NCERT Textbook: Number Play | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

 Page 1


5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the 
following —
Now, he is wondering —
•  “Can I write every natural number as a sum of consecutive 
numbers?” 
•  “Which numbers can I write as the sum of consecutive numbers in 
more than one way?”
•  “Ohh, I know all odd numbers can be written as a sum of two 
consecutive numbers. Can we write all even numbers as a sum of 
consecutive numbers?”
•  “Can I write 0 as a sum of consecutive numbers? Maybe I should 
use negative numbers.”
Explore these questions and any others that may occur to you. 
Discuss them with the class.
Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place ‘+’ and 
‘–’ signs in between the numbers. How many different possibilities exist? 
Write all of them.
7   = 3 + 4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
     = 4 + 5 + 6
     = 1 + 2 + 3 + 4 + 5
Math 
Talk
3 + 4 – 5 + 6 
3 – 4 – 5 – 6  
NUMBER PLAY
5
Chapter 5 Letter Number Play 06-07-2025.indd   112 Chapter 5 Letter Number Play 06-07-2025.indd   112 10-07-2025   15:07:56 10-07-2025   15:07:56
Page 2


5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the 
following —
Now, he is wondering —
•  “Can I write every natural number as a sum of consecutive 
numbers?” 
•  “Which numbers can I write as the sum of consecutive numbers in 
more than one way?”
•  “Ohh, I know all odd numbers can be written as a sum of two 
consecutive numbers. Can we write all even numbers as a sum of 
consecutive numbers?”
•  “Can I write 0 as a sum of consecutive numbers? Maybe I should 
use negative numbers.”
Explore these questions and any others that may occur to you. 
Discuss them with the class.
Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place ‘+’ and 
‘–’ signs in between the numbers. How many different possibilities exist? 
Write all of them.
7   = 3 + 4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
     = 4 + 5 + 6
     = 1 + 2 + 3 + 4 + 5
Math 
Talk
3 + 4 – 5 + 6 
3 – 4 – 5 – 6  
NUMBER PLAY
5
Chapter 5 Letter Number Play 06-07-2025.indd   112 Chapter 5 Letter Number Play 06-07-2025.indd   112 10-07-2025   15:07:56 10-07-2025   15:07:56
Number Play
113
Eight such expressions are possible. You can use the diagram below to 
systematically list all the possibilities.
Evaluate each expression and write the result next to it. Do you notice 
anything interesting?
Now, take four other consecutive numbers. Place the ‘+’ and ‘–’ signs 
as you have done before. Find out the results of each expression. 
What do you observe?
Repeat this for one more set of 4 consecutive numbers. Share your 
findings.
Some sums appear always no matter which 4 consecutive 
numbers are chosen. Isn’t that interesting?
Do these patterns occur no matter which 4 consecutive numbers are 
chosen? Is there a way to find out through reasoning?
Hint: Use algebra and describe the 8 expressions in a general form.
You might have noticed that the results of all expressions are even 
numbers. Even numbers have a factor of 2. Negative numbers having 
a factor 2 are also even numbers, for example, – 2, – 4, – 6, and so on. 
Check if anyone in your class got an odd number.
When 4 consecutive numbers are chosen, no matter how the ‘+’ and 
‘–’ signs are placed between them, the resulting expressions always have 
even parity.
3
4
4
5
5
5
5
6
6
6
6
6
6
6
6
+
–
–
+
+
–
–
–
–
–
+
+
+
+
3 + 4 + 5 + 6
3 + 4 + 5 – 6
Math 
Talk
3 + 4 – 5 + 6 = 8
3 – 4 – 5 – 6  = – 12
.
.
.
5 + 6 – 7 + 8 = 12
5 – 6 – 7 – 8  = – 16
.
.
.
__ + __ – __ + __  = __
__ – __ – __ – __  =  __
.
.
.
