📚 Practice With Worksheets
Continue practicing the order of operations with grouping symbols, whole numbers, decimals, fractions, and multi-step numerical expressions.
▶️ Learn With Videos
Review how to evaluate numerical expressions by following grouping symbols, exponents, multiplication or division, and addition or subtraction in the correct order.
🛠️ How to Use This Tool
Select one or more number types, including whole numbers, decimals, and fractions. You may also include mixed numbers and negative values.
Choose the expression length and optional features such as parentheses, exponents, or nested grouping. Selecting several number types creates a varied practice session.
Enter an exact value and select Check Answer. After making a valid attempt, use Show Solution to see which operation should be completed at each stage.
🧠 Things to Notice
Multiplication does not automatically come before division. Complete whichever of those operations appears first when reading from left to right.
Addition does not automatically come before subtraction. Complete addition or subtraction from left to right after multiplication and division are finished.
When grouping symbols are nested, begin with the innermost group. Simplify one operation at a time and rewrite the remaining expression carefully.
📘 Vocabulary
Numerical expression: A mathematical phrase containing numbers, operations, and sometimes grouping symbols, but no equality sign.
Evaluate: To find the numerical value of an expression.
Order of operations: The agreed-upon sequence used to evaluate a numerical expression correctly.
Grouping symbols: Parentheses, brackets, or braces used to identify parts of an expression that should be evaluated together.
Parentheses: Curved grouping symbols, \( (\ ) \), used to group part of an expression.
Brackets: Square grouping symbols, \( [\ ] \), often used outside parentheses.
Braces: Curved grouping symbols, \( \{\ \} \), sometimes used as the outermost grouping symbols.
Nested grouping: A grouping symbol placed inside another set of grouping symbols.
Exponent: A number that tells how many times a base is used as a factor.
Base: The number or expression being raised to a power.
Power: An expression consisting of a base and an exponent.
Product: The result of multiplication.
Quotient: The result of division.
Sum: The result of addition.
Difference: The result of subtraction.
Left to right: The direction used when evaluating operations that have equal priority.
Equivalent expressions: Expressions that have the same value even though they may look different.
Exact value: A value written without unnecessary rounding or approximation.
📐 The Order of Operations Process
Work through an expression one stage at a time. Rewrite the entire expression after each step so the operations that remain stay in their original order.
Grouping Symbols
Evaluate expressions inside parentheses, brackets, or braces first. With nested grouping, begin with the innermost group.
Exponents
Evaluate powers after grouping symbols and before multiplication, division, addition, or subtraction.
Multiply or Divide From Left to Right
Multiplication and division have equal priority. Complete whichever operation appears first when reading from left to right.
Do not: Multiply first simply because multiplication is listed before division in a mnemonic.
Add or Subtract From Left to Right
Addition and subtraction also have equal priority. Complete whichever operation appears first when reading from left to right.
Do not: Add first simply because addition is listed before subtraction in a mnemonic.
Complete Example
Step 1: Complete \(3\times4\) and \(16\div2\).
Step 2: The remaining operations are subtraction and addition.
Step 3: Work from left to right: \(18-12=6\), followed by \(6+8=14\).
Important reminder: A mnemonic such as PEMDAS can be helpful, but multiplication and division form one shared stage, and addition and subtraction form another shared stage. Within each stage, always work from left to right.
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