The Order of Grouping Agreement
by Shelley Walsh ©2000-2001
This article is about arithmetic problems where there is more than one operation to do. But it isn't really. If it were about that, the clearest way to write it would probably be somewhat like it is on a tax form. Add this number to this number and put the result on line 1. Then multiply line 1 by this number and put the result on line 2, etc., etc., etc.. But in order to write multiple operations in a compact form for writing formulas and doing algebra, and to be able to better get a big picture, we write them another way, with the numbers and operations all written together. This notational system is a product of a long evolution and is far from the only way it could be done, and probably not even the best way, but it has become the custom of the mathematical and scientific world, so it is important to learn it.
Within this system there are rules or agreements on what order to do the operations, which are in some textbooks called the order of operations agreement. I like the idea of the word 'agreement' here because it emphasizes the human aspect of it. I think it is important to keep in mind that these are just notational conventions, and therefore not anything that you would be expected to be able to figure out for yourself if you were smart enough, but on the other hand they are a custom very much worth knowing.
Order of Operations Agreement
- Parentheses
- Exponents
- Multiplication and Division
- Left to Right
Also the order is not really such an arbitrary thing to be memorized with the aid of the various ditties that have been recently invented. There is sort of a sense to it. Parentheses are the manual way of indicating grouping. They even sort of look like a kind of container. One could simply indicate all groupings with them, but it would be hard on the eyes, so customs have developed that certain operations that are most often done first are assumed to be done first unless otherwise indicated. Roughly speaking the pattern is to do the more complicated operations first. This pretty much works also when you learn about other fancy functions like logs and trig functions, and to the extent that it doesn't, you will learn at the time the operations are introduced. The left to right part you don't really have to think about too much, because it is the way you would think it would be if we didn't have any parentheses or priority of operation. So basically the priority of operation is exponents over multiplication and division, and multiplication and division over addition and subtraction.
I have found a system that is helpful for keeping the order of grouping agreement straight and to see the advantages of seeing it as groupings rather than just as a lot of arithmetic drill, which I think my students have found quite useful. You circle each grouping in a different color, and then do its computation in the same color. By having the groupings circled you can look the problem over before doing any computing and see it at various levels. Again, this may not be the most efficient method for computing, but that is not what these problems are about. Nobody is going to ask you in a job to sit down all day and work out problems like this by hand. The quickest way to quick lots of these problems worked out is to get a machine to do them, but still you need to understand what it is that the machine would be doing if you gave it instructions to compute expressions like these, and for that it is far more beneficial to do problems like this slowly and think about them than quickly and automatically.
In the following examples the instruction is to evaluate the expressions according to the order of grouping agreement. I have written them by hand with felt tip pens and photographed them with the digital camera, because it was easier to deal with the symbols and the circling that way, and also it makes them look more like my lecture white board.
Whole Number Examples
Example 1:
Solution:
Explanation:
The red circle is there because powers come before multiplying and the green circle is there because multiplying comes before adding. Notice that after doing this circling we can look at the problem in several different ways. The big picture view is that it is 5 plus the green thing. Then inside the green circle is 3 times the red thing, and finally the red thing is raising 2 to the 3rd power. Even though when you actually do the computations you work from the inside out, it is important to also be able to look at the problem from the outside in, which is what we have just been doing.To do the operations what we have to always remember is that everything inside any of the circles must be done before mixing it with the rest. So the stuff inside the green circle must be done before it can be added to the 5. Inside the green circle the stuff in the red circle has to be done before it can be multiplied by the 3. So first we must compute 2 to the 3rd, which I have shown in red to match its circle. Then we multiply 3 times it, shown in green to match its circle. Finally the 5 can be added to the green 24, the result of the operations in the green circle.
Example 2:
Solution:
Explanation:
The red circle is there because powers come before subtraction. The
blue circle is there because multiplication comes before addition.
Parentheses are the manual way of showing grouping, so they get a
circle too, the orange one. Then looking at this from the outside in we
can see this as 2 plus the blue and then the blue is 3 times the orange
and the orange is the red minus 5 and finally the red is 3 squared.
To compute this we need to go from the inside out, because before adding 2 to it the blue has to be computed, and to compute the blue we need to multiply 3 times the orange, so we need to first compute the orange and to compute the orange we need to know what the red is, so finally we know that in order to get started we are going to have to compute the red. Again I am writing all of the computations in the color of their circles. We compute the red by raising 3 to the 2nd power and getting 9. Then we compute the orange by subtracting five from the result of the red 9 to get 4. Then to get the blue we multiply 3 times this to get 12. Then finally we add 2 plus 12 to get the answer.
