⭐ Plus Book a free demo

📙 Class 8 maths · session 1 of 13 · 60 minutes · free

Rational numbers

Welcome aboard the Math Cruiser, hero! I am Captain Calc, and today we are zooming into the world of rational numbers.

🎯 Practice mode🖨 Print a worksheet

🤖 Written with Google's Gemini AI. A second AI solved all 55 questions without seeing the answers, and both agree. One of our teachers has not read it yet, so if something looks wrong, please tell us.

A complete lesson · 60 minutes

Teach it, see it, practise it, all on this page.

Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.

See the teacher's plan
  1. Warm-up3 min
  2. Warm-up games5 min
  3. What is a Rational Number?8 min
  4. Properties of Rational Numbers7 min
  5. Operations on Rational Numbers7 min
  6. Rational Numbers on Number Line & Mean Method7 min
  7. Hero Vault Code6 min
  8. Captain Calc's Market Mission5 min
  9. Number Line Navigator6 min
  10. Quiz5 min
  11. Set the project1 min

By the end you can

  • Identify rational numbers and express them in p/q standard form
  • Apply additive and multiplicative inverses to simplify expressions
  • Perform addition, subtraction, multiplication, and division on rational numbers
  • Locate rational numbers on a number line and find numbers between them

Before you start

  • Basic operations on positive and negative integers
  • How to simplify fractions using common factors

Have ready

  • Notebook and pencil for calculation steps
  • Ruler for drawing clear number lines

Class 8 · CBSE, ICSE and state boards

Warm-up · think first

Priya has ₹100. She spends 1/4 of her money on a comic book. How much money does she have left?

Show the answer

₹75. Priya spent 1/4 of 100, which is ₹25. Subtracting ₹25 from ₹100 leaves ₹75.

Teacher tip: Ask students how they converted the fraction 1/4 into a rupee amount.

More warm-ups for every class →

🎮 Warm-up games · 5 quick games

Play with Captain Calc

Six minutes with Captain Calc. Vote, say why, then show the answer.

  1. Which one does not belong: 3/4, -5/2, 0, 7/0? (3/4 · -5/2 · 0 · 7/0)
    Answer: 7/0 does not belong because division by zero is not defined, so it is not a rational number.
  2. Would you rather add 1/3 + 1/6 or multiply (1/3) × (1/6)? (Add 1/3 + 1/6 · Multiply (1/3) × (1/6))
    Answer: Multiplication is often easier because you just multiply numerators and denominators without finding a common denominator!
  3. A rational number is a whole number. (Always · Sometimes · Never)
    Answer: Sometimes. Numbers like 4/1 = 4 are whole numbers, but 1/2 is a rational number that is not a whole number.
  4. I am a negative rational number. My numerator is -3. When you multiply me by -5/3, the answer is 1. What number am I?
    Answer: -3/5
  5. Why do you think zero is considered a rational number?
    Answer: Zero can be written as 0/1, 0/2, or 0/(-5). Since it fits the form p/q with q not equal to 0, it is rational!

Part 1 of 4 · learn, see, practise

What is a Rational Number?

A rational number is any number that can be written in the form p/q, where p and q are integers and q is not equal to 0.

Positive fractions, negative fractions, whole numbers, and zero are all rational numbers.

A rational number is in standard form when its denominator is positive and p and q share no common factor other than 1.

Example 1

Express -8 as a rational number in p/q form.

  1. Any integer can be placed over 1.
  2. Write -8 as -8/1.

Answer: -8/1

Example 2

Reduce -15/20 to standard form.

  1. Find the greatest common factor of 15 and 20, which is 5.
  2. Divide numerator and denominator by 5: (-15 ÷ 5) / (20 ÷ 5) = -3/4.

Answer: -3/4

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. Yes
  2. -6/1
  3. -5
  4. -2/3
  5. 3/4
  6. -12

Part 2 of 4 · learn, see, practise

Properties of Rational Numbers

The additive identity is 0 because adding 0 to any rational number leaves it unchanged.

The additive inverse of p/q is -p/q because their sum equals 0.

The multiplicative inverse or reciprocal of p/q is q/p because their product equals 1.

Example 1

Find the additive inverse of 5/9.

  1. Change the sign of the number.
  2. The additive inverse of 5/9 is -5/9.

Answer: -5/9

Example 2

Find the multiplicative inverse of -4/7.

  1. Flip the numerator and denominator.
  2. Keep the negative sign: -7/4.

Answer: -7/4

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. -4/7
  2. -8/3
  3. 0
  4. 1
  5. -9
  6. -2/3

Part 3 of 4 · learn, see, practise

Operations on Rational Numbers

To add or subtract rational numbers, find a common denominator first.

To multiply rational numbers, multiply top by top and bottom by bottom.

To divide by a rational number, multiply by its reciprocal.

