preparation of a unit for 8th std students.a unit on "IDENTITIES"

1. 8 = 4 + 3
6 = 4 + 2
4 x 4 3 x 4
4 x 2 3 x 2

2. Product of sums
Here is a rectangle of length 12 centimeters and breadth 8
centimeters.
12
8
Rinu is trying to find out its area.
Can you help Rinu to calculate its area ?
What is it ?
To find area, we want to multiply the rectangle’s length and
breadth.
So it is 12 x 8 .
Now, Rinu divide the rectangle into two rectangles.
He split the length of the square as shown below.
7 5
8
which is the largest rectangle ?
What are their length and breadth ?
What about the other one ?

3. Is there any change in its breadth ?
The area of the first rectangle is 7 x 8.
What about the second one ?
It is 5 x 8.
We can calculate the area of the original rectangle as
(7 +5 ) x 8
Now , suppose in the original rectangle, we split the breadth.
12
6
2
Now if we calculate the area of the original rectangle, it is
12 x (6 + 2)
Suppose we split the length and breadth simultaneously.
6
2
7 5
Then what is the area of the largest rectangle ?

4. +
It is ( 7 +5 ) x ( 6 + 2 )
Consider all the small rectangles inside the rectangle.
Let us compute all the area.
Area of the first rectangle is ( 7 x 6 )
What about the other three rectangles ?
So the area of the original rectangle is
( 7 + 5 ) x (6 + 2 )
Which is same as
( 7 x 6 ) + (7 x 2 ) + ( 5 x 6 ) + (5 x 2 )
Thus ,
( 7 + 5 ) x (6 + 2 ) = ( 7 x 6 ) + ( 7 x 2 ) + (5 x 6 ) x (5 x 2 )
What did we do here ?
To get 12 x 8
• 12 x 8 is split as ( 7 + 5 ) x ( 6 + 2 )
• 6 and 2 are multiplied by 7
• 6 and 2 are multiplied by 3
• All these products are added
Let us see how we can get 18 x 22 from 17 x 20
(18 x 22 ) = (17 + 1 ) x (20 + 2 )
= (17 x 20 ) + (17 x 2 ) + (1 x 20 ) + (1 x 2 )
Identities
The equation
2x+3 = 3x+2 is
true only for
x=1.what about
the equation
x+(x+1) = 2x+1.
It is true whatever
number we take as
x. such equations
which are true for all
numbers are called
identities.
In both problems, we multiplied
a sum by another sum. What is
the general method to do this ?
To multiply a sum of
positive numbers by a sum
of positive numbers,
multiply each number in the
second sum by each number
in the first sum and add.

5. Let us do 32 x 44 using this.
32 x 44 = ( 30 +2 ) ( 40+ 4 )
= ( 30 x 40 ) + ( 30 x 4 ) + ( 2 x 40 ) + (2 x 4 )
= 1200 + 120 + 80 + 8
= 1408
What is about 1004 x 301 ?
1004 x 301 = (1000 + 4 ) (300 + 1 )
= 300000 + 1000 + 1200 + 4
= 302204
Using algebra we can write,
( x + y ) ( u + v ) = xu + xv + yu + yv ,for any four positive
numbers x, y, u, v.
If we consider ( y + 1 ) ( x + 1 ) , what is its product ?
Can you state this as a general principle in ordinary
language ?
Can we do some multiplication in head using this ?
What if we take 2 instead of 1 in this ?

6. :
1. 6½ x 8⅓
2. Write numbers like this
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
i. Mark four numbers in a square and find the difference of
diagonal products. Is it same for all squares of four
numbers ?
ii. Explain why this is so ?
iii. Instead of a square of four numbers, take a square of nine
numbers and mark only the numbers at the four corners.
what is the difference of diagonal products ? Explain using
algebra.
8 9 10
13 14 15
18 19 20

7. 3.
3. Look at this :
1 x 4 = ( 2 x 3 ) – 2
2 x 5 = ( 3 x 4 ) – 2
3 x 6 = ( 4 x 5 ) – 2
4 x 7 = ( 5 x 6 ) – 2
i. Write the next two lines in the pattern.
ii. If we take four consecutive natural numbers, what is the
relation between the products of the first and the last, and
the product of the middle two ?
iii. Write this as a general principle in algebra and explain it ?
Square of a sum
we have seen in class 7, that
the square of a product is
the product of squares.
Is the square of a sum equal
to the sum of squares ?

