Order of Operations

The order of operations (BIDMAS) tells us that multiplication is performed before addition. So we calculate \(4 \times 2 = 8\) first, then add: \(3 + 8 = 11\). The answer 14 comes from incorrectly working left to right: \(3 + 4 = 7\), then \(7 \times 2 = 14\).

Think of it as: “3 plus four lots of 2.” The multiplication creates a group — you wouldn’t split it up by adding the 3 first. If we wanted to add first, we’d need brackets: \((3 + 4) \times 2 = 14\).

Analogy: Operations have different strengths of “gravity”. Indices pull numbers together tighter than multiplication, and multiplication pulls tighter than addition. Brackets are like a forcefield that shields a calculation, forcing it to be resolved completely before the outside gravity can affect it.

Addition and subtraction have equal priority, so we work left to right. That gives \(8 \;-\; 3 = 5\), then \(5 + 2 = 7\). The wrong answer of 3 comes from doing the addition first: \(3 + 2 = 5\), then \(8 \;-\; 5 = 3\).

Students who get 3 are misreading BIDMAS as “A comes before S,” but the A and S are at the same level. A helpful way to remember: in BIDMAS, the D M pair and the A S pair are each done left to right, not in the order the letters appear.

Indices (powers) come before multiplication in BIDMAS. The power applies only to the 3, so we evaluate \(3^2 = 9\) first, then \(2 \times 9 = 18\). The incorrect answer of 36 comes from multiplying first: \(2 \times 3 = 6\), then squaring: \(6^2 = 36\).

Read \(2 \times 3^2\) as “2 lots of 3 squared,” not as “2 times 3, squared.” If we wanted to square the product, we’d write \((2 \times 3)^2 = 6^2 = 36\) — the brackets tell us to multiply first.

Division has higher priority than subtraction, so we evaluate \(12 \div 4 = 3\) first, then subtract: \(20 \;-\; 3 = 17\). The common wrong answer of 2 comes from working left to right: \(20 \;-\; 12 = 8\), then \(8 \div 4 = 2\).

Think of it as: “20 minus a quarter of 12.” The division creates a group that must be resolved before the subtraction can happen. If we wanted to subtract first, we’d need brackets: \((20 \;-\; 12) \div 4 = 2\).

The radical symbol (the square root line) acts as a grouping symbol, just like brackets. This means that the addition inside the root must be completely resolved before the root is applied.

So, we first calculate \(9 + 16 = 25\), and then evaluate \(\sqrt{25} = 5\). The incorrect answer comes from distributing the root to each number separately (\(\sqrt{9} + \sqrt{16} = 3 + 4 = 7\)), which violates the order of operations.

Example: \(2 + 3 \times 4\) — left to right gives 20, but the correct answer is 14.

Another: \(10 \;-\; 6 \div 2\) — left to right gives 2, but the correct answer is 7.

Creative: \(1 + 1 \times 0\) — left to right gives 0, but the correct answer is 1. The left-to-right error wipes out the entire expression!

Trap: \(6 \div 2 \times 3\) — a student might offer this, but division and multiplication have equal priority, so left to right is the correct method here — both approaches give 9. This doesn’t satisfy the condition because there’s no difference between left-to-right and the correct order of operations. (A student with the “M before D” misconception would get 1, but that’s a different kind of error.)

Example: \((2 + 3) \times 4 = 20\), versus \(2 + 3 \times 4 = 14\). Brackets force the addition first.

Another: \((10 \;-\; 6) \div 2 = 2\), versus \(10 \;-\; 6 \div 2 = 7\).

Creative: \(5 \times (3 \;-\; 3) = 0\), versus \(5 \times 3 \;-\; 3 = 12\). The brackets collapse the answer to zero!

Trap: \((2 \times 3) + 4 = 10\), and \(2 \times 3 + 4 = 10\). The brackets don’t change anything because multiplication already has higher priority than addition. Brackets only change the answer when they override the natural order.

Example: \(4 \times 6 \;-\; 8 \div 2 = 24 \;-\; 4 = 20\)

Another: \(2 + 3 \times 6 = 2 + 18 = 20\)

Creative: \(2^4 + 12 \div 3 = 16 + 4 = 20\) — uses a power and division. Or \((7 \;-\; 3)^2 + 4 = 16 + 4 = 20\) — uses brackets and a power.

Trap: \(6 + 4 \times 2\) — a student working left to right gets \((6 + 4) \times 2 = 20\) and confidently offers it. But the correct evaluation is \(6 + 8 = 14\), not 20. The expression doesn’t satisfy the condition at all — this is exactly the kind of error the order of operations is designed to prevent.

Example: \(5 \times 2^3\) — correct: \(5 \times 8 = 40\). Mistake (multiplying first): \(10^3 = 1000\).

