Learn about Patterns in Mathematics from NCERT Class 6 Maths Chapter 1. Understand number patterns, sequences, and logical reasoning with easy explanations.
NCERT CLASS 6TH MATHS CHAPTER 1 PATTERNS IN MATHEMATICS
- NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES
- NCERT CLASS 6TH MATHS CHAPTER 3 NUMBER PLAY
- NCERT CLASS 6TH MATHS CHAPTER 4 DATA HANDLING AND PRESENTATION
- NCERT CLASS 6TH MATHS CHAPTER 5 PRIME TIME
- NCERT CLASS 6TH MATHS CHAPTER 6 PERIMETER AND AREA
- NCERT CLASS 6TH MATHS CHAPTER 7 FRACTION
- NCERT CLASS 6TH MATHS CHAPTER 8 PLATING WITH CONSTRUCTIONS
- NCERT CLASS 6TH MATHS CHAPTER 9 SYMMETRY
- NCERT CLASS 6TH MATHS CHAPTER 10 THE OTHER SIDE OF ZERO
Topic 1.1 What is mathematics ?
Figure It Out
Question 1
Can you think of other examples where mathematics helps us in our everyday lives?
Answer. Yse from the morning wake up to sleep at night, there is mathematics all around us.
For example.
- To count given article.
- To add or subtract some quantities.
- To recognize figures.
- To understand different pattern in mathematics.
etc.
Question 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. Mathematics played a major role for human development. It is unsung hero behind many of innovation, achivements, progress and uncountable aspects of our lives. It played a great role to propel humanity forward in many field like
- Technological advancements
- Medical breakthroughs
- Weather forcasting
- Artificial intelligence
- To understand pattern in mathematics
- Infrastructure development etc.
Topic 1.2 Patterns in Numbers
Figure It Out
 |
| Class 6th Maths Chapter 1 |
Question 1
Can you recognize the pattern in each of the sequences in Table 1?
Answer. Yes,we recognized the pattern in each of the sequences in the table 1.
Question 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.
Sequence 1 : 1,1,1,1,1,1,1,.....
Answer. Next three terms are 1,1,1 because all terms are 1.
Sequence 2 : 1,2,3,4,5,6,7,......
Answer. Next three terms are 8,9,10 because every next term increases by 1.
Sequence 3 : 1,3,5,7,9,11,13,.....
Answer. Next three terms are 15,17,19 because every next term increases by 2 OR It is consecutive odd number sequence.
Sequence 4 : 2,4,6,8,10,12,14,.....
Answer. Next three terms are 16,18,20 because It is consecutive even number sequence.
Sequence 5 : 1,3,6,10,15,21,28,....
Solution: 1=1
1+2=3
1+2+3=6
1+2+3+4=10
1+2+3+4+5=15
1+2+3+4+5+6=21
1+2+3+4+5+6+7=28
therefore next three terms are
1+2+3+4+3+6+7+8=36
1+2+3+4+5+6+7+8+9=45
1+2+3+4+5+6+7+8+9+10=55
This sequence is called triangular number sequence.
Sequence 6 : 1,4,9,16,25,36,49,....
Solution: 1×1=1
2×2=4
3×3=9
4×4=16
5×5=25
6×6=36
7×7=49
therefore next three terms are
8×8=64
9×9=81
10×10=100
This sequence is called square number sequence.
Sequence 7 : 1,8,27,64,125,216,...
Solution: 1×1×1=1
2×2×2=8
3×3×3=27
4×4×4=64
5×5×5=125
6×6×6=216
therefore next three terms are
7×7×7=343
8×8×8=512
9×9×9=729
This sequence is called cube number sequence.
Sequence 7 : 1,2,3,5,8,13,21,...
Solution: 1=1
1+1=2
1+2=3
2+3 =5
3+5=8
5+8=13
8+13=21
therefore next three terms are
13+21=34
21+34=55
34+55=89
This sequence is called virahanka number sequence.
Sequence 7 : 1,2,4,8,16,32,64,....
Solution:
This sequence is called power of 2 number sequence.
Sequence 8 :1,3,9,27,81,243,729,....
Solution: 
Topic 1.3 Visualising Number Sequences
Figure It Out
Question 1
Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
 |
| Class 6th Maths Chapter 1 |
Question 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: 1,3,6,10,15.... are triangular numbers because we can represent this number by help of dots in the shape of triangle . For example
1,8,27,64,125.... are cube numbers because we can represent this numbers in the shape of cube. For example
Question 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:
36 as a triangular number and we can arrange 36 dots like this
36 as a square number and we can arrange 36 dots like this
Question 4
What would you call the following sequence of numbers?