8. What is 71² ?
To see it, let us split 71².
71² = 71 x 71
= (70 + 1 ) (70 + 1 )
We can write this as the sum of four products.
( 70 + 1 ) ( 70 + 1 ) = ( 70 x 70 ) + ( 70 x 1 ) + (1 x 70 ) + ( 1 x 1 )
= 4900 + 70 + 70 + 1
= 5041
We can split any square like this.
How do we write this in algebra ?
To get ( x + 1 )² from x², we must add x and the next number x + 1
to x².
That is,
( x + 1 )² = ( x + 1 ) ( x + 1 )
= x.x + x.1 + 1.x + 1.1
= x² + x + ( x + 1 )
We know that x + ( x + 1 ) = 2x + 1. so,
( x + 1 )² = x² + 2x + 1.
Using this calculate 51² ?
Now suppose we want to compute 65².
If we try to do this by writing it as ( 64 + 1 )², we could have to
compute 64².
Suppose we write it as ² = ( 60 + 5 )², how we get 65² ?
65² = (60 + 5) ( 60 + 5 )
= 60² + (60 x 5 ) + ( 5 x 60 ) + 5²
= 3600 + 300 + 300 + 25
= 4225
How about 203² ?
Let us generalize this idea as :
The square of sum of two positive numbers is sum of the squares
of the two numbers and twice their product.

9. We can write it in algebra like this :
( x + y )² = x² + y² + 2xy, for any two positive numbers x, y.
In another way ,
Consider a square of length ( x + y ). We can split this x + y as x
and y.
So we obtain four rectangles.
As we have seen in the beginning of
this chapter, area of the square of side
length ( x + y ) is equal to the sum of
Areas of four rectangles.
Thus,
( x + y )² = x² + xy + xy + y²
That is, ( x + y )² = x² + y² + 2xy.
x
y
x y
x² xy
xy y²
1. Compute the squares of these numbers.
( 10½)² , 45² , 20.4
2. Is there a general method to compute the squares of numbers
like 1½, 2½, 3½, …? Explain it using algebra.
3. Given below is a method to calculate 37².
3² = 9 9 x 100 900
2 x ( 3 x 7 ) = 42 42 x 10 420
7² 49
37² 1369
i. check this for some more two digit numbers.
ii. Explain why this is correct, using algebra.
iii. Find an easy method to compute squares of number
ending in 5.

10. Look at this :
1 x 3 = 3 = 2² - 1
2 x 4 = 8 = 3² - 1
3 x 5 = 15 = 4² - 1
Does the pattern continue like this ?
Product of differences
We have seen how some products can be split into sums.
For example,
204 x 103 = ( 200 + 4 ) ( 100 + 3 )
= 20000 + 600 + 400 + 12
= 21012
Now, suppose we want to calculate 196 x 97 ?
We can split it as
196 x 97 = ( 200 – 4 ) ( 100 – 3 )
How do we split this as before, into four products ?
First we write
196 x 103 = (200 – 4 ) x 97
= (200 x 97 ) – ( 4 x 97 )
Now let us write 97 = 100 – 3, and split each product.
200 x 97 = 200 x ( 100 – 3 ) = 20000 – 600 = 19400
4 x 97 = 4 x ( 100 – 3 ) = 400 – 12 = 388
Putting all these together gives,

11. 5-
196 x 97 = ( 200 – 4 ) x 97
= ( 200 x 97 ) – ( 4 x 97 )
= 19400 – 388
= 19012
Let us try multiplication of 298 x 195 like this :
298 x 195 = ( 300 – 2 ) x 195
= ( 300 x 195 ) – ( 2 x 195 )
300 x 195 = 300 x ( 200 – 5 )
= ( 300 x 200 ) – ( 300 x 5 )
= 60000 – 1500
= 58500
2 x 195 = 2 x ( 200 – 5 )
= ( 2 x 200 ) – ( 2 x 5 )
= 400 – 10
= 390
What do we get on putting these together ?
298 x 195 = 58500 – 390
= 58110
It is not easy to write the general principle
in ordinary language. In algebra :
Cutting down
Suppose we shorten by
2 centimeters, the
length of a rectangle of
length 10 centimeters
and breadth 6
centimeters.
6
10
6
8 2
By how much the area
decreased ? What is the
calculation done here ?
( 10 – 2 ) x 6
= ( 10 x 6 ) ( 2 x 6 )
( x – y ) ( u – v ) = xu – xv – yu + yv
for all positive numbers x, y, u, v with
x > y and u > v
To find ( x – y )², we put x and y
respectively instead of u and v in the
above principle.
( x – y )² = ( x – y ) ( x - y )
= ( x . x ) – ( x . y ) – ( y . x )
+ ( y . y )
= x² - xy – yx + y²
= x² - xy – xy + y²
xy + xy = 2xy. So,
( x – y )² = x² - 2xy + y² .

