What are the different types of numbers in the Cambridge Lower Secondary Mathematics curriculum?

In the Cambridge Lower Secondary Mathematics curriculum, students learn about various types of numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Natural numbers are counting numbers starting from 1, whole numbers include all natural numbers and zero, integers extend whole numbers to include negative numbers, rational numbers are fractions or ratios of integers, and irrational numbers cannot be expressed as simple fractions.

How do you compare and order numbers effectively?

To compare and order numbers, start by looking at the number of digits; the number with more digits is larger. If they have the same number of digits, compare the digits from left to right. For decimals, compare the whole number part first, then the decimal part digit by digit. Use number lines for visual comparison, especially when dealing with negative numbers or fractions.

What is the importance of understanding place value in Mathematics?

Understanding place value is crucial in Mathematics as it helps in reading, writing, and comparing numbers. It allows students to comprehend the value of each digit in a number based on its position. This understanding is fundamental for performing arithmetic operations such as addition, subtraction, multiplication, and division, and is a key concept in the Cambridge Lower Secondary curriculum.

How can students effectively learn to perform operations with integers?

Students can effectively learn to perform operations with integers by using number lines to visualize addition and subtraction, practicing with real-life scenarios, and using rules for operations. For example, when adding integers, if the signs are the same, add the numbers and keep the sign; if different, subtract and take the sign of the larger absolute value. Consistent practice and using visual aids can enhance understanding.

What strategies can be used to solve problems involving fractions and decimals?

To solve problems involving fractions and decimals, students should first ensure they understand how to convert between the two. For fractions, find a common denominator for addition or subtraction, and multiply or divide directly. For decimals, align the decimal points for addition or subtraction, and use place value understanding for multiplication or division. Practice with word problems and real-life applications can also improve proficiency.

Why is "Treating 'subtract a negative' as a subtraction, e.g. writing $6 - (-7) = -1$." a common mistake in Integers and Order of Operations?

Why it happens: The two minus signs next to each other are read as one big subtraction. How to avoid it: Two negative signs together make a plus. Rewrite first: 6 - (-7) = 6 + 7 = 13.

Why is "Writing $-4^2 = 16$, treating it the same as $(-4)^2$." a common mistake in Integers and Order of Operations?

Why it happens: Without brackets, the minus looks like it is part of the number being squared. How to avoid it: -4^2 squares only the 4: -(4 × 4) = -16. Only (-4)^2 squares the whole -4 to give 16.

Why is "Doing all division before all multiplication, e.g. $36 \div 6 \times 3 = 2$." a common mistake in Integers and Order of Operations?

Why it happens: BIDMAS is read as a strict top-to-bottom list, so D looks like it always beats M. How to avoid it: Division and multiplication rank equally — work them left to right: 6 × 3 = 18.

Why is "Working a nested bracket from the outside in, or ignoring the inner pair." a common mistake in Integers and Order of Operations?

Why it happens: Long expressions look daunting, so the most obvious bracket gets done first. How to avoid it: Always start with the innermost bracket and work outwards, one layer at a time.

Why is "Getting the sign of a product wrong, e.g. writing $-2 \times -3 \times -1 = 6$." a common mistake in Integers and Order of Operations?

Why it happens: Only the last pair of signs is considered, not the whole chain. How to avoid it: Count every negative sign: three negatives is odd, so the answer is negative — here -6.

Cambridge Lower Secondary Mathematics (Grade 8)

Integers and Order of Operations

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Short Notes - Integers and Order of Operations

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Detailed Study Notes

Detailed notes on Number for Cambridge Lower Secondary Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Integers and Order of Operations — Cambridge Lower Secondary Maths, Grade 8

You can already calculate with negatives and follow BIDMAS. Stage 9 asks for genuine fluency — handling layered brackets, indices on negative bases, and signed numbers woven through multi-step problems without ever losing track of a sign.

At a glance

An integer is a whole number — positive, negative or zero, never a fraction.
Two signs sitting together combine, so $a - (-b) = a + b$ every time.
Same signs multiply or divide to a positive; different signs give a negative.
Count the negatives in a product: even count → positive, odd count → negative.
$(-4)^2 = 16$ but $-4^2 = -16$ — brackets decide what the index acts on.
BIDMAS sets the order: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Division and multiplication rank equally — always work strictly left to right.
A quick sense-check on the sign and size of an answer catches most slips.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Stage 9 — Calculate fluently with positive and negative integers across all four operations and longer chains.
Stage 9 — Apply the order of operations to multi-step calculations with several layers of brackets and indices.
Stage 9 — Use negative numbers confidently inside word problems and reasoning tasks.
Stage 9 — Estimate and sense-check signed calculations for a sensible sign and size.

Fluent arithmetic with integers

At Stage 9 the four sign rules should run automatically — even inside long calculations.

