GCSE: Number Stairs, Grids and Sequences
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Maths number grid
34 x 45 = 1530 44 x 35 = 1540 d = 10 As you can see by my calculations it is not just a 2 x 2 grid in the corner that gives a difference of 10 when the diagonally opposite corners are multiplied. The position of the 2 x 2 grid on a 100 square grid does not change the difference. I am now going to investigate if the difference is the same in a 2 x 3 grid when multiplying diagonally opposite corners.
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number grid investigation]
The top left number in the grid (letter (a) in the above example) will be represented by the term 'n', which will be referred to in this manner in all proceeding investigations also. This is only the first section of the investigation. In calculating the formula, enough information will be gained to progress and investigate other factors, variables and measurements that affect the difference between products. In order to see a trend (in the form of an n x n grid) from the sample numeric grids (2 x 2, 3 x 3 etc.) it is necessary to use summary tables.
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number grid
The top left number in the grid (letter (a) in the above example) will be represented by the term 'n', which will be referred to in this manner in all proceeding investigations also. This is only the first section of the investigation. In calculating the formula, enough information will be gained to progress and investigate other factors, variables and measurements that affect the difference between products. In order to see a trend (in the form of an n x n grid) from the sample numeric grids (2 x 2, 3 x 3 etc.) it is necessary to use summary tables.
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Investigate Borders  a fencing problem.
Then, I will get those three formulas and get one Universal formula in the end. Diagram of Borders of square: 1x1 Table of results for Borders of square: 1x1 Formula You can always find 'the nth term' using the Formula: 'a' is simply the value of THE FIRST TERM in the sequence. 'd' is simply the value of THE COMMON DIFFERENCE between the terms. To get the nth term you just need to find the values of 'a' and 'd' from the sequence and stick them in the formula. Formula to find the number of squares needed for each border (for square 1x1): Common difference = 4 First term = 4 Formula = Simplification =
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number grid
N x (N + 11). Then multiply top right box with bottom left box e.g. (N +1) (N + 10). Do the same as with the arithmetic product when told to find the difference, by subtracting the N x (N + 11) from the (N +1) (N + 10) using the FOIL method. N (N + 11)  (N + 10) (N + 1) N2 +11N  N2 + 1N +10N + 10 N2 +11N  N2 + 11N + 10 Difference = 10 Cconclusion: the difference for a 2by2 squared box is always 10 Now I will investigate further, by finding the difference of
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Applied Statistics
Thus we have used different approaches to variable selection in order to obtain the final equation. We used Stepwise regression, Backward elimination and Forward selection. In each approach we can see that we obtained the same RSquare value of 0,952 but we also obtained a better equation than with the simple model. We can also see that in each approach the variable Temp2 has been dropped. Omitting Temp2 must have had the least effect on the explanatory power of the model. (Appendix Q1a) We can conclude that the variable Temp2 isn't significant in the model. All of these approaches gave us the same final equation with significant variables (Sig.<0,05).
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Number Grids
Let's say the top left number was 'n'. If followed along the grid, the other numbers would come to be: (bottom right) n+11, (top right) n+1 and (bottom left) n+10. So if worked out, the sum would look like: After this, I tried squares of different sizes around the grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 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
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Open box. In this investigation, I will be investigating the maximum volume, which can be made from a certain size square piece of card, with different size sections cut from their corners. The types of cubes I will be using are all open topped boxes.
1cm by 1cm, piece of square card. Length of the section (cm) Height of the section (cm) Depth of the section (cm) Width of the section (cm) Volume of the cube (cm3) 0.1 0.1 0.8 0.8 0.064 0.2 0.2 0.6 0.6 0.072 0.3 0.3 0.4 0.4 0.048 0.4 0.4 0.2 0.2 0.016 0.5 2cm by 2cm, piece of square card Length of the section (cm) Height of the section (cm) Depth of the section (cm) Width of the section (cm)
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Stair shape maths GCSE coursework
First I will draw a 3 stair shape at the bottom left with the number 1. Then I am going to add up the numbers in the stairs to get the total. Then I am going to move the stairs one step right and find the total of the particular stair shape. Then I am going to repeat this once more moving it one step right and finding its total. Finally I am going to find the relationship between these stair shapes.
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Investigation into the patterns of mutiplication sqaures
3 X 21 = 63 35 36 37 45 46 47 55 56 57 35 X 57 = 1995 The difference is forty 37 X 55 = 2035 71 72 73 81 82 83 91 92 93 71 X 93 = 6603 The difference is forty. 73 X 91 = 6643 This time the difference is always forty. To prove my results were correct I did another example. 78 79 80 88 89 90 98 99 100 78 X 100 = 7800 My results were correct.
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maths stairs
How I got the formula is explained in part 1. (Diagrams are above) I am unable to do a '1 by 1', '2 by 2' as there is not enough room to get 1 result. n+n+1+n+2+n+3+n+4+n+7=6n+16 I have worked out that if the corner square is 1 in a 3 by 3 grid the total will be. 1+2+3+4+5+8=22 I worked this out by adding all of the numbers inside the stair and finding the total. I came up with the formula of 16+6n. As there is still the same number of 'n' in the diagram, the only change is that of the numbers.
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Open box problem
2.25 27.56250 8 8 2.5 22.50000 8 8 2.75 17.18750 8 8 3 12.00000 8 8 3.25 7.31250 8 8 3.5 3.50000 8 8 3.75 0.93750 8 8 4 0.00000 From the table above it can clearly be seen that the biggest volume gained from an 8 by 8 piece of square sheet is 37.81cm (2dp), this volume lies between the height of 1cm and 1.5cm. Therefore I am going to further investigate the height between these two points to gain the highest possible volume.