Chapter 5 Letter Number Play 06-07-2025.indd   113 Chapter 5 Letter Number Play 06-07-2025.indd   113 10-07-2025   15:07:57 10-07-2025   15:07:57
Page 3


5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the 
following —
Now, he is wondering —
•  “Can I write every natural number as a sum of consecutive 
numbers?” 
•  “Which numbers can I write as the sum of consecutive numbers in 
more than one way?”
•  “Ohh, I know all odd numbers can be written as a sum of two 
consecutive numbers. Can we write all even numbers as a sum of 
consecutive numbers?”
•  “Can I write 0 as a sum of consecutive numbers? Maybe I should 
use negative numbers.”
Explore these questions and any others that may occur to you. 
Discuss them with the class.
Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place ‘+’ and 
‘–’ signs in between the numbers. How many different possibilities exist? 
Write all of them.
7   = 3 + 4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
     = 4 + 5 + 6
     = 1 + 2 + 3 + 4 + 5
Math 
Talk
3 + 4 – 5 + 6 
3 – 4 – 5 – 6  
NUMBER PLAY
5
Chapter 5 Letter Number Play 06-07-2025.indd   112 Chapter 5 Letter Number Play 06-07-2025.indd   112 10-07-2025   15:07:56 10-07-2025   15:07:56
Number Play
113
Eight such expressions are possible. You can use the diagram below to 
systematically list all the possibilities.
Evaluate each expression and write the result next to it. Do you notice 
anything interesting?
Now, take four other consecutive numbers. Place the ‘+’ and ‘–’ signs 
as you have done before. Find out the results of each expression. 
What do you observe?
Repeat this for one more set of 4 consecutive numbers. Share your 
findings.
Some sums appear always no matter which 4 consecutive 
numbers are chosen. Isn’t that interesting?
Do these patterns occur no matter which 4 consecutive numbers are 
chosen? Is there a way to find out through reasoning?
Hint: Use algebra and describe the 8 expressions in a general form.
You might have noticed that the results of all expressions are even 
numbers. Even numbers have a factor of 2. Negative numbers having 
a factor 2 are also even numbers, for example, – 2, – 4, – 6, and so on. 
Check if anyone in your class got an odd number.
When 4 consecutive numbers are chosen, no matter how the ‘+’ and 
‘–’ signs are placed between them, the resulting expressions always have 
even parity.
3
4
4
5
5
5
5
6
6
6
6
6
6
6
6
+
–
–
+
+
–
–
–
–
–
+
+
+
+
3 + 4 + 5 + 6
3 + 4 + 5 – 6
Math 
Talk
3 + 4 – 5 + 6 = 8
3 – 4 – 5 – 6  = – 12
.
.
.
5 + 6 – 7 + 8 = 12
5 – 6 – 7 – 8  = – 16
.
.
.
__ + __ – __ + __  = __
__ – __ – __ – __  =  __
.
.
.
Chapter 5 Letter Number Play 06-07-2025.indd   113 Chapter 5 Letter Number Play 06-07-2025.indd   113 10-07-2025   15:07:57 10-07-2025   15:07:57
Ganita Prakash | Grade 8 
114
Now take any 4 numbers, place ‘+’ and ‘–’ signs in the eight different 
ways, and evaluate the resulting expression. What do you observe about 
their parities?
Repeat this with other sets of 4 numbers.
Is there a way to explain why this happens?
Hint:  Think of the rules for parity of the sum or difference of two 
numbers.
Explanation 1: Let us consider any of the 8 expressions formed by four 
numbers a, b, c, and d. When one of its signs is switched, its value always 
increases or decreases by an even number! Let us see why.
Consider one of the expressions: a + b – c – d. 
Replacing +b by – b, we get
a – b – c – d.
By how much has the number changed? It has changed by
(a + b – c – d) –  (a – b – c – d)
= a + b – c – d – a + b + c + d (notice how the signs changed when we 
opened the second set of brackets)
= 2b (this is an even number).
If the difference between two numbers is even, can they have different 
parities? No! So either both are even or both are odd.
