Class 6 Maths Ganita Prakash Chapter 1: Patterns in Mathematics Questions

📘 1.1 What is Mathematics? (Easy Explanation)

Mathematics is the study of patterns and numbers. It helps us understand how things work and why they happen.

We see math all around us — in nature 🌳, at home 🏠, at school 🎓, and even in space 🌙. Math is used when we cook 🍲, play games 🎯, go shopping 🛒, or tell time ⏰.

Math is like a puzzle 🧩. Mathematicians look for patterns and try to explain them. These explanations can help people make new things — like rockets 🚀, phones 📱, bridges 🌉, and more!

🎉 Fun Fact:
Math helped scientists understand how the moon and planets move. This is how we were able to send astronauts to space!

🧩 Figure it Out – Answers

1. Can you think of other examples where mathematics helps us in our everyday lives?

Answer:

• Count money 💰
• Tell time on a clock ⏰
• Cook using measurements 🍲
• Share things equally 🍫
• Plan trips or travel 🚗
• Play games and keep score 🏏🎯
• Use our phone or calculator 📱

2. How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)

Answer:

• Build houses, bridges, and tall buildings 🏠🌉
• Make cars, trains, and airplanes 🚗🚆✈️
• Create computers, TVs, and mobile phones 💻📺📱
• Send rockets into space 🚀
• Discover new medicines and treat diseases 💊🧪
• Count votes and run the country fairly 🗳️
• Make calendars and clocks to manage time 🗓️⌛

📘1.2 Patterns in Numbers

In math, numbers often follow patterns. These patterns help us understand how numbers work. When numbers are arranged in a special order, they form a number sequence.

The part of math that studies these patterns in whole numbers is called number theory.

Table 1: Examples of Number Sequences and Their Patterns

🧩 Figure it Out – Answers

1. Can you see the pattern in the sequences?

Answer:

Yes! Each list of numbers follows a special rule to get the next numbers.

2. Write the sequences with next 3 numbers and the easy rule:

Answer:

📘1.3 Visualising Number Sequences

Many number sequences can be shown with pictures. Seeing numbers as shapes or dots helps us understand their patterns better.

Table 2: Pictorial representation of some number sequences

🧩 Figure it Out – Answers

1. Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!

Answer:

2. Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes?

Answer:

Triangular numbers like 1, 3, 6, 10, 15 are called so because you can arrange dots to form a triangle.

Square numbers like 1, 4, 9, 16, 25 are called square numbers because dots can be arranged in a square shape (like 2×2, 3×3, etc.).

Cubes like 1, 8, 27, 64, 125 come from multiplying a number by itself three times (e.g., 2×2×2 = 8) and represent dots arranged in a cube shape.

3. You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways

Answer:

36 is both a triangular number and a square number because:

• As a square number, 36 dots can be arranged in a 6 by 6 square (6 × 6 = 36).
• As a triangular number, 36 dots can be arranged in a triangle with 8 rows where the number of dots in each row increases by 1 (1 + 2 + 3 + … + 8 = 36).

4. What would you call the following sequence of numbers?

That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?

Answer:

The numbers 1, 7, 19, 37, … are hexagonal numbers because you can arrange dots to form hexagon shapes.

The number of dots added each time grows by 6:
6, 12, 18, …

So, after 37, add 24 dots:
37 + 24 = 61

5. Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3? Here is one possible way of thinking about Powers of 2:

Answer:

Powers of 3:

Numbers: 1, 3, 9, 27, 81…

Each number is 3 times the previous one.

📘1.4 Relations Among Number Sequences – Easy Explanation

Sometimes, different number patterns are linked in amazing ways!

1️⃣ Adding Odd Numbers = Square Numbers

Let’s add odd numbers:

• 1 = 1
• 1 + 3 = 4
• 1 + 3 + 5 = 9
• 1 + 3 + 5 + 7 = 16
• 1 + 3 + 5 + 7 + 9 = 25

👉 The total becomes 1², 2², 3², 4², 5²… (Square Numbers)!

🎨 You can draw this using dots in square shapes to see how it works.

2️⃣ Why Does This Happen?

Each time you add an odd number, you’re building the next bigger square.
For example:

• Start with 1 dot
• Add 3 more to get a 2×2 square
• Add 5 more to get a 3×3 square
  And so on…

3️⃣ Try This!

What is the sum of the first 10 odd numbers?
👉 Answer: 100 (10²)

• What about the first 100 odd numbers?
  👉 Answer: 10,000 (100²)

4️⃣ Another example of such a relation between sequences:

Look at this:

• 1 = 1
• 1 + 2 + 1 = 4
• 1 + 2 + 3 + 2 + 1 = 9
• 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
• 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
• 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36

Going up and down gives square numbers too!

🧩 Figure it Out – Answers

1. Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, …, gives square numbers?

Answer:

Yes, adding counting numbers up and down (like 1, 1+2+1, 1+2+3+2+1, etc.) gives square numbers because the pattern forms a symmetrical pyramid. When you stack the numbers upward and then mirror them downward, the total number of dots fits into a perfect square shape.

For example:

• 1 + 2 + 1 = 4 = 2²
• 1 + 2 + 3 + 2 + 1 = 9 = 3²
• 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4²

So, the total always becomes a square number due to the mirror-like arrangement.

2. By  imagining  a  large  version  of  your  picture,  or  drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1?

Answer:

We are adding numbers in this pattern:

1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1

👉 Step 1: Break it into two parts

• First half (1 to 100):

Sum = 100 × 101/2 = 5050

• Second half (99 to 1):

Sum = 99 × 100/2 = 4950

5050 + 4950 = 10, ⁣000

So, the total value is 10,000.