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Class 6 Maths Chapter 1 – Patterns in Mathematics | Notes, Solutions & Worksheet

Class 6 Maths Chapter 1 Patterns in Mathematics – easy notes, concepts, NCERT book solutions, important questions, worksheet and answers. Learn about number patterns, visual patterns, sequences, shapes and mathematical patterns with simple explanations and step-by-step solutions.

Class 6 Mathematics Chapter 1 Patterns in Mathematics | Notes and Solutions

PATTERNS IN MATHEMATICS

Class 6 Mathematics

Ganita Prakash | Chapter 1

Detailed Notes, Figure it Out Solutions, Formulas, Examples and Visual Learning

Chapter 1 — Patterns in Mathematics

What will we learn?

Mathematics is not only about calculations. It is also about finding patterns and understanding why those patterns happen.
  • Patterns in everyday life
  • Patterns in numbers
  • Number sequences
  • Visualising numbers using pictures
  • Relations between different number sequences
  • Patterns in shapes
  • Relations between shapes and number sequences

Page No. 1 — Section 1.1

1.1 What is Mathematics?

Mathematics is, in large part, the search for patterns, and for the explanations as to why those patterns exist.

Such patterns indeed exist all around us—in nature, in our homes and schools, and in the motion of the sun, moon, and stars. They occur in everything that we do and see, from shopping and cooking, to throwing a ball and playing games, to understanding weather patterns and using technology.

The search for patterns and their explanations can be a fun and creative endeavour. It is for this reason that mathematicians think of mathematics both as an art and as a science.

Easy example:
We see patterns when we count numbers, arrange objects, observe the days of a week, make designs, measure things, or calculate money.

It is important to keep in mind that mathematics aims to not just find out what patterns exist, but also the explanations for why they exist.

Page No. 2 — Section 1.1 — Figure it Out

Q1. Can you think of other examples where mathematics helps us in our everyday lives?
Solution
Some examples are paying for fruits, vegetables, groceries etc. calculation of speed of vehicles, designs or patterns in different buildings, finding area of any plot or out own home. There could be many more such contexts in our everyday lives. Discuss for other examples also.
Q2. 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.)
Solution
Teacher student discussion is required.
Think about: Mathematics helps people measure, design, calculate, compare and predict. These ideas are useful in science, engineering, technology, transport and many other areas.

Page No. 2–3 — Section 1.2

1.2 Patterns in Numbers

Among the most basic patterns that occur in mathematics are patterns of numbers, particularly patterns of whole numbers:

0, 1, 2, 3, 4, …

The branch of Mathematics that studies patterns in whole numbers is called number theory.

Number sequences are the most basic and among the most fascinating types of patterns that mathematicians study.

Important Number Sequences

SequenceNameEasy Rule
1, 1, 1, 1, 1, 1, 1, …All 1’s1 is repeated.
1, 2, 3, 4, 5, 6, 7, …Counting numbersAdd 1 each time.
1, 3, 5, 7, 9, 11, 13, …Odd numbersAdd 2 each time.
2, 4, 6, 8, 10, 12, 14, …Even numbersAdd 2 each time.
1, 3, 6, 10, 15, 21, 28, …Triangular numbersAdd 2, 3, 4, 5, …
1, 4, 9, 16, 25, 36, 49, …Squares1×1, 2×2, 3×3, …
1, 8, 27, 64, 125, 216, …Cubes1×1×1, 2×2×2, 3×3×3, …
1, 2, 3, 5, 8, 13, 21, …Virahānka numbersAdd the previous two numbers.
1, 2, 4, 8, 16, 32, 64, …Powers of 2Multiply by 2.
1, 3, 9, 27, 81, 243, 729, …Powers of 3Multiply by 3.

Page No. 3 — Section 1.2 — Figure it Out

Q.1. Can you recognize the pattern in each of the sequences in Table 1?
Solution
Yes, There is a pattern in each case.

Powers of 2- 1,2,4 = 2 x 2, 8 = 2 x 2 x2 ,16 = 2 x 2 x2 x 2 x 2…

Powers of 3 – 1, 3, 9 = 3 x 3 = 27 =3 x 3 x3,…

Virahanka numbers- 1,2, 3, 5 = 2 + 3, 8 = 3 + 5, 13 = 5 + 8,….

