Order of operations, brackets and BODMAS are common Mathematics search topics because students can know every individual operation and still get a mixed expression wrong by applying them in the wrong sequence. The difficulty becomes larger in Secondary Mathematics when brackets, negative numbers, indices, fractions and algebraic substitution appear together.
This Mathematics Improvements in Punggol guide treats order of operations as a convention for making mathematical expressions unambiguous. Major Mathematics resources such as Khan Academy and Maths Is Fun teach the same core hierarchy under names such as PEMDAS, BIDMAS or BODMAS. The letters differ by region; the underlying structure is the same: grouping and powers first, multiplication/division next, addition/subtraction last, with operations of equal priority handled from left to right.
At eduKate Punggol, Mathematics tutorials are held at 83 Punggol Central, Singapore 828761, near Punggol MRT and Waterway Point, in small groups of up to three students. In a small group, the tutor can see whether an error comes from the operation hierarchy, negative signs, bracket handling, fraction bars or simply weak arithmetic after the structure was chosen correctly.
Why order of operations exists
Without a shared convention, an expression such as 3 + 4 × 5 could produce different answers depending on which operation is done first. The agreed order makes the expression have one consistent value.
The correct evaluation is 3 + 20 = 23 because multiplication is completed before addition.
BODMAS, BIDMAS and PEMDAS describe the same hierarchy
- Brackets or grouping symbols first.
- Orders, indices or exponents next.
- Division and multiplication at the same level, from left to right.
- Addition and subtraction at the same level, from left to right.
A major misconception is thinking multiplication always comes before division because the M appears before D in one acronym. They have equal priority. The same is true for addition and subtraction.
Left-to-right matters for equal-priority operations
Consider 24 ÷ 6 × 2. Division and multiplication have equal priority, so work from left to right: 24 ÷ 6 = 4, then 4 × 2 = 8.
Doing 6 × 2 first would change the structure and produce the wrong result.
Brackets change the structure
Compare 3 + 4 × 5 with (3 + 4) × 5. The first equals 23; the second equals 35 because the bracket changes which operation is completed first.
Students should read brackets as grouping instructions, not decoration.
Fraction bars also act as grouping symbols
A fraction such as (8 + 4)/(3 + 1) means the entire numerator and entire denominator are grouped. Evaluate the top and bottom before dividing.
Students who ignore this grouping can make serious errors in algebraic fractions and formula substitution.
Negative numbers create another layer
In −3², the exponent applies to 3 before the leading negative, so the value is −9. In (−3)², the bracket makes the negative number the base, so the value is +9.
This distinction is important in Secondary Mathematics and links directly to the Negative Numbers and Integers guide.
Worked example: mixed operations
Expression: 6 + 3 × 4 − 2.
Multiply first: 3 × 4 = 12. Then work addition/subtraction left to right: 6 + 12 − 2 = 16.
Worked example: brackets and powers
Expression: 2(3 + 5²).
Evaluate the power: 5² = 25. Then the bracket: 3 + 25 = 28. Then multiply: 2 × 28 = 56.
Worked example: division and multiplication left to right
Expression: 48 ÷ 8 × 3.
48 ÷ 8 = 6, then 6 × 3 = 18.
Worked example: nested structure
Expression: 4 + 2[3(5 − 2) + 1].
Start inside the innermost bracket: 5 − 2 = 3. Then 3 × 3 = 9. Add 1 to get 10. Multiply by 2 to get 20. Finally 4 + 20 = 24.
Order of operations in formula substitution
If a formula contains several operations, substitute with brackets around negative or compound values. For example, if x = −2 in 3x² + 4, write 3(−2)² + 4 before simplifying.
Brackets protect the sign and make the structure visible.
Order of operations in Algebra
Algebraic simplification depends on the same hierarchy. In 2 + 3x, multiplication between 3 and x is part of one term. In 2(3x + 4), the bracket controls distribution.
The companion Algebra improvement guide extends these ideas into equations and expressions.
Calculator use does not remove the need for structure
A calculator follows its programmed order of operations, but the student still has to enter the expression correctly. Missing brackets can change the meaning of the input.
Estimate before pressing equals. If the display is wildly different from the expected scale, inspect the expression and bracket placement.
The order-of-operations error taxonomy
- Priority error — addition or subtraction is performed too early.
- Left-to-right error — multiplication is always forced before division or addition before subtraction.
- Bracket error — grouping symbols are ignored or removed incorrectly.
- Exponent error — powers are confused with multiplication by the exponent.
- Negative-sign error — the base of a power is misunderstood.
- Fraction-grouping error — numerator or denominator structure is lost.
- Calculator-entry error — brackets are missing from the input.
The reliable BODMAS routine
- Rewrite the expression clearly if needed.
- Mark grouping symbols.
- Evaluate powers or indices.
- Complete multiplication and division from left to right.
- Complete addition and subtraction from left to right.
- Check the sign and approximate magnitude.
Why memorising the acronym is not enough
A student may recite BODMAS perfectly and still mishandle 24 ÷ 6 × 2 because the acronym does not explain equal priority. The learner needs the hierarchy and left-to-right rule, not just the letters.
How to practise order of operations efficiently
Start with simple mixed arithmetic. Add brackets. Add powers. Add negative numbers. Then use formula substitution and algebraic expressions.
Each layer should be reliable before the expression becomes visually dense.
How to know the skill is improving
- Mixed expressions are evaluated consistently.
- Equal-priority operations are handled left to right.
- Negative bases are bracketed correctly.
- Fraction bars are treated as grouping.
- Calculator entries preserve the original structure.
- Algebraic substitution produces fewer sign errors.
- The student can explain why one step must come before another.
How small-group tuition can help
One student may know the hierarchy but misread negative powers; another may mishandle division/multiplication order; another may enter calculator expressions without brackets. A small group lets the tutor target those specific failure points.
Frequently asked questions
Is multiplication always before division?
No. Multiplication and division have equal priority and are handled from left to right.
Is addition always before subtraction?
No. Addition and subtraction also have equal priority and are handled from left to right.
Why is −3² different from (−3)²?
Because the bracket changes the base of the exponent. Without brackets, the square applies to 3 before the leading negative.
Continue the Mathematics Improvements in Punggol lane
- How to Master Factors, Multiples, Prime Numbers, HCF and LCM.
- How to Improve Number Patterns, Sequences and nth-Term Thinking.
- How to Improve Coordinates, Linear Graphs, Gradient and Intercepts.
- How to Master Negative Numbers and Integers.
Order of operations becomes reliable when students understand the hierarchy rather than merely chant an acronym. Group first, powers next, then multiplication and division left to right, then addition and subtraction left to right. Preserve brackets and signs, and the same logic carries directly into Algebra and formula work.
Further learning: Khan Academy Order of Operations · Maths Is Fun BODMAS.
