Sec 1 Maths Chapter Guide

Integers, Rational Numbers and Real Numbers

Learn how the number system expands from whole numbers to negative numbers, integers, fractions, decimals, rational numbers, irrational numbers and real numbers.

This chapter guide is written for Secondary 1 students and teachers. It gives the main ideas, worked examples, common mistakes and links to practice pages for the full Chapter 2 cluster.

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Infographic showing integers rational numbers irrational numbers and real numbers on a number line

Chapter overview

This chapter helps students move beyond positive whole numbers. Students learn how negative numbers work, how integers behave under the four operations, and how fractions, decimals and irrational numbers fit into the real number system.

The big idea is that different types of numbers can still be compared, placed on a number line and used in real-world situations.

Start Negative numbers and integers
Operate Add, minus, multiply and divide
Extend Fractions, decimals and mixed numbers
Connect Rational, irrational and real numbers
Learning section

Learning goals and prerequisites

By the end of this chapter, students should be able to compare and operate with negative numbers, understand rational and irrational numbers, and explain how different number types fit inside the real number system.

Understand

Explain negative numbers, integers, rational numbers, irrational numbers and real numbers using examples and number lines.

Calculate

Add, minus, multiply and divide negative integers, fractions, mixed numbers and decimals using correct rules.

Connect

Recognise how number types are related, and decide whether answers are reasonable in real-world contexts.

Prerequisite knowledge

Students should already be comfortable with whole numbers, basic operations, fractions, decimals and simple square roots or cube roots.

whole numbers four operations number order fractions decimals square roots

Key vocabulary

IntegerA whole number, zero, or negative whole number.
Negative numberA number less than zero.
Rational numberA number that can be written as a fraction.
Irrational numberA number that cannot be written exactly as a fraction.
Real numberA number that can be placed on the number line.
Number lineA line that shows numbers in order from left to right.
Concept 1

Negative numbers

Negative numbers are numbers less than zero. They are useful when a quantity goes below a reference point, such as temperatures below 0°C, heights below sea level, basement floors, or money owed.

−8°C A temperature 8 degrees below zero.
−35 m A position 35 metres below sea level.
B3 = −3 A basement level 3 floors below ground floor.

Positive and negative direction

The sign tells us which side of zero the number is on. Positive numbers are greater than zero. Negative numbers are less than zero.

+6 means 6 units above or to the right of zero.

−6 means 6 units below or to the left of zero.

The number 0 is neither positive nor negative.

Distance from zero

Some numbers are the same number of steps away from zero, but on different sides.

−8 is 8 steps below zero.

+8 is 8 steps above zero.

They are on opposite sides of 0.

Real-life model

Using an elevator to understand negative numbers

Many Sec 1 students understand negative numbers better when they can picture going up and going down. An elevator is useful because it shows clearly that going up means add, and going down means minus.

In this model, ground floor is 0. Floors above ground are positive numbers. Basement floors are negative numbers. So B1 means −1, B2 means −2, and B3 means −3.

Teri using an elevator to understand negative numbers and integer subtraction from B3 to Thinkel tuition
Teri’s elevator story connects integer addition and minus to movement between basement floors, ground floor and upper floors.

Teri’s elevator story

Teri starts at the B3 carpark. In Maths, B3 corresponds to −3. She takes the elevator up to her tuition lesson at Thinkel on Level 5.

Teaching rule:
Go up = add
Go down = minus
1
Start at B3 B3 means −3.
2
Go up to Level 5 From −3 to 5, Teri goes up 8 floors. So, −3 + 8 = 5.
3
Go down to B1 After class, Teri goes down 6 floors to get a snack. So, 5 − 6 = −1.
4
Go down to B3 Then Teri goes down 2 floors to her parents’ car. So, −1 − 2 = −3.

Why −7 − 5 = −12

Students often ask why −7 − 5 gives −12. The elevator model helps.

Start at −7.

Minus 5 means go down 5 floors.