Example 1

Add: 1/4 + 2/3.

  1. Find common denominator for 4 and 3, which is 12.
  2. Convert fractions: 1/4 = 3/12 and 2/3 = 8/12.
  3. Add numerators: 3/12 + 8/12 = 11/12.

Answer: 11/12

Example 2

Divide: (-4/5) ÷ (2/15).

  1. Multiply by reciprocal of second fraction: (-4/5) × (15/2).
  2. Simplify cross terms: (-2/1) × (3/1) = -6.

Answer: -6

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 5/7
  2. -3/10
  3. 1/2
  4. -6
  5. 5
  6. -1/12

Part 4 of 4 · learn, see, practise

Rational Numbers on Number Line & Mean Method

Rational numbers can be plotted between integers on a number line.

To find a rational number between two numbers a and b, calculate their mean: (a + b) / 2.

There are infinitely many rational numbers between any two given rational numbers.

Example 1

Find a rational number halfway between 1/3 and 1/2.

  1. Add the two numbers: 1/3 + 1/2 = 2/6 + 3/6 = 5/6.
  2. Divide the sum by 2: (5/6) ÷ 2 = 5/12.

Answer: 5/12

Example 2

Which is larger: -1/2 or -1/4?

  1. On a number line, numbers further to the right are larger.
  2. -1/4 is closer to 0 than -1/2.
  3. Therefore -1/4 is larger than -1/2.

Answer: -1/4

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up

Warm-up

Core

Core

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 1/2
  2. -1/4
  3. 5/12
  4. 1/10
  5. Infinite
  6. 0

Watch out

Common mistakes

Thinking zero is not a rational number.

Remember that 0 can be written as 0/1, so it fits the definition p/q.

Forgetting to change division to multiplication by reciprocal.

Always flip the second fraction when dividing rational numbers!

Assuming there are only a fixed number of rational numbers between two fractions.

Remember there are infinitely many rational numbers between any two given numbers.

Thinking -3/4 is different from 3/-4.

The negative sign belongs to the entire rational number, so -3/4, 3/-4, and -(3/4) are equal.

Class activity 1 of 3

Hero Vault Code

Captain Calc needs you to crack the secret vault by solving rational number puzzles!

  1. Solve each problem step by step in your notebook.
  2. Simplify your answer to lowest terms.
  3. Type the simplified answer into the control panel.

Answers (for the teacher)
  1. 3/5
  2. 9/7
  3. 11/3
  4. -3/5
  5. 1/2

Class activity 2 of 3

Captain Calc's Market Mission

Help Ananya and Vivaan budget their market purchases using rational numbers!

  1. Read each word problem carefully.
  2. Set up the correct fraction operation.
  3. Calculate the final numerical value.

Answers (for the teacher)
  1. 90
  2. 30
  3. 11/12
  4. 2/5
  5. -1/5

Class activity 3 of 3

Number Line Navigator

Navigate through space by finding missing coordinates on the number line!

  1. Identify the start and end values on the line.
  2. Use the mean formula (a+b)/2 or common denominators.
  3. Type your exact simplified answer.

Answers (for the teacher)
  1. 2/5
  2. 1/2
  3. 0
  4. 0
  5. 1

After class · project

Rational Number Comic Strip Mission

Create a 4-panel comic strip featuring Captain Calc solving a real-life problem using rational numbers.

What to make: A 4-panel comic strip on paper or digital canvas with clear mathematical calculations.

Stretch: Include negative rational numbers and find three rational numbers between two fractions in your story!

How it is marked

Look forDone well when
Mathematical AccuracyAll rational numbers and operations are computed correctly.
Clarity & PresentationSteps are clearly shown with visual comic elements.
CreativityOriginal comic storyline featuring Captain Calc and real-life context.

Project template

Copy it onto paper or print the practice sheet.

Panel 1: Introduce hero and problem (e.g., sharing food or money)
Panel 2: Express the problem using rational numbers in p/q form
Panel 3: Perform the calculation using properties or operations
Panel 4: Display the final solution clearly with Captain Calc

Homework

Practise at home

Try each on paper, then tap to see the answer and working. Or print the practice sheet.

  1. Represent -3/5 as a rational number with denominator 20.

    -12/20 because multiplying numerator and denominator by 4 gives -12/20.

  2. Find the additive inverse of -7/11.

    7/11 because (-7/11) + (7/11) = 0.

  3. Find the multiplicative inverse of -13/5.

    -5/13 because (-13/5) × (-5/13) = 1.

  4. Simplify: 2/5 - 3/10.

    1/10 because 4/10 - 3/10 = 1/10.

  5. Simplify: (-5/6) × (3/10).

    -1/4 because (-5×3)/(6×10) = -15/60 = -1/4.

