1.1 Calculations

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: adding and subtracting, negative numbers, the order of operations and inverse operations. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Add and subtract whole numbers using written column methods
• Add, subtract, multiply and divide with negative numbers
• Use the order of operations (BIDMAS) to get the right answer
• Use inverse operations to check answers and find missing numbers

Key words

• Integer: A whole number. It can be positive, negative or zero, e.g. −3, 0, 12.
• Negative number: A number less than zero, e.g. −5. It sits to the left of 0 on a number line.
• Sum / difference: The sum is the answer to an addition. The difference is the answer to a subtraction.
• Product / quotient: The product is the answer to a multiplication. The quotient is the answer to a division.
• Inverse operation: The operation that undoes another. + undoes −, × undoes ÷, and the other way round.

Written addition and subtraction

Line up the digits in their place-value columns: units under units, tens under tens. Start with the units column and work to the left. When a column adds up to 10 or more, carry 1 into the next column.

• ; Th 3; H 8; T 4; U 7
• +; Th 2; H 5; T 9; U 6
• carry; Th 1; H 1; T 1; U
• sum; Th 6; H 4; T 4; U 3

Worked example: column subtraction with zeros

Work out 6005 − 2378
Units: 5 − 8 will not go. There is nothing to borrow in the tens or hundreds, so exchange from the thousands: 6005 becomes 5 thousands, 9 hundreds, 9 tens and 15 units.
15 − 8 = 7, 9 − 7 = 2, 9 − 3 = 6, 5 − 2 = 3
Check with the inverse: 3627 + 2378 = 6005 ✓

Answer: 3627

Adding and subtracting negative numbers

Use a number line. Adding a positive number moves you to the right. Subtracting a positive number moves you to the left.

Rule: Two signs next to each other: + (−) is the same as −, and − (−) is the same as +. So 5 + (−8) = 5 − 8 = −3 and 2 − (−6) = 2 + 6 = 8.

Worked example: temperatures

At 5 am the temperature is −6 °C. By noon it has risen by 9 degrees. Then it falls by 11 degrees by midnight.
Noon: −6 + 9 = 3 °C
Midnight: 3 − 11 = −8 °C

Answer: noon 3 °C, midnight −8 °C

Multiplying and dividing negative numbers

Work out the numbers as normal, then decide the sign.

• positive and positive; Answer positive; Example 6 × 4 = 24
• negative and negative; Answer positive; Example −6 × −4 = 24, −35 ÷ −7 = 5
• one positive, one negative; Answer negative; Example −6 × 4 = −24, 36 ÷ −9 = −4

Worked example: three numbers

Work out −2 × −3 × −5
−2 × −3 = 6 (same signs, positive)
6 × −5 = −30 (different signs, negative)

Answer: −30

Squaring a negative number gives a positive answer: (−7)² = −7 × −7 = 49.

The order of operations

BIDMAS: Brackets → Indices (powers and roots) → Division and Multiplication → Addition and Subtraction. Division and multiplication have equal priority: work left to right. The same goes for addition and subtraction.

Worked example: with and without brackets

(a) 7 + 3 × 4 = 7 + 12 = 19
(b) (7 + 3) × 4 = 10 × 4 = 40
(c) 20 − 12 ÷ 4 + 2² = 20 − 12 ÷ 4 + 4 = 20 − 3 + 4 = 21

Answer: (a) 19 (b) 40 (c) 21

A fraction line acts like a bracket. In (18 + 6) / (11 − 3) work out the top and the bottom first: 24 ÷ 8 = 3.

Inverse operations

Inverse operations undo each other. Use them to check an answer, or to work backwards to a missing number.

Worked example: missing number

I think of a number, multiply it by 6 and subtract 5. I get 37. What was my number?
Undo − 5: 37 + 5 = 42. Undo × 6: 42 ÷ 6 = 7
Check: 7 × 6 − 5 = 42 − 5 = 37 ✓

Answer: 7

Worked example: checking a division

Hana says 247 ÷ 13 = 19. Check her answer with an inverse operation.
19 × 13 = 190 + 57 = 247, so she is right.

Answer: correct

Common mistakes

• Doing a calculation from left to right instead of using BIDMAS: 7 + 3 × 4 is 19, not 40.
• Thinking “two negatives make a negative”. When you multiply or divide, two negatives make a positive.
• Treating −3 − 5 as 2 or 8. Start at −3 and move 5 to the left: −8.
• Undoing operations in the wrong order. Work backwards: undo the last step first.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out -8 + 13.

Answer: 5

Working: From −8, moving 8 right reaches 0, then 5 more reaches 5. So -8 + 13 = 5.

Question 2. Work out -6 × -7.

Answer: 42

Working: 6 × 7 = 42. Both numbers are negative, so the answer is positive: 42.

Question 3. Work out 18 - 6 ÷ 3 + 1. (Choose from: 5, 17, 3, 13.)

Answer: 17

Working: 6 ÷ 3 = 2 first, then 18 - 2 + 1 = 17.

Related lessons

• Next lesson: 1.2 Decimal Numbers
• In this chapter: 1.3 Place Value
• In this chapter: 1.4 Factors and Multiples
• In this chapter: 1.5 Squares, Cubes and Roots

Continue in the full course

This lesson is section 1.1 (EGCMF0101) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.