Order of Operations, Expressions and Patterns: Grade 5 Study Guide
Grade 5 introduces the rules that make every expression mean the same thing to everyone.
An expression is a mathematical phrase with numbers and operations, such as 3 × (4 + 5). To make sure everyone gets the same answer, mathematicians follow the order of operations: first do anything in parentheses (or brackets), then multiplication and division from left to right, then addition and subtraction from left to right.
Students also write expressions for words (add 8 and 7, then multiply by 2 is 2 × (8 + 7)) and describe patterns made from two rules.
The order of operations
Work inside brackets.
Left to right.
Left to right.
Does the answer fit an estimate?
Examples
6 + 4 × 2 = 6 + 8 = 14, not 20, because multiplication comes first. But (6 + 4) × 2 = 10 × 2 = 20. 20 − 12 ÷ 3 = 20 − 4 = 16. Parentheses can be nested: 2 × [3 + (8 − 5)] = 2 × 6 = 12.
| Expression | Steps | Value |
|---|---|---|
| 8 + 2 × 5 | 8 + 10 | 18 |
| (8 + 2) × 5 | 10 × 5 | 50 |
| 24 ÷ 6 + 3 | 4 + 3 | 7 |
| 2 × (9 − 4) + 1 | 2 × 5 + 1 | 11 |
Words to expressions
Phrases translate directly. 'Three times the sum of 5 and 2' is 3 × (5 + 2). 'The difference of 10 and 4, divided by 2' is (10 − 4) ÷ 2. You do not have to work out the answer to write the expression; the expression 4 × (12 + 6) already describes the calculation.
Patterns from two rules
Start both patterns at 0. Rule 1: add 3 gives 0, 3, 6, 9, 12. Rule 2: add 6 gives 0, 6, 12, 18, 24. Each term in the second is double the matching term in the first, because the rule adds double each time. Writing the ordered pairs (0, 0), (3, 6), (6, 12) and graphing them shows the relationship.
Common mistakes
Example: 4 + 3 × 2 is 10, not 14. Common mistakes: working left to right without respecting multiplication, treating addition as always before subtraction (they are equal rank and go left to right) and forgetting to finish all steps inside brackets.
Check your understanding: model answers
1. 17. 2. 2 × (6 + 9) = 30. 3. 36 ÷ 6 = 6.
✨ Key Takeaways
- Parentheses, then × ÷, then + −.
- Same-rank operations go left to right.
- Expressions describe a calculation.
- Patterns can be linked by a rule.
💬 Questions for Discussion & Reflection
- Evaluate 5 + 3 × 4.
- Write: add 6 and 9, then multiply by 2.
- Evaluate 36 ÷ (2 + 4).