You do not move towards 0.

You reach −12.

So, −7 − 5 = −12.

Teacher note: avoid saying “more” when explaining this step, because many students immediately connect “more” with addition. Use “go down”, “move down”, “lower” and “reach”.
Concept 2

Integers and the number line

Integers are whole numbers, zero and negative whole numbers. They do not include fractions or decimals.

−4
−3
−2
−1
0
1
2
3
4

Ordering integers

On a horizontal number line, numbers become greater as we move to the right and smaller as we move to the left.

−1 is greater than −3 because −1 is closer to zero.

−5 is less than −2 because −5 is further left.

3 is greater than −4 because every positive number is greater than every negative number.

Common comparison mistake

Students sometimes think that −8 is greater than −3 because 8 is greater than 3. That is not correct.

Compare their positions on the number line.

−8 is further left than −3.

So, −8 < −3.

Concept 3

Add and minus negative integers

Adding and minusing negative integers becomes clearer when students think about movement on a number line or elevator.

Adding integers

Adding a positive number moves right or up. Adding a negative number moves left or down.

3 + 4 = 7

3 + (−4) = −1

−3 + (−4) = −7

When two negative numbers are added, the answer moves lower on the number line.

Minusing integers

Minus means taking steps down or left, depending on the model used.

5 − 2 = 3

5 − 6 = −1

−1 − 2 = −3

−7 − 5 = −12

The important question is: where do you start, and which direction do you move?

Practice link

The first practice page focuses on negative numbers, integers, number lines, ordering, going up/down models, addition and minus with negative integers.

Go to Negative Numbers and Integer Operations
Concept 4

Multiplication and division of negative integers

Multiplication and division involving negative integers follow clear sign rules. Students should understand the rules, then practise until the sign of the answer becomes automatic.

Operation
Signs
Result
Multiplication or division
positive and positive
positive
Multiplication or division
positive and negative
negative
Multiplication or division
negative and positive
negative
Multiplication or division
negative and negative
positive

Worked examples

6 × (−3) = −18

(−6) × (−3) = 18

(−24) ÷ 6 = −4

(−24) ÷ (−6) = 4

Teaching shortcut

Count the number of negative signs in multiplication or division.

Odd number of negative signs → negative answer.

Even number of negative signs → positive answer.

Example: (−2) × (−3) × (−4) has three negative signs, so the answer is negative.

Concept 5

Combined operations with negative numbers

Combined operations require students to manage both operation order and signs. This is where many careless mistakes happen.

Order of operations

1
BracketsWork out expressions inside brackets first.
2
Powers and rootsEvaluate powers, square roots and cube roots next.
3
Multiplication and divisionWork from left to right.
4
Addition and minusWork from left to right.

Worked example

Evaluate 6 − 2 × (−4).

Do multiplication first.

2 × (−4) = −8

6 − 2 × (−4) = 6 − (−8)

6 − (−8) = 14

Answer: 14

Do not work from left to right blindly. Multiplication must be done before minus.

Practice link

The second practice page covers multiplication, division, combined operations, fractions, decimals and real number classification.

Go to Integer Operations, Fractions, Decimals and Real Numbers
Concept 6

Fractions, mixed numbers and decimals

Once students can work with negative integers, the same ideas extend to negative fractions, mixed numbers and decimals. The arithmetic becomes more detailed, but the sign rules remain consistent.

Fractions

Add or minus fractions using a common denominator. Multiply and divide fractions using the usual fraction rules, while also checking the sign.

Mixed numbers

Convert mixed numbers to improper fractions when multiplication or division is involved. This keeps working cleaner.

Decimals

Line up decimal points when adding or minusing. For multiplication and division, handle the sign and decimal placement carefully.

Example with signs

−0.8 + 1.5 = 0.7

−0.8 − 1.5 = −2.3

(−0.4) × 3 = −1.2

(−1.2) ÷ (−0.3) = 4

Teacher note

Students often make two mistakes at once: one with the sign and one with the fraction or decimal calculation. Ask them to decide the sign first, then complete the numerical calculation.