Rest of the patterns have been shown on page 4 ,Table 2
Q.2. Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.
Easy Solution
All 1’s: 1, 1, 1, 1, 1, 1, 1, …
Next three: 1, 1, 1
Rule: Keep repeating 1.

Counting numbers: 1, 2, 3, 4, 5, 6, 7, …
Next three: 8, 9, 10
Rule: Add 1.

Odd numbers: 1, 3, 5, 7, 9, 11, 13, …
Next three: 15, 17, 19
Rule: Add 2.

Even numbers: 2, 4, 6, 8, 10, 12, 14, …
Next three: 16, 18, 20
Rule: Add 2.

Triangular numbers: 1, 3, 6, 10, 15, 21, 28, …
Next three: 36, 45, 55
Rule: Add 2, then 3, then 4, then 5, and so on.

Squares: 1, 4, 9, 16, 25, 36, 49, …
Next three: 64, 81, 100
Rule: 1×1, 2×2, 3×3, 4×4, …

Cubes: 1, 8, 27, 64, 125, 216, …
Next three: 343, 512, 729
Rule: 1×1×1, 2×2×2, 3×3×3, …

Virahānka numbers: 1, 2, 3, 5, 8, 13, 21, …
Next three: 34, 55, 89
Rule: Add the previous two numbers.

Powers of 2: 1, 2, 4, 8, 16, 32, 64, …
Next three: 128, 256, 512
Rule: Multiply by 2.

Powers of 3: 1, 3, 9, 27, 81, 243, 729, …
Next three: 2187, 6561, 19683
Rule: Multiply by 3.

Page No. 3–4 — Section 1.3

1.3 Visualising Number Sequences

Many number sequences can be visualised using pictures. Visualising mathematical objects through pictures or diagrams can be a very fruitful way to understand mathematical patterns and concepts.

Solution:

Page No. 5 — Section 1.3 — Figure it Out

Q.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?
Solution
Refer Table 2, page 4, and check for yourself.
Easy explanation:
Triangular numbers can be arranged as dots in a triangle. Square numbers can be arranged in a square. Cube numbers come from multiplying the same number three times.
Q.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!
Solution
Refer Table 2, page 4, and draw. [2]
36 = 6 × 6 36 = triangular arrangement
36 can be represented both as a square arrangement and as a triangular arrangement.
Q.4. What would you call the following sequence of numbers?

1 7 19 37

That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?
Solution
61.

Q.5. Can you think of pictorial ways to visualise the sequence of powers of 2? powers of 3?
Solution
Powers of 3:
1, 3, 9, 27, 81, …

Page No. 6 — Powers of 2

Here is one possible way of thinking about Powers of 2:

1 → 2 → 4 → 8 → 16 → 32 → 64 → …
Each term = Previous term × 2
Example:
4 × 2 = 8
8 × 2 = 16
16 × 2 = 32

Important idea

When we multiply by the same number again and again, we get a power sequence.

Page No. 7 — Section 1.4

Relations among Number Sequences

Sometimes, number sequences can be related to each other in surprising ways.

Adding Odd Numbers

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
Sum of the first n odd numbers = n²
Examples:
First 3 odd numbers:
1 + 3 + 5 = 9 = 3²

First 5 odd numbers:
1 + 3 + 5 + 7 + 9 = 25 = 5²
By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?
Solution
100
Sum of first n odd numbers = n²

For n = 10:
10² = 100
Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers?
Solution
10000
Sum of first n odd numbers = n²

For n = 100:
100² = 10,000

Adding up and down

1 = 1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16

This gives another way of getting square numbers — by adding the counting numbers up and then down.

Page No. 8 — Section 1.4 — Figure it Out

Q.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?
Solution

Q.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?
Solution
10,000
1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1 = 100²
For a smaller example:

1 + 2 + 3 + 2 + 1 = 9
9 = 3²

Therefore, when the middle number is 100:
100² = 10,000
Q.3. Which sequence do you get when you start to add the All 1’s sequence up? What sequence do you get when you add the All 1’s sequence up and down?
Solution
• 1 + (1 +1) + (1 +1 +1) + (1 + 1+1+1) …

• 1, 1+(1+1) + 1, 1+(1+1) +(1+1+1) + (1+1) +1, 1+(1+1) +(1+1+1) +(1+1+1+1) + (1+1+1) + (1+1) +1, ……….
Q.4. Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?
Solution

Q.5. What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, … ? Which sequence do you get? Why? Can you explain it with a picture?
Solution
We get: 4, 9, 16, 25, ……… (Square numbers).

