GCSE: Number Stairs, Grids and Sequences
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Number Stairs
Every time it goes up by 6. Stair Number: 71 72 73 74 Stair total: 470 476 482 488 Difference: 6 6 6 Now I know this pattern I can predict that the stair total for stair 75 will be 492 as 488+6. To make sure that is right I have to test prediction. 75+76+77+85+86+95=494 my predict is right We can therefore say that every time you move the stair shape one square to the right you increase the stair total by 6 and every time you move the stair shape one square to the left you decrease the stair total by 6.
 Word count: 2022

Investigate the relationships between the numbers in the crosses.
Finding numbers: 36 (a) 36 (b11) (a+9) 45 46 47 (a+11) (b2) 45 46 47 (b) (a+10) 56 (a+20) (b1) 56 (b+9) 36 (c20) 36 (l9) (c11) 45 46 47 (c9) (l) 45 46 47 (l+2) (c10) 56 (c) (l+1) 56 (l+11) 36 (x10) (x1) 45 46 47 (x+2) (x) 56 (x+10) Finding numbers in equations: * If a is known: = (a+9)(a+11)  (a)(a10) = (a +11a+9a+99)  (a +20a) = (a +20a+99)  (a +20a) = 99 * If b is known: = (b)(b2)
 Word count: 2530

Investigating the number of patterns in a certain grid.
y+5 Square 6 extensions Rules for different shapes Change pattern in grid 1 3 5 7 etc. This is the number grid I am going to be investigating: 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
 Word count: 2608

Marked on the grid is a stair shape; this is called a 3Step Stair.The total of the numbers inside the stair shape is:
However, on larger or smaller number grids some blocks may not exist or more blocks may exist. The formulae in the furthest right hand side column are always in the furthest right hand side column on every size of grid as these define what the number is at the end of each row. However, the height and width of the number grid may be more than, or less than, five. Additional blocks on a larger grid will follow the same pattern as that which can be seen above and if on a smaller number grid there will just be less
 Word count: 2084

Number stairsMy aim is to investigate the relationship between the stair total and the position of the stair shape on the grid for 3 step
For the first term n=1, so 6n=6. But the first term is 50 which is 44 more than 6n. This suggests that the formula is 6n+44. Trying a few values of n will help prove that my formula is correct. (6 multiplied by 1) +44 =50. (6 multiplied by 2) +44 =56. (6 multiplied by 3) +44 =62. (6 multiplied by 4) +44 =68. (6 multiplied by 5) +44 =72. I will now pick a random 3 step stair and test my formula.
 Word count: 2734

Number Stairs Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid, for ot
(Apart from the last column, which only consists of 1 number.) The difference between these numbers is 10 because in the grid, each row contains 10 numbers. Table of Results: snumber stotal 1 50 + 6 2 56 + 6 3 2 + 6 4 68 25 194 To go from one term to the next, simply add 6 each time. However, this does not show the relationship between the snumber and the stotal. This sequence (adding/subtracting a number each time), is known as an arithmetic sequence.
 Word count: 2015

Maths Coursework  Number Grid
Once I have fully investigated the patterns within the squares and found an algebraic formula for the patterns I will look at rectangles. I will start by looking at a 3 by 2 rectangle and looking for patterns there; if I find a pattern I will try to work out a formula for this pattern. 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.
 Word count: 2683

Number Grids Coursework
1 32 x 54 = 1728 34 x 52 = 1768 Difference = 1768  1728 = 40 Here, however I found that the difference was 40 to find out if the difference was the same for every 3 x 3 square, I decided to test another 3 squares, and here are the results I received; 35 36 37 45 46 47 55 56 57 Square 2 35 x 57 = 1995 37 x 55 = 2035 Difference = 2035  1995 = 40 38 39 40 48 49 50 58 59 60 Square 3 38 x 60 = 2280
 Word count: 2696

Number Grid Coursework
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 2 x 2 12x23 = 276 13x22 = 286 difference = 10 64x75 = 4800 65x74 = 4810 difference = 10 38x49 = 1862 39x48 = 1872 difference = 10 N N+1 N+10 N+11 X = N x (N + 11) = (N x N)
 Word count: 2920

Investigate the product of the top left number and the bottom right number of a box drawn around four numbers in a 10 x 10 grid
= 8900 >10 > 10 82 x 91 = 7462 90 x 99 = 8910 Page 2 Michelle Solley My investigation has shown me that any 2 x 2 box in a 10 x 10 grid has a difference of 10. I am now going to try larger boxes in a 10 x 10 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
 Word count: 2333

Number Stairs
on a 9�9 grid 6x + 40 3step stair on an 8�8 grid 6x + 36 4step stair on a 10�10 grid 10x + 110 Table 1 Step size 8�8 grid 9�9 grid 10�10 grid 11�11 grid 1 x + 0 x + 0 x + 0 x + 0 2 3x + 9 3x + 10 3x + 11 3x + 12 3 6x + 36 6x + 40 6x + 44 6x + 48 4 10x + 90 10x + 100 10x + 110 10x + 120 5 15x + 180 15x + 200 15x + 220 15x
 Word count: 2132

