 Level: International Baccalaureate
 Subject: Maths
 Word count: 3125
Stellar numbers. This internal assessment has been written to embrace one of the rather inquisitive aspects of mathematics, stellar numbers
Extracts from this document...
Introduction
Last Name Sample
IB Math SL
Mr. Doty
11 September 2011
Internal Assessment 1: Type 1
Stellar Numbers
This internal assessment has been written to embrace one of the rather inquisitive aspects of mathematics, stellar numbers.A stellar number is a numerical value for the total number of dots that can be positioned within a designated geometric shape with a determined number of vertices.This paper will consist of a panoptic yet concise set of instructions involving the use of technology, primarily a graphic calculator and a computer, for the reader to fully appreciate how this candidate analyzed the patterns of various examples of triangular and stellar numbers to formulate a conclusive general statement regarding the variables p, the number of vertices, and n, the term of the stellar number.
In mathematics, it is essential that the learner is able to distinguish a pattern within a set of numbers, define the pattern using words, and carry on the pattern. A series of numbers in which a pattern is present is called a number sequence. The numbers in the sequence are called its members or its terms. A number sequence can be defined with the use a formula which represents the general term or also known as the nth term.The formula is useful for determining the value of the nth term without actually having to count out and write down every single member of the sequence. With this concept of number patterns and sequences of numbers, a general statement will be produced which will generate the sequence of p stellar numbers for any value of p at any term, n.
First, this investigation will deal with triangular numbers. A triangular number is the total sum of the dots in an equilateral triangular pattern of evenly spaced dots.
Middle

S3
37
24
12
S4
73
36
12
S5
121
48
12
S6
181
60
12
Table 3 shows the tabulated data of the total number of dots, 1st differences, and the 2nd differences of 6stellar numbers. Similar to the method used in triangular numbers, the data was collected by counting all the dots within each stellar and calculating the differences. After already discovering the general statement for the triangular numbers, the constant change in 1st differences or constant 2nd difference suggests that the general statement for 6stellars will also be a quadratic function. Using the TI84 Plus graphic calculator and following the same method mentioned in the task of triangular numbers, the following equation was given, y=6x26+1, where x∈N. Although the general statement is present, where do these numbers come from? After some thorough analysisof the relationship between 6stellar numbers, an interesting observation was discovered.
Table 4
Sn  Sum of a Stellar Number  
S1  S1  1  1  1+0(12) 
S2  S1+12  1+12  (1)+12  1+1(12) 
S3  S2+24  13+24  (1+12)+24  1+3(12) 
S4  S3+36  37+36  (1+12+24)+36  1+6(12) 
S5  S4+48  73+48  (1+12+24+36)+48  1+10(12) 
S6  S5+60  121+60  (1+12+24+36+48)+60  1+15(12) 
With the help of Table 4, where the different patterns have been organized, the pattern is much easier to visualize and interpret. The last 4 columns show the sum of the stellar numbers expressed in different forms and notations. The 1st column shows the sum in terms of adding the 1st difference to the previous term. The 2nd column replaces the previous term with a numerical value. The 3rd column takes that value and substitutes it with an expression of the sum of the previous term. After this process of logical reasoning, the pattern is now clearer to see. With exception to the first number, everything else can be factored out to a common factor of 12. By factoring everything out, the expression is recorded in the 4th column. At a first glance, nothing particular stands out, but the red bolded numbers correspond to the exact same numbers from the previous task of triangular numbers.
Table 5
n  y  Sn  Total Sum of Sn 
0  0  1  1+0(12) 
1  1  2  1+1(12) 
2  3  3  1+3(12) 
3  6  4  1+6(12) 
4  10  5  1+10(12) 
5  15  6  1+15(12) 
Table 5 helps illustrate the pattern more clearly. With this set of data combined with the general statement regarding triangular numbers, the general statement for 6stellars can be formulated by substituting corresponding variables. Let s=x because they will represent the same variable and by looking at the numbers from the table it can be inferred that n=s1=x1. For 6stellars, the general statement would be in the form of y=1+Tn12, where Tn is the triangular number. Knowing that the general statement for triangular numbers is Tn=(n2+n2), and thatn can be substituted with x1, everything can be combined into one andgive the following equation:
y=1+(x1)2+(x1)212
By expanding (x1)2+(x1) and multiplying it by 12, the equation simplifies to
y=1+(12x224x+12)+(12x12)2=1+12x212x2=6x26x+1
Hence, it’s been derived that the general statement for a 6stellar numbers is
y=6n26n+1 
where{nn>0,n∈N}.y represents the total number of dots and n represents the term of the 6stellar. This formula is equivalent to the equation found if one were to use the calculator to determine the quadratic function of the 6stellars.
A valid theory cannot be formed with only one trial/sample, so therefore several more trials will be tested for other values of p. The explanation of the process of obtaining the general statement for the next pstellars will be swift as the reader is now accustomed to the concept of triangular numbers and one concrete example of a stellar number.
Figure 4
A sample of a 4stellar will be tested first. The image of what a 4stellar looks like is given to the left in Figure 4. Notice, that a 4stellar is not a square, a misinterpretation that is very commonlymade. After drawing all the diagrams using the computer^{[2]} and counting all the dots, the data was tabulated into Table 6 shown below.
Table 6
Sn  y  1st Difference  2nd Difference  Total Sum of Sn 
S1  1      1+0(8) 
S2  9  8    1+1(8) 
S3  25  16  8  1+3(8) 
S4  49  24  8  1+6(8) 
S5  81  32  8  1+10(8) 
S6  121  40  8  1+15(8) 
The data is organized into aneat combination of the tables made for the 6stellar numbers. Once again, a constant 2nd difference means that the general statement will be a quadratic function and similar to the 6stellar the total sum of Sninvolves the triangular numbers. As a matter of fact the relationship between Sn and n is exactly the same as it was for the 6stellar, n=s1=x1. Now, by substituting the general statement for the triangular numbers into y=1+Tn8 this equation is formulated:
y=1+n2+n28=1+(x1)2+(x1)28
By expanding (x1)2+(x1) and multiplying it by 8, the equation simplifies to
y=1+(8x216x+8)+(8x8)2=1+88x2=4x24x+1
Hence, it’s been derived that the general statement for a 4stellar number is
y=4n24n+1 
QuadReg y=ax2+bx+c a=4 b=4 c=1 
Conclusion
What this investigation has showed is that it is possible to derive a pattern out seemingly ordinary objects and shapes. At first no particular pattern can be seen, but after some thorough analysis one might notice something that ordinarily would be unnoticeable. The mathematical concepts of number sequences and quadratic functionsreally helped making this investigation a fluid task. Given more time, further investigation could be completed with regard to other possible relationships between triangular numbers and some other geometric shapes such as square numbers or perhaps thetotal number of lines in each stellar. The possibilities are endless and the model in this investigation can be applied to many other aspects of mathematics. Nonetheless, this investigation shows that with tedious analysis and effort, a pattern can be generated regarding even the most mundane objects in life provided there is consistency with the variables.
[1] The diagram was created using GeoGebra. All the diagrams in this paper were created with GeoGebra.
[2] With the aid of GeoGebra
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
Found what you're looking for?
 Start learning 29% faster today
 150,000+ documents available
 Just £6.99 a month