STELLAR NUMBERS
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 stellar numbers sequences. Triangular numbers are those which follow a triangular pattern, these numbers can be represented in a triangular grid of evenly spaced dots.

The sequence of triangular numbers is shown in the diagrams above. The first stage has 1 dot; the second stage has 3 dots (1+2); the third stage has 6 dots (1+2+3); the fourth stage has 10 dots (1+2+3+4); the fifth stage has 15 dots(1+2+3+4+5); the sixth stage has 21 dots (1+2+3+4+5+6) ; the seventh stage has 28 dots(1+2+3+4+5+6+7) and the eighth stage has36 dots(1+2+3+4+5+6+7+8). As it could be noticed, there is a sequence where in every stage the number of dots is obtained by adding up all the positive integers that correspond to the previous stages and every time one more number is added.

In terms of n, where n matches up to the stage number, it is accurate to establish an equation so that when trying to find the number of dots in stage 592, it is easy and fast by simply applying the following formulae:

Now it is possible to find the nth number by using the formula, going back to the example where n is 592, so we replace n by 592 and solve the equation as follows: so the 592nd term will contain 175528 dots. =175528

Furthermore, to prove my equation I will use different values for n but they have to be positive integers otherwise if I use negative or irrational or fractions it would not have any common sense. thus i will replace n by 6 to prove that the result is 21 as shown in the first diagram and also by 10:

=21 =55

Subsequently considering stellar numbers which are stars with 6 vertices, the number of dots in each stage will represent each stellar number just as it was done before with the...

...The aim of this task is to investigate geometric shapes, which lead to special numbers. The simplest example of these are square numbers, such as 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3, and 4.
Triangular numbers are defined as “the number of dots in an equilateral triangle uniformly filled with dots”. The sequence of triangular numbers are derived from all natural numbers and zero, if the following number is always added to the previous as shown below, a triangular number will always be the outcome:
1 = 1
2 + 1 = 3
3 + (2 + 1) = 6
4 + (1 + 2 + 3) = 10
5 + (1 + 2 + 3 + 4) = 15
Moreover, triangular numbers can be seen in other mathematical theories, such as Pascal’s triangle, as shown in the diagram below. The triangular numbers are found in the third diagonal, as highlighted in red.
The first diagrams to be considered show a triangular pattern of evenly spaced dots, and the number of dots within each diagram represents a triangular number.
Thereafter, the sequence was to be developed into the next three terms as shown below.
The information from the diagrams above is represented in the table below.
Term Number (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Triangular Number (Tn) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 |
Establishing...

...– 6 – MATME/PF/M11/N11/M12/N12
For 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 (1,3,6,)….
1 3 6 10 15
Complete the triangular numbers sequence with three more terms.
Find a general statement that represents the nth triangular number in terms of n.
Consider stellar (star) shapes with p vertices, leading to p-stellarnumbers. The first four representations for a star with six vertices are shown in the four stages S1–S4 below. The 6-stellarnumber at each stage is the total number of dots in the diagram.
S1 S2 S3 S4
Find the number of dots (i.e. the stellarnumber) in each stage up to S6. Organize the data so that you can recognize and describe any patterns.
Find an expression for the 6-stellarnumber at stage S7.
Find a general statement for the 6-stellarnumber at stage Sn in terms of n.
Now repeat the steps above...

...
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 other graphic packages to find the general pattern to support their result
The general term in this stage is
2. 6- stellarnumbers
Stage
No of dots
1
1
2
13
3
37
4
73
5
121
6
181
Students are expected to make the stellar shape for the next to stages and count the no of dots to get the 6-stellarnumber in 5th and 6th stage
Diagrams can be hand made or using technology
Communication or observation of the number pattern has to be given
From the observation, the expression of the terms of this sequence has to be identified
Expression for the 7th term
General expression 6 – stellar shape
or
p n (n-1) +1
Other stellar shapes – based on the no of vertices the students choose...

...Math SL Investigation Type 2
StellarNumbers
This is an investigation about stellarnumbers, it involves geometric shapes which form special number patterns. The simplest of these is that of the square numbers (1, 4, 9, 16, 25 etc…)
The diagram below shows the stellar triangular numbers until the 6th triangle.
The next three numbers after T5 would be: 21, 28, and 36.
A general statement for nth triangular numbers in terms of n is:
The 6-stellar star, where there are 6 vertices, has its first four shapes shown below:
The number of dots until stage S6: 1, 13, 37, 73, 121, 181
Number of dots at stage 7: 253
Expression for number of dots at stage 7:
Since the general trend is adding the next multiple of 12 (12, 24, 36, 48 etc…) for each of the stars, so for S2 it would be 1+12=13, and for S3 it would be 13+24=37
General statement for 6-stellar star number at stage Sn in terms of n:
For P=9:
Since S1 must equal 1 then we can prove this formula by showing that:
So the first six terms are: 1, 19, 55, 109, 181, 271
Therefore the equation for the 9-Stellar star at
For P=5:
Since S1 must equal 1 then we can prove this formula by showing that:
So the first six terms are: 1, 11, 31, 61, 101,...

