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GCSE: Number Stairs, Grids and Sequences
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To find out what size squares must be cut out of the corners of square and rectangle pieces of card to give the box it would create its optimum volume.
I predict that after we have found the optimum volume that the volumes after that will drop very rapidly. I also think that the size square that we have to cut out of the corner to create the box will always be quite a low number around two or three because as you cut bigger squares off the corner you lose a lot more card that could be used to make the box bigger, but when you cut a little off the corners you get much depth so the volume is compromised.
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We will first choose the number in the bottom left hand corner of the step stair. I will call it n. I will then move the stair sequence systematically, with n increasing in tens, starting from five. This is because it is impossible to have a three step stair where n is any of the non-highlighted numbers below. Table of results n 5 15 25 35 45 55 65 75 total 74 134 194 254 314 374 434 494 The first difference between all of these is 60. We therefore know that the formula is to be a linear one.
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- (1 x 36) 186 - 36 = 150 Answer = 150 I do not need to draw any more diagrams because it is clear what is being done to get the answer and I now have enough data to make a table of results. This is another method where I can look for any obvious patterns in the data. Table of Results Grid size Number of Product of top right Product of top left Answer (Products Squares number and bottom number and bottom Subtracted)
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67 68 77 78 I did (67 X 78) - (68 X 77) = 10 This shows that the product difference is 10. Below are two more examples to prove it will work anywhere on the grid. 42 43 52 53 I did (42 X 53) - (43 X 52) = 10 TL X BR - TR X BL 12 13 22 23 I did (12 X 23) - (13 X 22) = 10 TL X BR - TR X BL I am now going to prove algebraically that in a 10 X 10 grid, with a 2 X 2 square the difference will always be 10.
- Word count: 6037
3 x 25 = 75 27 x 49 = 1323 5 x 23 = 115 29 x 47 = 1363 D = 40 D = 40 n(n+22) = n2 + 22n (n+2)(n+20) = n2 + 22n + 40 Squares of equal sizes always have the same difference. E.g. 2 x 2, D=10, 3 x 3, D=40. From now on I will only use n grids in my diagrams as they show the numbers that produce the difference. 4 x 4 Length of side = X n(n+33)
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is 25 (and circled.) I am now going to investigate how the position of the stair shape (stair number) effects the stair total, I intend to do this by systematically moving the stair number, horizontally. I will find the stair total of each stair shape and looking at the results try to notice a pattern. I will firstly do this with a 3-step stair. This is my first stair shape I used this particular stair shape, as its stair number is one, which seems a sensible place to start. Stair no. - 1 Stair total - 1+2+3+11+12+21= 50 I have moved along the stair number by one and I will continue to do this so the pattern will be simpler to detect.
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e.g. stair number 55: 55+56+57+65+66+75 = 374 (stair total) I will then proceed not to just rotate the stair but change its size and see from these results if I can find one general equation which sums up the project. Then I will plot these results onto a table and look to find an equation to solve them. Stair no. 55 56 57 58 59 Stair total 374 380 386 392 398 To find an equation from this I will have to translate a stair into algebraic form: Proof: (6x55)
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78 x 100 =7800 80 x 98 =7840 7800-7840= 40 On a 3by3 grid the answer is 40. I will do some more just to make sure. 45 46 47 55 56 57 65 66 67 45 x 67 = 3015 47 x 65 = 3055 3015-3055 = 40 71 72 73 81 82 83 91 92 93 71 x 93 = 6603 73 x 91 = 6643 6603-6643= 40 The answer is 40. This means that the answer isn't going to change. 4by4 Grid 65 66 67 68 75 76 77 78 85 86 87 88 95 96 97 98 Although this grid looks quite complicated it's done exactly the same ways as the ones before, it's the 4 corner numbers we are intrusted in.
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I will need to take 3, 4 or 5 readings (depending on grid size) and an nth term to prove that my calculations are correct. How will you know when you have got enough information? I will know I have got enough data when I am able to successfully and easily calculate an nth term. How accurate does your data have to be? My data will all be in whole numbers so it does not need to be accurate to any decimal places. How will you check that your data is accurate? I will use a calculator to verify my results so I know that they are accurate.
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As Australia is one of the most popular tourist destinations in the world it is necessary to observe the tourist precincts Australia has to offer.
There is lots of signage around the square and also a small information booth at the front corner of the square for information of where anything is situated and a general overview of what's happening in the square and around the city. One of the major attractions within the square is the art gallery. It is a great place for tourists to visit and explore the art inside Melbourne. The Ian Potter NGV has a lot of art pieces where locals and tourists can visit. Admission is free and they also have tours that show traditional Aboriginal and Australian art.
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To investigate the relationship between a 10 by 10-number grid and various stair shapes (as in the example below).
I began to look at different ways, I found that by working systematically on the grid i.e. starting by looking at horizontal, vertical and diagonal movements I would be able to find the common factors involved. In the example we are given the numbers: 25, 26, 27, 35, 36, 45 They total to 194 To work out the placement of the stair shape I used the smallest number of the stair to translate the shape from, in this case 25.
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3640 3640 - 3630 = 10 72 73 82 83 72 x 83 = 5976 73 x 82 = 5986 5986 - 5976 = 10 From doing these three checks I have found that the difference in a 2 x 2array is ten, to be completely sure that it is ten in any place in the 10 x 10 grid I am going to put it into a N x N formula. X X+1 X+10 X+11 This box is called an x-box and it is going to produce my N x N formula, which will then tell me if the difference is ten, or not.