By Stage 9 the number line is no longer something you need to picture step by step — the sign rules should run automatically. The aim now is fluency under pressure: keeping every sign correct even in a long, busy calculation.

Two ideas still carry all of it. For adding and subtracting, two signs sitting next to each other combine into one — a plus and a minus make a minus, two minuses make a plus. For multiplying and dividing, look only at the two signs: same signs give a positive, different signs give a negative.

Four small rules cover every signed calculation you will meet.

The skill that grows at Stage 9 is stamina — staying accurate across a calculation with five or six operations. Take it one step at a time and rewrite the whole line each time; that single habit removes nearly every careless slip.

Two signs together combine: $a - (-b) = a + b$ .
Same signs multiply or divide to a positive answer.
Different signs multiply or divide to a negative answer.
Rewrite the full line after each step to stay accurate.

Indices on negative bases

A negative base to an even power is positive; to an odd power it stays negative.

Raising a negative number to a power is repeated multiplication, so the sign-counting rule applies directly. $(-3)^4$ means $-3 \times -3 \times -3 \times -3$ — four negatives, an even count, so the answer is positive $81$ . But $(-3)^3$ has three negatives, an odd count, so it is negative $-27$ .

Even powers turn a negative base positive; odd powers keep it negative.

The trap to meet head-on is the difference between $(-4)^2$ and $-4^2$ . They look almost identical but mean different things:

$(-4)^2$ means the whole of $-4$ is squared: $-4 \times -4 = 16$ .
$-4^2$ means only the 4 is squared, and the minus is applied afterwards: $-(4 \times 4) = -16$ .

The brackets decide what the index acts on. Whenever you raise a negative number to a power, wrap it in brackets so there is no doubt at all.

A power is repeated multiplication, so the sign rule applies.
Even power of a negative base → positive answer.
Odd power of a negative base → negative answer.
$(-4)^2 = 16$ but $-4^2 = -16$ — brackets change the meaning.

Order of operations with layered brackets

BIDMAS still rules — the Stage 9 step up is several layers of brackets and indices at once.

The order of operations has not changed: BIDMAS — Brackets, Indices, Division and Multiplication, then Addition and Subtraction. What grows at Stage 9 is the number of layers in a single calculation.

You will meet nested brackets and indices acting on whole bracketed expressions. Always work from the innermost bracket outwards, settling each layer before you move on.

Peel the brackets from the inside, then indices, then the rest.

So $50 - 3 \times (12 - (5 + 2))^2$ goes: inner bracket $5 + 2 = 7$ ; outer bracket $12 - 7 = 5$ ; index $5^2 = 25$ ; multiplication $3 \times 25 = 75$ ; subtraction $50 - 75 = -25$ .

Remember that division and multiplication rank equally — you work them left to right, never all division first. The same holds for addition and subtraction. A fraction bar acts like a hidden bracket: in $\frac{8 + 4}{3}$ you finish the top before dividing.

BIDMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Work nested brackets from the innermost outwards.
An index can act on a whole bracketed expression.
Division and multiplication rank equally — go left to right.

Combining signs, powers and order

Real Stage 9 calculations weave all three ideas together — one step per line.

The toughest Stage 9 calculations weave together negative numbers, indices and a multi-step order. The cure is the same every time: one step per line, rewriting the whole calculation as you go.

Try $(-3)^3 + 24 \div (-4) - (-7)$ .

Indices first: $(-3)^3 = -27$ (three negatives, odd, so negative).
Division next: $24 \div (-4) = -6$ (different signs).
The line is now $-27 + (-6) - (-7)$ .
Tidy the signs: $-27 - 6 + 7$ .
Add and subtract left to right: $-27 - 6 = -33$ , then $-33 + 7 = -26$ .

Each line does one job — the answer is hard to lose this way.

When you finish, glance back: does the sign feel right, and is the size sensible? If you expected a small negative and got a large positive, a sign has slipped somewhere.

Mix of signs, powers and order — do one operation per line.
Indices and brackets are settled before ×, ÷, + and −.
Tidy double signs as soon as the line allows it.
Sense-check the final sign and size of the answer.

Negative numbers inside problems

Word problems hide signed arithmetic — read carefully and decide what each sign means.

Stage 9 questions often dress signed arithmetic up as a real situation. The skill is deciding what a negative number means in context, and which operation the question is really asking for.

Temperatures are a classic. A reading of $-6\degree\text{C}$ rises by $11\degree$ — that is $-6 + 11 = 5\degree\text{C}$ . The difference between $-6\degree\text{C}$ and $5\degree\text{C}$ is $5 - (-6) = 11\degree$ , a subtraction of one reading from the other.

Subtracting one reading from the other gives the gap between them.