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Number Stairs Coursework
from positions 15 (see green), vertically across the grid and calculate the total of the numbers inside it. This can indicate what the relationship between the totals and the position vertically across the grid could be, for example, a key pattern could emerge in the totals. Then I will change the position of the number stair horizontally up the grid from 25  65 and record the total of the numbers inside it. This can indicate what the relationship between the totals and the position horizontally up and down the grid could be. Furthermore, I will find a formula for the total of a 3step stair on every position possible on the number grid.
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Number Grid
This is a general hypothesis that I will aim to prove throughout the investigation. Investigate further I will start by developing the task and investigating different sized grids. I will calculate the square grids 2x2, 3x3, 4x4 and 5x5 to begin with to calculate the differences of the products in each grid. I will begin to investigate the numbers in a 2x2 square grid in the 10x10 master grid by using the above rule. 14 15 14x25 = 350 From this calculation there is a difference of 10.
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Opposite Corners
I will test any rules, patterns and theories I find by using predictions and examples. I will record any ideas and thoughts I have as I proceed. Plan: Firstly use a 2x2 box on a 10x10 grid in 5 positions Move on to 3x3 in 3 positions. Then 4x4 in 3 positions. Then predict what the difference will be for a 5x5 box. Test the prediction. Prove using algebra why the difference is always the same. Find formula for a square on a 10x10 grid. Prove formula works Investigate rectangles(with same method). Change grid size.
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Algebra Investigation  Grid Square and Cube Relationships
The top left number in the grid (letter (a) in the above example) will be represented by the term 'n', which will be referred to in this manner in all proceeding investigations also. This is only the first section of the investigation. In calculating the formula, enough information will be gained to progress and investigate other factors, variables and measurements that affect the difference between products. In order to see a trend (in the form of an n x n grid) from the sample numeric grids (2 x 2, 3 x 3 etc.) it is necessary to use summary tables.
 Word count: 8966

number stairs
37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 Therefore in order for me to see a pattern occurring, I will be moving this 3 step stair, systematically to find out a formula for the total of the three step stair, which works for the entire 3 step stair on a 10 by 10 grid.
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number stairs
54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 The first 3step stair is made up of 1 + 2 + 3 + 11 + 12 + 21 = 50 If I move the 3step stair 1 unit to the right, the 3step stair would be made up of 2 + 3
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number grid
For example take 2x2 square and multiply its corners diagonally. 25 26 35 36 25 x 36= 900 26 x 35= 910 910  900= 10 77 78 87 88 77 x 88 = 6776 78 x 87= 6786 6786  6776= 10 After a few results I observed that any box of 2 by 2 the difference will always be 10 if grid size is 10 by 10. 4 5 6 14 15 16 24 25 26 I decided to try out some square but this time I going to do it bigger like 3 by 3 and 4 by4.
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Maths algerbra
I done the same to 3/4  2/3 9/12  8/12 which equalled 1/12 and my second difference. I shall begin to search for an algebraic rule for the first differences Number 1 2 3 4 5 nth n+1th D1 1/6 1/12 1/30 1/42 1/56 To find the nth term I had to take away the n+1th term away from the nth term ; n+1  n n+2 n+1 I made the denominators the same and got ; (n+1)(n+2)  n(n+2) (n+1)(n+2) (n+1)(n+2) I then multiplied the brackets out by completing the square. For the numerator X n 1 n n � n 1 n 1 X n 2 n n � 2n (n � + 2n + 1)
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Investigating when pairs of diagonal corners are multiplied and subtracted from each other.
40 16 x 49 = 784 19 x 46 = 874 Difference = 90 51 x 84 = 4284 54 x 81 = 4374 Difference = 90 67 x 100 = 6700 70 x 97 = 6790 Difference = 90 In a 4 x 4 box on a 10 x 10 grid the difference is 90. Algebraic Method x x + 3 x + 30 x + 33 x(x + 33) = x� + 33x (x + 3)(x + 30)
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This piece of coursework is called 'Opposite Corners' and is about taking squares of numbers from different sized number grids
Prediction I predict that a 3x3 square from a 5 wide grid, it will have a final difference of 20. Proof: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Comparison: N N+2 x N+10 N+12 N (N+12) = N� + 12N (N+2) (N+10) = N� + 10N + 2N + 20 = N� + 12N + 20 Difference: (N� + 12N + 20)
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Number Stairs
The stairs position is always worked out by the number in the bottom left hand corner. The position= n Investigation for 3 step stairs on a 10 by 10 grid 21 11 12 1 2 3 22 12 13 2 3 4 23 13 14 3 4 5 24 14 15 4 5 6 n Total 1 50 2 56 3 62 4 68 As the stair position goes up by 1 the total goes up by 6 each time. So for the stair in position 5 I predict that its total will be 74. 25 15 16 5 6 7 n+2g n+g n+g+1 n n+1 n+2 The nth term can be worked out by changing
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OXO investigation
Also I will use some abbreviations in this investigation. W=Width L=length T= Number of squares needed for a winning line S= Number of winning lines. Investigation 1 I will start off by investigating square grids with the number of squares that the length and width is will be the number needed for a winning line i.e. L=W=T 3x3 8 winning lines 4x4 10 winning lines 5x5 12 winning lines Size of grid WxL N.o of squares needed for a winning line 'T' N.o of winning lines 'S' 3x3 3 8 4x4 4 10 5x5 5 12 Rule WxW W 2W+2 I got this rule because you can see there
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The purpose of this investigation is to look at diagonal differences on different sizes of grids
1x12=12 2x11=22 2212=10 2. 17x28=476 18x27=486 486476=10 3. 24x35=840 25x34=850 850540=10 4. 31x42=1302 32x41=1312 13121302=10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 3 by 3 grids 1.
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