What is:
8 + 3 × 4?
One learner works from left to right:
8 + 3 = 11.
11 × 4 = 44.
Another learner multiplies first:
3 × 4 = 12.
8 + 12 = 20.
The expression has only one conventional value:
20.
Order of operations is a reading rule for mathematical structure. It tells us how one line of symbols should be grouped before calculation begins.
This is why the topic matters. It is not about memorising a slogan such as “BODMAS” or “PEMDAS” for its own sake.
It is about preventing one expression from producing several incompatible answers simply because readers grouped its operations differently.
Under the current Singapore Primary Mathematics syllabus, order of operations without calculator and the use of brackets are explicit Primary 5 content. That placement is important: the topic builds on earlier fluency with all four operations and prepares learners for algebraic expressions, formulae and more compact symbolic mathematics.
The quick answer: multiplication and division bind before addition and subtraction
In an expression containing the four basic operations and no brackets:
- Do multiplication and division first, working from left to right among operations of equal priority.
- Then do addition and subtraction, working from left to right among operations of equal priority.
For:
18 − 12 ÷ 3 + 5
divide first:
12 ÷ 3 = 4.
The expression becomes:
18 − 4 + 5.
Now addition and subtraction have equal priority, so work left to right:
18 − 4 = 14.
14 + 5 = 19.
Why multiplication comes before addition
Consider:
8 + 3 × 4.
The multiplication 3 × 4 describes three groups of four, or 12.
The expression therefore means:
8 + twelve.
The product is one quantity embedded inside a larger sum.
If we added 8 + 3 first, we would change the grouping to:
(8 + 3) × 4.
That is a different expression.
So multiplication-before-addition is not arbitrary decoration. It is part of how standard notation encodes grouping without requiring brackets around every product.
Brackets explicitly change the grouping
Compare:
8 + 3 × 4 = 20.
and:
(8 + 3) × 4 = 44.
The same numbers and operation symbols appear.
The brackets change which calculation is treated as one unit first.
Brackets are structural punctuation. They tell the reader which sub-expression must be resolved as one object before the surrounding expression continues.
Division and multiplication have equal priority
Consider:
24 ÷ 6 × 2.
Division does not always come before multiplication merely because the letter D may appear before M in a mnemonic.
They have the same priority.
Work from left to right:
24 ÷ 6 = 4.
4 × 2 = 8.
If a learner multiplies 6 × 2 first, the expression has effectively been changed into:
24 ÷ (6 × 2),
which is not the same notation.
Addition and subtraction also have equal priority
Consider:
20 − 8 + 3.
Work left to right:
20 − 8 = 12.
12 + 3 = 15.
Doing 8 + 3 first would create:
20 − (8 + 3) = 9,
which again is a different grouped expression.
A useful structural reading: expression trees without the terminology
Take:
18 + 4 × 5 − 6.
Instead of reading this as a flat row, identify the embedded product:
4 × 5 = 20.
Now the larger expression is:
18 + 20 − 6.
This can then be evaluated left to right:
38 − 6 = 32.
The multiplication was a smaller calculation nested inside the larger additive structure.
This way of seeing expressions prepares learners for algebra, where products such as 3x are routinely treated as single terms inside sums.
Worked example: 36 − 4 × 5
Estimate the structure first.
The multiplication is 4 × 5 = 20.
Then:
36 − 20 = 16.
A left-to-right answer would be:
(36 − 4) × 5 = 160.
The enormous difference between 16 and 160 shows why standard grouping conventions matter.
Worked example: 48 ÷ 6 + 7 × 3
Resolve multiplication and division first.
48 ÷ 6 = 8.
7 × 3 = 21.
Now:
8 + 21 = 29.
Notice that the division and multiplication can be resolved independently because they belong to separate terms of the final addition.
Worked example with brackets
50 − (18 + 6) ÷ 4.
Brackets first:
18 + 6 = 24.
Then division:
24 ÷ 4 = 6.
Then subtraction:
50 − 6 = 44.
The brackets create a quantity of 24 that is then divided by 4.
Why calculators can produce surprising answers
Many scientific calculators follow standard operation precedence.
Entering:
8 + 3 × 4
typically returns 20.
A learner expecting 44 may think the calculator is wrong.
The more useful conclusion is that the learner and calculator grouped the expression differently.
MOE learning experiences for Primary 5 explicitly use a scientific calculator to help students discover the rules for order of operations and explain why the rules are necessary, even though the required order-of-operations calculations themselves are listed without calculator.
The calculator can therefore act as a comparison instrument, not a replacement for understanding.
BODMAS and PEMDAS are memory aids, not complete explanations
Different countries and classrooms use different mnemonics.
- BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.
- PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
At Primary 5, the relevant core is simpler:
- brackets control explicit grouping;
- multiplication and division share priority;
- addition and subtraction share priority;
- equal-priority operations are handled left to right.
The mnemonic should support this structure rather than override it.
Why “division before multiplication” is a dangerous misreading
In BODMAS, D appears before M.
That does not mean every division in an expression must be done before every multiplication.
Likewise, A before S does not give addition universal precedence over subtraction.
The correct reading groups them by equal priority:
- multiplication/division;
- addition/subtraction.
Then work left to right within each group.
The distributive property explains why brackets matter
Compare:
3 × (10 + 4).
The brackets say that 10 + 4 forms one grouped quantity.
So:
3 × 14 = 42.
Using distribution:
3×10 + 3×4 = 30 + 12 = 42.
This shows that brackets are not merely instructions to “do first”. They identify a grouped mathematical object whose structure can sometimes be transformed while preserving value.
Order of operations in word problems
Suppose four students each buy a $3 notebook, and the group also buys one $8 folder.
Total cost:
4 × 3 + 8.
The repeated notebook cost forms one product:
4 × 3 = 12.
Then add the folder:
12 + 8 = $20.
The standard order matches the real-world grouping.
This is an important connection: symbolic conventions often preserve a natural quantity structure already present in the situation.
Brackets can make word-problem structure explicit
Suppose three people each buy a meal costing $8 and a drink costing $2.
Total:
3 × (8 + 2).
The brackets show that one person’s combined meal-and-drink cost is the repeated unit.
3 × 10 = $30.
Without brackets:
3 × 8 + 2
would mean three meals plus only one drink.
The bracket changes the model, not merely the calculation order.
Common misconception 1: always work left to right
Left-to-right is used only among operations of equal priority after higher-priority structures have been resolved.
Repair: circle multiplication/division sub-expressions before calculating.
Common misconception 2: division always before multiplication
They have equal priority.
Repair: write “× and ÷: left to right” rather than reading mnemonic letters as a ranking.
Common misconception 3: addition always before subtraction
Again, they have equal priority.
Work left to right after multiplication and division are complete.
Common misconception 4: brackets are optional decoration
8 + 3 × 4 and (8 + 3) × 4 have different values.
Brackets encode a different structure.
Common misconception 5: calculator output proves understanding
A calculator can apply precedence rules correctly even when the learner cannot explain them.
Repair: predict which sub-expression the calculator will resolve first before pressing equals.
A diagnostic ladder
- Can the learner evaluate a single-operation expression?
- Can the learner distinguish multiplication/division from addition/subtraction in one expression?
- Can the learner explain why 8 + 3×4 is not read as (8+3)×4?
- Can the learner work left to right among multiplication and division?
- Can the learner work left to right among addition and subtraction?
- Can the learner interpret brackets as explicit grouping?
- Can the learner translate a word problem into an expression whose grouping matches the situation?
- Can the learner predict calculator behaviour before entering the expression?
- Can the learner detect when two different bracket placements create two different meanings?
A five-minute home investigation
Write these three expressions:
- 6 + 2 × 5;
- (6 + 2) × 5;
- 6 + (2 × 5).
Ask the learner to:
- predict which two are equivalent;
- calculate all three;
- explain what the brackets changed;
- invent a word problem for one expression;
- change the brackets to create a different word-problem meaning.
This moves the topic from rule-following into expression interpretation.
What parents should listen for
- “The multiplication is one grouped quantity inside the larger sum.”
- “Division and multiplication have equal priority, so I work left to right.”
- “The brackets change what counts as one unit.”
- “Without the brackets, this expression means three meals plus one drink, not three meal-and-drink sets.”
- “The calculator gave 20 because it followed the same precedence convention.”
How this fits Singapore Primary 5 Mathematics
The current MOE Primary Mathematics syllabus places order of operations without calculator and use of brackets without calculator in Primary 5. Its learning experiences include using a scientific calculator to discover the rules and explain why such rules are necessary.
This is a genuine year-level boundary. The topic follows Primary 4 arithmetic and problem solving, but it should not be mislabelled as explicit Primary 4 content.
The deeper lesson: notation compresses a tree of relationships into one line
A written expression looks flat.
Its mathematical structure is not.
Some operations form smaller units inside larger ones.
Brackets can override the default grouping.
Precedence rules allow mathematicians to write compact expressions without surrounding every product or quotient with brackets.
Order of operations is what lets a line of symbols carry one stable meaning for every careful reader.
Final thought
The point of order of operations is not to make arithmetic more complicated.
It is to make mathematical language less ambiguous.
Once learners see that, the rules stop looking like a referee’s arbitrary commands and begin to look like punctuation for structure.