Order of operations: make the grouping explicit
Before you start: Use addition, subtraction, multiplication, division and simple powers.
Why expressions need an agreed reading
Two loose counters plus three groups of four counters is 2 + 3 × 4. The grouped product represents 12 counters, so the total is 14. Calculating 2 + 3 first would describe a different grouping. Parentheses make that change explicit: (2 + 3) × 4 = 20.
The priority levels
Evaluate brackets from the inside out, then powers. Next perform multiplication and division at the same level from left to right. Finally perform addition and subtraction at the same level from left to right. PEMDAS and BODMAS are reminders of these levels, not instructions to do every multiplication before every division.
Worked example with several levels
Evaluate 18 − 2 × (3 + 1)² ÷ 4. First the bracket is 4, then its square is 16. The expression becomes 18 − 2 × 16 ÷ 4. Evaluate the equal-priority operations in order: 2 × 16 = 32 and 32 ÷ 4 = 8. Finish with 18 − 8 = 10. Rewrite the remaining expression after each step so that you do not lose an operation.
Left-to-right ties and negative signs
24 ÷ 6 × 2 = 4 × 2 = 8. If the intended divisor is the whole product 6 × 2, it must be grouped: 24 ÷ (6 × 2) = 2. Similarly 10 − 3 + 2 = 7 + 2 = 9, not 10 − 5.
Brackets also distinguish (−3)² = 9 from −3² = −9. In the first expression, the negative number itself is squared. In the second, the square is evaluated before the leading minus: −(3²). A fraction bar groups its entire numerator and denominator, so (6 + 2)/4 means 8/4 = 2.
Try it yourself
Evaluate 20 − 12 ÷ 3 × 2. Then evaluate (20 − 12) ÷ (3 × 2).
Show answer and explanation
For the first, 12 ÷ 3 = 4, then 4 × 2 = 8, then 20 − 8 = 12. For the second, the brackets give 8 ÷ 6 = 4/3. The grouping changes the value.
Use the activity to check your understanding
Identify the next permitted operation and explain its priority before clicking. Treat equal-priority operations as ties.