The same care helps with bank balances, heights above and below sea level and game scores. A balance of $-\$ 40 $that receives a$ $25 $payment becomes$ -40 + 25 = -$15 $— still overdrawn. A diver at$ -18 $m who rises$ 7 $m is at$ -18 + 7 = -11$ m. Read slowly, decide what the sign means, then choose the operation with confidence.

Decide what a negative number means in the situation.
A difference is one value subtracted from another.
Temperatures, balances, depths and scores all use signed arithmetic.
Read slowly, then choose the operation deliberately.

Where you'll use this next

Fluent integer work underpins almost every topic in Stage 9 and beyond.

Solid integer and order-of-operations skills make a huge amount of later maths easier:

Algebra substitutes negative values into expressions and formulae constantly — $2x^2 - 5x$ when $x = -3$ is a sign-and-power question in disguise.
Coordinate geometry uses negative coordinates on both axes, and gradients are differences that can be negative.
Standard form relies on negative indices, which are simply negative integers used as powers.
In everyday life, signed calculations describe temperatures, bank balances, elevation and any multi-step calculator problem.

If a later topic feels shaky, it is often a sign rule or a BIDMAS step underneath that needs a quick refresh. Come back to this guide whenever you need it — accuracy beats speed every time.

Algebra substitutes negative numbers into expressions and powers.
Coordinate geometry uses negative values and negative gradients.
Standard form uses negative integers as indices.
Strong integer fluency smooths every later topic.

Quick recap

An integer is a whole number — positive, negative or zero.
Two signs together combine, so subtracting a negative is the same as adding.
Same signs multiply or divide to a positive; different signs to a negative.
An even power of a negative base is positive; an odd power is negative.
$(-4)^2 = 16$ but $-4^2 = -16$ — brackets decide what is squared.
BIDMAS sets the order; work nested brackets from the inside out.
Sense-check the sign and size of every answer.

Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 9). Last reviewed 2026-05-19.

Take this whole topic with you

Step-by-step worked examples — Integers and Order of Operations

Step-by-step solutions for integers and order of operations, written exactly the way a tutor would explain them at the board.

1Tidying a chain of signed numbers
Getting started• negative numbers, double signs
Question
Work out $8 + (-5) - (-3) + (-6)$ .
Step-by-step solution
1. Step 1
  Replace each pair of signs before doing any arithmetic.
  $8 - 5 + 3 - 6$
2. Step 2
  Now add and subtract left to right.
  $8 - 5 = 3,\quad 3 + 3 = 6,\quad 6 - 6 = 0$
Answer
$0$
2BIDMAS with brackets and an index
Getting started• order of operations, bidmas
Question
Work out $4 + 2 \times (7 - 4)^2$ .
Step-by-step solution
1. Step 1
  Brackets first.
  $7 - 4 = 3$
2. Step 2
  Indices next: square the result.
  $3^2 = 9$
3. Step 3
  Multiplication before addition.
  $2 \times 9 = 18$
4. Step 4
  Finally the addition.
  $4 + 18 = 22$
Answer
$22$
3An index on a negative base
Building confidence• powers, negative numbers
Question
Work out $(-2)^4$ and $-2^4$ , and explain why they differ.
Step-by-step solution
1. Step 1
  $(-2)^4$ squares the whole of $-2$ four times. Four negatives is an even count, so the answer is positive.
  $(-2)^4 = -2 \times -2 \times -2 \times -2 = 16$
2. Step 2
  $-2^4$ raises only the 2 to the power, then applies the minus afterwards.
  $-2^4 = -(2 \times 2 \times 2 \times 2) = -16$
3. Step 3
  The brackets decide what the index acts on, so the two results have opposite signs.
Answer
$(-2)^4 = 16$ and $-2^4 = -16$
4The difference between two temperatures
Building confidence• negative numbers, word problem
Question
At dawn the temperature is $-7\degree\text{C}$ . By midday it is $6\degree\text{C}$ . How much warmer is midday?
Step-by-step solution
1. Step 1
  'How much warmer' is a difference — subtract the dawn reading from the midday reading.
  $6 - (-7)$
2. Step 2
  Two minus signs together make a plus.
  $6 - (-7) = 6 + 7$
3. Step 3
  Now add.
  $6 + 7 = 13$
Answer
$13\degree\text{C}$ warmer
5A multi-step integer challenge
Stretch• bidmas, negative numbers, nested brackets
Question
Work out $(-3)^3 + 40 \div (8 - (5 + 1)) - (-4)$ . Show each step.
Step-by-step solution
1. Step 1
  Inner bracket first.
  $5 + 1 = 6$
2. Step 2
  Outer bracket next.
  $8 - 6 = 2$
3. Step 3
  Indices: $(-3)^3$ has three negatives, an odd count, so it is negative.
  $(-3)^3 = -27$
4. Step 4
  Division: the result is now divided by 2.
  $40 \div 2 = 20$
5. Step 5
  Tidy the double sign, then add and subtract left to right.
  $-27 + 20 + 4 = -3$
Answer
$-3$

Key Definitions and Keywords — Integers and Order of Operations

The important words for integers and order of operations and what they mean — learn these so you can explain your thinking clearly.

