• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

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.

Extracts from this document...

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.

...read more.

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:

  1. We notice that the current values for d, the number of dots, seem to have to no apparent relation with their corresponding term.
  2. 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

  1. The values for the number of dots then become multiples of 12.
...read more.

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.

...read more.

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

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Maths Internal Assessment -triangular and stellar numbers

    Therefore the statement is: Question 3: Find the number of dots (i.e. the stellar numbers) in each stage up to S6. Organize the data so that you can recognize and describe any patterns. The table below organizes the data so that we can recognise and describe the patterns.

  2. Stellar numbers

    to the nth term. n will be defined as the stage number of the triangle F(n) will be defined as the function of the nth triangle In the above the equation: appears twice once as normally viewed and the other time as the reason is that they are being added

  1. Stellar Numbers Portfolio. The simplest example of these is square numbers, but over the ...

    While the first differences are 10, 20, 30, and so on with the same increasing increments of 10, the second differences are always 10, meaning that it is a quadratic. I noticed that the formula for 6-stellar numbers was 6n2-6n+1, and throughout that formula the number 6 is consistent, so

  2. Comparing the surface area of different shapes with the same volume

    2nd Column 5.000000000000000000000 78.539816339744800000000 12.732395447351600000000 478.539816339745000000000 5.000000000000000000000 25.000000000000000000000 40.000000000000000000000 825.000000000000000000000 Cylinder Prism 5. In the second last step of the project, the use of graphing computer software or a GDC (Graphic Display Calculator) is necessary for the completion of this step.

  1. Stellar Numbers. After establishing the general formula for the triangular numbers, stellar (star) shapes ...

    a, thereafter the sequence is extended to T0, as the number highlighted in red is c. To figure out the value of b, the values have been substituted into the quadratic formula as shown below, after which the equation has been solved using simple algebra.

  2. Stellar Numbers. In this task geometric shapes which lead to special numbers ...

    I will need to find the constant difference in the sequence in order to establish the type of equation. Again the second difference is the constant therefore the formula for the nth term contains n2 as in the quadratic equation: ax2 + bx + c The value of 'a' is half the constant difference.

  1. Stellar numbers. The aim of the current investigation is to consider different geometric ...

    3 n = 3 u3 = 6 u3 = u2 + n = 6 n = 4 u4 = 10 u4 = u3 + n = 10 n = 5 u5 = 15 u5 = u4 + n = 15 n = 6 u6 = 21 u6 = u5 +

  2. Logarithm Bases - 3 sequences and their expression in the mth term has been ...

    Term (u) Logarithm of 8 with base 2n (c) Logarithm of 81 with base 3n (b) Logarithm of 25 with base 5n (a) New points on the plane above y = . 1 3.00 4.00 2.00 5.00 2 1.50 2.00 1.00 2.50 3 1.00 1.33 0.66 1.63 4 0.75 1.00 0.50 1.25 5 0.60 0.80 0.40 1.00 6 0.50

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work