At first glance I have achieved the following information by adding each border and extending upon it.
It is clear from this first glance that the number of squares increases dramatically. The number of squares on each pattern increases like this.
Pattern 1 – 2 = 10
Pattern 2 - 3 = 14
Pattern 3 - 4 = 18
Pattern 4 - 5 = 22
Pattern 5 - 6 = 26
Obviously from this we can learn that the amount between each pattern, which has been compared, adds up to a total of 4. Therefore this is an already clear pattern into investigating borders
To extend our investigation of borders it would be useful to use algebra as I have done.
1. 2. 3. 4. 5.
6 10 14 18 22
4 4 4 4
Obviously from this we can learn that the amount between each pattern, which has been compared, adds up to a total of 4. Therefore this is an already clear pattern into investigating borders. The simplest name then given to this kind of pattern is linear so this is defiantly is a linear pattern. The number of squares in each pattern can be more closely analysed if I went into more depth with finding out the Mean, Median and Mode of the totals.
Mean = 46
Median = 41
Mode = N/A
From these we can explore more aspects of the original and find out if there is any connection between them. To go even in more detail in investigating borders it would be optional to explore borders on different shapes. I have done this and have this time used 3 template squares instead of 2 for the borders to occupy round. It proves to be a very different kind of pattern as it created a entirely different shape compared to the first, where only 2 squares were used. For this pattern I have only added 3 borders, which however will still gives us enough information. The results from this new pattern are shown below:
To go even further into the analysis of the borders I have created another shape to base borders around. This time I have included 4 template squares rather than the 3 and 2. This will obviously give a greater number in the results and would hopefully therefore give us a greater depth in our investigation. The results of these patterns are shown below.
Every single total of squares in this pattern are based in the 4 Times Table.
1. 12 = 3 X 4
2. 20 = 5 X 4
3. 28 = 7 X 4
4. 36 = 9 X 4
The numbers, which have been times by 4 also, have a pattern as they go up in 2s and they are all odd numbers.
To conclude my investigation I would have to say that there was indeed a connection, which was however only based in the third pattern with the four template squares. This was most certainly because of the shape itself as it made a clear square for each border to go round. Therefore the 4 Times Table obviously plays a great part as there is 4 sides to a square.
However in the other two different patterns and especially the second one there was no connection whatsoever made between each diagram on that sheet. This was mainly because of the shape itself as there were 3 squares used for the template and I have found out that using even numbers not odd numbers for the squares in the template makes the investigation much easier and clearer. That is indeed a connection.
Therefore I have found out a great deal because of the template. I believe that it can defiantly affect the entire pattern and it is much easier to find a connection between each diagram with the same template if it is of even numbers and not odd.
The first pattern with the template with two squares was indeed the easiest one to find a connection between. As I shown in the diagram the total between each pattern number went up by 4 each time making it the simplest connection to be found out of the entire 3.