Another: \(1 + 4^2\) — correct: \(1 + 16 = 17\). Mistake (adding first): \(5^2 = 25\).

Creative: \(-3^2\) — correct: \(-(3^2) = -9\). Mistake: \((-3)^2 = 9\). The negative sign is treated as subtraction \((0 \;-\; 3^2)\), so the power applies to 3 alone. This catches even confident students.

Trap: \((5 \times 2)^3 = 10^3 = 1000\) — a student might offer this, but the brackets force the multiplication to happen first, so there is no order-of-operations mistake to be made. The brackets remove the ambiguity entirely.

It depends on where the brackets go. Brackets change the answer when they override the natural order of operations: \((2 + 3) \times 4 = 20\), but \(2 + 3 \times 4 = 14\).

However, if the brackets simply enclose an operation that would already be done first, they have no effect: \((2 \times 3) + 4 = 10\), and \(2 \times 3 + 4 = 10\). The brackets are redundant because multiplication already has higher priority than addition.

Division and multiplication have equal priority. The fact that D appears before M in the acronym BIDMAS does not mean division is done first. When both appear, we work left to right.

This misconception causes real errors. Consider \(12 \div 2 \times 3\): the correct answer (left to right) is \(6 \times 3 = 18\). A student who does multiplication first gets \(2 \times 3 = 6\), then \(12 \div 6 = 2\) — completely wrong. Think of BIDMAS as B–I–DM–AS, where DM and AS are pairs at the same level.

The B in BIDMAS stands for Brackets, and they are always the first priority. Whatever is inside the brackets must be evaluated before the result is used in the rest of the calculation.

For example, in \(3 \times (2 + 5)\), we must compute \(2 + 5 = 7\) before multiplying, even though multiplication normally has higher priority than addition. That is precisely the purpose of brackets: they override the default order. Note that inside the brackets, the normal order of operations still applies: in \((3 + 2 \times 4)\), we do \(2 \times 4 = 8\) first, then \(3 + 8 = 11\).

Addition and multiplication are commutative — swapping the numbers doesn’t change the answer: \(3 + 5 = 5 + 3 = 8\) and \(4 \times 6 = 6 \times 4 = 24\).

But subtraction and division are not commutative: \(8 \;-\; 3 = 5\) but \(3 \;-\; 8 = -5\), and \(12 \div 4 = 3\) but \(4 \div 12 = \frac{1}{3}\). So whether swapping changes the answer depends entirely on which operation is involved. Students who assume all operations behave like addition may make errors when rearranging subtractions or divisions.

The student is using the “left-to-right” misconception — reading the calculation like a sentence and performing each operation as they reach it, ignoring the order of operations.

The correct approach: multiplication before addition. \(3 \times 2 = 6\), then \(5 + 6 = \mathbf{11}\). Working left to right only applies when operations have equal priority (e.g. addition and subtraction, or multiplication and division).

The answer is correct, but the reasoning is dangerously wrong. The student believes “A comes before S in BIDMAS, so addition has higher priority.” In reality, addition and subtraction have equal priority and are performed left to right.

Here, the student got lucky because the addition happened to come first when reading left to right. Apply the same faulty rule to \(10 \;-\; 3 + 5\) and they’d get \(3 + 5 = 8\), then \(10 \;-\; 8 = 2\) — but the correct answer is \(10 \;-\; 3 + 5 = 12\). A correct answer doesn’t validate a wrong method.

The student is doing multiplication before the index (power), which reverses the BIDMAS priority. They calculated \(2 \times 4 = 8\) and then squared the result, effectively computing \((2 \times 4)^2\) instead of \(2 \times 4^2\).

The correct approach: indices before multiplication. The power applies only to the 4: \(4^2 = 16\), then \(2 \times 16 = \mathbf{32}\). If we wanted to square the product, we would need brackets: \((2 \times 4)^2 = 64\).

The student thinks “do the brackets first” means “work left to right inside the brackets.” This is the “flat brackets” misconception — treating brackets as a signal to ignore the order of operations inside them. In fact, the full BIDMAS hierarchy still applies within brackets.

Inside the brackets, multiplication before addition: \(2 \times 4 = 8\), then \(3 + 8 = 11\). Finally, \(5 \times 11 = \mathbf{55}\). The brackets tell us to evaluate their contents first, but the contents must still be evaluated using the correct order of operations.

The student missed that a fraction bar acts as a hidden grouping symbol (brackets) for both the entire numerator and the denominator. The calculation is actually \((12 + 8) \div 4\).

By typing it into the calculator linearly without brackets, the calculator correctly followed BIDMAS and did the division first (\(8 \div 4 = 2\)) and then added 12. The correct approach is to resolve the numerator completely: \(12 + 8 = 20\), then \(20 \div 4 = \mathbf{5}\).