- Level: International Baccalaureate
- Subject: Maths
- Word count: 2332
Stellar Numbers. Aim: To deduce the relationship found between the stellar numbers and the number of vertices in the stellar shapes that dictate their value.
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Introduction
IB Standard Mathematics
Internal Assessment
Candidate Name: Alejandro Ortigas Vásquez
Candidate Number: -
Title: Stellar Numbers
D | DCP | CE | ||||||
Aim:
To deduce the relationship found between the stellar numbers and the number of vertices in the stellar shapes that dictate their value.
Procedure:
When considering the stellar numbers, it is important to understand that their origin can be traced back to geometric shapes. Consider, for instance, triangular numbers (1, 3, 6…):
1 3 6 10 15
As we can see, we are able to obtain the same sequence from the triangular pattern of evenly spaced dots. We can also use these diagrams to expand the sequence and validate this hypothesis:
21 28 36
We can put this information in a table:
Term | Dots |
1 | 1 |
2 | 3 |
3 | 6 |
4 | 10 |
5 | 15 |
6 | 21 |
7 | 28 |
8 | 36 |
We can then find the relation between the values in the table by using the computer program Microsoft Excel and plotting them in a graph:
As seen, we can also use the Display Equation on Chart option to obtain the equation of the line and thus a general statement for any term of the triangular numbers. Let us consider the n number of rows, and so the general expression becomes:
n2+n
2
We can test the validity of this statement by trying to arrive to the same expression without the use of technology and simply using mathematical reasoning.
Middle
Now, let us put this information in a table:
Term | Dots |
1 | 1 |
2 | 13 |
3 | 37 |
4 | 73 |
5 | 121 |
6 | 181 |
We can then find the relation between the values in the table by using the computer program Microsoft Excel and plotting them in a graph:
Once again, we can use the Display Equation on Chart option to obtain the equation of the line and thus a general statement for any term of the 6-stellar numbers. Let us consider the nth term, and so the general expression becomes:
d = 1 + 6(n2-n)
We can also test the validity of this statement by trying to arrive to the same expression without the use of technology and simply using mathematical reasoning. Let us consider the following table:
Term | Dots |
1 | 1 |
2 | 13 |
3 | 37 |
4 | 73 |
5 | 121 |
6 | 181 |
From this we can obtain the general expression:
- We notice that the current values for d, the number of dots, seem to have to no apparent relation with their corresponding term.
- However, by looking at the diagrams we are able to decipher that the dot in the middle of the stellar shapes is constant between the different terms. Thus, if we remove the constant we obtain the following table:
Term | Dots |
1 | 0 |
2 | 12 |
3 | 36 |
4 | 72 |
5 | 120 |
6 | 180 |
- The values for the number of dots then become multiples of 12.
Conclusion
Conclusion:
The general statement can be used in almost any situation since ‘n’ is a value decided by the individual, in the sense that when asking “how many d dots are there in the diagram of the nth term of the p-stellar numbers?” the ‘d’ value acts as a dependent variable while the ‘n’ and ‘p’ values act as independent.
The limitations of this experiment include that the most reliable way to validate the patterns is to draw the stellar shape diagrams. However, drawing a lot of them out would be far too time consuming. Additionally, counting dots is a rather inaccurate method of obtaining values for d which could, in the end, potentially alter the results.
This investigation was carried out by analyzing three different variables and trying to understand the mathematical relationship between them. Before starting, an analysis of other types of special numbers and their origin from geometric shapes was conducted so as to validate the method. The statement was found after first observing the effect of an increase in term in the number of dots present in the stellar shape and then doing the same for the number of vertices. Finally, an analysis of the variation in the general expression of the original p-stellar sequence allowed a pattern to be found that helped form an equation that included them both.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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