GCSE: Number Stairs, Grids and Sequences
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Number Grids
Without taking the different numbers on the top row into account, this would mean that the difference between the two products would now be the number in the bottom right hand corner. However, because the top left hand square is one less than the top right hand corner, you have to take away the number inside that square away from the higher product. This in theory always gives you a difference of 10. Here is an example: 5 6 15 16 Because you are multiplying 6 x 15 and 5 x 16 you are immediately making a large difference between the two products, as you are multiplying by different numbers.
 Word count: 3515

Maths  number grid
(r+1)(r+10) r (r+11) = r (r+10) +1 (r+10)  r 11r = r +10r +r+10  r 11r =r +11r+10 r 11r =10 This algebra proves my results that for any 2x2 square the answer for the defined difference will always be 10. Furthering my Investigation I am now going to further my investigation by increasing the size of the squares. I am going to be repeating my process used previously but will be looking at randomly selected 3x3 squares, my aim being to see if I can find a trend and a pattern in my results.
 Word count: 6680

Number Stairs
Therefore T = 56. 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 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 Stair number (N)
 Word count: 5404

Number grid
3 x 3 Box n n+1 n+2 n+10 n+11 n+12 n+20 n+21 n+22 4 x 4 Box n n+1 n+2 n+3 n+10 n+11 n+12 n+13 n+20 n+21 n+22 n+23 n+30 n+31 n+32 n+33 I am going to display my results in a table, where I would try finding any patterns within the results I have. 10 x 10 Grid Square Box sizes Difference Pattern 2 x 2 10 =1 x 10 12 = 1 3 x 3 40 = 4 x 10 22 = 4 4 x 4 90 = 9 x 10 32 = 9 Predict 5 x 5
 Word count: 3313

Numberical method
Decimal research For function f(x) = , there is a change sign between the interval of 2 and 3 means there is a root. In this method i am going to use Excel spreadsheet to do decimal search by take increments in of the size 0.1 between the interval 2 and 3 and work out the function value for each. To work out x I'll use the formula r+1= r+0.1 given function of f(x) ==> f(x)+1= x 3 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2 f(x)
 Word count: 771

Maths Grids Totals
3 x 3 = 10 x 22 = 10 x 4 = 40). This gives me the formula 10(n1)2. Using this formula, I predict that the difference for an 8 x 8 square will be 10(81)2 = 10 x 72 = 10 x 49 = 490. I will take a random 8 x 8 square from the 10 x 10 grid: 12 13 14 15 16 17 18 19 22 23 24 25 26 27 28 29 32 33 34 35 36 37 38 39 42 43 44 45 46 47 48 49 52 53 54 55 56 57 58
 Word count: 5017

Number grid
= n�+n+10n+10 = n�+11n+10 I will now subtract the two algebraic products to get an overall product. n�+11n+10 n�+11n  +10 This proves that the difference is always 10 I have proved that the above results hold for any 2 by 2 square. This shows that for any 2 by 2 square the difference will always be equal to 10. To extend this investigation I am going to investigate the effect of changing the shape of the box. This variable will look at the effect of changing the length of the rectangle by 1 square each time.
 Word count: 2272

Number Grid
For the 2nd part of the investigation, I will be using 2 variables to extend the task further. I am going to investigate what the difference between the opposite products inside a rectangular shaped box is. I will do this by using the 1st formula and then see if there are any connections or similarities I can make. For the 3rd part of the investigation, I will be using 3 variables to extend the task even further. I will be using the first two formulas to link them with the grid size. I will then find a formula which will relate the shape of the box inside the grid and the size of the main grid.
 Word count: 1006

maths grid coursework
n+g n+(w1)+g 3x3 sqaure on a 10 size grid n n+(w2) n+(w1) n+g n+(w2)+g n+(w1)+g n+2g n+(w2) +2g n+(w1)+ 2g I can simpifly this into a forumula G(n1)� and i realise i have 2 varibles that can be changed the number of sqaures grid size RESULTS Size of squares Difference between answers 2x2 10 3x3 40 4x4 90 5x5 160 From these results i can conclude that the formula for this pattern (when n is the number in and G equals grid size) G(n1)� i will now prove this by finding the square of 9 by 9 on a 12 sqaure grid example 9=n 12=G algerbraic equation Rectangles 10 squared grid 2x3 8 9 10 18 19
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Maths Coursework  Grid Size
'N' will represent the top left number in the square/rectangle. I will calculate the DPD of that formula. If the answer to both the numerical and the algebraic formulas are the same then I will prove my theory by changing the position of the square/rectangle. I will next change the size of the grid and see how this affects the DPD. I will manipulate the numbers again using 'g' to represent grid size. I will then test this on different grid sizes and square/rectangle positions. I will find a pattern between the grid sizes and also have a general formula for grid size and square/rectangle position.
 Word count: 2506

MathsNumber Grid
I will now use this grid to come up with a prediction for the other grids of the same length and width. I will do this by multiplying 63 and 54 together to give me a product of 3402. Afterwards, I will multiply 64 and 53 to give me a product of 3392. Lastly, I will subtract 3392 from 3402 to give me a product difference of 10. I will draw up 3 other grids, to test if my prediction is correct.
 Word count: 4188

GCSE Maths Sequences Coursework
1 2 3 4 5 6 Sequence 1 5 13 25 41 61 1st difference +4 +8 +12 +16 +20 2nd difference +4 +4 +4 +4 Unshaded I can see here that there is not much of a pattern in the 1st difference, but when I calculate the 2nd difference I can see that it goes up in 4's, therefore this is a quadratic sequence and has an Nth term. The second difference is 4 therefore the coefficient of N� must be half of 4 i.e.
 Word count: 3939

number grid
I will then try changing the size of the rectangles and looking for patterns there. I will look at 2 by 4 rectangles, 5 by 3 rectangles and 4 by 5 rectangles. I will also look at changing the size of the number grids to see if this has an affect on the patterns. I will look at a 9 by 9 grid, an 11 by 11 grid and a 5 by 5 grid. I will be looking for patterns in 2 by 2 squares within the different size grids and trying to find an algebraic formula to explain my findings.
 Word count: 7010

