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Introduction
Order of Operations (PEMDAS) helps you connect a mathematical rule to a result you can explain and check. Use parentheses first and respect left-to-right order within equal-priority operations. Explain why the same symbols in a different grouping can produce a different value. This lesson belongs to algebra foundations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in order of operations (pemdas).
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Add, subtract, multiply, divide, and square rational numbers.
Previous lesson: Evaluating Expressions
Key mathematical vocabulary
- exponent
- A number specifying a power.
- ratio
- A comparison by division.
- expression
- A mathematical phrase representing a value.
- solution
- A value making the original conditions true.
- input
- A value supplied to a rule.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Order of operations
Use parentheses first and respect left-to-right order within equal-priority operations. Explain why the same symbols in a different grouping can produce a different value.
Grouping symbols come first, working from the innermost group outward. Evaluate exponents next. Multiplication and division share one priority and are performed from left to right. Addition and subtraction also share a priority and proceed left to right.
PEMDAS is a memory aid, not an instruction to multiply before every division or add before every subtraction. A fraction bar groups the whole numerator and denominator. Write one change per line so that you can find an arithmetic mistake without restarting the entire problem.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. First group: 2+2=4.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Expanding the repeated multiplication gives the same value.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Evaluate 29 + 12(2 + 2)².
- First group: 2+2=4.
- Then square: (4)²=16.
- Multiply: 12·16=192. Finally add 29: 221.
Answer: 221
Verify: Expanding the repeated multiplication gives the same value.
Avoid this mistake: Doing addition before multiplication: apply operation priorities.
Example 2 · Basic
Evaluate 29 + 13(2 + 2)².
- First group: 2+2=4.
- Then square: (4)²=16.
- Multiply: 13·16=208. Finally add 29: 237.
Answer: 237
Verify: Expanding the repeated multiplication gives the same value.
Avoid this mistake: Multiplying before an earlier division: work left to right at equal priority.
Example 3 · Intermediate
Evaluate 29 + 14(2 + 2)².
- First group: 2+2=4.
- Then square: (4)²=16.
- Multiply: 14·16=224. Finally add 29: 253.
Answer: 253
Verify: Expanding the repeated multiplication gives the same value.
Avoid this mistake: Squaring only part of a group: apply the exponent to the entire parenthesized value.
Example 4 · Advanced
Evaluate 30 + 2(2 + 2)².
- First group: 2+2=4.
- Then square: (4)²=16.
- Multiply: 2·16=32. Finally add 30: 62.
Answer: 62
Verify: Expanding the repeated multiplication gives the same value.
Avoid this mistake: Doing addition before multiplication: apply operation priorities.
Example 5 · Challenge
Evaluate 30 + 3(2 + 2)².
- First group: 2+2=4.
- Then square: (4)²=16.
- Multiply: 3·16=48. Finally add 30: 78.
Answer: 78
Verify: Expanding the repeated multiplication gives the same value.
Avoid this mistake: Multiplying before an earlier division: work left to right at equal priority.
Common mistakes and corrections
- Doing addition before multiplication: apply operation priorities.
- Multiplying before an earlier division: work left to right at equal priority.
- Squaring only part of a group: apply the exponent to the entire parenthesized value.
Real-world applications
- Evaluate a pricing formula with setup charges. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Calculate the area of a square after increasing its side length. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Use parentheses first and respect left-to-right order within equal-priority operations. Explain why the same symbols in a different grouping can produce a different value.
Grouping symbols come first, working from the innermost group outward. Evaluate exponents next. Multiplication and division share one priority and are performed from left to right. Addition and subtraction also share a priority and proceed left to right.
Frequently asked questions
What should I know before order of operations (pemdas)?
Add, subtract, multiply, divide, and square rational numbers.
How can I check my work?
Expanding the repeated multiplication gives the same value. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Doing addition before multiplication: apply operation priorities. Multiplying before an earlier division: work left to right at equal priority. Squaring only part of a group: apply the exponent to the entire parenthesized value.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Evaluate a pricing formula with setup charges.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Use parentheses first and respect left-to-right order within equal-priority operations. Explain why the same symbols in a different grouping can produce a different value.
- Doing addition before multiplication: apply operation priorities.
- Multiplying before an earlier division: work left to right at equal priority.
- Squaring only part of a group: apply the exponent to the entire parenthesized value.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials