OCR Mathematics: Matchstick Patterns - Essay Example

OCR Mathematics work YOUR FULL THE OF YOUR Matchstick Patterns My investigation is called Matchstick Patterns. I have been given the following information, and asked to answer a series of questions. Gemma has made the following patterns with matchsticks: A: || B: |||| C: ||||||| |_| |_|_|_| |_|_|_|_|_|_| Picture A shows a pattern 1 matchstick wide, Picture B shows a pattern 3 matchsticks wide, and Picture C shows a pattern 6 matchsticks wide. With this information, I have been asked to work out the number of matchsticks in Gemma's pattern, and then to investigate the link between the width of each pattern and the number of matchsticks used to make it. After I have investigated this relationship, I have been tasked to extend my investigation to other possible patterns of matchsticks. In all investigations, I will explain the rules and methods I use to establish my conclusions. As I work through this task, I will hope to find a way that I can predict the number of matchsticks in any given formation through the use of a mathematical function. At first, I will simply count the matchsticks to determine the number in each of Gemma's patterns. I want to find a way to accurately establish the number of matchsticks in a set of patterns without having to physically count them, especially for large diagrams. I think this method will most likely result in a formula. To start with, I will answer the first task which instructs me to determine how many matchsticks are in each of Gemma's pictures. My method will be to simply count the number of matchsticks in each diagram, so that I can have a basis for comparing the number of matchsticks in each pattern and then investigate how they relate to the number of matchsticks in patterns with different widths. I can show my initial results as follows: Picture A = 1 matchstick wide = 6 matchsticks Picture B = 3 matchsticks wide = 14 matchsticks Picture C = 6 matchsticks wide = 26 matchsticks I can place this data in a table for ease of use, and include the new width that I will use to check my formula: Number of Units in Width Number of Matchsticks 1 6 3 14 6 26 8 I notice that there is a relationship between the width of the pattern and the number of matchsticks used. Obviously, as the picture gets wider, more matchsticks are used; but they are not in a direct relationship. In a direct relationship, if a diagram that is one matchstick wide has 6 matchsticks, then it could be expected that one that is 2 matchsticks wide would have 12, and one that is 3 matchsticks wide would have 18. Mathematically, this could be expressed as the number of matchsticks (n) is equal to 6 times the number of widths (w), or: n=6(w). This formula works for the first picture, but is not accurate for the other two. Clearly, there is a relationship of increasing linear proportions, but it is made more complicated by the fact that each pattern of matchsticks shares a common side. This explains why the sequence is not 6, 12, 18, 24, 30, and 36. Any mathematical or formula representation will have to account for the fact that after the first unit, each additional unit of width lacks the two matchsticks it has in common with its neighbor. I think I should use algebra to try and explain this relationship because it is useful in understanding quantitative relationships, and I think a simple linear function will work. The first unit of 1 width has six matchsticks. A second unit (or 2 widths) would share two of those matchsticks already in place and add four more. It would look like this: || ||| |_| 6 Matchsticks |_|_| 10 Matchsticks If the formula can account for the initial width having 6 matchsticks and all other additions having four, it would be a reliable expression. This could be accomplished by representing the total number of matchsticks as a function of the first width having six and all others having four. By simply adding the number of the first width (6 matchsticks) to the product of four times the number of widths in the picture (4 matchsticks each), minus the first width that has already been accounted for, a linear function should provide an accurate calculation. Thus, the total number of matchsticks (n) is equal to 6 plus 4 times the number of widths (w) minus one, or: n=6+4(w-1) where n is the total number of matchsticks and w is the total number of widths in the picture. Thus, if there are 2 widths as in the diagram above: n =6+4(2-1) =6+4 =10 Based on this, I am now in a position to test this formula on my control group of 3 widths and 6 widths with the following result: N=6+4(3-1) N=6+4(6-1) = 6+8 = 6+20 = 14 = 26 The formula results matches the hand counts and provides a reliable method for determining the number of matchsticks arranged in this pattern, no matter how many widths there are. Rather than count matchsticks in a picture, I can use this formula to accurately predict the number of matchsticks in this pattern regardless of the number of widths. If I add just two more widths to Gemma's widest picture, taking it to 8, the formula produces the same predicted result: n=6+4(8-1)=34, which I can confirm by adding the number of matchsticks in the 6 wide picture (26) with the number of two more widths (8) to achieve the same result: 