Q.6. What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, … ? Now add 1 to each of these numbers—what numbers do you get? Why does this happen?
Solution
• We get, 1, 3, 7, 15, 31, …………

• After adding 1 to each number, we get:2, 4, 8, 16, 32, ……

Refer the picture on page 6.
1 + 2 + 4 + 8 + … + 2ⁿ = 2ⁿ⁺¹ − 1
Detailed easy explanation:

Start with powers of 2:
1, 2, 4, 8, 16, …

Now add them step by step:
1 = 1
1 + 2 = 3
1 + 2 + 4 = 7
1 + 2 + 4 + 8 = 15
1 + 2 + 4 + 8 + 16 = 31

So we get:
1, 3, 7, 15, 31, …

Now add 1 to every answer:
1 + 1 = 2
3 + 1 = 4
7 + 1 = 8
15 + 1 = 16
31 + 1 = 32

Therefore we get:
2, 4, 8, 16, 32, …

These are powers of 2.
Why does this happen?
Each sum is exactly one less than the next power of 2. For example:

1 + 2 + 4 + 8 = 15
15 + 1 = 16
16 is 2⁴.

Page No. 9 — Section 1.4 — Figure it Out

Q.7. What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?
Solution
(1×6) +1, (3×6) +1, (6×6) +1, (10×6) +1, (15×6) +1, …

= 7, 19, 37, 61, 91, ….

For picture refer picture in Q.4 page 5.
6T + 1
Detailed easy explanation:

The triangular numbers are:
1, 3, 6, 10, 15, …

Multiply each by 6 and then add 1.

1 × 6 + 1 = 7
3 × 6 + 1 = 19
6 × 6 + 1 = 37
10 × 6 + 1 = 61
15 × 6 + 1 = 91

So the new sequence is:
7, 19, 37, 61, 91, …
Example:
For triangular number 10:
(10 × 6) + 1 = 60 + 1 = 61
Q.8. What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, … ? Which sequence do you get? Can you explain it using a picture of a cube?
Solution
1, 8, 27, 64, ………We get cube numbers.

For picture, refer Table 2, page 4
1 = 1³
8 = 2³
27 = 3³
64 = 4³
Add the hexagonal numbers:

1 = 1
1 + 7 = 8
1 + 7 + 19 = 27
1 + 7 + 19 + 37 = 64

So the results are cube numbers.
9. Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?
Easy Example Answer
One possible pattern is:

1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25

The results are square numbers.

Page No. 9–10 — Section 1.5

1.5 Patterns in Shapes

Other important and basic patterns that occur in Mathematics are patterns of shapes. These shapes may be in one, two, or three dimensions (1D, 2D, or 3D)—or in even more dimensions.

The branch of Mathematics that studies patterns in shapes is called geometry.

Examples of Shape Sequences

Shape SequenceRelated Number Pattern
Stacked Triangles1, 3, 6, 10, 15, …
Complete Graphs1, 3, 6, 10, 15, …
Stacked Squares1, 4, 9, 16, 25, …
Koch Snowflake3, 12, 48, 192, …
Regular Polygons3, 4, 5, 6, 7, 8, 9, 10, …

Page No. 10 — Section 1.5 — Figure it Out

Q.1. Can you recognise the pattern in each of the sequences in Table 3?
Solution
Yes.

➢ 3, 4, 5, 6, 7, 8, 9, 10
One of the ways to interpret this is that we get a sequence of number of sides of the pictures of the shapes

➢ 1, 3, 6, 10, 15

➢ 1, 4, 9, 16, 25

➢ 1, 4, 9, 16, 25

➢ 3, 3×4, 3×4×4, 3×4×4×4, 3×4×4×4×4 [5]
No picture is included here as requested.
Q.2. Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
Easy Solution
Yes. Each sequence follows a particular rule. The next shape can be drawn by continuing the same rule.
Examples:

Regular polygons: add one side at a time.
Triangle → Quadrilateral → Pentagon → Hexagon → …

Stacked squares: the number of small squares follows 1, 4, 9, 16, 25, …

Koch Snowflake: each line segment is replaced by four smaller line segments.