Now, let us see what happens when a negative sign is switched to a 
positive sign.
Replace any negative sign in the expression a + b – c – d with a positive 
sign and find the difference between the two numbers.
What do you conclude from this observation?
Starting from any expression, we can get 7 expressions by switching one 
or more ‘+’ and ‘–’ signs. Thus, all the expressions have the same parity!
Explanation 2: We know that
odd ± odd = even
even ± even = even
odd ± even = odd.
We have seen that the parity of a + b and a – b is the same, regardless 
of the parities of a and b.
In short, a ± b have the same parity. By the same argument, a ± b + c 
and a ± b – c have the same parity. Extending this further, we can say that 
all the expressions a ± b ± c ± d have the same parity.
Math 
Talk
Chapter 5 Letter Number Play 06-07-2025.indd   114 Chapter 5 Letter Number Play 06-07-2025.indd   114 10-07-2025   15:07:57 10-07-2025   15:07:57
Page 4


5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the 
following —
Now, he is wondering —
•  “Can I write every natural number as a sum of consecutive 
numbers?” 
•  “Which numbers can I write as the sum of consecutive numbers in 
more than one way?”
•  “Ohh, I know all odd numbers can be written as a sum of two 
consecutive numbers. Can we write all even numbers as a sum of 
consecutive numbers?”
•  “Can I write 0 as a sum of consecutive numbers? Maybe I should 
use negative numbers.”
Explore these questions and any others that may occur to you. 
Discuss them with the class.
Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place ‘+’ and 
‘–’ signs in between the numbers. How many different possibilities exist? 
Write all of them.
7   = 3 + 4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
     = 4 + 5 + 6
     = 1 + 2 + 3 + 4 + 5
Math 
Talk
3 + 4 – 5 + 6 
3 – 4 – 5 – 6  
NUMBER PLAY
5
Chapter 5 Letter Number Play 06-07-2025.indd   112 Chapter 5 Letter Number Play 06-07-2025.indd   112 10-07-2025   15:07:56 10-07-2025   15:07:56
Number Play
113
Eight such expressions are possible. You can use the diagram below to 
systematically list all the possibilities.
Evaluate each expression and write the result next to it. Do you notice 
anything interesting?
Now, take four other consecutive numbers. Place the ‘+’ and ‘–’ signs 
as you have done before. Find out the results of each expression. 
What do you observe?
Repeat this for one more set of 4 consecutive numbers. Share your 
findings.
Some sums appear always no matter which 4 consecutive 
numbers are chosen. Isn’t that interesting?
Do these patterns occur no matter which 4 consecutive numbers are 
chosen? Is there a way to find out through reasoning?
Hint: Use algebra and describe the 8 expressions in a general form.
You might have noticed that the results of all expressions are even 
numbers. Even numbers have a factor of 2. Negative numbers having 
a factor 2 are also even numbers, for example, – 2, – 4, – 6, and so on. 
Check if anyone in your class got an odd number.
When 4 consecutive numbers are chosen, no matter how the ‘+’ and 
‘–’ signs are placed between them, the resulting expressions always have 
even parity.
3
4
4
5
5
5
5
6
6
6
6
6
6
6
6
+
–
–
+
+
–
–
–
–
–
+
+
+
+
3 + 4 + 5 + 6
3 + 4 + 5 – 6
Math 
Talk
3 + 4 – 5 + 6 = 8
3 – 4 – 5 – 6  = – 12
.
.
.
5 + 6 – 7 + 8 = 12
5 – 6 – 7 – 8  = – 16
.
.
.
__ + __ – __ + __  = __
__ – __ – __ – __  =  __
.
.
.
Chapter 5 Letter Number Play 06-07-2025.indd   113 Chapter 5 Letter Number Play 06-07-2025.indd   113 10-07-2025   15:07:57 10-07-2025   15:07:57
Ganita Prakash | Grade 8 
114
Now take any 4 numbers, place ‘+’ and ‘–’ signs in the eight different 
ways, and evaluate the resulting expression. What do you observe about 
their parities?
