do with “Common Core” or “new math” or anything like that. The Common Core State Standards say nothing about changing the order of operations from what it has been for centuries.

Misinterpretation #1:

6 ÷ 2(1 + 2)
Do what is in parentheses first: 6 ÷ 2(3)
2(3) still has parentheses in it, so do it before the division: 6 ÷ 6
This equals 1.
The rule, and the fundamental concept of what parentheses are, is that what is INSIDE the parentheses is done first. Parentheses go AROUND the operation that needs to be done first (or out of order). The parentheses are not AROUND the multiplying by 2, so the fact that 2 is next to the parentheses has nothing to do with changing the usual order.

Misinterpretation #2: “I typed it into my calculator and it said 1.” Your calculator was programmed by a human being who apparently interpreted the rules differently from someone who got 9 as the answer. I have actually seen photographic evidence that some calculators give 1 as the answer and some give 9. Calculators will do whatever their users and programmers tell them to.

Misinterpretation #3: In my observations, this one is the most common. The issue here is a lack of understanding of the acronym PEMDAS:

6 ÷ 2(1 + 2)
P: Parentheses first: 6 ÷ 2(3)
E: there are no exponents.
M: Multiplication comes next, so do 2 times 3. 6 ÷ 6
D: Division comes next, and 6 divided by 6 is 1.
A, S: there is no addition or subtraction, so the problem is done, and the answer is 1.

As I explained above, multiplication and division are evaluated on the same step; it is not correct to evaluate all of the multiplication and then all of the division. Addition and subtraction work the same way. Evaluating all of the multiplication, then all of the division, then all of the addition, and finally all of the subtraction introduces other problems as well. Multiplication and division have to be evaluated on the same step, because every division can be changed to a multiplication using fractions (e.g., 6 divided by 2 is the same as 6 times ½). Addition and subtraction have to be evaluated on the same step, because every subtraction can be changed to an addition using negative numbers (e.g., 6 minus 4 is the same as 6 plus -4).

More than once, I have pointed out this error to people, and the response is something like “then my teacher/textbook/etc. taught it wrong, because I’m doing what I learned.” Now that is possible; I have seen teachers inadvertently teach incorrect subject matter. A more likely explanation, however, is that the person making this statement either never learned it properly or remembered it wrong. One who only remembers the acronym PEMDAS and what the letters stand for, with no context, would be very likely to make this error, because they follow the order of the letters. The rules for order of operations written as complete English sentences say something very different.

For this reason, I have often encouraged students to write “P-E-MD-AS” to emphasize that some of the operations have equal predence. I have also started to hear some students in middle school say that their elementary teachers told them not to just memorize “PEMDAS,” probably for this reason. And that is what good teachers should be doing, teaching students to understand rather than just memorize out of context.

Misinterpretation #4: “Multiplication without a times sign, just writing two numbers or letters next to each other, creates an ‘implied grouping,'” which would then make the work identical to misinterpretation #1 above. In a simpler example of this thinking, these people would say that “10 ÷ xy” would mean to divide 10 by whatever you get when you multiply x by y, instead of to follow PEMDAS strictly and divide 10 by x first, then multiply this answer by y.

I find this hardest to argue against. Although the order of operations rules say nothing of this implied grouping, it doesn’t always look right. I read “10 ÷ xy,” and my first thought would be to find out what xy is, and then divide 10 by this answer. This violates PEMDAS, but maybe it looks right because x and y are written more closely together. There is mathematical precedent for this implied grouping as well; to find the cosine of 3x, one usually writes “cos 3x,” and this is almost universally interpreted as the cosine of 3x, rather than whatever the cosine of 3 is, multiplied by x.

Ultimately, there is a simple solution to this issue of implied grouping: be like every mathematician ever, and every high school and college textbook, and don’t use the ÷ symbol in the first place. Many people don’t remember math beyond elementary school and buttons on a calculator, so they may have forgotten that in more advanced math, division is usually written as a fraction. So now, if division is written as a fraction, it can be made extremely clear which order of operations the author of the problem intended, and the so-called implied grouping becomes an expression within the numerator or denominator, which is grouped explicitly by the fraction bar.

And if you are typing instead of writing by hand, use an extra set of parentheses to make sure that your intended order is clear. In other words, even though the answer to the original problem is 9, it is written in a way as to be intentionally ambiguous. So stop arguing about it and do something more productive with your time.