Example 3:
Solution:
Explanation:
Looking at this from the outside in, it is 15 minus orange. Orange is 2 times red. Red is green minus blue. Green is 2 to the 4th. Okay, we can compute that. It is 16. Blue is 3 times 4. We can compute that. It is 12. So now we can compute red, and it is 4, and after computing red, we can compute orange, and it is 8. Then finally we can compute 15-8 to get an answer of 7.Example 4:
Solution:
Explanation:
I'm not showing the groupings for left to right, because we write from left to right, so it should be pretty natural to follow this order. Looking at it this way, at the highest level we are dividing 60 by orange and then multiplying that result by red, but first we must figure out what orange and red are. Orange can be computed right away. Red is 6 minus blue, so blue needs to be computed. Blue is 2 times 2, which is 4, so now red can be computed, and it is 2. Now we can get the final answer by computing from left to right.Example 5:
Solution:
Explanation:
This problem is yellow plus red minus blue. Figure out what each one is and then go from left to right.Integer Examples
Example 1
Solution:
Explanation:
Division comes before addition, so we circle the division. Now the problem is orange plus 5, and to find orange we have to do a division. Remember here that a negative divided by a positive is a negative. Then we have to add a negative to a positive. 5 is bigger than 2, so the positives win and the answer is positive.Example 2:
Solution:
Explanation:
This is red minus blue, so first we have to figure out what red and blue are, and then subtract. Blue turned out to be bigger than red, so we have to think about this subtraction in terms of signed numbers. For a simple one like this you can just think of it as that when you subtract a bigger number from a smaller number you get the same thing as the reverse except that the answer is negative, but you can also be safe and use the simple rule that subtraction always means adding the opposite. Once you have changed the problem into addition, since 24 is larger than 16 the negatives win, so the answer is negative.Example 3:
Explanation:
In this problem there is another convention to deal with and that is that minus signs are the on the same level as multiplying in the order of grouping, I suppose because it is the same as multiplying by minus 1, so in the expression in the orange circle it is 3 that is being squared, and then the minus sign is being tacked on, but in the second expression it is -3 that is being squared. At the end I have circled the -9 and the 9 to make it clear what is being subtracted. This is sometimes a helpful thing to do in subtraction problems to keep clear what are the two numbers that are being subtracted, so that subtraction signs and signs of numbers don't get confused. The two numbers here that are being subtracted are -9 and 9. To change the problem to addition you change the sign of the second one, the 9.Example 4:
Solution:
Explanation:
After working out what the red and the blue are, we go from left to right, but this time no that we have the use of plus and minus numbers, it is convenient to change all of the operations to addition, so that the order won't really matter. After that I usually find it is simplest to add up all of the positives and add up all of the negatives and then add the positives and negatives by subtracting and using the sign of the one which is larger in magnitude.Example 5:
Solution:
Explanation:
The same thing with this one as the last one, we don't have to be strict about going from left to right in the last stage if you change it into one big addition problem. Then we can add up all of the positives and all of the negatives and then balance things out between them.Example 6:
Solution:
Explanation:
It is green minus light blue here. Green is orange times dark blue. I have circled the -2 in yellow to show that it is all of the -2 that is being squared, but I haven't written anything in yellow, because there is nothing to compute for it. Orange is -2 times -2. To get dark blue we first need to compute red. Notice also that thinking about the problem this way we don't really have to keep strictly to the prescribed 'order of operations'. For example, there would have been nothing wrong with light blue first, and I wasn't strictly following the order either when I squared the -2 before subtracting the 7 from the 4. But it didn't matter, because I was still respecting the order of grouping. At the end we have a subtraction, 36 minus a negative 4. Again here I have circled the two numbers that are being subtracting in order to tell the signs on the numbers from the subtraction sign. All you need to do to do this is circle the first number and then the sign immediately after that is the operation sign and any further sign after that must be attached to the number. So according to the rule for changing subtraction to addition, you change the sign of the second number and add, so this becomes 36+4=40.Rational Number Examples:
Example 1:
Solution:
Explanation:
Division comes before addition, so it is circled. Then to do the division, we multiply by the reciprocal and in the multiplying it is a very good idea to cross cancel if possible. Then we have to add, and to add we need a common denominator.Example 3:
Solution:
Explanation:
This is 5/14 plus pink and pink is 3/7 times purple, so first we need to work out what purple is. For that we need a common denominator and it is always good to reduce to lowest terms. Then we get pink by multiplying 3/7 times purple and cross canceling is a good idea. Then finally we get to add, and for that it is necessary to write 1/7 as 2/14 to have a common denominator, and again always reduce to lowest terms.Example 3:
Solution:
Explanation:
In this problem we have a little exception to our rule. With division that is shown by a fraction line the top and bottom are automatically grouped and there is an automatic grouping around the fraction as well.I think you are supposed to rely on the visually suggestive nature of this. Mathematical notation was invented by people who have eyes, so it is intended to be at least partly visually suggestive. You can't totally rely on this, though. Use it when it works and ignore it when it doesn't. Think with you mind, not with your eyes, but every now and then the eyes can be helpful.
It is also best here to see the division as a fraction rather than dividing it out. The priority of grouping is still there, even though you may not do the operation. Then only thing I have done with the green is to give the minus sign a better home.
Example 4:
Solution:
Explanation:
There is a complex fraction in this problem, which indicates division, but still the convention holds that the top and bottom of the fraction are grouped because it looks like it, so first we have to subtract the 3/4 from the 2. Then we work out the complex fraction before mixing it with the rest of the world and invert and multiply to get 3/4.Looking at the problem from the outside in, it is 1/2 plus dark blue. Dark blue is light blue divided by 5/8. Light blue is 5/15 divided by red. And finally red is 2 minus 3/4.