I see it happen all the time in my math classroom: students attempt math problems and get the wrong answer even though they are performing each math operation correctly. How can this be!? Isn’t math supposed to have a single correct answer?
Well, as it turns out, there is a set of rules that tells us the right order and wrong order that operations can be performed in a mathematical expression. And without knowing this very important set of rules, you can’t be sure that you will always get the correct answer! Let’s take a look at this set of rules and dig into some practice with order of operations problems and answers so that you can make sure this doesn’t happen to you!
In math and algebra, the order of operations is an important set of rules that tell us the correct order that arithmetic operations should be performed in when working with a numerical expression. Performing operations in the right order using a standard method makes it so that two people will always get the same correct answer when solving a given problem.
In order to help remember the standard order of operations, we can use the acronym PEMDAS.
The PEMDAS rule (sometimes known as BEDMAS or the BODMAS rule) works by matching the first letter of each operation to each of the mathematical operations.
To help my students, I have told them to think of the acronym PEMDAS as standing for “Please Excuse My Dear Aunt Sally”. There are many different ways to remember the PEMDAS acronym, but I have found that this mnemonic device is a great way to help my students remember the order of the PEMDAS acronym.
If you follow the rules of the order of operations, you should find that arriving at the correct answer isn’t as hard as you once thought!
In order to understand when order of operations matters, take a look at this simple 2-step order of operations problem. Consider the following expression:
There are two approaches that you could take here, and only one of them will give you the correct answer! Which one do you think is correct?
As you can see, each strategy results in different answers. The strategy on the left adds 2 + 4 first, while the strategy on the right multiplies 4 x 3 first. Remember that we use the PEMDAS rule to help us identify the right order.
PEMDAS tells us that multiplication must be performed before addition. This tells us that the second solution is correct!
Let’s take a look at a few more examples of order of operations problems and answers! I’ll start by introducing you to some simpler problems with two basic operations, and we’ll work our way up to more complex 4-step order of operations problems! Just be sure to review the answer key for each problem to make sure you get the same answer!
Example #1: \(5 – 3 \times 2\)
In this first example, following order of operations tells us to perform multiplication before subtraction. Taking a look at the given values, we know that this will result in:
\begin &5 – 3 \times 2 \\ \\ =&5 – 6 \\ \\ =& -1 \end
Remember, performing subtraction first is a common mistake that will prevent you from obtaining the correct answer!
Example #2: \((4 \div 2) + 7\)
The first step of the PEMDAS rule is to tackle any math expressions inside parentheses. After that, we can add 7 to the result.
\begin &(4 \div 2) + 7 \\ \\ =&2 + 7 \\ \\ =& 9 \end
Example #3: \(3^3 – 4\)
The first step in this example is to work out our exponent. After that, we can subtract 4 from the result.
\begin &3^3 – 4 \\ \\ =&27 – 4 \\ \\ =&23 \end
Example #4: \(\sqrt \times 3 – 5\)
Remember that the ‘E’ in the PEMDAS acronym also includes the square root operation. As such, we need to evaluate the square root of 4 before multiplying by 3 and subtracting 5.
\begin &\sqrt \times 3 – 5 \\ \\ =&2 \times 3 – 5 \\ \\ =&6 – 5 \\ \\ =&1 \end
Example #5: \((6 ÷ 2) + 3 × 2\)
In this example we must remember to tackle the parentheses first. While your instinct may be to add the 3 next, remember that you need to multiply 3 times 2 first!
\begin &(6 ÷ 2) + 3 × 2 \\ \\ =&3 + 3 × 2 \\ \\ =&3 + 6\\ \\ =&9 \end
Example #6: \(7 – 2 × 3 ÷ 2\)
In this example, we multiply 2 by 3 first, then divide the result by 2. Remember that multiplication and division operations are performed in the order in which they appear from left to right.
\begin &7 – 2 × 3 ÷ 2 \\ \\ =&7 – 6 ÷ 2 \\ \\ =&7 – 3 \\ \\ =&4 \end
Example #7: \(2 + 4 × 3 – 4 ÷ 2\)
\begin &2 + 4 × 3 – 4 ÷ 2 \\ \\ =&2 + 12 – 4 ÷ 2 \\ \\ =&2 + 12 – 2 \\ \\ =&12 \end
Example #8: \(2^3 + (4 × 3) – 6 ÷ 2\)
\begin &2^3 + (4 × 3) – 6 ÷ 2 \\ \\ =&2^3 + 12 – 6 ÷ 2 \\ \\ =&8 + 12 – 6 ÷ 2 \\ \\ =&8 + 12 – 3 \\ \\ =&17 \end
When I teach order of operations in my classes, I always encourage my students to keep the PEMDAS rule handy for every problem. Sometimes these problems can seem very simple, but may actually require more thinking. In particular, problems with both multiplication and division operations tend to confuse students!
Take the time that you need to fully understand this very important concept. You will find it comes up often in your studies of math, particularly when working with algebraic formulas!
Like all math concepts, mastering the use of order of operations takes practice and critical thinking. I am hopeful that these order of operations problems and answers have helped you feel more comfortable with this very important algebra skill!
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