Algebraic Thinking and Expressions
The Algebraic Thinking domain tests whether you can read, write, and evaluate numerical expressions and recognize patterns. Grade 5 FAST items use whole-number expressions with parentheses, brackets, and braces; they ask you to interpret a written description as an expression, evaluate an expression using order of operations, and extend or describe a numerical pattern from a rule or table.
Order of operations and grouping
- Evaluate inside grouping symbols first, working from the innermost set outward: parentheses ( ), then brackets [ ], then braces { }.
- Multiply and divide from left to right.
- Add and subtract from left to right.
- Grouping symbols can change the value. Compare 3 × 8 + 4 = 28 with 3 × (8 + 4) = 36.
Writing expressions
Translate words into operations carefully. Phrases such as “the sum of 8 and 5, multiplied by 4” become 4 × (8 + 5). “Subtract 6 from 20” means 20 − 6, not 6 − 20. “Three times the difference of 9 and 2” means 3 × (9 − 2).
Patterns and rules
A pattern rule describes how to get from one term to the next. In 4, 7, 10, 13, the rule is “add 3.” In an input-output table, test one operation at a time. If input 2 gives output 7 and input 5 gives output 16, the rule 3n + 1 fits both: 3(2) + 1 = 7 and 3(5) + 1 = 16.
Worked example
Evaluate 3 × (14 − 8) + 12 ÷ 4.
- Parentheses: 14 − 8 = 6.
- Multiply and divide left to right: 3 × 6 = 18 and 12 ÷ 4 = 3.
- Add: 18 + 3 = 21.
Traps that cost marks
- Working strictly left to right and adding before multiplying in 3 × 6 + 3.
- Ignoring brackets or braces after evaluating the parentheses inside them.
- Reversing subtraction phrases such as “less than” or “subtract from.”
- Assuming a pattern is additive when it is multiplicative; check consecutive terms before naming the rule.
- Using the input as the output in a table rule, or applying the rule backward when the question asks for the next term.