Math 20 Portfolio: Matrix

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By: Bob Bao

March 10, 2010




Triangular Pattern

Stellar Shape Pattern



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:  

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. In another form, the sum can be expressed as (50)(100 + 1). By using this information, we can deduce the number of dots in each diagram of the triangular pattern in which follows a similar pattern.

The variables are as follows:   tn = triangular number  (numbers of dots)

                                                      n =  the stage

In order to modify the expression for the previous series of 100 consecutive positive numbers into the general formula for the triangular pattern, we can simply replace the number 100 within the expressing (50)(100+1) by n.  The reason for which is because within the context of the series of 100 consecutive numbers, there is 100 terms and therefore the 100th stage. Also, as you can see the 50 within the expression is derived by dividing 100 by 2. As such, it can be modified as being (n/2). After the two modifications, we can rewrite the expression for adding consecutive a hundred positive numbers into the general statement: tn = (n / 2)(n + 1). After simplifying this multiplication of a monomial by a binomial, we get the general statement as: tn = 0.5n2 + 0.5n. This general statement can also express the pattern of triangular geometric shapes as it poses the same pattern as the series of 100 consecutive positive numbers.

         Nevertheless, there is also an entirely new method of finding a general statement that represents the nth triangular number in terms of n that is enabled by the triangular geometric shape. Considering the follow arrays of rectangular diagrams shown below:

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:

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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 + ...

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