Repeat this with other sets of 4 numbers.
Is there a way to explain why this happens?
Hint:  Think of the rules for parity of the sum or difference of two 
numbers.
Explanation 1: Let us consider any of the 8 expressions formed by four 
numbers a, b, c, and d. When one of its signs is switched, its value always 
increases or decreases by an even number! Let us see why.
Consider one of the expressions: a + b – c – d. 
Replacing +b by – b, we get
a – b – c – d.
By how much has the number changed? It has changed by
(a + b – c – d) –  (a – b – c – d)
= a + b – c – d – a + b + c + d (notice how the signs changed when we 
opened the second set of brackets)
= 2b (this is an even number).
If the difference between two numbers is even, can they have different 
parities? No! So either both are even or both are odd.
Now, let us see what happens when a negative sign is switched to a 
positive sign.
Replace any negative sign in the expression a + b – c – d with a positive 
sign and find the difference between the two numbers.
What do you conclude from this observation?
Starting from any expression, we can get 7 expressions by switching one 
or more ‘+’ and ‘–’ signs. Thus, all the expressions have the same parity!
Explanation 2: We know that
odd ± odd = even
even ± even = even
odd ± even = odd.
We have seen that the parity of a + b and a – b is the same, regardless 
of the parities of a and b.
In short, a ± b have the same parity. By the same argument, a ± b + c 
and a ± b – c have the same parity. Extending this further, we can say that 
all the expressions a ± b ± c ± d have the same parity.
Math 
Talk
Chapter 5 Letter Number Play 06-07-2025.indd   114 Chapter 5 Letter Number Play 06-07-2025.indd   114 10-07-2025   15:07:57 10-07-2025   15:07:57
Number Play
115
Explanation 3: This can also be explained 
using the positive and negative token 
model you studied in the chapter on 
Integers. Try to think how.
The number of ways to choose 4 
numbers a, b, c, d and combine them using 
‘+’ and ‘–’ signs is infinite. Mathematical 
reasoning allows us to prove that all the 
combinations a ± b ± c ± d always have the 
same parity, without having to go through 
them one by one.
Is the phenomenon of all the expressions having the same parity limited 
to taking 4 numbers? What do you think?
Breaking Even
We know how to identify even numbers. Without computing them, find 
out which of the following arithmetic expressions are even.
Using our understanding of how parity behaves under different 
operations, identify which of the following algebraic expressions give 
an even number for any integer values for the letter-numbers.
3g + 5h 2a + 2b 2u – 4v 4m + 2n
6m – 3n 13k – 5k b
2
 + 1
4k × 3j
x
2
 + 2
Several problems in mathematics can be thought about and 
solved in different ways. While the method you came up with 
may be dear to you, it can be amusing and enriching to know 
how others thought about it. Two tidbits: ‘share’ and ‘listen’.
‘What if …?’, ‘Will it always happen?’— Wondering and posing 
questions and conjectures is as much a part of mathematics as 
problem solving.
672 – 348 43 + 37 708 – 477 4 × 347 × 3
119 × 303 809 + 214 513
3
543 – 479
a
b
b
c
c
c
c
d
d
d
d
d
d
d
d
+
–
–
+
+
–
–
–
–
–
+
+
+
+
Chapter 5 Letter Number Play 06-07-2025.indd   115 Chapter 5 Letter Number Play 06-07-2025.indd   115 10-07-2025   15:07:58 10-07-2025   15:07:58
Page 5


5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the 
following —
Now, he is wondering —
•  “Can I write every natural number as a sum of consecutive 
numbers?” 
•  “Which numbers can I write as the sum of consecutive numbers in 
more than one way?”
•  “Ohh, I know all odd numbers can be written as a sum of two 
consecutive numbers. Can we write all even numbers as a sum of 
consecutive numbers?”
•  “Can I write 0 as a sum of consecutive numbers? Maybe I should 
use negative numbers.”
Explore these questions and any others that may occur to you. 
