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Maths Foundations

Multi-Step Arithmetic and Order of Operations

This lesson teaches you to evaluate multi-step calculations accurately by following the order of operations — brackets, then multiplication and division, then addition and subtraction — and to estimate as a check, rather than working left to right.

  • Apply the order of operations to a multi-step calculation.
  • Work one clean step at a time.
  • Estimate to check the answer is a sensible size.
Free sample lesson — reading only

Lesson overview

What this free sample teaches

Evaluate the calculation in the correct order and check it with an estimate.

Focus

  • Order of operations: brackets, orders, x and /, + and -.
  • One step at a time: keep each result clean.
  • Estimate check: round and roughly recompute.

What gets tested

  • Applying the order of operations.
  • Working multi-step sums accurately.
  • Estimating to check the answer.

Quick guide

  • Do brackets first, then x and /, then + and -.
  • Write each step down; don't rush in your head.
  • Estimate to catch a big slip.

Success criteria

  • You can state the order of operations.
  • You can evaluate a multi-step sum correctly.
  • You can explain why left-to-right gives the wrong answer.

teach

Teach: do the steps in the right order, and check the size

Multi-step sums follow a fixed order — brackets, then x and /, then + and - — and an estimate guards against slips.

Arithmetic questions look simple but reward accuracy: doing operations in the correct order, keeping each step clean, and checking the answer is a sensible size. The order of operations is not optional — it changes the answer.

When a calculation mixes operations, the order you work in decides the answer. Mathematicians agree on one order so everyone gets the same result: brackets first, then powers, then multiplication and division (left to right), and finally addition and subtraction (left to right). Working strictly left to right is the most common way to get a 'simple' sum wrong.

A helpful name for the order is BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction); BIDMAS and PEMDAS are the same rule. Division and multiplication share a level and are done left to right; so do addition and subtraction. Treat anything inside brackets as a mini-problem to finish first, then bring its single value into the rest.

Work one step at a time and write each result down. Rushing several operations in your head is where careless errors creep in under time pressure. After each step the calculation gets shorter and simpler, until a single number is left. Neat, one-step-at-a-time working is faster overall because you do not have to redo a slip.

Finish with an estimate to catch big mistakes. Round the numbers and do a rough calculation: if your exact answer is wildly different from the estimate, you have slipped somewhere — often the order of operations. An answer that is far too large usually means you added before multiplying, or ignored the brackets.

Order of operations

the fixed order: brackets, orders, x and /, + and -.It changes the answer; it is not optional.

BODMAS

Brackets, Orders, Division/Multiplication, Addition/Subtraction.A name for the order (same as BIDMAS / PEMDAS).

Brackets first

finish anything inside brackets before the rest.Brackets override the normal order.

Estimate check

round and roughly recompute to test the answer's size.Catches order-of-operations slips.
Anatomy of a multi-step calculation
  1. Question cluework out, calculate, what is the value of, or evaluate
  2. Core evidencethe operations present and their order: brackets, then orders, then x and /, then + and -
  3. Reasoning movedo brackets first, then x and / (left to right), then + and - (left to right), one step at a time
  4. Trap checkworking left to right and ignoring the order of operations
  5. Answer shapeWorking in order: ... = ... (estimate confirms).

The moveMove from the calculation through the correct order of operations to the answer, then estimate-check.

  • You can apply the order of operations to a multi-step sum.
  • You can work one clean step at a time.
  • You can estimate to check your answer is sensible.

show

Show: a worked example

Watch a strong mathematician prove the answer, extract the method, and prepare to reuse it.

Read the problem and the question, then follow the worked solution. First prove the correct answer. Then name the reusable method, recognise its common traps, and turn the solution into actions for the next question.

Full problem: follow the order

Question: What is the value of 6 + 4 x (8 - 3)?

Order of Operations

Calculate the value of:

6 + 4 x (8 - 3)

There are three operations here — an addition, a multiplication, and a subtraction inside brackets. The order you do them in decides the answer.

  1. Step 1 - Decode the question and name what to look for

    Name the exact thinking the question wants before you start.

    This mixes brackets, multiplication and addition, so it is an order-of-operations problem. I must follow BODMAS — brackets first, then multiply, then add — not work left to right.

    • What the question is really askingThe student names what to look for first, so the search has a clear target.
    • What not to confuseNaming the question type heads off the nearest wrong move before it starts.

  2. Step 2 - Reasoning chain: clue -> rule -> step -> answer

    Follow the strong reasoner's path from the facts to the answer.