This sequence of numbers are called hexagonal numbers. The next number in the sequence is 61.
here, 1+6x0=1, 1+6x1=7, 1+6x3=19, 1+6x6=37, 1+6x10=61
`
{Note:0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,......}
Question 5
Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
Solution:
yes we can think visualise the sequence of power of 2 and power of 3.
Power of of 2:1,2,4,8,......
Power of 3:1,3,9,27,.....
Topic 1.4 Relations among Number Sequences
Figure It Out
Question 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:
Yes we can find a similar pictorial explanation for this
Question 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?
Since,
1=1✖ 1 = 1
1+2+1 = 2✖2 =4
1+2+3+2+1 = 3 ✖ 3 =9
1+2+3+4+3+2+1 = 4 ✖ 4 =16
Therefor 1+2+3+....+99+100+99+...+3+2+1 = 100✖100 =10000
Question 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: We know that 1's sequence are 1,1,1,1,1,.....
1 = 1
1+1 = 2
1+1+1 = 3
Hence, when we start to add the All 1’s sequence up, we get counting numbers.
Add all 1's sequence up and down,
1 = 1
1+1+1 = 3
1+1+1+1+1 = 5
1+1+1+1+1+1+1 = 7 and so on
Hence, when we start to add the All 1’s sequence up and down, we get odd numbers.
Question 4
Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?
Solution: Add counting numbers up,
1 = 1
1+2 = 3
1+2+3 = 6
1+2+3+4 = 10 and so on
we can represent these numbers in the form of triangle by help of dots there for these numbers are triangular numbers.
Question 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: Triangular numbers are 1,3,6,10,15,.....
now, 1 = 1
1+3 = 4
` 3+6 = 9
6+10 = 16
10+15 = 25 and so on
Hence,when we add up pairs of consecutive triangular numbers, we get square numbers sequence.
Question 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?
Now, 1 = 1
1+2 = 3
1+2+4 = 7
1+2+4+8= 15
1+2+4+8+16 =31 and so on
Add one with each number
1+1 = 2
3+1 =4
7+1 = 8
15+1 = 16
31+1 = 32
When we followed question's instructions we obtained again sequence of power of 2.
Its happen because adding of 1 increase power of 2.
Question 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: Triangular numbers are 1,3,6,10,15,.....
According to the question
1 ✖ 6 +1 =7
3 ✖ 6 + 1 = 19
6 ✖ 6 + 1 = 37
10 ✖ 6 + 1= 61 and so on
Hence,when we multiply the triangular numbers by 6 and add 1 we get hexagonal numbers sequence.
Question 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: Hexagonal numbers are 1,7,19,37,61,.....
1 = 1
1 + 7 = 8
1 + 7 + 19 =27
1 + 7 + 19 +37 =64
1 + 7 + 19 + 37 + 61 = 125 and so on
Hence when you start to add up hexagonal numbers we get cubes.
Topic 1.5 Patterns in Shapes
Figure It Out
Question 1
Can you recognise the pattern in each of the sequences in Table 3?
Solution:
Yes,we can recognise the pattern in each of the sequences in Table 3.
When we write a number sequence according to their number squares used in figures we get 1,4,9,16,25,..... these all are square number so this sequence is square number sequence.
Question 3
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.
Solution: 
For rules see just above solution
Topic 1.6 Relation to Number Sequences
Figure It Out
Question 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:
Question 2
Question 3
Question 4
Yes, we get same number of sequence because in regular polygons number of sides are equal to number of corners.
Question 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:
K2 =1
K3 =3
K4 = 6
K5 = 10
K6 = 15 and so on
Sequence: 1,3,6,10,15,.... These all are triangular numbers and this is triangular numbers sequence.
Question 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:
Figure 1 = 1 square
Figure 2 = 4 squares
Figure 3 = 9 squares
Figure 4 = 16 squares
Figure 5 = 25 squares and so on
Sequence: 1,4,9,16,25,.... These all are square numbers and this is square numbers sequence.
Question 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: 
Question 5
Figure 1 = 1 triangle
Figure 2 = 4 triangles
Figure 3 = 9 triangles
Figure 4 = 16 triangles
Figure 5 = 25 triangles and so on
Sequence: 1,4,9,16,25,.... These all are square numbers and this is square numbers sequence.
Question 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:
Figure 1 =3 line segments
Figure 2 = 3 ✖ 4 =12 line segments
Figure 3 = 12 ✖ 4 = 48 line segments
Figure 4 = 48 ✖ 4 =192 line segments
Figure 5 =192 ✖4=768 line segments
Sequence:3,12,48,192,768,.... here each next step is product of previous step and 4 and for next figure each line segment become 4 line segments. This is very tough pattern in mathematics to draw .
0 Comments