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Stellar numbers. This internal assessment has been written to embrace one of the rather inquisitive aspects of mathematics, stellar numbers

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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.

...read more.

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 6-stellar 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 6-stellars will also be a quadratic function. Using the TI-84 Plus graphic calculator and following the same method mentioned in the task of triangular numbers, the following equation was given, y=6x2-6+1, where x∈N. Although the general statement is present, where do these numbers come from? After some thorough analysisof the relationship between 6-stellar 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 6-stellars 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=s-1=x-1. For 6-stellars, 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 x-1, everything can be combined into one andgive the following equation:

y=1+(x-1)2+(x-1)212

By expanding (x-1)2+(x-1) and multiplying it by 12, the equation simplifies to

y=1+(12x2-24x+12)+(12x-12)2=1+12x2-12x2=6x2-6x+1

Hence, it’s been derived that the general statement for a 6-stellar numbers is

y=6n2-6n+1

where{n|n>0,n∈N}.y represents the total number of dots and n represents the term of the 6-stellar. This formula is equivalent to the equation found if one were to use the calculator to determine the quadratic function of the 6-stellars.

        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 p-stellars 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 4-stellar will be tested first. The image of what a 4-stellar looks like is given to the left in Figure 4. Notice, that a 4-stellar 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 6-stellar numbers. Once again, a constant 2nd difference means that the general statement will be a quadratic function and similar to the 6-stellar 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 6-stellar, n=s-1=x-1. Now, by substituting the general statement for the triangular numbers into y=1+Tn8 this equation is formulated:

y=1+n2+n28=1+(x-1)2+(x-1)28

By expanding (x-1)2+(x-1) and multiplying it by 8, the equation simplifies to

y=1+(8x2-16x+8)+(8x-8)2=1+8-8x2=4x2-4x+1

Hence, it’s been derived that the general statement for a 4-stellar number is

y=4n2-4n+1

QuadReg

   y=ax2+bx+c

   a=4

   b=-4

   c=1

...read more.

Conclusion

n and y were pasted). The table wasselected, and then a graph was created by following these steps: select the Insert tab and then choose the option shown in Figure 8, the 2-D Line Graph.On the graph it can be seen that as n increases, or in other words as it approaches infinity, the y value is always increasing showing no signs of a limit. Even though quadratic functions’ domain includes negative numbers, negative numbers and 0 are excluded from the domain, because it is impossible to have a negative term of a sequence and a 0 term does not exist.With regard tothe number of vertices,p>0 because it is not possible to have a stellar with a negative number of vertices or a stellar without any vertices.Figure 8

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        

...read more.

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