    Work in the correct order, one step at a time.

    • Brackets first: 8 - 3 = 5. The calculation becomes 6 + 4 x 5.
    • Multiplication before addition: 4 x 5 = 20. The calculation becomes 6 + 20.
    • Addition last: 6 + 20 = 26.
    • Estimate check: roughly 6 + 4 x 5 is about 26, which matches.
    • Answer: 26.
    • Clue to ruleThe proof step names the exact fact or rule, not a general impression.
    • Step and effectThe student shows what each step forces and how it decides the answer.

  3. Step 3 - Common wrong answer: spot the trap

    See why a tempting answer is wrong before choosing.

    Common wrong answer: 50.

    Why students choose it: they work left to right — 6 + 4 = 10, then 10 x 5 = 50 — which feels natural.

    Why it is wrong: addition must come after multiplication, and the brackets must be done first. Working left to right ignores the order of operations. The correct order gives 6 + (4 x 5) = 26.

    Corrected reasoning: brackets (5), then 4 x 5 = 20, then 6 + 20 = 26.

    • Why it tempts studentsA trap usually uses a real fact, which is why students choose it too quickly.
    • Corrected reasoningThe fix shows which fact or rule changes the answer, not just that it is wrong.

  4. Step 4 — The answer, and why it is correct

    State the answer and give the shortest complete proof.

    The value is 26. Following the order of operations, do the brackets first (8 - 3 = 5), then the multiplication (4 x 5 = 20), then the addition (6 + 20 = 26). A quick estimate, 6 + 4 x 5, confirms an answer around 26 — not 50, which comes from wrongly adding before multiplying.

    • The answerThe result is stated directly, with its unit when the question requires one.
    • Why it is correctEvery important step is tied to the relevant property, operation, relationship or check.

  5. Step 5 — Going deeper: the reusable method and the traps

    Name the method, explain why it works, and turn mistakes into warnings.

    The answer is 26, and the reliable route is to apply the order of operations explicitly. Brackets are resolved first (8 - 3 = 5), reducing the problem to 6 + 4 x 5; multiplication outranks addition, so 4 x 5 = 20 is done before the final 6 + 20 = 26. The tempting 50 comes from a strict left-to-right reading (6 + 4, then x 5), which the convention specifically forbids. A rounded estimate (about 26) is the cheap safeguard: it instantly flags the left-to-right error, because 50 is far larger than a sum that adds 6 to roughly twenty.

    • Reusable methodThe explanation names the mathematical structure that makes the method work again.
    • Trap patternsTempting mistakes become recognisable patterns that can be caught next time.

  6. Step 6 — How to use this method next time

    Turn this solution into concrete actions for a new question.

    In this questionOn the next questionWhy it helps
    Evaluated brackets, then multiplication, then addition.Mark the operation order before calculating when a line mixes brackets, multiplication and addition.The marked structure prevents an automatic left-to-right calculation.
    Rewrote the expression after each completed operation.Carry forward the untouched parts of the expression on every new line.This preserves the original structure and reduces dropped-operation errors.
    Estimated a result near 26 and rejected 50.Make a quick magnitude estimate before accepting the choice.An estimate can expose a wrong operation order without repeating the calculation.

compare

Compare: left-to-right vs the order of operations

The difference between reading the sum and following the rule.

Both students looked at the same problem. One stops at a first impression; the other proves the answer from the facts and rules. Markers reward the second.

The whole response

Left to right

6 + 4 = 10, then 10 x 5 = 50.

Order of operations

Brackets (5), then 4 x 5 = 20, then 6 + 20 = 26.

Point-by-point breakdown3
  • Reading the sumWeaker: Works through it left to right.Stronger: Spots the brackets and multiplication to do first.The order of operations, not reading order, decides the answer.
  • BracketsWeaker: May ignore the brackets.Stronger: Resolves the brackets before anything else.Brackets override the normal order.
  • CheckingWeaker: Accepts 50 without a check.Stronger: Estimates (about 26) and rejects 50 as too large.An estimate catches order slips.

guide

Guide: student checkpoint

Do one small reasoning move before independent practice.

Reread the worked problem and question. Now pause like a strong reasoner: find the rule, name the trap, or upgrade a basic answer before you work on your own.

Work out 12 / (1 + 2) + 5. In one sentence, give the answer and the order you used.

Want feedback on your own answer? Get started to practise with instant marking.

  • Apply brackets, then x and /, then + and -.
  • Work one step at a time.
  • Estimate to check the answer.