5 NUMBER PLAY
5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the
following —
7 =3+4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
=4+5+6
=1+2+3+4+5
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. Math
Discuss them with the class. Talk
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.
3+4–5+6
3–4–5–6
Reprint 2026-27
, Number Play
Eight such expressions are possible. You can use the diagram below to
systematically list all the possibilities.
+ 6 3+4+5+6
5
+ – 6 3+4+5–6
4 +
+ 6
– 5
– 6
3
+ 6
– + 5
4 – 6
– + 6
5
– 6
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?
Math
Repeat this for one more set of 4 consecutive numbers. Share your Talk
findings.
3+4–5+6=8 5 + 6 – 7 + 8 = 12 __ + __ – __ + __ = __
3 – 4 – 5 – 6 = – 12 5 – 6 – 7 – 8 = – 16 __ – __ – __ – __ = __
. . .
. . .
. . .
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.
113
Reprint 2026-27
, Ganita Prakash | Grade 8
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.
Math
Is there a way to explain why this happens? Talk
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.
114
Reprint 2026-27
, Number Play
+ d
Explanation 3: This can also be explained c
using the positive and negative token + – d
model you studied in the chapter on b
Integers. Try to think how. + d
+ – c
The number of ways to choose 4 – d
numbers a, b, c, d and combine them using a
‘+’ and ‘–’ signs is infinite. Mathematical + d
– + c
reasoning allows us to prove that all the b – d
combinations a ± b ± c ± d always have the
same parity, without having to go through – + d
them one by one. c
– d
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’.
Is the phenomenon of all the expressions having the same parity limited
to taking 4 numbers? What do you think?
‘What if …?’, ‘Will it always happen?’— Wondering and posing
questions and conjectures is as much a part of mathematics as
problem solving.
Breaking Even
We know how to identify even numbers. Without computing them, find
out which of the following arithmetic expressions are even.
43 + 37 672 – 348 4 × 347 × 3 708 – 477
809 + 214 119 × 303 543 – 479 513 3
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.
2a + 2b 3g + 5h 4m + 2n 2u – 4v
13k – 5k 6m – 3n x2 + 2 b2 + 1 4k × 3j
115
Reprint 2026-27
5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the
following —
7 =3+4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
=4+5+6
=1+2+3+4+5
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. Math
Discuss them with the class. Talk
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.
3+4–5+6
3–4–5–6
Reprint 2026-27
, Number Play
Eight such expressions are possible. You can use the diagram below to
systematically list all the possibilities.
+ 6 3+4+5+6
5
+ – 6 3+4+5–6
4 +
+ 6
– 5
– 6
3
+ 6
– + 5
4 – 6
– + 6
5
– 6
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?
Math
Repeat this for one more set of 4 consecutive numbers. Share your Talk
findings.
3+4–5+6=8 5 + 6 – 7 + 8 = 12 __ + __ – __ + __ = __
3 – 4 – 5 – 6 = – 12 5 – 6 – 7 – 8 = – 16 __ – __ – __ – __ = __
. . .
. . .
. . .
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.
113
Reprint 2026-27
, Ganita Prakash | Grade 8
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.
Math
Is there a way to explain why this happens? Talk
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.
114
Reprint 2026-27
, Number Play
+ d
Explanation 3: This can also be explained c
using the positive and negative token + – d
model you studied in the chapter on b
Integers. Try to think how. + d
+ – c
The number of ways to choose 4 – d
numbers a, b, c, d and combine them using a
‘+’ and ‘–’ signs is infinite. Mathematical + d
– + c
reasoning allows us to prove that all the b – d
combinations a ± b ± c ± d always have the
same parity, without having to go through – + d
them one by one. c
– d
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’.
Is the phenomenon of all the expressions having the same parity limited
to taking 4 numbers? What do you think?
‘What if …?’, ‘Will it always happen?’— Wondering and posing
questions and conjectures is as much a part of mathematics as
problem solving.
Breaking Even
We know how to identify even numbers. Without computing them, find
out which of the following arithmetic expressions are even.
43 + 37 672 – 348 4 × 347 × 3 708 – 477
809 + 214 119 × 303 543 – 479 513 3
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.
2a + 2b 3g + 5h 4m + 2n 2u – 4v
13k – 5k 6m – 3n x2 + 2 b2 + 1 4k × 3j
115
Reprint 2026-27
