This is a habit, not a rule. The rule is BODMAS or PEMDAS — you already know it. The habit is the thing almost nobody teaches: use the rule as a scanning tool every single time, before your pen touches the paper. Before you pick up the pen, scan the expression and locate the operator hierarchy. Students who skip this step solve left-to-right, in reading order, and get wrong answers on problems they perfectly well know how to do.

The BODMAS/PEMDAS ladder as a scanning tool

You have been told BODMAS is an acronym to memorise. That is not what it is. It is a priority queue — a ladder of operator importance, with the top rung computed first and the bottom rung computed last. When you look at an expression, your eye should sweep down the ladder looking for the highest rung that is actually present.

BODMAS priority ladder with operator symbols stacked by precedenceA vertical ladder with five rungs, top to bottom. The top rung, labelled highest priority, contains parentheses or brackets. The second rung contains exponents and roots. The third rung contains multiplication and division tied at the same level. The fourth rung contains addition and subtraction tied at the same level. An arrow labelled scan top to bottom runs down the right side of the ladder. To the left of the ladder, a sample expression three plus four times two is shown with the times operator circled and a note saying resolve first. Priority ladder — scan top to bottom ( ) brackets highest x² , √ exponents × , ÷ (tied) + , − (tied) lowest scan sample: 3 + 4 · 2 × beats +, so 4·2 first Answer: 3 + 4·2 = 3 + 8 = 11. (Not (3+4)·2 = 14.) Same-priority operators (× with ÷, or + with −) are evaluated left to right.
BODMAS is not a spelling list to recite — it is a priority queue to scan. Your eye runs down the ladder from top to bottom and stops at the highest rung with an operator actually present in the expression. That is where the first move happens.

The cleanest demonstration is 3 + 4 \cdot 2. Read left to right, it looks like "three plus four is seven, times two is fourteen." That answer, 14, is wrong. Multiplication sits higher on the ladder than addition, so 4 \cdot 2 resolves first, giving 8, and then 3 + 8 = 11. Two students, same line of symbols, one scanning and one not — three points' difference on a test.

The scan is a habit

Before you write a single symbol of simplification, run this three-step pre-flight. It takes ten seconds.

  1. Find the innermost parentheses. Your eye hunts for the deepest, tightest bracket — the one with no other brackets nested inside it. If there are several at the same depth, you will handle them in parallel.
  2. Inside each bracket (or, if no brackets, in the whole expression), find the highest-priority operator. Scan the ladder: is there an exponent? If yes, that is the highest. No exponents? Look for a \times or \div. None? Then your first move is at the +/- level.
  3. That is the first move. Not the easiest-looking one, not the leftmost one — the one the ladder picks out.

Only after the pre-flight do you write the next line. Everything that feels like "intuition" in algebra is actually this habit done so often that it has gone underground.

Worked example: 2 \cdot (x + 3)^2 - 4 \cdot (x + 1) + 5

Let us simplify this carefully with the scan running. No calculator, no guessing — just the ladder.

Pre-flight.

Move 1. (x + 3)^2 = x^2 + 6x + 9. Why: this is the identity (a+b)^2 = a^2 + 2ab + b^2 applied to a = x, b = 3, giving x^2 + 2 \cdot x \cdot 3 + 3^2 = x^2 + 6x + 9. We deal with the exponent before any outside multiplication touches it — the ^2 is higher on the ladder than the 2\cdot.

Now the expression reads:

2 \cdot (x^2 + 6x + 9) - 4 \cdot (x + 1) + 5.

Pre-flight, round two. No exponents are left. Highest priority now is multiplication. There are two: 2 \cdot (\ldots) and 4 \cdot (\ldots). Both need distributing.

Move 2. Distribute the 2: 2 \cdot (x^2 + 6x + 9) = 2x^2 + 12x + 18. Distribute the 4: 4 \cdot (x + 1) = 4x + 4. Why: the distributive law says a(b+c) = ab + ac. Each term inside the bracket gets multiplied by the outside factor, one by one. Sign matters — since the 4 \cdot (x+1) has a minus in front of it, the whole product will be subtracted.

Expression becomes:

2x^2 + 12x + 18 - (4x + 4) + 5 \;=\; 2x^2 + 12x + 18 - 4x - 4 + 5.

Pre-flight, round three. Only additions and subtractions remain. Combine like terms, left to right.

Move 3. 12x - 4x = 8x, and 18 - 4 + 5 = 19. Final answer:

2 \cdot (x + 3)^2 - 4 \cdot (x + 1) + 5 \;=\; 2x^2 + 8x + 19.

Here is what goes wrong if you do not scan. A common student error is to distribute the outer 2 before squaring — writing "2 \cdot (x+3) = 2x + 6, then square it." But 2 \cdot (x+3)^2 does not mean (2(x+3))^2; the ^2 attaches only to the bracket, not to the 2. Squaring first, then multiplying by 2, is the correct order because the exponent is higher on the ladder. Get the hierarchy wrong and you end up with 4x^2 + 24x + 36 instead of 2x^2 + 12x + 18 — a silently wrong answer that looks perfectly reasonable.

Why left-to-right fails

It is worth seeing the two answers side by side. Consider 8 - 3 \cdot 2 + 4.

The right way. Multiplication is higher than addition or subtraction, so 3 \cdot 2 = 6 resolves first. Then left to right: 8 - 6 + 4 = 6. Answer: \boxed{6}.

The wrong, left-to-right way. 8 - 3 = 5; then 5 \cdot 2 = 10; then 10 + 4 = 14. Answer: 14.

Same ink on the page. Two completely different numbers. The only difference is whether your eye scanned the ladder before your pen moved.

Same-priority ties go left to right

When two operators share a rung on the ladder — two multiplications, or a multiplication and a division, or a plus and a minus — you evaluate left to right. The ladder does not pick a winner; position on the page does.

Take 10 - 4 + 1. A subtraction and an addition, tied at the bottom rung. Scan left to right: 10 - 4 = 6, then 6 + 1 = 7. Answer: \boxed{7}.

A common student error treats subtraction as if it binds the whole right-hand side: 10 - (4 + 1) = 5. Wrong. Subtraction is not right-associative; it does not invisibly wrap its right neighbour in brackets. The same trap appears with \div: 12 \div 4 \div 2 is (12 \div 4) \div 2 = 1.5, not 12 \div (4 \div 2) = 6. Left to right, every time.

Implicit multiplication tightens things

One convention worth a warning. When a number sits directly next to a variable — 2x, 5a, 3\pi — the multiplication is implicit. In strict BODMAS, this implicit multiplication often binds tighter than an explicit \times or \div. So 6 / 2x is usually read, in school and exam conventions, as 6 / (2x), not (6/2) \cdot x = 3x. Some calculators disagree. The safe rule: whenever implicit multiplication meets division on the same line, bracket it yourself. Write \dfrac{6}{2x} or \dfrac{6}{2}x explicitly. Ambiguity is what loses marks.

The takeaway

Before your first move, you spend ten seconds reading priority. You ask: where are the innermost brackets? What is the highest-priority operator in each region? Only then do you write. It costs nothing — you are going to do the algebra anyway. It prevents roughly eighty percent of the silly mistakes on simplification problems, because almost all of those mistakes are priority errors disguised as arithmetic errors.

The students who look "naturally good at algebra" are not computing faster. They are doing a ten-second scan that has become invisible through repetition. Every time you see an expression, before you pick up the pen, ask: where does the ladder start? Then start there.