Operations and Algebraic Thinking: Expressions, Grouping Symbols, and Patterns
Operations and Algebraic Thinking in Grade 5 asks you to evaluate expressions that contain parentheses, brackets, and braces; to translate a written calculation into an expression; and to generate and compare two numerical patterns. Work is limited to the four operations - exponents belong to Grade 6.
Order of operations with grouping symbols
Work from the innermost grouping symbol outward: parentheses ( ), then brackets [ ], then braces { }. Inside any grouping symbol, multiply and divide from left to right, then add and subtract from left to right.
- 2 + 3 × 4 = 14, not 20.
- 20 − 8 + 2 = 14, because subtraction and addition are handled left to right.
Writing expressions
Words such as sum, difference, product, and quotient name the operation, and the grouping must match the meaning. “The sum of 12 and 9, divided by 3” is (12 + 9) ÷ 3, which equals 7 - not 12 + 9 ÷ 3 = 15. A letter can stand for an unknown number: “8 more than 5 times n” is 5n + 8.
Patterns
Given two rules, list each pattern, pair corresponding terms, and look for a relationship. If Pattern A starts at 0 and adds 3 (0, 3, 6, 9, …) and Pattern B starts at 0 and adds 6 (0, 6, 12, 18, …), each B term is twice the matching A term. The ordered pairs (0, 0), (3, 6), and (6, 12) can be graphed in the first quadrant.
Worked example
Evaluate 4 × [18 ÷ (6 − 3) + 5]. Parentheses first: 6 − 3 = 3. Inside the brackets, divide: 18 ÷ 3 = 6. Add inside the brackets: 6 + 5 = 11. Multiply by 4: 4 × 11 = 44.
Traps that cost marks
- Adding before multiplying when no grouping symbol requires it.
- Subtracting in the wrong direction in a left-to-right chain.
- Dropping needed parentheses when writing an expression from words.
- Pairing pattern terms incorrectly - match term 1 with term 1, not with the nearest number on the list.