Activity 15

OBJECTIVE MATERIAL REQUIRED

To construct a Pascal's Triangle and to Drawing board, white paper,
write binomial expansion for a given matchsticks, adhesive.
positive integral exponent.

METHOD OF CONSTRUCTION
1. Take a drawing board and paste a white paper on it.
2. Take some matchsticks and arrange them as shown in Fig.15.

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 3. Write the numbers as follows:
1 (first row)
1 1 (second row)
1 2 1 (third row)
1 3 3 1 (fourth row), 1 4 6 4 1 (fifth row) and so on (see Fig. 15).
4. To write binomial expansion of (a + b)n, use the numbers given in the
(n + 1)th row.

DEMONSTRATION
1. The above figure looks like a triangle and is referred to as Pascal’s Triangle.
2. Numbers in the second row give the coefficients of the terms of the binomial
expansion of (a + b)1. Numbers in the third row give the coefficients of the
terms of the binomial expansion of (a + b)2, numbers in the fourth row give
coefficients of the terms of binomial expansion of (a + b)3. Numbers in the
fifth row give coefficients of the terms of binomial expansion of
(a + b)4 and so on.

OBSERVATION
1. Numbers in the fifth row are ___________, which are coefficients of the
binomial expansion of __________.
2. Numbers in the seventh row are _____________, which are coefficients
of the binomial expansion of _______.
3. (a + b)3 = ___ a3 + ___a2b + ___ab2 + ___b3
4. (a + b)5 = ___ +___+ ___+ ___ + ___+ ___.
5. (a + b)6 =___a6 +___a5b + ___a4b2 + ___a3b3 + ___a2b4 + ___ab5 + ___b6.
6. (a + b)8 = ___ +___ +___+ ___ + ___+ ___ + ___ + ___+ ___.
7. (a + b)10 =___ + ___ + ___+ ___ + ___+ ___ + ___+___+ ___+ ___+ __.

APPLICATION
The activity can be used to write binomial expansion for (a + b)n, where n is a
positive integer.

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Activity 16
OBJECTIVE MATERIAL REQUIRED
To obtain formula for the sum of Wooden/plastic unit cubes,
squares of first n-natural numbers. coloured papers, adhesive and nails.

METHOD OF CONSTRUCTION
1. Take 1 ( = 12) wooden/plastic unit cube Fig.16.1.
2. Take 4 ( = 22) wooden/plastic unit cubes and form a cuboid as shown in
Fig.16.2.
3. Take 9 ( = 32) wooden/plastic unit cubes and form a cuboid as shown in
Fig.16.3.
4. Take 16 (= 42) wooden/plastic unit cubes and form a cuboid as shown in
Fig. 16.4 and so on.
5. Arrange all the cube and cuboids of Fig. 16.1 to 16.4 above so as to form an
echelon type structure as shown in Fig.16.5.
6. Make six such echelon type structures, one is already shown in Fig. 16.5.
7. Arrange these five structures to form a bigger cuboidal block as shown in
Fig. 16.6.

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DEMONSTRATION
1. Volume of the structure as given in Fig. 16.5

= (1 + 4 + 9 + 16) cubic units = (12 + 22 + 32 + 42) cubic units.

2. Volume of 6 such structures = 6 (12 + 22 + 32 + 42) cubic units.
3. Volume of the cuboidal block formed in Fig. 16.6 (which is cuboid of
dimensions = 4 × 5 × 9) = 4 × (4 + 1) × (2 × 4 + 1).
4. Thus, 6 (12 + 22 + 32 + 42) = 4 × (4 + 1) × (2 × 4 + 1)
1
i.e., 12 + 22 + 32 + 4 2 = [4 × (4 + 1) × (2 × 4 +1)]
6
OBSERVATION
1
1. 12 + 22 + 32 + 42 = ( _____ ) × ( _____ ) × ( _____ ).
6
1
2. 12 + 22 + 32 + 42 + 52 = ( _____ ) × ( _____ ) × ( _____ ).
6
1
3. 12 + 22 + 32 + 42 + ... + 102 = ( _____ ) × ( _____ ) × ( _____ ).
6

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 1
4. 12 + 22 + 32 + 42 ... + 252 = ( _____ ) × ( _____ ) × ( _____ ).
6

1
5. 12 + 22 + 32 + 42 ... + 1002 = ( _____ ) × ( _____ ) × ( _____ ).
6

APPLICATION
This activity may be used to obtain the sum of squares of first n natural numbers
1
as12 + 22 + 32 + ... + n2 = n (n + 1) (2n + 1).
6

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Activity 17
OBJECTIVE MATERIAL REQUIRED
An alternative approach to obtain Wooden/plastic unit squares,
formula for the sum of squares of first coloured pencils/sketch pens,
n natural numbers. scale.

METHOD OF CONSTRUCTION
1. Take unit squares, 1, 4, 9, 16, 25 ... as shown in Fig. 17.1 and colour all of
them with (say) Black colour.

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2. Take another set of unit squares 1, 4, 9, 16, 25 ... as shown in Fig. 17.2 and
colour all of them with (say) green colour.
3. Take a third set of unit squares 1, 4, 9, 16, 25 ... as shown in Fig. 17.3 and
colour unit squares with different colours.
4. Arrange these three set of unit squares as a rectangle as shown in Fig. 17.4.

DEMONSTRATION
1. Area of one set as given in Fig. 17.1

= (1 + 4 + 9 + 16 + 25) sq. units

= (12 + 22 + 32 + 42 + 52) sq. units.

2. Area of three such sets = 3 (12 + 22 + 32 + 42 + 52)

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  5× 6 
Area of rectangle = 11 × 15 = [2 (5) + 1] 
 2 
3.

1
∴ 3 (12 + 22 + 32 + 42 + 52) = [5 × 6] [2 (5) + 1]
2

1
or 12 + 22 + 32 + 42 + 52 = [5 × (5 + 1)] [2 (5) + 1].
6

OBSERVATION

1
3 (12 + 22 + 32 + 42 + 52) = ( ____ × ____) ( ___ + 1)
2

1
⇒ 12 + 22 + 32 + 42 + 52 = ( ____ × ____) ( ___ + 1)
6

1
∴ 12 + 22 + 32 + 42 + 52 + 62 + 72 = ( ____ × ____) ( ___ + 1)
6

1
12 + 22 + 32 + 42 +...+ 102 = ( ____ × ____) ( ___ + 1).
6

APPLICATION
This activity may be used to establish
1
12 + 22 + 32 + ... + n2 = n (n + 1) (2n + 1).
6

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Activity 18
OBJECTIVE MATERIAL REQUIRED
To demonstrate that the Arithmetic Coloured chart paper, ruler, scale,
mean of two different positive sketch pens, cutter.
numbers is always greater than the
Geometric mean.
METHOD OF CONSTRUCTION
1. From chart paper, cut off four rectangular pieces of dimension a × b (a > b).
2. Arrange the four rectangular pieces as shown in figure. 18.

DEMONSTRATION
1. ABCD is a square of side (a + b) units.
2. Area ABCD = (a + b)2 sq. units.
3. Area of four rectangular pieces = 4 (ab) = 4ab sq. units.

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