26+8=34. Therefore, even looking at a picture with 100 widths, I can easily determine the number of matchsticks using this formula: N=6+4(100-1) = 402. From my results, I know that the number of matchsticks in Gemma's pattern is related in a way that can be expressed by a linear equation. I have found a rule which always works for this type of pattern in the formula n=6+4(w-1) where n is the total number of matchsticks in the pattern and w is the number of widths in the picture. This rule works because it accurately expresses the relationships of adding widths and matches the provided data, as well as it is confirmed using new data. Understanding the relationship between patterns of this nature, I wish to extend my investigation into other patterns as well. I wonder what would happen if I constructed a pattern of matchsticks only one matchstick high, did not allow them to share a common side, but formed them in a stair-step pattern. Would a linear equation allow me to accurately predict the number of matchsticks Here will be my initial four patterns: A: || B: || C: || D: || |||| |||| |||| |||||| |||||| |||||||| I know from manually counting the matchsticks that Picture A has 4 matchsticks, Picture B has 12 matchsticks, Picture C has 24, and Picture D has 40. Similar to my previous investigation, I can create a table for this data as well: Number of Units in Width Number of Matchsticks 1 4 2 12 3 24 4 40 5 I want to use the number of widths as the basis of my prediction just as I did in the last investigation. I can infer that since I have added rows on top of the columns, my simple linear formula will not work. To check that, however, I will change the formula to account for a smaller number of matchsticks in each picture by reducing the number of matchsticks in the first unit to 4 and the number of each successive unit to 4; thus n=4+4(w-1). Now, I will apply that formula and see what the results will show for Picture B and C. For B: For C: n=4+4(2-1) n=4+4(3-1) =8 =12 Clearly, this is not a workable formula because it does not line up with the control data. I notice that the number of matchsticks in this pattern increase much faster with each successive width than the first set of patterns. This is because I am not just working with width, but also with height. This fact causes me to consider that a simple linear formula is not going to yield accurate results. I can observe from my table that each successive count is increasing at a constant rate. The increase from 1 width to 2 is 8 matchsticks, from 2 widths to 3 is 12 matchsticks, and from 3 widths to 4 is 16 matchsticks. Each increase, or secondary output, is by a factor of 4. This causes me to consider that a quadratic equation might be the best way to express the relationships because that type of formula is the best fit for a situation where the successive or secondary outputs increase by a constant rate. To determine how to set up the formula, I will have to consider the way in which additional pictures are constructed. Each time the width is increased, it adds height-but as the width increases beyond 2, it also adds units in the middle of the picture. This suggests that an exponential relationship might be present. At the same time, the width is going to continue to expand in a somewhat linear fashion and that will have to be accounted for in the formula as well. I think that the total number of matchsticks is going to be calculated by considering the expansion of width as a linear expression, and the height/middle additions as an exponential one. Since the secondary output is increasing by a factor of 4, I will start by using 2 as my formula base. I think that the number of matchsticks will be accurately expressed by considering the sum of the linear growth and the product of the exponential growth in height. I will construct a formula where the total number of matchsticks (n) is produced by multiplying the width (w) times 2, and adding the exponential growth of the units above the first line times 2 as well. This meets the need to account for the growth factor of 4. The formula I will test is: n=2w+2w. For the control data, the calculations are as follows: 1 width: n=2(1)+2(1)=4 2 widths: n=2(2)+2(2)=12 3 widths: n=2(3)+2(3)=24 4 widths: n=2(4)+2(4)=40 This formula fits the control data, so now I will apply it to the next logical step, which is 5 widths: n=2(5)+2(5)=60. I can confirm this by adding the next logical secondary growth rate of 20 (8, 12, 16...20) to the previous number in the control group (40) and arrive at the same total number of matchsticks. This formula would work no matter how many widths there were in the picture. From my results, I know that the number of matchsticks in my stair-step pattern is related in a way that can be expressed by a quadratic equation. I have found a rule which always works for this type of pattern in the formula n=2(w)+2(w) where n is the total number of matchsticks in the pattern and w is the number of widths in the picture. This rule works because it accurately expresses the relationships of adding widths and the exponential growth that takes place in the picture each time a width is added. The formula matches the control data, as well as it is confirmed using new data and a different method of calculation. Read More