Integer
Key word
A whole number, which can be positive, negative or zero — for example $-8$ , 0 and 41. Integers do not include fractions or decimals.
Negative number
Key word
A number less than zero, written with a minus sign. It sits to the left of zero on the number line.
Example
A balance of $-\$ 15 $means you are \$ 15 overdrawn.
Order of operations
Key word
The agreed sequence for working out a calculation with several steps, so everyone reaches the same answer. Often remembered as BIDMAS.
BIDMAS
Key word
A way to remember the order of operations: Brackets, Indices, Division and Multiplication, Addition and Subtraction.
Nested brackets
Brackets placed inside other brackets. Always work from the innermost pair outwards.
Example
In $12 - (8 - (3 + 1))$ you work out $3 + 1$ first.
Index (power)
Key word
A small raised number showing how many times a base is multiplied by itself.
Example
In $(-2)^3$ the index 3 means $-2$ is used three times.
Base
The number being multiplied repeatedly in a power. In $(-5)^2$ the base is $-5$ .
Difference
The result of subtracting one number from another. With integers, subtract the lower value, signs and all.
Example
The difference between $-4$ and $9$ is $9 - (-4) = 13$ .
Product
The result of multiplying numbers together.
Example
The product of $-3$ , $2$ and $-4$ is $24$ .
Double sign
Key word
Two operation signs sitting next to each other. They combine into one: two minuses make a plus, a plus and a minus make a minus.
Operation
An action carried out on numbers — addition, subtraction, multiplication or division.
Estimate
A rough answer used to check that a calculation has a sensible sign and size.

Common Mistakes and Misconceptions — Integers and Order of Operations

The slip-ups students most often make with integers and order of operations — and simple ways to avoid them.

✕Treating 'subtract a negative' as a subtraction, e.g. writing $6 - (-7) = -1$ .
Why it happens
The two minus signs next to each other are read as one big subtraction.
How to avoid it
Two negative signs together make a plus. Rewrite first: $6 - (-7) = 6 + 7 = 13$ .
✕Writing $-4^2 = 16$ , treating it the same as $(-4)^2$ .
Why it happens
Without brackets, the minus looks like it is part of the number being squared.
How to avoid it
$-4^2$ squares only the 4: $-(4 \times 4) = -16$ . Only $(-4)^2$ squares the whole $-4$ to give $16$ .
✕Doing all division before all multiplication, e.g. $36 \div 6 \times 3 = 2$ .
Why it happens
BIDMAS is read as a strict top-to-bottom list, so D looks like it always beats M.
How to avoid it
Division and multiplication rank equally — work them left to right: $6 \times 3 = 18$ .
✕Working a nested bracket from the outside in, or ignoring the inner pair.
Why it happens
Long expressions look daunting, so the most obvious bracket gets done first.
How to avoid it
Always start with the innermost bracket and work outwards, one layer at a time.
✕Getting the sign of a product wrong, e.g. writing $-2 \times -3 \times -1 = 6$ .
Why it happens
Only the last pair of signs is considered, not the whole chain.
How to avoid it
Count every negative sign: three negatives is odd, so the answer is negative — here $-6$ .

Integers and Order of Operations — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Integers and Order of Operations

Short Notes - Integers and Order of Operations

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Take this whole topic with you

1Tidying a chain of signed numbers

2BIDMAS with brackets and an index

3An index on a negative base

4The difference between two temperatures

5A multi-step integer challenge

Integer

Negative number

Order of operations

BIDMAS

Nested brackets

Index (power)

Base

Difference

Product

Double sign

Operation

Estimate

✕Treating 'subtract a negative' as a subtraction, e.g. writing 6−(−7)=−16 - (-7) = -16−(−7)=−1.

✕Writing −42=16-4^2 = 16−42=16, treating it the same as (−4)2(-4)^2(−4)2.

✕Doing all division before all multiplication, e.g. 36÷6×3=236 \div 6 \times 3 = 236÷6×3=2.

✕Working a nested bracket from the outside in, or ignoring the inner pair.

✕Getting the sign of a product wrong, e.g. writing −2×−3×−1=6-2 \times -3 \times -1 = 6−2×−3×−1=6.

Integers and Order of Operations — frequently asked questions

✕Treating 'subtract a negative' as a subtraction, e.g. writing $6 - (-7) = -1$ .

✕Writing $-4^2 = 16$ , treating it the same as $(-4)^2$ .

✕Doing all division before all multiplication, e.g. $36 \div 6 \times 3 = 2$ .

✕Getting the sign of a product wrong, e.g. writing $-2 \times -3 \times -1 = 6$ .