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Math 20 Portfolio: Matrix

Extracts from this document...

Introduction

MATH 20IB SL PORTFOLIO TYPE 1

STELLAR NUMBERS

By: Bob Bao

March 10, 2010


TABLE OF CONTENTS

Introduction

Triangular Pattern

Stellar Shape Pattern

Conclusion

Introduction:

In this portfolio, I will investigate a pattern of numbers that can be represented by a regular geometric arrangement of equally spaced points. The simplest examples of these are square numbers, 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3 and 4. Other geometric shapes which can lead to special numbers are the triangular shapes and the stellar (star) shapes. Our calculations will be based on the sets of triangular and stellar diagrams that are already provided and those that will be constructed. The aim of this investigation is to examine and determine the general statement for geometric patterns that lead to special numbers as well as to demonstrate a good and clear understanding of patterns and the operations that can be done with them. At the end of the project, we should be able to generate expressions and recognise patterns of other various geometric arrangements.

Triangular Pattern:

In the following, I will examine how we can derive general statements of patterns within triangular shapes. To begin, we can use a sequence of diagrams to better illustrate the pattern:  image05.pngimage06.png

image17.png

image26.pngimage00.png

image28.pngimage29.png

image30.png

image01.pngimage31.png

image32.png

image04.png

Firstly, looking at the pattern, we can see that the number of dots is the sum of a series of consecutive positive numbers. Considering this pattern within the triangular diagrams, we can use it to formulate the general formula. In the Math 10 pure curriculum, we were taught the sum of the series of consecutive 100 positive numbers. Accordingly, the sum of the first 100 positive numbers = 1 + 2 + 3... + 100 = (1+100) + (2+99) + ...+ (50+51) = (50)(101) = 5050.

...read more.

Middle

image03.pngimage03.pngimage03.pngimage03.pngimage03.pngimage03.pngimage03.pngimage02.pngimage02.pngimage02.pngimage02.png

image02.pngimage02.pngimage03.pngimage03.pngimage03.pngimage02.pngimage02.pngimage02.pngimage02.pngimage03.pngimage02.pngimage03.pngimage03.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.pngimage02.png

In each diagram, there are two copies of the triangular diagram, a black one and a white one. The original triangular diagrams from stage 1 to stage 8 now transformed into rectangles. It can be realized then that the number of dots of the original triangular diagram can be derived through the simple triangular area formula:

image33.png

It is easy to see then that:

The 1st triangular number= (1 X 2) / 2 = 1

The 2nd triangular number= (2 X 3) / 2 = 3

The 3rd triangular number= (3 X 4) / 2 = 6

The 4th triangular number= (4 X 5) / 2 = 10

The 5th triangular number= (5 X 6) / 2 = 15

 The 6th triangular number= (6 X 7) / 2 = 21

 The 7th triangular number= (7 X 8) / 2 = 28

The 8th triangular number = (8 X 9) / 2 = 36

In general, then:  

tn = [n X (n + 1)] / 2

We can simplify the above general statement and arrive at:

tn = 0.5n2 + 0.5n

To have a correct general expression, it is essential that the statement be verified. In order to verify, we can use the already known value of 21 dots for the 6th stage. Therefore, simply substitute 6 for n into the derived general statement:

LS= 0.5(6)2 + 0.5(6)

RS= 21

LS = RS

For further insurance, simply substitute 3 for n into the general statement:

LS= 0.5(3)2 + 0.5(3)

RS= 6

LS = RS

By using advanced technology such as a graphing calculator, we can derive the general statement that represents the nth triangular number in terms of n much faster. The following table gives the number of evenly spaced dots in relation to the first seven stages of the pattern.

image07.png

By plotting the first seven stages of the pattern on a graph, we can study the relation within the pattern much more effectively.

Graph settings are as follows:         X:[ 0, 10, 1]           Y:[0, 30, 1]

image08.png

...read more.

Conclusion

Sn is limited to be greater and equal than 1, and is also a set of natural numbers. Since p is a parameter, it generally does not have a limitation. However, in the case of a geometric pattern, p > 0 since we cannot have a negative number of vertices in a geometric shape. As well, p cannot be 0 because the general formula would not have worked as the stellar number of every stage will become 1. To illustrate, when we substitute 0 for p, the general statement becomes 0n2 – 0n + 1 = 1. No matter what n is, the result will always be 1. Therefore, the derived general formula is invalid in a situation where p is 0.

Conclusion:

Through analysing the patterns, the resourcefully use of a graphing calculator, the studying of the graphs and undertaking regression of the data, we can easily deduce the general statement of the two geometric patterns. The general statement of the triangular pattern is  tn = 0.5n2 + 0.5n where tn is the number of evenly spaced dots, and n is the nth stage. The general statement of the stellar pattern is Sn = pn2 – pn + 1 where Sn is the stellar number (number of dots), p is the number of vertices, and n is the nth stage. From the above investigation, we come to a conclusion that the general statements for any geometric shapes which lead to special numbers are quadratic functions with an upward concaving parabola that is only valid in the first quadrant of the graph where the Domain: image11.png, and Range: image27.png.

...read more.

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