...SL Math Internal Assessment: StellarNumbers
374603
Mr. T. Persaud
Due Date: March 07, 2011
Part 1: Below is a series of triangle patterned sets of dots. The numbers of dots in each diagram are examples of triangular numbers.
Let the variable ‘n’ represent the term number in the sequence.
n=1 n=2 n=3 n=4 n=5
1 3 6 10 15
n=6 n=7 n=8
21 28 36
Term Number (n) | Number of Dots (tn) | First Difference | Second Difference |
1 | 1 | - | - |
2 | 3 | 2 | - |
3 | 6 | 3 | 1 |
4 | 10 | 4 | 1 |
5 | 15 | 5 | 1 |
6 | 21 | 6 | 1 |
7 | 28 | 7 | 1 |
8 | 36 | 8 | 1 |
As we can see from the chart above, there is a growing increase in the differences between each consecutive set of numbers of dots. The difference between the number of dots in sets 2 and 1 is 2, the difference between the...

...
IB Math: Studies Statistics
Portfolio:
What is the relationship between the numbers of goals the top sixteen players of the FIFA world cup 2014 score with the height of the players?
Due date: Friday November 23, 2013
School Name: Franklin Delano Roosevelt
Course: IB Math Studies
Name: Valerie Philco
What is the relationship between football player’s height that is participating in FIFA and has scored more than fourteen goals with the number of goals they have scored?
Introduction
The FIFA world cup is one of the most celebrated soccer tournaments around the world today. Not only does it serve as entertainment to everyone that watches the tournament but to participate teams have to have play a game against every team of there area teams classified this year include country’s like Argentina, Spain, Bosnia, Brazil, USA, Italy, France and many more. This being said for teams to win games they have to score more goals than other teams in this investigation the purpose is to verify if there is a correlation between the height of soccer players and the amount of goals they score.
Statement of task
The main purpose of this investigation is to determine whether there is a relationship between the height of the player and the amount of scores that they scored through out the period of qualifications for the 2014 FIFA world...

...Assignment 1: Number and Operations
. . . Math 1901 .
1.
−3 + 2 =
a. The temperature in the morning when you leave to come to school is -3 degrees. When the sun comes out, the temperature warms up by 2 degrees. What is the temperature after the sun comes out?
1 0 -1 -2 So by moving up 2 degrees, we see that we end up at -1 degrees. -3 To solve this problem, start by finding -3 degrees on our thermometer/ number line. We know from before, that when we are adding numbers, we move up the number line.
-3 + 2 = -1
-4 Morning temp Rise in temp
b. A frog is sitting on the stairs, on the 3rd step down from the main floor. It jumps up 2 steps. Which step is it on now?
Main Floor -1 step -2 step -3 step -4 step -5 step So by having our frog jump up 2 steps, we see that we end up on 1 step below the main floor, or -1. To solve this problem, start by putting our frog on the 3rd step down from the main floor (-3). We know from before, that when we are adding numbers, we move up the number line, or in this case the stairs.
-3 + 2 = -1
2.
5 − (−3) =
a. Black tokens are positive numbers, red tokens are negative numbers. There are 5 black tokens in the cup. I want to take out 3 red tokens. How many tokens are left in the cup? I have no red (negative) tokens in my cup, so I need to add some. By adding some, that changes how many I have...

...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 is shown in the diagram below.
As shown on the assignment sheet, r=OA. We therefore need to find the values of OPʹ when r=OA=1 for the following of the values of OP: OP=2, OP=3 and OP=4.
We first of all extract the triangle OPA from the above diagram and since we have the values for all the three sides we can finds the angle AOP which will later on help to get the value of length OPʹ. The circle C2 and triangle OPA are shown below with all side of OPA indicated.
OP=AP since they are the radii of the same circle, C2. Having all the three sides, we can now calubulate the angle AOP using the cosine rule. Angle AOP is calculated below:
Cos AOP=(2^(2 _ ) 2^(2 _ ) 1^2)/(-2×2×1)
Cos AOP=0.25
∴ AOP =COS-10.25
=75.52248781
≈75.5˚...