Page No. 11 — Section 1.6

1.6 Relation to Number Sequences

Often, shape sequences are related to number sequences in surprising ways. Such relationships can be helpful in studying and understanding both the shape sequence and the related number sequence.

Regular Polygon: A regular polygon has equal-length sides and equal angles.
Number of sides of regular polygons:
3, 4, 5, 6, 7, 8, 9, 10, …

Page No. 11 — Section 1.6 — Figure it Out

Q.1. Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Solution
• Number of sides = 3,4,5,6,7,8,9,10… We get counting number sequence, starting with 3.

• Number of Corners = 3,4,5,6,7,8,9,10… Yes, we get the same number sequence
• in any closed figure, number of sides = number of corners (vertices).
Q.2. Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
Solution
• 1, 3, 6, 10, 15. This is a Triangle number sequence
Easy idea:
As the number of points increases, every new point is joined to all the earlier points. Therefore the number of lines grows as:

1, 3, 6, 10, 15, …
Q.3. How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
Solution
• 1, 4, 9, 16, 25. This is a Square number sequence.
• Squares can be drawn using these number of dots.
No picture is included for Page 12 Q3 as requested.
Q.4. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
Solution
1, 4, 9, 16, 25
= 1, 1+2+1, 1+2+3+2+1, ……….
Square number sequence (As adding up and down) gives us square number sequence.
No picture is included for Page 12 Q4 as requested.
Q.5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment‘—’ by a ‘speed bump’ . As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, …, i.e. 3 times Powers of 4; this sequence is not shown in Table 1)
Solution
• Total line segments in each shape: 3, 12, 48, 192, 768

• Corresponding sequence: 3, 3×4, 3×4×4, 3×4×4×4×, 3×4
Easy correction/check:
The multiplication pattern is by 4 each time:

3
3 × 4 = 12
3 × 4 × 4 = 48
3 × 4 × 4 × 4 = 192
3 × 4 × 4 × 4 × 4 = 768

So the sequence is:
3, 12, 48, 192, 768, …

Page No. 12 — Summary

Mathematics may be viewed as the search for patterns and for the explanations as to why those patterns exist.
Among the most basic patterns that occur in mathematics are number sequences.
Some important examples of number sequences include the counting numbers, odd numbers, even numbers, square numbers, triangular numbers, cube numbers, Virahānka numbers, and powers of 2.
Sometimes number sequences can be related to each other in beautiful and remarkable ways. For example, adding up the sequence of odd numbers starting with 1 gives square numbers.
Visualising number sequences using pictures can help to understand sequences and the relationships between them.
Shape sequences are another basic type of pattern in mathematics. Some important examples of shape sequences include regular polygons, complete graphs, stacked triangles and squares, and Koch snowflake iterations. Shape sequences also exhibit many interesting relationships with number sequences.
Quick Revision:
Patterns → Number Sequences → Pictures → Relations → Shape Patterns → Number and Shape Relations

Important Formulas and Patterns — Quick Revision

Sum of first n odd numbers = n²
1 + 2 + 3 + … + n = n(n + 1) ÷ 2
1 + 2 + 3 + … + n + … + 3 + 2 + 1 = n²
1 + 2 + 4 + 8 + … + 2ⁿ = 2ⁿ⁺¹ − 1
Square number = n²
Cube number = n³
Koch Snowflake line segments: 3, 3×4, 3×4², 3×4³, …

Chapter 1 — Quick Revision

PatternSequence
Counting numbers1, 2, 3, 4, 5, …
Odd numbers1, 3, 5, 7, 9, …
Even numbers2, 4, 6, 8, 10, …
Triangular numbers1, 3, 6, 10, 15, …
Square numbers1, 4, 9, 16, 25, …
Cube numbers1, 8, 27, 64, 125, …
Powers of 21, 2, 4, 8, 16, …
Powers of 31, 3, 9, 27, 81, …
Koch Snowflake line segments3, 12, 48, 192, 768, …
Class 6 Mathematics | Ganita Prakash | Chapter 1 — Patterns in Mathematics

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