Making e Snowflakes

Dear students,
Some of you have choosen Paper folding and cutting as the strategy for making mathematics project. I was exploring some ideas of making paper snowflakes and I noticed the following link :

Making e snowflakes

Use this for making snowflakes and save images and upload them on the KHMS e Mathematics.It is very enjoyable and creative project.

Here is one example

Make this jigsaw

Puzzle for you
Click on the link

Join the pieces and make the above figure

Arithmetic Progression

Consider the following arrangement of numbers 3,6,9,12,.....

These numbers are arranged according to a specific rule. If you observe the rule, you will find that every next number to the previous one is obtained by adding 3(except the first one).

What is a sequence?

A sequence is an arrangement of numbers in specific order like the above one.

Now, what is an arithmetic progression?

A special type of sequence in which every term except the first is obtained by adding a fixed number which may be positive/negative to the preceding term.

A general A.P. (Arithmetic Progression) is given by a , a+d , a+2d , .....a+(n-1)d

where a is the first term , d is the common difference, n is the total number of terms.

General /nth term/last term of an A.P is given by an = a +(n-1) d

Example :Find the common difference and nth term of the given A.P.

-5 , -1, 3 ,7 ......

Here a = -5

d = -1-(-5) = -1+5 = 4

So, an = a+ (n-1)d

= -5+(n-1)4

= -5+4n-4

= 4n-9

Visual representation of sequences....

Let us consider the following sequences

(1) 4,6,8,10,......... Geometrically its representation could be like a ladder in which the height of each step from the first step is same.

(2) 3,6,8,10,.......... Geometrically, its representation is like steps as shown below, but the difference of heights between two consecutive steps ia not always same.

Graphs of cubic polynomials

This is the graph of a cubic polynomial x^3 + 2. Observe carefully, that it cuts the x-axis at 1 point only. So, it has only one zero which is the value of x-coordinate of the point where the curve cuts the x-axis. In the following graph, the curve intersects the x-axis at three points, so the given cubic polynomial has 3 zeroes.

A cubic polynomial has degree 3.It has therefore atmost 3 zeroes.