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Establishing a connection In order to establish a link between n (n=stair number) and the st (st=step total) I must conduct a series of tests on various different three step stairs in different positions on the ten by ten grid. I will, whilst examining the results I gain from different stairs, attempt to find a formula that can be used to predict the st of any three step stair on the grid. As an example of what I am going to do: E.g.
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This stair is called stair2 as the number in the bottom left hand corner is 1. Stair2 is a translation of stair1 one square to the right. 22 12 13 2 3 4 The stair total for this stair shape is 2 + 3+ 4 + 12 + 13 + 22 = 56. Now we are going to find the step total for stair3 (a translation of stair2 one step to the right). 23 13 14 3 4 5 The stair total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62.
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Why did the operation keep resulting in the number seven, no matter what data I processed through it? My earlier recognition of a potential relationship was now strengthened; it seemed to me that the pattern would produce a value that was equal to the number of columns involved in the grid as a whole. In order to prove this I would have to create an algebraic formula for this particular event. First I assembled an algebraic representation of the box: n n+1 n+7 n+8 n = root number Using this I could concoct a generic formula that could be used with every box in a calender to find the difference.
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24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 I calculated up to seven three by three L-Shapes and found their L-Sum (as this is the last L-Shape that will fit across a nine by nine grid).
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You can tell that soldiers stayed in the gatehouse because of the latrines on the second floor. There was also a fireplace on the second floor. This must have been where the soldiers boiled the oil or water to pour down the murder holes in the ceiling. THE CASTLE WALLS The castle had curtain walls. One wall surrounded the keep. The other was a much thicker, taller wall surrounding the entire castle. This had all sorts of arrow slits in them.
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15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 For this coursework I have been asked to investigate; *The relationship between the stair total and the position of the stair shape on the grid. *Investigate further the relationship between the stair totals and the other step stairs on the number grids.
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Number Stairs - For part one, I am investigating the relationship between the stair total and the position of the stair shape on the grid.
I have studied these answers and concluded that they are all even. To investigate further, I looked at what would happen if I moved the 3-step stairs to the left and right and up and down. I predict that if I move the 3-step stair upwards, the stair total will be greater and if I was to move the 3-step stair downwards, the total will be less. If I moved the 3-step stair to the left, it would be less than moving it to the right.
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work out as all that you need to do is look at the n figures and see how the other numbers relate to them. N 2x3 squares (3 = width) 1x13= 13 11x3= 33 18x10= 180 8x20= 160 66x58= 3828 56x68= 3808 The difference is always 20. Nn n+1 n+2 n+3 Ln+10 n+11 n+12 n+13 Hn(n+13) Difference is 20 (n+3) (n+10) 3x3 Squares 1x23= 23 21x3= 63 43x65= 2795 63x45= 2835 68x90= 6120 88x70= 6160 The difference is always 40.
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Due to the fact that the difference is always 10 we have decided to show this in a table algebraically. If we multiply the bottom left digit by the top right digit and the top left digit by the bottom right digit we can see that the N's disappear to leave just 10 which is the difference. As is shown below. N(n+1)=n2+1 =see below N(n+21)=n2+21 =N2+21-N2+1 =10 For a two by two grid we can see that there is an algebraic table This is shown below also: N N+1 N+10 N+11 N2+1 N2+21 When subtracted the n cancels out leaving the number 10 which is actually the sized number grid I am working with.
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To find any relationships and patterns in the total of all the numbers in 'Number Stairs', that might occur if this stair was in a different place on the grid.
+ (x + 2) + (x + 3) + (x + 10) + (x + 11) + (x + 20) = 6x + 1 + 2 + 3 + 10 + 11 + 20 = 6x + 44 25+26+27+35+36+45= 194 The stair total for this three-step stair is 194 This formula should work with every number stair that can fit onto a 10 by 10 grid. To support my judgement I will repeat the formula another two times on the same three-stair grid, but in a different position to the previous three stairs.
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Can it be true for any square ? I tried to investigate further if it can be true while choosing a 3 3 square in 10 10 grid following the same four steps. The results are in the table below. BOX MULTIPLIED NUMBER PRODUCT DIFFERENCE 5 6 7 5 27 135 15 16 17 7 25 175 40 25 26 27 12 13 14 12 34 408 22 23 24 14 32 448 40 32 33 34 36 37 38 36 58 2088 46 47 48 38 56 2128 40 56 57 58 It shows that in a diagnol square of 3 3 in 10 10 grid , the difference between the products is 40.It is three times to the difference of 2 2 square.
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= 455 difference = 40 48 x 39 = 1872 82 x 64 = 5248 38 x 49 = 1862 difference = 10 62 x 84 = 5208 difference = 40 96 x 87 = 8352 88 x 70 = 6160 86 x 97 = 8342 difference = 10 68 x 90 = 6120 difference = 40 4 x 4 5 x 5 41 x 14 = 574 46 x 10 = 460 11 x 44 = 484 difference = 90 6 x 50 = 300 difference = 160 67 x 40 = 2680 62 x 26 = 1612 37
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For 5 x 5. 24 25 26 27 28 34 35 36 37 38 44 45 46 47 48 54 55 56 57 58 64 65 66 67 68 So (24 x 68)-(28 x 64) 1632-1792 = -160 Just like the normal 5 x 5 grid the answer is the same; -160. This shows that no matter what the numbers are, the answer, always, for a square will always be the same, as long as there is the same number of columns, and therefore rows.
- Word count: 2622