Order of Arithmetic Matters
Addition and subtraction go together. Multiplication and division go together. What happens when you combine them? How do calculations change when you have addition, subtraction, multiplication, and division in the same problem? It turns out that there are two common ways to calculate such problems: the left-to-right method and the PEMDAS method. These two methods sometimes produce different answers. For example, consider the problem
$$1 + 2 \times 3 -14 \div 2 = \: ?$$
If you work this problem out in left-to-right order (the way you read), the calculation looks like:
$$3\times 3 -14\div 2 = 9-14\div 2 = \frac {-5}{\: 2}\:\:\text{Incorrect!}$$
In contrast, using PEMDAS order (the way math should be calculated), it looks like:
$$1+ 6 – 7 = 0\:\:\text{Correct!}$$
These answers are not the same. Order matters!
What is PEMDAS?
Order matters even more when parentheses and exponents (sometimes called “powers”) appear. Algebra students learn the PEMDAS order of operations, yet elementary students could benefit from knowing this topic. The PEMDAS order means that calculation are done in this order:
- P = parentheses
- E = exponents (sometimes called P = powers)
- MD = multiplication and division
- AS = addition and subtraction
PEMDAS is also called PPMDAS where the 2nd P represents “powers” (another word for exponents). To remember PPMDAS, student may memorize the phrase “Pretty Please My Dear Aunt Sally.”
In PEMDAS, the first step is parentheses. Parentheses and exponents (otherwise known as powers) must be done in the correct order. Otherwise, the calculations don’t work out consistently. For example,
$$\begin{align}
(1+3)^2 &= 4^2 = 16 \:\:\:\text{parentheses before powers/exponents}\\
&\not = 1^2+3^2 = 10\:\text{powers/exponents first}
\end{align}$$
Here’s a more complicated example:
$$(3+2\times 3)^2 \div 3^3 \times 4= \: ?$$ For practice, calculate this using PEMDAS. Then check below to see if your solution followed the same steps. First, we use PEMDAS inside the parentheses : 3+2×3 = 3 + 6 = 9. Second, we calculate the exponent:
$$\begin{align}
(3+2\times 3)^2 \div 3^3 \times 4
&= 9^2 \div 3^3 \times 4\\
&= 81 \div 27 \times 4\\
&= 3\times 4\\
&=12
\end{align}$$
When Order Doesn’t Matter
In the “MD” step of PEMDAS, multiplication and division are performed. If a problem only requires multiplication and division, then the order can be changed without affecting the problem. For example,
$$\begin{align}
3\times 5\div 10\times 2\div 3
&= 3\div 3\times 5\times 2\div 10\\
&= (3\div 3)\times (5\times 2\div 10)\\
&= 1 \times 1\\
&= 1
\end{align}$$
Similarly, in the “AD” step of PEMDAS, addition and subtraction can be done in any order.
$$\begin{align}
123,456 &+45,090 -365 -123,456 +365 -45,090\\
&= 123,456-123,456 + 45,090 -45,090 +365 -365\\
&= (123,456-123,456) + (45,090 -45,090) +(365 -365)\\
&= 0 + 0 + 0\\
&= 0
\end{align}$$
Changing the order of an addition/subtraction problem can make it much easier to calculate. We will see more examples in a future post on Grouping.
A Step Before PEMDAS: The Distributive Property
One valid way to get around the PEMDAS order of calculation is to use the Distributive Property first.
Rather than giving an abstract formula, I’ll present a few examples:
$$\begin{align}
(40 + 3) \times 52 &= 40\times 52 + 3\times 52\\
4\times(100+3) &= 4\times 100 + 4\times 3\\
(100 + 20 + 3) \times 14 &= 100\times 14 + 20\times 14 + 3\times 14
\end{align}$$
The general formula is (a+b) x c = a x c + b x c. The FOIL method of multiplication uses this property. While this level of calculation may seem extreme for elementary school math, PEMDAS and the distributive property are required knowledge for algebra and all further math classes. Students who do not learn PEMDAS while learning arithmetic will attempt to mast the concept in algebra class at the same time they are learning to manipulate variables. Learning both concepts at once may be difficult for most students.
PEMDAS prepares elementary students for later mathematics, as well as opens up new options for calculation of arithmetic. Arithmetic methods of calculation such as Grouping, FOIL, and Approximate Values depend on students knowing the proper order of calculation.