Order of operations and long division

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Lesson notes

What is the Order of Operations?

The order of operations is a set of rules that tells us which calculation to do first in a maths problem.
It helps everyone get the same answer, so there is no confusion.
The order is: Brackets first, then Indices (powers), then Division and Multiplication, and finally Addition and Subtraction.
A simple way to remember it is the word BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction).
If you don't follow this order, you might get a different answer.

Why the Order Matters

Different orders can give different answers for the same problem.
For example, 1 + 2 × 3 = 7 because we multiply first: 2 × 3 = 6, then add 1.
If we added first, we would get (1 + 2) × 3 = 9, which is wrong.
The order of operations is a convention – a rule that everyone agrees to follow.
Using the correct order keeps maths clear and consistent.

Brackets First

Brackets are symbols like ( ) that group numbers together.
Always work out what is inside the brackets first.
For example, (2 + 3) × 4 = 5 × 4 = 20.
If there are brackets inside brackets, work from the inside out.
Example: [2 × (3 + 4)] − 5 = [2 × 7] − 5 = 14 − 5 = 9.

Indices (Powers)

An index (plural: indices) is a small number that tells you how many times to multiply a number by itself.
For example, 5² means 5 × 5 = 25.
Indices come after brackets but before multiplication, division, addition, and subtraction.
So in 3 + 5², we first do 5² = 25, then add 3 to get 28.
In 3 × 5², we first do 5² = 25, then multiply by 3 to get 75.

Multiplication and Division

After brackets and indices, do multiplication and division next.
They have the same importance, so work from left to right.
For example, 10 ÷ 2 × 3 = 5 × 3 = 15 (do division first because it is on the left).
If you did multiplication first, you would get 10 ÷ 6 = 1.67, which is wrong.
Remember: multiplication and division are done before addition and subtraction.

Addition and Subtraction

Addition and subtraction are done last.
They also have the same importance, so work from left to right.
For example, 10 − 3 + 2 = 7 + 2 = 9 (do subtraction first because it is on the left).
If you did addition first, you would get 10 − 5 = 5, which is wrong.
Always check your work by following the order step by step.

Long Division

Long division is a method for dividing big numbers step by step.
It helps you find the answer (quotient) and the leftover part (remainder).
For example, 847 ÷ 7: 7 goes into 8 once (1), remainder 1; bring down 4 to make 14; 7 goes into 14 twice (2), remainder 0; bring down 7; 7 goes into 7 once (1), remainder 0. So 847 ÷ 7 = 121.
You write the steps in a special layout with the divisor on the left and the quotient on top.
Practice with smaller numbers first, like 96 ÷ 4 = 24.

Checking Your Answers

Always check your answers to avoid mistakes.
Use estimation – round the numbers to get a rough answer.
For example, 847 ÷ 7 is about 840 ÷ 7 = 120, so 121 is close.
Use inverse operations – do the opposite operation to check.
For division, multiply the quotient by the divisor to see if you get the original number: 121 × 7 = 847.

Slides

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Practice questions

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1.What is the correct order of operations?
Easy
- ABrackets, then indices, then multiplication and division, then addition and subtraction
- BAddition, then subtraction, then multiplication, then division
- CLeft to right, no matter what
- DIndices, then brackets, then addition and subtraction, then multiplication and division
2.In the expression 6 + 2 × 3, you should do the addition first.
Easy
True or false?
3.What is the value of 2³ + 1?
Medium
- A9
- B7
- C10
- D8
4.Which of the following expressions equal 12? (select all that apply)
Medium
- A2 × (3 + 3)
- B2 × 3 + 3
- C3 × (2 + 2)
- D3 + 3 × 2
5.Put the steps of the order of operations in the correct order:
Easy
- Addition and subtraction
- Brackets
- Indices
- Multiplication and division
6.Match each expression with its value.
Medium
- 5 + 2 × 3
- (5 + 2) × 3
- 5 × 2 + 3
- 21
- 11
- 13
7.Match each expression with its correct value.
Medium
- 2 + 3 × 4
- (2 + 3) × 4
- 2 × 3²
- 14
- 20
- 18
8.Put the steps of the order of operations in the correct order (first to last).
Easy
- Brackets
- Indices (powers)
- Multiplication and division
- Addition and subtraction