Concept 7

Rational, irrational and real numbers

The final part of the chapter connects different number types. Whole numbers, integers, fractions, decimals and irrational numbers can all be understood as part of the real number system.

Real numbers

All numbers that can be placed on the number line.

Rational numbers Numbers that can be written as a fraction, where the denominator is not zero.
−4 3/4 0.25 −2.5 7
Integers …, −3, −2, −1, 0, 1, 2, 3, …
Irrational numbers Numbers that cannot be written exactly as a fraction.
√2 π √5

Rational numbers

Rational numbers include integers, fractions, terminating decimals and recurring decimals.

7 is rational because 7 = 7/1.

0.25 is rational because 0.25 = 1/4.

−2.5 is rational because −2.5 = −5/2.

Irrational numbers

Irrational numbers cannot be written exactly as a fraction. Their decimal forms do not terminate or repeat.

√2 is irrational.

π is irrational.

These numbers still belong to the real numbers because they can be placed on the number line.

Chapter practice pathway

Practice pages for this chapter

Use the main page for teaching. Then move students through the practice pages in order. Each practice page can be used for classwork, self-study, revision or teacher-led review.

Practice Page 1 Negative Numbers and Integer Operations Practice negative numbers, integers, number lines, ordering, elevator movement, addition and minus. Practice Page 2 Integer Operations, Fractions, Decimals and Real Numbers Practise multiplication, division, combined operations, fractions, decimals, rational and irrational numbers. Review Page Integers, Rational Numbers and Real Numbers Summary and Review Review the full chapter with summary notes, mixed revision questions, common mistakes and answers.
Teacher support

Common mistakes and teacher notes

This chapter has many calculation rules, so students need both conceptual understanding and careful practice.

Common mistakes

1
Thinking −8 is greater than −3Use the number line to show that −8 is further left and therefore smaller.
2
Thinking minus always moves towards zeroUse the elevator example: −7 − 5 means start at −7, go down 5 floors, and reach −12.
3
Mixing up sign rules for multiplication and additionNegative × negative gives positive, but negative + negative gives a lower negative answer.
4
Calling every decimal rationalTerminating and recurring decimals are rational. Non-terminating, non-recurring decimals are irrational.

Teaching notes

A
Use the elevator model earlyIt helps students visualise zero, positive levels and negative basement levels.
B
Use “go down” for minusThis helps avoid confusion when students see examples like −7 − 5 = −12.
C
Link representationsConnect fractions, decimals and number-line positions whenever possible.
D
Build accuracy firstLet students explain direction before asking them to work quickly.
FAQs

Frequently asked questions

These questions clarify the common points students meet in this chapter.

What is the difference between a negative number and an integer?
A negative number is any number less than zero. An integer is a whole number, zero, or negative whole number. For example, −3 is both negative and an integer, but −2.5 is negative and not an integer.
Why is −2 greater than −5?
On the number line, −2 is to the right of −5. Numbers further right are greater, so −2 is greater than −5.
Why does −7 − 5 become −12?
Think of an elevator. Start at −7. Minus 5 means go down 5 floors. You reach −12. So −7 − 5 = −12.
Are all fractions rational numbers?
Yes, a fraction with a non-zero denominator is rational because it is already written in the form of one integer divided by another integer.
Are all decimals rational numbers?
Terminating decimals and recurring decimals are rational. Non-terminating, non-recurring decimals are irrational.
Why are irrational numbers still real numbers?
Irrational numbers such as √2 and π cannot be written exactly as fractions, but they still have positions on the number line. Therefore, they are real numbers.

Ready to practise Chapter 2?

Start with negative numbers and integer operations first. Then move to combined operations, fractions, decimals and the full chapter review.

Start Practice Page 1 Back to Sec 1 Maths
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