  6. Find a rational number halfway between 1/5 and 2/5.

    3/10 because (1/5 + 2/5) ÷ 2 = (3/5) ÷ 2 = 3/10.

🔥 Challenge round · finished early?

The hardest questions of the lesson, no notes

Cover the notes and try these from memory. They are meant to make you think hard: draw, try small cases, and check your answer.

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Stretch 🔥

Answers (for the teacher)
  1. 3/4
  2. -12
  3. -9
  4. -2/3
  5. 5
  6. -1/12
  7. Infinite
  8. 0

🧠 Think-hard question

Find five rational numbers between -1/3 and 1/2 using equivalent fractions with common denominator 30.

Talk it through or use the drawing board. There is more than one way to get there.

🎯 Practice mode: one question at a time🧩 Brain warm-ups🎮 Maths and logic games

Quiz · 10 questions

Final quiz: check what you learned

Pick an answer to see if it is right and why. Your best score is saved on this device.

1Which of the following is a rational number?
Show the answer

3/5. 3/5 is written in the form p/q with non-zero denominator.

2What is the additive inverse of -4/9?
Show the answer

4/9. The sum of -4/9 and 4/9 is zero.

3What is the multiplicative inverse of -7?
Show the answer

-1/7. Multiplying -7 by -1/7 gives 1.

4What is the product of any rational number and its reciprocal?
Show the answer

1. By definition, a number multiplied by its reciprocal equals 1.

5Simplify: 2/3 + (-1/3).
Show the answer

1/3. Since denominators are equal, 2 + (-1) = 1, so the answer is 1/3.

6Simplify: (-3/5) × (-5/3).
Show the answer

1. Negative times negative is positive, and 15/15 reduces to 1.

7Which rational number lies exactly halfway between 1/4 and 1/2?
Show the answer

3/8. (1/4 + 2/4) ÷ 2 = (3/4) ÷ 2 = 3/8.

8What happens when any rational number is multiplied by 0?
Show the answer

It becomes 0. Zero multiplied by any number always equals zero.

9Simplify: (3/4) ÷ (-9/8).
Show the answer

-2/3. (3/4) × (-8/9) = -24/36 = -2/3.

10Which number has no multiplicative inverse?
Show the answer

0. 1/0 is undefined, so zero has no multiplicative inverse.

🖨 Print what we learned

Make a worksheet from this lesson

Pick what to practise, then print it or choose Save as PDF in the print window to download it.

Include
Options

For parents and teachers

The 60-minute lesson plan

Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.

TimePartThe teacherThe learner
0–3 minWarm-upAsk the question and wait for a guess before any hint.Guess, then explain.
3–8 minWarm-up gamesPlay three or four games. Ask for a reason each time.Vote and say why.
8–16 minWhat is a Rational Number?Explain, then reveal the examples one step at a time.Practise up to the stretch question.
16–23 minProperties of Rational NumbersExplain, then show the examples.Practise.
23–30 minOperations on Rational NumbersExplain, then show the examples.Practise.
30–37 minRational Numbers on Number Line & Mean MethodExplain, then show the examples.Practise; the stretch can be homework.
37–43 minHero Vault CodeLet learners work alone; hint only after a wrong check.Solve and check.
43–48 minCaptain Calc's Market MissionDo the first item together.Finish the activity.
48–54 minNumber Line NavigatorSet as homework if time is short.Solve the mission.
54–59 minQuizRead each question aloud; vote, then check.Answer and read why.
59–60 minSet the projectShow the template and agree a due date.Start the project.

Questions to ask

  • Why is 7/0 not a rational number?
  • What is the difference between an additive inverse and a multiplicative inverse?
  • How many rational numbers can you find between 0 and 1?

If it feels hard

If a student is struggling, draw a concrete number line from -1 to 1 divided into equal parts, and use simple whole numbers before moving to fractions.

If it feels easy

Find five rational numbers between -1/3 and 1/2 using equivalent fractions with common denominator 30.

FAQ

Rational numbers: questions

Why isn't division by zero allowed in rational numbers?

Dividing by zero has no logical answer because no number multiplied by 0 gives a non-zero number.

Are all fractions rational numbers?

Yes, all standard fractions with non-zero integer denominators are rational numbers.

How can I quickly find rational numbers between two fractions?

Convert both fractions to equivalent fractions with a larger common denominator, then pick numerators in between!

Start a conversation

Want a teacher to take this class?

Tell us the learner's age and level. A teacher from our network teaches with these lessons; we confirm timing and price first.

  • Tell us the learner’s age or grade and goals
  • We’ll suggest a suitable class or learning path
  • We’ll confirm format, timing and fees with a parent or guardian

For a child’s class, please have a parent or guardian send the enquiry. Avoid sharing passwords or sensitive personal details.

We’ll use your details only to respond to this request.

Chat on WhatsApp
Esc

Try