Mathematics  Number Stairs
number, for example, 20 T = 6 x 31 + 44 = 230 So: 51 41 42 31 32 33 31 + 32 + 33 + 42 + 41 + 51 = 230 So therefore my prediction works but I must prove it algebraically: n+20 n+10 n+11 n n+1 n+2 n + (n+1) + (n+2) + (n+10) + (n+11) + (n+20) = 6n + 44 T = 6n + 44 has been proved to work. 8 9 10 11 12 1 2 3 T = 6n + 44 4 5 So after the first part of the investigation, I will
 Word count: 4455

Math Grid work
2 x 2 square 24 25 34 35 46 47 56 57 89 90 99 100 I notice that I get the same number for any same size square so to prove this I will use X and prove why the answer is always 10. I can use a square in which the boxes are the edges of the square on the grid. X X + 1 X + 10 X + 11 If I multiply this out if I were doing what I was doing with the numbers in the grid I would, (X+1)(X+10)...which is = X2+11X+10...and...
 Word count: 1729

Number Stairs
shape is 26 +27 + 28 + 36 + 37 + 46 = 200 The stairtotal for this stair shape is 27 + 28+ 29 + 37 + 38 + 47= 206 The stairtotal for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62 The stairtotal for this stair shape is 4 + 5 + 6 + 14 + 15 + 23 = 68 This table summarizes these results : Stair number 24 25 26 25 3 4 Stair Total 188 194 200 206 62 68 In order to find a formula
 Word count: 587

Mathematics Borders
Now I will find the formula for the square numbers of the sigma notation. Sigma Notation 12 12+22 12+22+32 12+22+32+42 12+22+32+42+52 12+22+32+42+52+62 Value 1 5 14 30 55 91 1 5 14 30 55 91 4 9 16 25 36 2nd difference > 5 7 9 11 3rd difference > 2 2 2 The 3rd difference is denoted by the term 6a. Therefore in order to find a, I will replace the 3rd difference value 6a = 2 a = 2 = 1 6 3 By the given formula, Un = an3 + bn2 +cn + d, I will now be able to arrive to the generalized formula.
 Word count: 861

Personal Exercise Program
It is for all these reasons that I will be trying to push myself to my limit. The two areas of specific fitness I will be working on to make my performance levels in football higher are AGILITY and UPPER BODY STRENGTH. Agility is a major component of football. Being able to dodge your opponent and use quick feet to beat them to the ball is vital. Agility is not a weakness of mine, infact I think I am pretty good at it but, it is something I would like to improve.
 Word count: 2263

Number Grid and Stairs
Here are some rules I have found for a 3x3 stair in a 10x10 grid so far. Rules 1 = +6 1 = 60 1 = 6 1 = +60 I am now going to find the algebraic rule for finding the total of the numbers encased in a 3x3 stair anywhere in a 10x10 grid. 6n+44 I will now test my algebraic rule on random places in the grid to see if the rule works in on all the numbers.
 Word count: 2067

Mathematical Coursework: 3step stairs
12 13 14 15 16 17 18 19 10 1 2 3 4 5 6 7 8 9 10 Because I haven't discovered the algebraic equation, this would allow me to find the total stair number in less than minutes. I would have to find the total by using the basic method of adding. The Calculations 1+2+3+11+12+21= 50 5+6+7+15+16+25=74 83+73+63+64+65+74=422 47+48+49+57+58+67=326 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69
 Word count: 8908

rectangles. I will be trying to develop a formula that will enable me to calculate the sum of all the numbers in a rectangle given
We would get the value by calculating the sum of all the numbers. A way we could find this value out could be by using a formula. But what formula could we use? n n+1 n+2 n+10 n+11 n+12 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
 Word count: 1612

100 Number Grid
36 x 47 = 1692 37 x 46 = 1702 Product difference = 10 F. 49 x 60 = 2940 50 x 59 = 2950 Product difference = 10 G. 74 x 85 = 6290 75 x 84 = 6300 Product difference = 10 H. 85 x 96 = 8160 86 x 95 = 8170 Product difference = 10 By continuing this investigation, my prediction proves to be correct. As a result, an algebraic formula can be used to work out the product difference.
 Word count: 4202

Staircase Coursework
I will now find a formula for the total, which will make it easier to calculate the sum. I will label a stair algebraically This is an example of stair 1 n stands for the number in the left bottom corner of the step. In order of that if : 1 = n 2 = n + 1 and so on ................ n+20 n+10 n+11 n n+1 n+2 21 11 12 1 2 3 = I am now going to add up the single squares, which gives me the formula n + (n + 1) + (n + 2)
 Word count: 3422

Number grid coursework
I shall find the product of the top right and bottom left numbers. Now I shall find difference between these products. Let's now move the square and do same calculations. 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
 Word count: 2284

Number Stairs
On my 10x10 grid, I will first use an snumber of 25, which makes the stotal = 194. I will then work systematically, increasing the snumber by 1 every time. However, the snumber is limited in its placement  it cannot be a multiple of 10, or a multiple of 10 1  else it would not fit on the grid. I will record results in the following table: snumber (n)  1 2 3 4 5 stotal (S)  50 56 62 68 74 I can see from these results I can see that for every increase in n, S increases by 6.
 Word count: 1467