12. Let us write it as a general principle.
( x – y )² = x² - 2xy + y²
For all positive numbers x, y with x > y.
Geometrical explanation
Look at this picture of reducing a rectangle by shortening both
sides.
Look at this pictures :
If we subtract the areas of both the rectangle on the top and
on the right, the rectangle at the top corner would be
subtracted twice.
To compensate, we have to add this rectangle once.
( x – y ) ( u – v ) = xu – xv –yu + yv
v
u-v
x-y y
(x-y)(u-v)
v
u-v
x-y y
xv
(x-y)(u-v)
v
u-v
x-y y
(x-y)(u-v) uy
v
u-v
x-y y
(x-y)(u-v)
y
u
yv

13. This we can say in ordinary language also :
The square of the difference of two positive numbers is twice their
product subtracted from the sum of their squares.
For example,
199² = ( 200 – 1 )² = 200² - ( 2 x 100 x 1 ) + 1²
= 40000 – 400 + 1 = 39601
Look at this pattern :
2 ( 2² + 1² ) = 10 = 3² + 1²
2 ( 3² + 2² ) = 26 = 5² + 1²
2 ( 5² + 1² ) = 52 = 6² + 4²
2 ( 4² + 6² ) = 104 = 10² + 2²
Take some pairs of natural numbers and calculate the sum of
the squares ; can you write twice this sum again as a sum of a
pair of perfect squares ?
What is the relation between the starting pair and final pair ?
Find the sum and difference of the starting pair
What is the reason for this ?

14. 1. Compute the squares of these numbers.
89, 48, 5⅓, 8.25, 7⅔
2. Consider the following :
8 = 4 x 2 x 1 = 3² - 1²
12 = 4 x 3 x 1 = 4² - 2²
16 = 4 x 4 x 1 = 5² - 3²
20 = 4 x 5 x 1 = 6² - 4²
Can we written this as a general principle using algebra ?
3. Look at this pattern :
( ½ )² + ( 1 ½ )² = 2 ½ 2 = 2 x 1²
( 1 ½ )² + ( 2 ½ )² = 8 ½ 8 = 2 x 2²
( 2 ½ )² + ( 3 ½ )² = 18 ½ 18 = 2 x 3²
Explain the general principle using algebra.
Sum and difference
We have calculated products by splitting numbers as sums or
differences.
For example :
304 x 205 = ( 300 + 4 ) ( 200 + 5 )
198 x 195 = ( 200 – 2 ) ( 200 – 5 )
In this way how we compute 304 x 198 ?
304 x 198 = ( 300 + 4 ) ( 200 – 2 )
To compute this , first we split 304 alone, as before :
304 x 198 = ( 300 + 4 ) x 198 = ( 300 x 198 ) + ( 4 x 198 )
Now we split 198 and compute the two products separately.
300 x 198 = 300 x ( 200 – 2 ) = 60000 – 600 = 59400
4 x 198 = 4 x ( 200 – 2 ) = 800 – 8 = 792
Putting these together gives,
304 x 198 = 59400 + 792 = 60192
To understand the general principle, let us write
304 x 198 = 60000 – 600 + 800 – 8

15. That as
( 300 + 4 ) ( 200 – 2 ) = ( 300 x 200 ) – ( 300 x 2 ) + ( 4 x 200 ) –
( 4 x 2 )
Like this find out 197 x 103 ?
We can write the algebraic form of this as,
( x + y ) ( u – v ) = xu – xv + yu – yv for all positive numbers x, y,
u, v with u > v
If three numbers are such that the sum of the squares of two of
them is equals to the square of the third, we call them a pythagoras
triple.
For example,
3, 4, 5 form a pythagoras triple
because
3² + 4² = 5².
There is a method to find all such pythagorean triples.
Take any two natural numbers m, n and compute x, y, z as follows :
x = m² - n²
y = 2mn
z = m² + n²
It can be seen that
x² + y² = z²
Greek mathematicians knew this technique as early as 300 BC.

