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

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NUMBER PLAY
Page 2


NUMBER PLAY
Sum of Consecutive Numbers
Natural numbers can often be written as sums of consecutive 
numbers.
E x a m p l e s:
 7 = 3
 10 = 1 + 2 + 3 + 4
 12 = 3 + 4 + 5
P a t t e r n s :
Every odd number = sum of two consecutive numbers.
Some numbers can be written in multiple ways.
Page 3


NUMBER PLAY
Sum of Consecutive Numbers
Natural numbers can often be written as sums of consecutive 
numbers.
E x a m p l e s:
 7 = 3
 10 = 1 + 2 + 3 + 4
 12 = 3 + 4 + 5
P a t t e r n s :
Every odd number = sum of two consecutive numbers.
Some numbers can be written in multiple ways.
Patterns with 
Consecutive Numbers
Take any 4 consecutive numbers and place '+' and '3' 
signs between them. How many different possibilities 
exist?
Eight expressions are possible. Let's evaluate some:
3 + 4 3 5 + 6 = 8
5 + 6 3 7 + 8 = 12
3 3 4 3 5 3 6 = 312
5 3 6 3 7 3 8 = 316
Observation: Results are always even numbers.
Page 4


NUMBER PLAY
Sum of Consecutive Numbers
Natural numbers can often be written as sums of consecutive 
numbers.
E x a m p l e s:
 7 = 3
 10 = 1 + 2 + 3 + 4
 12 = 3 + 4 + 5
P a t t e r n s :
Every odd number = sum of two consecutive numbers.
Some numbers can be written in multiple ways.
Patterns with 
Consecutive Numbers
Take any 4 consecutive numbers and place '+' and '3' 
signs between them. How many different possibilities 
exist?
Eight expressions are possible. Let's evaluate some:
3 + 4 3 5 + 6 = 8
5 + 6 3 7 + 8 = 12
3 3 4 3 5 3 6 = 312
5 3 6 3 7 3 8 = 316
Observation: Results are always even numbers.
Explaining the Pattern
Why do we always get even numbers?
Explanation 1
When one sign is switched in any expression with four 
numbers a, b, c, and d, the value always increases or 
decreases by an even number!
Explanation 2
The parity of a ± b is the same, regardless of the parities 
of a and b. Extending this, all expressions a ± b ± c ± d 
have the same parity.
Using parity rules: odd ± odd = even, even ± even = 
even.
Page 5


NUMBER PLAY
Sum of Consecutive Numbers
Natural numbers can often be written as sums of consecutive 
numbers.
E x a m p l e s:
 7 = 3
 10 = 1 + 2 + 3 + 4
 12 = 3 + 4 + 5
P a t t e r n s :
Every odd number = sum of two consecutive numbers.
Some numbers can be written in multiple ways.
Patterns with 
Consecutive Numbers
Take any 4 consecutive numbers and place '+' and '3' 
signs between them. How many different possibilities 
exist?
Eight expressions are possible. Let's evaluate some:
3 + 4 3 5 + 6 = 8
5 + 6 3 7 + 8 = 12
3 3 4 3 5 3 6 = 312
5 3 6 3 7 3 8 = 316
Observation: Results are always even numbers.
Explaining the Pattern
Why do we always get even numbers?
Explanation 1
When one sign is switched in any expression with four 
numbers a, b, c, and d, the value always increases or 
decreases by an even number!
Explanation 2
The parity of a ± b is the same, regardless of the parities 
of a and b. Extending this, all expressions a ± b ± c ± d 
have the same parity.
Using parity rules: odd ± odd = even, even ± even = 
even.
Breaking Even (Algebraic Expressions)
Without computing them, can we identify which arithmetic expressions are even?
Using our understanding of parity, we can determine which algebraic expressions always give even 
numbers for any integer values.
Example: 4m + 2q
This expression is always even because:
4m is even and 2q is even, so their sum is 
even
It can be written as 2(2m + q), where 2 is a 
factor
Example: x² + 2
This expression is n o t always even because:
If x is even, x² is even, so x² + 2 is even
If x is odd, x² is odd, so x² + 2 is odd