🔢 MathGrade 6Lesson 7
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Order of Operations with Integers

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Learning Objectives

  • 1Apply PEMDAS to evaluate integer expressions with parentheses, exponents, and four operations
  • 2Identify the correct next step in a partially evaluated expression
  • 3Explain why multiplication and division share a level and work left to right
  • 4Avoid the common error of adding before multiplying
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Mini Lesson

When an expression mixes parentheses, exponents, and the four basic operations, mathematicians need to agree on the order so everyone gets the same answer. The rule is called the order of operations, and most learners remember it as PEMDAS (or GEMDAS, where G stands for grouping symbols).

PEMDAS / GEMDAS — One Step at a Time

  • P — Parentheses (or any grouping symbol such as brackets) come first.
  • E — Exponents next: 3^2 means 3 x 3, and 2^3 means 2 x 2 x 2.
  • M and D — Multiplication and Division share a level. Work them left to right in the order they appear.
  • A and S — Addition and Subtraction share the bottom level. Work them left to right too.
  • Do ONE operation per step. Rewrite the whole expression after each step.

Common Error: Adding Before Multiplying

In 3 + 4 x 2, multiplication beats addition. The correct path is 3 + 4 x 2 = 3 + 8 = 11. Someone who adds first would get 7 x 2 = 14, which is wrong. The order of operations is what stops two people from getting two different answers.

Worked Example: 3 + 4 x 2^2

  1. Exponent first: 2^2 = 4. Expression becomes 3 + 4 x 4.
  2. Multiplication next: 4 x 4 = 16. Expression becomes 3 + 16.
  3. Addition last: 3 + 16 = 19.
  4. Final answer: 19.

Working with Integers

  • Negative numbers follow the same order rules — handle the sign as you go.
  • (-3)^2 means (-3) x (-3) = 9 (parentheses make the whole -3 the base).
  • Without the parentheses, -3^2 means the opposite of 3^2, which is -9.
  • When a multiplication and a division sit next to each other, work LEFT to RIGHT — not multiplication first.
PEMDAS Order of OperationsA four-level ladder listing Parentheses, Exponents, Multiply and Divide, then Add and Subtract, from highest to lowest priority.PEMDAS Order of OperationsPParentheses(grouping symbols first)EExponentspowers and rootsM DMultiply and Divideleft to right (equal priority)A SAdd and Subtractleft to right (equal priority)Top to bottom. M and D tie. A and S tie. Work each tie left to right.
Four levels, top to bottom. Ties resolve left to right.
Worked Example using PEMDASStep-by-step evaluation of 3 plus 4 times 2 squared: handle exponent, then multiply, then add to get 19.Worked Example: 3 + 4 x 2^2One operation per step. Follow PEMDAS top to bottom.Start3 + 4 x 2^2expressionStep 13 + 4 x 4E: 2^2 = 4Step 23 + 16M: 4 x 4 = 16Step 319A: 3 + 16 = 19Final answer: 19Common error: adding before multiplying. Multiply first.
Work one operation at a time, rewriting the rest each line.
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Tip: do ONE operation per line and rewrite the rest of the expression. This prevents you from forgetting a number and makes mistakes easy to spot.

Vocabulary
order of operations
The agreed order for evaluating an expression: parentheses, exponents, multiply and divide left to right, add and subtract left to right.
parentheses
Grouping symbols ( ) that tell you to do whatever is inside first.
exponent
A small raised number that says how many times to multiply the base by itself, like the 2 in 5^2.
evaluate
To find the value of an expression by performing all of its operations in the correct order.
expression
A combination of numbers, operations, and grouping symbols that has a value but no equals sign.
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← Lesson 6: Powers and Exponents (Intro)Lesson 8: Multi-Step Word Problems →