Sunday, July 16, 2023

Class 09 To verify the algebraic identity :(a – b)2 = a2 – 2ab + b2

Activity 4

OBJECTIVE

To verify the algebraic identity :(a – b)² = a² – 2ab + b²

MATERIAL REQUIRED

Drawing sheets, cardboard, coloured papers, scissors, ruler and adhesive.

METHOD OF CONSTRUCTION

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

2. Cut out a square EBHI of side b units (b < a) from a drawing sheet/cardboard [see Fig. 2].

3. Cut out a rectangle GDCJ of length a units and breadth b units from a drawing sheet/cardboard [see Fig. 3].

4. Cut out a rectangle IFJH of length a units and breadth b units from a drawing sheet/cardboard [see Fig. 4].

5. Arrange these cut outs as shown in Fig. 5.

DEMONSTRATION

According to figure 1, 2, 3, and 4, Area of square ABCD = a², Area of square EBHI = b²

Area of rectangle GDCJ = ab, Area of rectangle IFJH = ab

From Fig. 5, area of square AGFE = AG × GF = (a – b) (a – b) = (a – b)²

Now, area of square AGFE = Area of square ABCD + Area of square EBHI

– Area of rectangle IFJH – Area of rectangle		Fig. 5
– Area of rectangle IFJH – Area of rectangle
GDCJ
= a²	+ b² – ab – ab
= a²	– 2ab + b²

Here, area is in square units.

OBSERVATION

On actual measurement:

a = ..............	,	b = ..............		, (a – b) = ..............	,
So, a² = ..............	,	b² =	..............	, (a – b)² = ..............	,
ab = ..............	,	2ab	= ..............

Therefore, (a – b)² = a² – 2ab + b²

APPLICATION

The identity may be used for

1. calculating the square of a number expressed as a difference of two convenient numbers.

2. simplifying/factorisation of some algebraic expressions.

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

Activity 3

OBJECTIVE

To verify the algebraic identity : (a + b)² = a² + 2ab + b²

MATERIAL REQUIRED

Drawing sheet, cardboard, cello-tape, coloured papers, cutter and ruler.

METHOD OF CONSTRUCTION

1. Cut out a square of side length a units from a drawing sheet/cardboard and name it as square ABCD [see Fig. 1].

2. Cut out another square of length b units from a drawing sheet/cardboard and name it as square CHGF [see Fig. 2].

Fig. 1 Fig. 2

3. Cut out a rectangle of length a units and breadth b units from a drawing sheet/cardbaord and name it as a rectangle DCFE [see Fig. 3].

Cut out another rectangle of length b units and breadth a units from a drawing sheet/cardboard and name it as a rectangle BIHC [see Fig. 4].

5. Total area of these four cut-out figures

= Area of square ABCD + Area of square CHGF + Area of rectangle DCFE

+ Area of rectangle BIHC

= a² + b² + ab + ba = a² + b² + 2ab.

Join the four quadrilaterals using cello-tape as shown in Fig. 5.

Clearly, AIGE is a square of side (a + b). Therefore, its area is (a + b)². The combined area of the constituent units = a² + b² + ab + ab = a² + b² + 2ab.

Hence, the algebraic identity (a + b)² = a² + 2ab + b² Here, area is in square units.

OBSERVATION

On actual measurement:

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

So, a² = .............. b² = .............., ab = ..............

(a+b)² = .............., 2ab = ..............

Therefore, (a+b)² = a² + 2ab + b² .

The identity may be verified by taking different values of a and b.

APPLICATION

The identity may be used for

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

2. simplifications/factorisation of some algebraic expressions.

Class 09 To represent some irrational numbers on the number line.

Activity 2

OBJECTIVE

To represent some irrational numbers on the number line.

MATERIAL REQUIRED

Two cuboidal wooden strips, thread, nails, hammer, two photo copies of a scale, a screw with nut, glue, cutter.

METHOD OF CONSTRUCTION

Two cuboidal wooden strips, thread, nails, hammer, two photo copies of a scale, a screw with nut, glue, cutter.

1. Make a straight slit on the top of one of the wooden strips. Fix another wooden strip on the slit perpendicular to the former strip with a screw at the bottom so that it can move freely along the slit [see Fig.1].

2. Paste one photocopy of the scale on each of these two strips as shown in Fig. 1.

3. Fix nails at a distance of 1 unit each, starting from 0, on both the strips as shown in the figure.

4. Tie a thread at the nail at 0 on the horizontal strip.

DEMONSTRATION

11. Take 1 unit on the horizontal scale and fix the perpendicular wooden strip at 1 by the screw at the bottom.

2. Tie the other end of the thread to unit ‘1’ on the perpendicular strip.

the horizontal strip to represent			_√2on the horizontal strip [see Fig. 1].
Similarly, to represent		_√3, fix the perpendicular wooden strip at √₂ and
repeat the process as above. To represent				_√a, a > 1, fix the perpendicular
scale at	_√a_{– 1}and proceed as above to get				√a ^.
OBSERVATION
On actual measurement:
a – 1 = ...........			_√a= ...........

APPLICATION

The activity may help in representing some irrational numbers such as √2, √3, √4, √5, √6,√7.