Introduction: In this following assignment, I will be considering geometric shapes that lead to special numbers. The simplest examples of these are square numbers (1, 4, 9, 16, etc), which are derived from squaring 1, 2, 3, and 4. From this I got the equation y= x2. This equation is illustrated in the table below.

y=x2
|x |y |
|1 |1 |
|2 |4 |
|3 |9 |
|4 |16 |

In the table on the left, I observe that from the y value 1 to the y value 4 there is an increase of 3. From the y values 4 to 9, there is an increase of 5. From the y values 9 to 16, there is an increase of 7. This shows that it goes: +3, +5, +7, which is then increasing by 2 between each of those numbers.

Below, is the graph of y=x2

[pic]

The equation y=x2 comes from the general equation y= ax2-bx+c. Y=x2 is the same as y=x2+0x+0. Therefore, a=1, b=0, and c=0.

The next example I am going to show you is similar to the one above. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1, 3, 6, 10, 15, 21, 28, 36).

In the table on the left, I observe that from the y value 1 to the y value 3 there is an increase of 2. From the y values 3 to 6, there is an increase of 3. From the y values 6 to 10, there is an increase of 4. This shows that it goes: +2, +3, +4, which is then increasing by 1 between each of those numbers.

This follows the form of an arithmetic sequence, because each new term is equal to the sum of the previous term plus the term number. A formula for this sequence can then be derived form the formula of an arithmetic sequence, shown below. n represents the term number, a represents the number of dots in the first term, and d represents the common difference between the terms.

[pic]
Since the first team has one dot, this means that for this sequence, a=1. The difference between each term is represented by d. Therefore, d=1. By substituting these values into the arithmetic sequence formula, I get the equation: [pic]

Which is then simplified to:
[pic]
This formula above can be used to calculate the number of dots of any triangular number is the sequence, n being greater than 0.

Graphing the points from the table above, gives me this graph:

The equation:
[pic]

is correct because the points in the table fall exactly on the graphed line.

Above, the equation: [pic] also follows the general equation y=ax2+bx+c. If I move around the equation to fit the general form, [pic]. Therefore, a= ½, b= ½, and c=0.

Next, I will be considering stellar shapes with p vertices, leading to p-stellar numbers. I will be showing you stages S1-S6 in diagrams below.

|Stage number |Number of dots |Dot increase between stages | |S1 |1 | | |S2 |13 |+12 | |S3 |37...

...STELLARNUMBERS
In order to develop this mathematics SL portfolio, I will require the use of windows paint 2010 and the graphic calculator fx-9860G SD emulator, meaning that I will use screenshots from this software with the intention of demonstrating my work and process of stellarnumbers sequences.
Triangular numbers are those which follow a triangular pattern, these numbers can be...

...MATHS PORTFORLIO SL TYPE I
CIRCLES
In this portfolio I am investigating the positions of points in intersecting circles. (These are shown on the following page.
The following diagram shows a circle C1 with centre O and radius r, and any point P.
The circle C2 has centre P and radius OP. Let A be one of the points of intersection of C1 and C2. Circle C3 has the centre A, and radius r. The point Pʹ is the intersection of C3 with (OP). This...

...Solving problem
Define the variables and parameters, we choose t as the time human beings could tolerant and g as the acceleration. The constraints of data are as follows:
Because t must be a positive number, when g is positive number it means a forward acceleration, if g is a negative number, it represent a backward acceleration.
Using Mat lab, we plot the data points as follows:
As time (t) increases, the tolerance of G-force (g) is...

...Math SL Portfolio – Tips and Reminders Checklist
Notation and Terminology
Check for the following:
• I did not use calculator notation. (I didn’t include things like ‘x^2’ for or Sn for Sn)
• I used appropriate mathematical vocabulary.
Communication
Check for the following:
• The reader will not need to refer to the list of questions in order to understand my work.
• My responses are not numbered.
• I have an introduction, conclusion,...

...final assessment in 2011 and 2012
STELLARNUMBERS SL TYPE I
Aim: In this task you will consider geometric shapes which lead to special numbers. The simplest example of these are square numbers, 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3 and 4.
The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers...

...
StellarNumbers
Results
1. Triangular Numbers
Observation of the number pattern of polynomial type or different pattern needed.
Identifying the order of the general term by using the difference between the succeeding numbers.
Students are expected to use mathematical way of deriving the general term for the sequence.
Students are expected use technology GDC to generate the 7th and 8th terms also can use...

...Leonardo Souilljee Alberton
Personal Statement
I am a passionate, persistent individual who wills to become a bachelor in British Law, in order to, further on, exert the profession of a lawyer. I am and have always been deeply interested in the principles of justice, which I try to put into action in my everyday life.
My intention on entering a Law School is acquiring a deeper knowledge on the subject which has always been my main interest, both in my social and in my personal...

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