Math Lab Activity

– Verify the Identity (a + b)² = (a² + 2ab + b²)

Objective

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

Materials Required

1. A piece of cardboard
2. A sheet of glazed paper
3. A sheet of white paper
4. A pair of scissors
5. A geometry box

Procedure
We take distinct values of a and b.

Step 1: On the glazed paper construct a square of side ‘a’ units. Construct two rectangles, each having length ‘a’ units and breadth ‘b’ units. Construct a square of side ‘b’ units.

Step 2: Paste the sheet of white paper on the cardboard. Draw a square ABCD having each side (a + b) units.

Step 3: Cut the two squares and the two rectangles from the glazed paper and paste them on the white paper. Arrange these inside the square ABCD as shown in Figure 10.1.

Step 4: Record your observations.

Observations and Calculations

1. Area of the square ABCD drawn on the white paper = (a + b)² square units.
2. Area of the square having each side a units (drawn on the glazed paper) = a² square units.
   Area of the square having each side b units (drawn on the glazed paper) = b² square units.
   Area of each rectangle (drawn on the glazed paper) = (ab) square units.
   ∴ total area of the four quadrilaterals (drawn on the glazed paper)
   =(a² +b² + ab + ab) square units = (a² + 2ab + b²) square units.
   Now, the area of the square ABCD = sum of the areas of the four quadrilaterals.
   ∴ we have, (a + b)² = (a² + 2ab + b²).

Result
The identify (a + b)² = (a² + 2ab + b²) is verified.

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– Verify the Algebraic Identity (a – b)² = a² – 2ab + b²

Objective

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

Materials Required

1. Drawing sheet
2. Pencil
3. Coloured papers
4. Scissors
5. Ruler
6. Adhesive

Prerequisite Knowledge

1. Square and its area.
2. Rectangle and its area.

Theory

1. For square and its area refer to Activity 3.
2. For rectangle and its area refer to Activity 3.

Procedure

1. From a coloured paper, cut a square PQRS of side a units, (see Fig. 4.1)
   
2. Further, cut out another square TQWX of side b units such that b < a. (see Fig. 4.2)
   
3. Now, cut out a rectangle USRV of length a units and breadth b units from another coloured paper, (see Fig. 4.3)
   
4. Now further, cut out another rectangle ZVWX of length a units and breadth b units, (see Fig. 4.4)
   
5. Now, arrange figures 4.1, 4.2, 4.3 and 4.4, according to their vertices and paste it on a drawing sheet, (see Fig. 4.5)
   

Demonstration
From the figures 4.1,4.2, 4.3 and 4.4, we have Area of square PQRS = a²
Area of square TQWX = b²
Area of rectangle USRV = ab and Area of rectangle ZVWX – ab
Area of square PUZT = Area of square PQRS + Area of square TQWX – Area of rectangle ZVWX – Area of rectangle USRV
= a² + b² – ba-ab
= (a² -2ab + b²) …(i)
Also, from Fig. 4.5, it is clear that PUZT is a square whose each side is (a – b).
Area of square PUZT = (Side)²
= [(a-b)]² =(a-b)² …(ii)
From Eqs. (i) and (ii), we get (a – b)² = (a² – 2ab + b²)
Here, area is in square units.

Observation
On actual measurement, we get
a =  ………… ,
b=  ………… ,
(a-b) =  ………… ,
a² = ………… ,
b² =  ………… ,
(a² – b²) = ………… ,
ab =  ………… ,
and 2ab =  ………… ,
Hence, (a – b)² = a² – 2ab + b²

Result
Algebraic identity (a – b)² = a² – 2ab + b² has been verified.

Application
The identity (a – b)² = a² -2ab + b² may be used for

1. calculating the square of a number which can be expressed as a difference of two convenient numbers.
2. simplification and factorization of algebraic expressions.

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– Verify the Algebraic Identity a² – b² = (a + b) (a – b)

Objective

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

Materials Required

1. Drawing sheet
2. Pencil
3. Colored papers
4. Scissors
5. Sketch pen
6. Ruler
7. Adhesive

Prerequisite Knowledge

1. Square and its area.
2. Rectangle and its area.
3. Trapezium.

Theory

1. For square and its area refer to Activity 3.
2. For rectangle and its area refer to Activity 3.
3. Trapezium is a quadrilateral whose two sides are parallel and two sides are non-parallel. In the trapezium ABCD, sides AB and CD are parallel while sides AD and BC are non-parallel.
4. Area of trapezium =½ (Sum of parallel sides x Distance between parallel sides)
   = ½ (AB + CD) x DE

Procedure

1. Cut out a square WQRS of side a units from a coloured paper, (see Fig. 5.2)
   
2. Cut out a square WXYZ of side b units (b < a) from another coloured paper, (see Fig. 5.3)
   
3. Paste the smaller square WXYZ on the bigger square WQRS as shown in Fig. 5.4.
   
4. Join the points Y and R using sketch pen. (see Fig. 5.4)
5. Cut out the trapeziums XQRY and ZYRS from WQRS (see Fig. 5.5 and 5.6).
   
6. 
7. Paste both trapeziums obtained in step 5th on the drawing sheet as shown in Fig. 5.7
   

Demonstration
From Fig. 5.2 and Fig. 5.3, we have Area of square WQRS = a²
Area of square WXYZ = b² Now, from Fig. 5.4, we have
Area of square WQRS – Area of square WXYZ = Area of trapezium XQRY + Area of trapezium ZYRS
=Area of rectangle XQZS [from Fig. 5.7]
= XS . SZ [∴ Area of rectangle = Length x Breadth]
So, a² – b² = (a + b) (a – b)
Here, area is in square units.

Observation
On actual measurement, we get
a =…….. ,  b = …….. ,
So, a² =…….. ,
b² = …….. ,
a + b = …….. ,
a-b = …….. ,
a² -b² = …….. ,
and (a + b)(a-b) = …….. ,
Hence, a² – b² = (a + b) (a – b)

Result
The algebraic identity a² – b² = (a + b) (a – b) has been verified.

Application
The identity (a² – b²) = (a + b)(a-b) can be used for

1. calculating the difference of squares of two numbers.
2. getting some products involving two numbers.
3. simplification and factorization of algebraic expressions.