PASCAL TRIANGLE

Pascal's triangle is a geometric arrangement of the binomial cofficients in a triangle.The triangle is bordered by ones on the right and left sides, and each interior entry is the sum of the two entries above.

Construction:-
At the top of Pascal's Triangle it have number 1 which is the zeroth row. Add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place.
First Row 0+1=1,1+0=1
Second Row 0+1=1,1+1=2,1+0=1
Third Row 0+1=1,1+2=3,2+1=3,1+0=1
And so on the rows go infinitely.
Some properties:-
Sums of the numbers in Row
The sum of the numbers in any row is equal to nth power of 2, when n is the number of the row.
For example:

and so on
Prime Numbers:-

The 0th element of every row is 1 and if the first element in any row is prime, then all the numbers in that row excluding 1’s are divisible by it .
For example:
In row5( 1 5 10 10 5 1)5,10 are divisible by 5.

Hockey Stick Pattern:-
If a diagonal of numbers of any length is selected starting at any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself.

For example:

1+6+21+56 = 84

1+7+28+84+210+462+924 = 1716

1+12 = 13
Powers of 11’s:-

SLIDE SHOW - CURVE STITCHING

(Activity 4)






Aim: To find the coordinates of a general point on x-axis and y- axis by an activity method.
Material Required: Graph Paper, Geometry box,Pencil ,Ruler
Procedure:
1) On a graph paper, draw coordinate axes and label the x-axis as X'OX and y-axis as YOY'.
2) Take 8 points on x-axis and write their coordinates.
3) Observe the coordinates and write the coordinate of any general point on x- axis.
4) Similarly take 8 points on the y-axis and repeat the procedure explained above.
5. Write the conclusion.

Observation :

Pt. on x- axis

A( 1,0)
B(2,0)
C(3,0)
D(4,0)
E(-1,0)
F(-2,0)
G(-3,0)
H(-4,0)

Pt. on y- axis

I(0,1)
J(0,2)
K(0,3)
L(0,4)
M(0,-1)
N(0,-2)
Q(0,-3)
P(0,-4)

Result:
1) Any general point on x-axis is of the form (x, 0).
2) Any general point on y-axis is of the form (0, y).

Activity(3)
Aim: Using the unit cubes verify the algebraic identity
(a-b)^3 = a^3 -3a^2b+3ab^2- b^3.
Material required: Unit cubes
Procedure:
1) Let a=3 and b=1.
(a-b)^3 = (2) ^3
To represent (a-b) ^3 make a cube of dimension (a-b) x (a-b) x (a-b) i.e. 2x2x2 cubic units

2) To represent (a) ^3 make a cube of dimension 3x3x3 cubic units



3) To represent 3ab^2 make 3 cuboids of dimension 3x1x1 cubic units.

4) To represent a^3 + 3ab^2, join the cube and the cuboids formed in steps 2 and 3.

5) To represent a^3 + 3ab^2- 3a^2b extract from the shape formed in the previous step 3 cuboids of dimension 3x3x1.

6) To represent a^3 + 3ab^2- 3a^2b-b^3 extract from the shape formed in the previous step 1 cube of dimension 1x1x1.



7) Arrange the unit cubes left to make a cube of dimension2x2x2 cubic units.

Observation:
The number of unit cubes in a^3 = …27…..
The number of unit cubes in 3ab^2 =…9……
The number of unit cubes in 3a^2b=…27…
The number of unit cubes in b^3 =…1…
The number of unit cubes in a^3 - 3a^2b + 3ab^2- b^3 =..8..
The number of unit cubes in (a-b) ^3 =…8...
The number of unit cubes in (a-b) ^3 = number of unit cubes

in a^3 -3a^2b+3ab^2-b^3.

Result:
(a-b)^3 = a^3 -3a^2b+3ab^2- b^3.

ACTIVITY(2)
(Activity 2)
Aim: - To verify the algebraic identity
(a+b)^3=a^3+3a^2b+3ab^2+b^3 using unit cubes.
Material required: - Unit Cubes
Procedure:-
Step1:-Let a=2 units and b=1unit.
Therefore, (a+b) = 3 units(a+b)^3 = 3^3=3 x3x3 cubic units.Make a cube of dimension 3x3x3 cubic units.

Step2:-a=2 unitsTherefore, a^3 =2^3=2x2x2 cubic unitsMake a cube of dimension 2 x2x2 .It represents a^3

Step3:- b=1 unitTherefore, b^3 =1^3=1x1x1 cubic units.

Step4:-3a^2b =3(2x2x1)Make 3 cuboids of this dimension .These 3 pieces will represent 3a^2b.

Step5:-3ab^2 = 3(2x1x1)
Make 3 cuboids of this dimension .These 3 pieces will represent 3ab^2.

Observation: -
1)The number of unit cubes in (a+b) ^3 =…27…
2)The number of unit cubes in a^3 = …8…..
3)The number of unit cubes in 3a^2b =…12……
4)The number of unit cubes in 3ab^2 =……6…
5)The number of unit cubes in b^3 =……1…
6)The number of unit cubes in a^3 + 3a^2b + 3ab^2 + b^3= …27…..

Result:- (a+b)^3=a^3+3a^2b+3ab^2+b^3

ACTIVITY(1)

Activity(1)
Aim: Making square root spiral