Revision notes

## Order of Operations and Long Division — Quick Revision ### Order of Operations (BIDMAS) When you see a long calculation, do the steps in this order: 1. **B**rackets (parentheses) first 2. **I**ndices (powers, like 2^3) 3. **D**ivision and **M**ultiplication (left to right) 4. **A**ddition and **S**ubtraction (left to right) **Why?** The order matters! For example: - 1 + 2 × 3 = 1 + 6 = 7 (multiply first) - (1 + 2) × 3 = 3 × 3 = 9 (brackets first) **Remember:** - Always work out brackets first, even if there are powers inside. - If you have multiplication and division together, do them from left to right. - If you have addition and subtraction together, do them from left to right. - Use brackets to change the order if you want. **Examples:** - 3 + 5 × 2 = 3 + 10 = 13 - (3 + 5) × 2 = 8 × 2 = 16 - 2^3 + 4 = 8 + 4 = 12 - 2^(3+1) = 2^4 = 16 ### Long Division Long division helps you divide big numbers step by step. **Steps:** 1. Divide the first digit (or first few digits) by the divisor. 2. Write the answer above the line. 3. Multiply the divisor by that answer and write it below. 4. Subtract to find the remainder. 5. Bring down the next digit and repeat. **Example:** 432 ÷ 3 - 4 ÷ 3 = 1 (write 1) - 1 × 3 = 3, 4 − 3 = 1 - Bring down 3 → 13 - 13 ÷ 3 = 4 (write 4) - 4 × 3 = 12, 13 − 12 = 1 - Bring down 2 → 12 - 12 ÷ 3 = 4 (write 4) - 4 × 3 = 12, 12 − 12 = 0 So 432 ÷ 3 = 144. **Check your answer:** Use multiplication. 144 × 3 = 432 ✔ ### Decimals When you add, subtract, multiply, or divide decimals, line up the decimal points (for + and −) and be careful with place value. **Example:** 3.5 + 2.75 = 6.25 ### Key Tips - Always estimate first to see if your answer is sensible. - Use inverse operations to check: addition ↔ subtraction, multiplication ↔ division. Good luck on your exam!

Order of operations: The rules that tell you which calculation to do first in a maths expression. The order is: Brackets, Indices (powers), Division and Multiplication, Addition and Subtraction (BIDMAS).
Brackets: Symbols like ( ) that group numbers together. You must work out what's inside brackets first.
Indices: Powers or exponents, like the 3 in 2^3. They tell you how many times to multiply a number by itself.
Long division: A step-by-step method for dividing large numbers. You divide, multiply, subtract, and bring down digits until you finish.
Inverse operations: Operations that undo each other. Addition and subtraction are inverses; multiplication and division are inverses. You can use them to check answers.

Flashcards

What is the order of operations?

The order is: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Do them in that order, left to right for same level.

Why do we need the order of operations?

So everyone gets the same answer. Without it, expressions like 1 + 2 × 3 could be 7 or 9.

What does 'precedence' mean?

Precedence means which operation is done first. Higher precedence operations are done before lower ones.

Calculate: 1 + 2 × 3

7. Multiply first: 2 × 3 = 6, then add: 1 + 6 = 7.

Calculate: (1 + 2) × 3

9. Parentheses first: 1 + 2 = 3, then multiply: 3 × 3 = 9.

What is an exponent?

An exponent tells how many times to multiply a number by itself. For example, 5² means 5 × 5 = 25.

Calculate: 3 + 5²

28. Exponents first: 5² = 25, then add: 3 + 25 = 28.

Calculate: 3 × 5²

75. Exponents first: 5² = 25, then multiply: 3 × 25 = 75.

Calculate: (3 + 5)²

64. Parentheses first: 3 + 5 = 8, then square: 8² = 64.

Calculate: 1 - 2 × 3⁴

-161. First exponent: 3⁴ = 81, then multiply: 2 × 81 = 162, then subtract: 1 - 162 = -161.

Calculate: 1 + 2^(3+4)

129. Parentheses first: 3 + 4 = 7, then exponent: 2⁷ = 128, then add: 1 + 128 = 129.

How do you divide multi-digit numbers using long division?

Write the dividend inside the division bracket and the divisor outside. Divide, multiply, subtract, bring down the next digit, and repeat until done.

Divide 84 by 3 using long division.

28. 3 goes into 8 two times (2 × 3 = 6), subtract to get 2, bring down 4 to make 24, 3 goes into 24 eight times, so 28.

What is the dividend in a division problem?

The number being divided. In 84 ÷ 3, 84 is the dividend.

What is the divisor in a division problem?

The number you divide by. In 84 ÷ 3, 3 is the divisor.

What is the quotient in a division problem?

The answer. In 84 ÷ 3 = 28, 28 is the quotient.

What is a remainder in division?

The amount left over when a number doesn't divide evenly. For example, 7 ÷ 2 = 3 with remainder 1.

How do you check a division problem?

Multiply the quotient by the divisor and add the remainder. You should get the dividend. For example, 28 × 3 = 84.

What is an inverse operation?

An operation that undoes another. Addition and subtraction are inverses; multiplication and division are inverses.

How can estimation help check your answer?

Round the numbers and do the calculation to get an approximate answer. If your exact answer is close, it's likely correct.

Add: 12.5 + 3.75

16.25. Align the decimal points and add: 12.50 + 3.75 = 16.25.

Subtract: 10.2 - 4.35

5.85. Align decimals: 10.20 - 4.35 = 5.85.

Multiply: 2.5 × 4

10. Multiply as whole numbers: 25 × 4 = 100, then place one decimal place: 10.0.

Divide: 7.5 ÷ 3

2.5. Divide as whole numbers: 75 ÷ 3 = 25, then place the decimal: 2.5.

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