Class 09 To verify the algebraic identity :(a+b)3 = a3 + b3 + 3a2b + 3ab2

 Activity 7

OBJECTIVE                

                                            

To verify the algebraic identity :(a+b)³ = a³ + b³ + 3a²b + 3ab²

 MATERIAL REQUIRED

Acrylic sheet, coloured papers, glazed papers, saw, sketch pen, adhesive, Cello-tape.

METHOD OF CONSTRUCTION

1.   Make a cube of side a units and one more cube of side b units (b < a), using acrylic sheet and cello-tape/adhesive [see Fig. 1 and Fig. 2].

 Similarly, make three cuboids of dimensions a×a×b and three cuboids of dimensions a×b×b [see Fig. 3 and Fig. 4].

3. Arrange the cubes and cuboids as shown in Fig. 5.

DEMONSTRATION

 Volume of the cube of side a = a×a×a = a³, volume of the cube of side b = b³ Volume of the cuboid of dimensions a×a×b = a²b, volume of three such cuboids

=   3a²b

 Volume of the cuboid of dimensions a×b×b = ab², volume of three such cuboids

=   3ab²

 Solid figure obtained in Fig. 5 is a cube of side (a + b)

 Its volume = (a + b)³

 Therefore, (a+b)³ = a³ + b³ + 3a²b + 3ab²

 Here, volume is in cubic units.

 OBSERVATION

 On actual measurement:

 a = ..............,         b = ............., a³  = ..............,

 So, a³ = ..............,       b³  = ............., a²b = ..............,            3a²b= ..............,

 ab² = ..............,   3ab² = ..............,          (a+b)³ = ..............,

 Therefore, (a+b)³ = a³ + b³ +3a²b + 3ab²

 APPLICATION

 The identity may be used for

 1.   calculating cube of a number expressed as the sum of two convenient numbers

 simplification and factorisation of algebraic expressions

Class 09 To verify the algebraic identity :(a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

 

Activity 6 

OBJECTIVE	MATERIAL REQUIRED
To verify the algebraic identity :	Hardboard, adhesive, coloured
(a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca	papers, white paper.
METHOD OF CONSTRUCTION

 1.   Take a hardboard of a convenient size and paste a white paper on it.

 2.   Cut out a square of side a units from a coloured paper [see Fig. 1].

 3.   Cut out a square of side b units from a coloured paper [see Fig. 2].

 4.   Cut out a square of side c units from a coloured paper [see Fig. 3].

 5.   Cut out two rectangles of dimensions a× b, two rectangles of dimensions b × c and two rectangles of dimensions c × a square units from a coloured paper [see Fig. 4].

6.   Arrange the squares and rectangles on the hardboard as shown in Fig. 5.

DEMONSTRATION

From the arrangement of squares and

 rectangles in Fig. 5, a square ABCD is

 obtained whose side is (a+b+c) units.

 Area of square ABCD = (a+b+c)² . 

Therefore, (a+b+c)2  = sum of all the
squares and rectangles shown in Fig. 1 to
Fig. 4.	Fig. 5

 =   a² + ab + ac + ab + b² + bc + ac + bc + c²

 =  a² + b² + c² + 2ab + 2bc + 2ca

 Here, area is in square units.

 OBSERVATION

 On actual measurement:

a = ..............	,	b = ..............	, c = ..............	,
So, a2 = ..............	,	b2 = ..............	, c2= ..............	, ab=	..............	,
bc= ..............	,	ca = ..............	,2ab = ..............	,	2bc =	..............,
2ca= ..............	,	a+b+c = ..............	,	(a+b+c)2 =	..............	,

 Therefore, (a+b+c)² = a² + b² +c² +2ab + 2bc + 2ca

 APPLICATION

 The identity may be used for

 1.   simiplification/factorisation of algebraic expressions

 calculating the square of a number expressed as a sum of three convenient numbers.

Class 09 To verify the algebraic identity :a2 – b2 = (a + b)(a – b)

 Activity 5

 OBJECTIVE                                                                    

To verify the algebraic identity :a² – b² = (a + b)(a – b)

METHOD OF CONSTRUCTION

 MATERIAL REQUIRED

Drawing sheets, cardboard, coloured papers, scissors, sketch pen, ruler, transparent sheet and adhesive.

1.   Take a cardboard of a convenient size and paste a coloured paper on it.

2.    Cut out one square ABCD of side a units from a drawing sheet [see Fig. 1].

3.3.Cut out one square AEFG of side b units (b < a) from another drawing sheet [see Fig. 2].

4.   Arrange these squares as shown in Fig. 3.

 5.   Join F to C using sketch pen. Cut out trapeziums congruent to EBCF and GFCD using a transparent sheet and name them as EBCF and GFCD, respectively [see Fig. 4 and Fig. 5].

6. Arrange these trapeziums as shown in

Fig. 6.

 DEMONSTRATION

 Area of square ABCD = a²

 Area of square AEFG = b²

 In Fig. 3,

 Area of square ABCD – Area of square

 AEFG

 = Area of trapezium EBCF + Area of

 trapezium GFCD

 =  Area of rectangle EBGD [Fig. 6].

 =  ED×DG

 Thus, a² – b² = (a+b) (a–b)Fig. 6

Here, area is in square units.

OBSERVATION

On actual measurement:

 a = ..............,         b = .............., (a+b) = ..............,

 So, a2 = ..............,       b2 = .............., (a–b) = ..............,

 a2–b2 = .............., (a+b) (a–b) = ..............,

 Therefore, a²–b² = (a+b) (a–b)

 APPLICATION

 The identity may be used for

 1.   difference of two squares

 2.   some products involving two numbers

 3.   simplification and factorisation of algebraic expressions.