Discuss them with the class.
Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place ‘+’ and 
‘–’ signs in between the numbers. How many different possibilities exist? 
Write all of them.
7   = 3 + 4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
     = 4 + 5 + 6
     = 1 + 2 + 3 + 4 + 5
Math 
Talk
3 + 4 – 5 + 6 
3 – 4 – 5 – 6  
NUMBER PLAY
5
Chapter 5 Letter Number Play 06-07-2025.indd   112 Chapter 5 Letter Number Play 06-07-2025.indd   112 10-07-2025   15:07:56 10-07-2025   15:07:56
Number Play
113
Eight such expressions are possible. You can use the diagram below to 
systematically list all the possibilities.
Evaluate each expression and write the result next to it. Do you notice 
anything interesting?
Now, take four other consecutive numbers. Place the ‘+’ and ‘–’ signs 
as you have done before. Find out the results of each expression. 
What do you observe?
Repeat this for one more set of 4 consecutive numbers. Share your 
findings.
Some sums appear always no matter which 4 consecutive 
numbers are chosen. Isn’t that interesting?
Do these patterns occur no matter which 4 consecutive numbers are 
chosen? Is there a way to find out through reasoning?
Hint: Use algebra and describe the 8 expressions in a general form.
You might have noticed that the results of all expressions are even 
numbers. Even numbers have a factor of 2. Negative numbers having 
a factor 2 are also even numbers, for example, – 2, – 4, – 6, and so on. 
Check if anyone in your class got an odd number.
When 4 consecutive numbers are chosen, no matter how the ‘+’ and 
‘–’ signs are placed between them, the resulting expressions always have 
even parity.
3
4
4
5
5
5
5
6
6
6
6
6
6
6
6
+
–
–
+
+
–
–
–
–
–
+
+
+
+
3 + 4 + 5 + 6
3 + 4 + 5 – 6
Math 
Talk
3 + 4 – 5 + 6 = 8
3 – 4 – 5 – 6  = – 12
.
.
.
5 + 6 – 7 + 8 = 12
5 – 6 – 7 – 8  = – 16
.
.
.
__ + __ – __ + __  = __
__ – __ – __ – __  =  __
.
.
.
Chapter 5 Letter Number Play 06-07-2025.indd   113 Chapter 5 Letter Number Play 06-07-2025.indd   113 10-07-2025   15:07:57 10-07-2025   15:07:57
Ganita Prakash | Grade 8 
114
Now take any 4 numbers, place ‘+’ and ‘–’ signs in the eight different 
ways, and evaluate the resulting expression. What do you observe about 
their parities?
Repeat this with other sets of 4 numbers.
Is there a way to explain why this happens?
Hint:  Think of the rules for parity of the sum or difference of two 
numbers.
Explanation 1: Let us consider any of the 8 expressions formed by four 
numbers a, b, c, and d. When one of its signs is switched, its value always 
increases or decreases by an even number! Let us see why.
Consider one of the expressions: a + b – c – d. 
Replacing +b by – b, we get
a – b – c – d.
By how much has the number changed? It has changed by
(a + b – c – d) –  (a – b – c – d)
= a + b – c – d – a + b + c + d (notice how the signs changed when we 
opened the second set of brackets)
= 2b (this is an even number).
If the difference between two numbers is even, can they have different 
parities? No! So either both are even or both are odd.
Now, let us see what happens when a negative sign is switched to a 
positive sign.
Replace any negative sign in the expression a + b – c – d with a positive 
sign and find the difference between the two numbers.
What do you conclude from this observation?
Starting from any expression, we can get 7 expressions by switching one 
or more ‘+’ and ‘–’ signs. Thus, all the expressions have the same parity!
Explanation 2: We know that
odd ± odd = even
even ± even = even
odd ± even = odd.
We have seen that the parity of a + b and a – b is the same, regardless 
of the parities of a and b.