16. We can find a general method to compute the product of the sum and
difference of two numbers.
( x + y ) ( x – y ) = ( x . x ) – ( x. y ) + ( y . x ) – ( y . y )
= x² - xy + yx – y²
= x² - y²
Hence the product of the sum and difference of two positive
numbers is the difference of their squares.
( x + y ) ( x – y ) = x² - y²
for all positive numbers x, y with x > y
For example,
204 x 196 = ( 200 + 4 ) ( 200 – 4 )
= 200² - 4² = 40000 – 16
= 39984
Differenceofsquares x x²
x
x
y² y
x - y
x x²-y²
y
x+y
(x+y)(x-y)
x-y
We can also apply this in reverse :
The difference of the squares of the two
positive numbers is the product of their
sum and difference.
For example,
234² - 230² = ( 234+230 ) ( 234 - 230 )
= 464 x 4 = 1856
We have seen that some natural numbers
can be written as the difference of two
perfect squares. The above principle can
be used to do it.
For example, consider 45. we want to
find numbers x, y such that x² - y² = 45.
This we can write
45 = ( x+ y ) ( x – y )
This means ( x + y ) and ( x – y ) must
be factors of 45.
45 can be written as a product of two
factors in various ways.

17. 45 = 45 x 1
45 = 15 x 3
45 = 9 x 3
Taking the factors 45 and 1, let us write
x + y = 45 and x – y = 1.
So x = 23 and y = 22 ( how ? )
So,
45 = 23² - 22²
Similarly let us take 45 = 15 x 3. Think about x and y.
We can write
45 = 9² - 6²
What about 45 = 9 x 5 ?
Can we write any natural number as the difference of two squares,
using this method ?
For example, let us take 10. we have 10 = 10 x 1.
By solving we get
10 = ( 5 ½ )² - ( 4 ½ )²
But these are not squares of natural numbers ; that is, not perfect
squares.
How about using 10 = 5 x 2 ?
What kind of natural
numbers can not be
written as the
difference of two
perfect squares ?

18. Sometimes, writing a product as the difference of two squares
makes the computation easier.
For example, consider 26.5 x 23.5. can we write it as the difference
of two squares ?
We need only find two numbers whose sum is 26.5 and product is
23.5, right ?
And for that we need only find half the sum and half the difference
of 26.5 and 23.5. that is 25 and 1.5, so,
26.5 = 25 + 1.5 23.5 = 25 – 1.5
Using this,
26.5 x 23.5 = ( 25 + 1.5 ) ( 25 – 1.5 )
= 25² - 1.5² = 625 – 2.25 = 622.75
1. Compute the following in head
i. 68² - 32²
ii. 3.6² - 1.4²
iii. ( 3 ½ )² - ( 2 ½ )²
iv. 10.7 x 9.3
v. 198 x 202
vi. 3 ⅔ x 1 ⅓
2. Find out the larger product of each pair below, without actual
multiplication.
i. 25 x 75, 26 x 74
ii. 76 x 24 83 x 17
iii. 20.4 x 19.6 10.5 x0.5
iv. 11.3 x9.7 10.7 x 19.3
v. 126 x 74 124 x 76

19. 3. Mark four numbers forming a square in a calender :
6 7
13 14
Add the squares of the diagonal pair and find the difference of these
sums :
6² + 14² = 232 13² + 7² = 218 222 – 218 = 14
4. Look at this pattern :
( 1 ½ )² - ( ½ )² = 2
( 2 ½ )² - ( 1 ½ )² = 4
( 3 ½ )² - ( 2 ½ )² = 6
Explain the general principle using
algebra.
5. Take nine numbers forming a
Square in a calendar and mark the
four numbers at the corners.
Multiply the diagonal pairs and find the difference of these products.
3 x 19 = 57 17 x 5 = 85 85 – 57 = 28
i. Do this for other four numbers.
ii. Explain using algebra, why the difference
is 14 always.
?Take some pairs of
numbers with the same
sum and find their
products. How does the
product change with the
difference of the number?
What is an easy method
to find the largest
product ?
3 4 5
10 11 12
17 18 19
i. Do this for other such squares
ii. Explain using algebra, why the difference is always
28 ( it is convenient to take the number at the center
as x ).

20. Lookingback
Learning outcomes
What I
can
With
teacher’s
help
Must
improve
• explaining the method of
multiplying a sum by a sum.
• explaining the square of a
sum geometrically and
algebraically.
• explaining the square of a
difference geometrically and
algebraically.
• explaining the characteristics
of natural numbers which can
be expressed as the differences
of perfect squares.
• finding out the pair with the
largest product from pairs of
numbers with the same sum.
• express number relations in
general using algebra.