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1.2: Order of Operations

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

At the end of this lesson, students will be able to:

  • Evaluate algebraic expressions with grouping symbols.
  • Evaluate algebraic expressions with fraction bars.
  • Evaluate algebraic expressions with a graphing calculator.


Terms introduced in this lesson:

order of operations
grouping symbols
nested parentheses

Teaching Strategies and Tips

As the lesson illustrates, a string of symbols is meaningless without an order to the operations established beforehand. The usual order of operations – evaluate expressions inside parentheses, exponents, multiplication and division from left to right before addition, and subtraction from left to right – is introduced alongside the options of grouping symbols available to students.

Examples in this lesson illustrate the different kinds of situations that can arise involving grouping symbols: expressions without parentheses; expressions with parentheses (and other grouping symbols); inserting parentheses manually that are otherwise not inherent in the expression; parentheses within parentheses (nested parentheses – grouping symbols several layers deep); working with a complicated-looking expression (working from inner grouping symbol to outer grouping symbol), thereby being convinced that by sticking to the order of operations, its simplification does not need to be difficult.

Students also learn to consider fraction bars as grouping symbols. The fraction bar is an invisible bracket – the numerator and the denominator need to be simplified before proceeding.

Consider evaluating the following with and without adding parentheses:

\begin{align*}-x^2 + 3, \ \ \ \text{when}\ \ \ x = –1\end{align*}

Teachers are advised to use caution when presenting the mnemonic PEMDAS (Parenthesis, Exponent, Multiplication, Division, Addition, Subtraction). Though it is a highly effective way to get students to memorize the order of operations, the horizontal nature of our writing is suggestive of having \begin{align*}M\end{align*} performed before \begin{align*}D\end{align*}, and \begin{align*}A\end{align*} before \begin{align*}S\end{align*}. As you know, it’s not multiplication before division or addition before subtraction. Students have responded positively to the vertical schematic:

\begin{align*}& \text{P}\\ & \text{E} \\ & \text{M-D} \\ & \text{A-S} \end{align*}

Error Troubleshooting

As discussed previously, there is a potential for error in the horizontally written mnemonic, PEMDAS. Students, especially those who have consciously or unconsciously shut out mathematics and refuse to participate in it in any meaningful way, will inevitably ignore the left-to-right precept and go with what is suggested: \begin{align*}M\end{align*}. before \begin{align*}D\end{align*}, and \begin{align*}A\end{align*} before \begin{align*}S\end{align*}.

Students should be reminded occasionally that subtracting negatives is equivalent to adding positives.

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