In short, a ± b have the same parity. By the same argument, a ± b + c 
and a ± b – c have the same parity. Extending this further, we can say that 
all the expressions a ± b ± c ± d have the same parity.
Math 
Talk
Chapter 5 Letter Number Play 06-07-2025.indd   114 Chapter 5 Letter Number Play 06-07-2025.indd   114 10-07-2025   15:07:57 10-07-2025   15:07:57
Number Play
115
Explanation 3: This can also be explained 
using the positive and negative token 
model you studied in the chapter on 
Integers. Try to think how.
The number of ways to choose 4 
numbers a, b, c, d and combine them using 
‘+’ and ‘–’ signs is infinite. Mathematical 
reasoning allows us to prove that all the 
combinations a ± b ± c ± d always have the 
same parity, without having to go through 
them one by one.
Is the phenomenon of all the expressions having the same parity limited 
to taking 4 numbers? What do you think?
Breaking Even
We know how to identify even numbers. Without computing them, find 
out which of the following arithmetic expressions are even.
Using our understanding of how parity behaves under different 
operations, identify which of the following algebraic expressions give 
an even number for any integer values for the letter-numbers.
3g + 5h 2a + 2b 2u – 4v 4m + 2n
6m – 3n 13k – 5k b
2
 + 1
4k × 3j
x
2
 + 2
Several problems in mathematics can be thought about and 
solved in different ways. While the method you came up with 
may be dear to you, it can be amusing and enriching to know 
how others thought about it. Two tidbits: ‘share’ and ‘listen’.
‘What if …?’, ‘Will it always happen?’— Wondering and posing 
questions and conjectures is as much a part of mathematics as 
problem solving.
672 – 348 43 + 37 708 – 477 4 × 347 × 3
119 × 303 809 + 214 513
3
543 – 479
a
b
b
c
c
c
c
d
d
d
d
d
d
d
d
+
–
–
+
+
–
–
–
–
–
+
+
+
+
Chapter 5 Letter Number Play 06-07-2025.indd   115 Chapter 5 Letter Number Play 06-07-2025.indd   115 10-07-2025   15:07:58 10-07-2025   15:07:58
Ganita Prakash | Grade 8 
116
The expression 4m + 2q will always evaluate to an even number for 
any integer values of m and q. We can justify this in two different ways —
• We know 4m is even and 2q is even for any integers m and q. 
Therefore, their sum will also be even.
• The expression 4m + 2q is equal to the expression 2(2m + q). Here, 
the expression 2(2m + q) means 2 times 2m + q. In other words, 2 is 
a factor of this expression. Therefore, this expression will always 
give an even number for any integers m and q.
For example, if m = 4 and q = – 9, the expression 4m + 2q becomes 
4 × 4 + 2 × (–9) = – 2, which is an even number.
In the expression x
2
 + 2, x
2
 is even if x is even, and x
2
 is odd if x is odd. 
Therefore, the expression x
2
 + 2 will not always give an even number. 
An example and a non-example for when the expression evaluates to an 
even number — (i) if x = 6, then x
2
 + 2 = 38, and (ii) if x = 3, then x
2
 + 2
 
 = 11.
Similarly, determine and explain which of the other expressions always 
give even numbers. Write a couple of examples and non-examples, as 
appropriate, for each expression.
Write a few algebraic expressions which always give an even number.
Pairs to Make Fours
Take a pair of even numbers. Add them. Is the sum divisible by 4? 
Try this with different pairs of even numbers.
When is the sum a multiple of 4, and when is it not? 
Is there a general rule or a pattern?
Even numbers can be of two types based on the remainders they leave 
when divided by 4.
.
.
.
.
.
.
.
Even numbers that are multiples 
of 4 leave a remainder of 0 when 
divided by 4.
Even numbers that are not 
multiples of 4 leave a remainder 2 
when divided by 4.
Chapter 5 Letter Number Play 06-07-2025.indd   116 Chapter 5 Letter Number Play 06-07-2025.indd   116 10-07-2025   15:07:58 10-07-2025   15:07:58