First, try to solve this math problem:
6 ÷ 2(1 + 2) =
Even though it seems simple, there is lots of debate about the answer on social networking sites. Click here to view the article.
When I first saw the problem, I thought the answer was undoubtably 1. When I did the problem, I had multiplied by two while I was simplifying the parentheses. So, I assumed that people who got 9 as the answer were forgetting the order of operations.
Before I talk about the article, I was excited to see that this was a debate between people. Any time where the general public is talking about mathematics, especially in a argumentative way, is a huge accomplishment for society.
Reading the article, I was intrigued to find myself in the minority. I did not think of the parentheses as a substitute for the multiplication symbol, but a quick way to write two times the quantity one plus two.
What if the problem was written like this?
6 ÷ 2 • (1 + 2) =
If the problem were written like that, I would have gotten 9 as my answer. Since the 2 was separated from the parentheses, "Please Excuse My Dear Aunt Sally" would put the division first and then the multiplication.
However, without the dot there (more formally known as an interpunct), I do feel that the answer should be one. It seems like the most practical perspective on the problem.
Take an expression like 2a. It is implied that the 2 and the a are a single quantity. If you were using it in an actual problem, it is far more likely that you would be dividing, say 8, by the whole quantity 2a than 8 by 2, and then multiplying this by a.
Similarly, let's say you had 2(a + 3). Again, there would be no doubt that you should distribute the 2. This problem would then become 2a + 6, which seems like the correct thing.
If you ended up with 1 ÷ 2(a + 3), there would be no practical case where the one were to be divided by the two first.
Also, I think that the order of operations was put into place to make algebra and geometry more structured. For instance, say you had rhombus where the diagonals cross, or bisect, at a (3x3 + 9)° angle, and you had to find the measure of the sides which were of length 25 - 2x2. As you probably know, a rhombus is a quadrilateral with all four sides equal (a square has all four sides and all four angles equal).
There is a theorem that states that the intersection of the diagonals of a rhombus is always equal to 90°. I think this is pretty cool on its own, to know this angle all the time. So, we can assume that 3x3 + 9 must equal 90°.
3x3 + 9 = 90
3x3 = 81
x3 = 27
x = 3
We must then plug this into the expression equal to the side measure to find our answer. First, let's look at the expression.
25 - 2x2
What are we being asked to do? Since two is a coefficient for x2, we would square the x first, then multiply by 2, and then subtract that from 25. This would be agreed upon by all algebra teachers. What happens if we plug the 3 back in?
25 - 2(3)2
If there was no order of operations, we would probably do this problem from left to right. This would give us:
25 - 2(3)2
When I first saw the problem, I thought the answer was undoubtably 1. When I did the problem, I had multiplied by two while I was simplifying the parentheses. So, I assumed that people who got 9 as the answer were forgetting the order of operations.
Before I talk about the article, I was excited to see that this was a debate between people. Any time where the general public is talking about mathematics, especially in a argumentative way, is a huge accomplishment for society.
Reading the article, I was intrigued to find myself in the minority. I did not think of the parentheses as a substitute for the multiplication symbol, but a quick way to write two times the quantity one plus two.
What if the problem was written like this?
6 ÷ 2 • (1 + 2) =
If the problem were written like that, I would have gotten 9 as my answer. Since the 2 was separated from the parentheses, "Please Excuse My Dear Aunt Sally" would put the division first and then the multiplication.
However, without the dot there (more formally known as an interpunct), I do feel that the answer should be one. It seems like the most practical perspective on the problem.
Take an expression like 2a. It is implied that the 2 and the a are a single quantity. If you were using it in an actual problem, it is far more likely that you would be dividing, say 8, by the whole quantity 2a than 8 by 2, and then multiplying this by a.
Similarly, let's say you had 2(a + 3). Again, there would be no doubt that you should distribute the 2. This problem would then become 2a + 6, which seems like the correct thing.
If you ended up with 1 ÷ 2(a + 3), there would be no practical case where the one were to be divided by the two first.
Also, I think that the order of operations was put into place to make algebra and geometry more structured. For instance, say you had rhombus where the diagonals cross, or bisect, at a (3x3 + 9)° angle, and you had to find the measure of the sides which were of length 25 - 2x2. As you probably know, a rhombus is a quadrilateral with all four sides equal (a square has all four sides and all four angles equal).
There is a theorem that states that the intersection of the diagonals of a rhombus is always equal to 90°. I think this is pretty cool on its own, to know this angle all the time. So, we can assume that 3x3 + 9 must equal 90°.
3x3 + 9 = 90
3x3 = 81
x3 = 27
x = 3
We must then plug this into the expression equal to the side measure to find our answer. First, let's look at the expression.
25 - 2x2
What are we being asked to do? Since two is a coefficient for x2, we would square the x first, then multiply by 2, and then subtract that from 25. This would be agreed upon by all algebra teachers. What happens if we plug the 3 back in?
25 - 2(3)2
If there was no order of operations, we would probably do this problem from left to right. This would give us:
25 - 2(3)2
23(3)2
692
4761
You can probably already tell that 4761 is way bigger than this side length would be intended to be. Even if you didn't make this estimate, this attempt at the simplification was nowhere close to the agreed order. By implementing the order of operations, there is no debate that the order should be PEMDAS. This clarifies lots of algebra when you are substituting terms into equations.
What if the expression were 108 ÷ 2x2? I still think most of us would agree that the 2x2 is meant to be its own term, and would therefore be divided as a quantity into the 108.
But after substitution, we get the following:
108 ÷ 2(3)2
108 ÷ 2(9)
If we did the 108 ÷ 2 first, we would end up with the wrong answer. Even though the surface definition of PEMDAS would ask for this to be done first, it is not a practical approach to the problem.
However, I did see that the SAT or ACT would expect students to receive 9 as an answer. Since there is clearly not a correct perspective to have, I would encourage you to comment what you thought the answer to be and why. Just like with Pi vs Tau, this is a post that is actually a lot of fun to debate about.
Without parenthesis to guide you and all operations are either multiplication and division or all operations are adding and subtracting then the grouping is left to right.
ReplyDeleteIn the referenced problem it works like this.
6 / 2 (2+1)
6 / 2 X 3
3 X 3
9
The key to this problem is the first two lines are equivalent and the second line is clearly a left to right order of operations problem which yields 9.
At first I got 12:
ReplyDelete6+2(1+2)=12
1=2=3
2*3=6
6=6=12
looking more carefully, I see that it was really:
6/2(1+2)=
Many of the errors might come from mistaking ÷ for + in the case of people using small fonts for increasing the amount of text on screen. I recommend using / instead of ÷ for legibility. Let us try again:
6/2(1+2)=1
1+2=3
3*2=6
6/6=1
It is important that symbols are clear.