Number Grid
(Could try another example here but the difference will be 10) Move to the general case to see if the answer for a 2 x 2 grid is always 10 (general case means true for any 2 x 2 square on this number grid) n n+1 n+10 n+11 n means starting any where in the grid n+1 because it is one more further on n+10 because each row goes up 10 at a time n+11 because this is one more than the previous number So multiplying as before we get; Top left hand number x Bottom right hand number n x (n+11)
 Word count: 2120

My coursework task is to investigate why, in a number grid square of 1100, when a section of two by two squares is extracted and the two opposite squares are multiplied and then subtracted the result is always 10.
Now that I have proven the formula with 2x2 squares extracted from a 1100 grid square, will the formula still apply to a 3x3 square and will the result be 10? (3x21) (1x23) = 40 (36x24) (34x56) = 40 (67x89) (69x87) = 40 (80x98)(78x100)=40 As we can see the result is clearly 40, no matter what section of the number square we use. We can once again show why the result is always 40 by using algebra as formulas. (P+20) (P+2)  P (P+22) Now we can put the formula to the test by using it with numbers: (57x75) (55x77)
 Word count: 2571

Maths Coursework on Diagonal Differences
a 5 by 5 grid and an 11 by 11 grid, and find the formula to see if they are the same for a 10 by 10 grid. I will also do an extension by doing a rectangle as well as a square and then find the diagonal difference and the formula for this. I am going to find the formula by finding the diagonal difference of all the sizes within the 10 by 10 grid, and then try to find any pattern, which would help me in finding the formula by drawing a grid.
 Word count: 2332

Number Grid
Prediction I predict that square number 5 will have a difference of 10. 5x16 = 80 6x15 = 90 Difference = 9080 = 10 Square Number Difference 5 10 I will now try this with an 8x8 grid, to help with finding the formula for any size grid. Grid Two 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
 Word count: 2816

Find relationships between the stair total and the position of the stair shape on the grid for a three step stair.
+ (x + 2) + (x + 10) + (x + 11) + (x + 20) Total is T = (xR+c) where R is the reference number and C is the coefficient Therefore gathering the terms I found the formula to be 6x + 44 Other Step Stairs on A 10X10 Grid By knowing this now I then went onto work out other steps on a 10x10 grid. I started by working out a 2 step stair using the method I used to work out the 3 step stair. 11 1 2 x + (x + 1)
 Word count: 2296

Mathematics Layers Coursework
Algebra: The formula for this problem is: n This is were n equals the size of the grid. When we substitute numbers in it is: 6 Part Two A For part two A I will investigate the relationship between number of arrangements and the size of the grid when there are two layers of cubes on a grid of six squares. Method: I am, like part 1, going to draw out all of the combinations for the specifications above to find out how many combinations there are.
 Word count: 2679

Number Grids.
I will need a new way to get around this problem so I try to introduce algebra. N N+1 N+10 N+11 This grid (above) represents a 2x2 grid; it can be used with the correct formula to work out any difference for a 2x2 grid. I then tried the formula to fin the two numbers multiplied together. N?(n+11), but could this be improved? I tried tuning the formula into one formula. (n+1)(n+10) when I expanded the brackets, it reveals the difference for any 2x2 grid.
 Word count: 2409

Number grid.
and then found the product of the top left number: 12 x 23 = 276 I then repeated this with the top right number and the bottom left number: 13 x 22 =286 On completion of this I found the difference: 286  276 = 10 I repeated this process and recorded the results in a table: Box size: 2 x 2 Box numbers difference 1 12 x 23 = 276 13 x 22 = 286 286  276 = 10 2 27 x 38 = 1026 28 x 37 = 1036 1036  1026 = 10 Box numbers difference
 Word count: 2909

He analysis of number patterns on various types of number grids.
The Rule that would have to be applied to the above numbers is as follows: A box is drawn around the four or more numbers as shown in the above diagram. Find the product of the top left number and the bottom right number in the box. Do the same with the top right and the bottom left numbers, Calculate the difference between these products, making sure the number is a positive number. Note that in the above grid of 10 x 10, we have the following boxes and the specific numbers in the boxes.
 Word count: 2790

Investigate the difference of the product of the diagonally opposite corners of a certain shape, drawn on a 10x10 grid with the individual squares numbered off 1 to 100.
46 x 55 = 2,530 45 x 56 = 2,520 Diff = 10 It seems so. I now predict that all 2x2 squares will have a difference of 10. I shall test this prediction: 70 x 79 = 5,530 69 x 80 = 5,520 Diff = 10 I am correct; all 2x2 squares have a difference of 10 This brings up the obvious question: * Is the difference different when the square changes size? In answer to this I will do a test for a 3x3 square and a 4x4 square 3x3 square 1 2 3 4 5 6 7
 Word count: 2603

Step Stairs  number stairs.
However, on larger or smaller number grids some blocks may not exist or more blocks may exist. The formulae in the furthest right hand side column are always in the furthest right hand side column on every size of grid as these define what the number is at the end of each row. However, the height and width of the number grid may be more than, or less than, five. Additional blocks on a larger grid will follow the same pattern as that which can be seen above and if on a smaller number grid there will just be less
 Word count: 2047

Diagonal Difference.
I will also do an extension by doing a rectangle and a square and then find the diagonal difference and the formula for this. I am going to find the formula by finding the diagonal difference of all the sizes within the 10 by 10 grids, and then try to find any patterns, which would help me in finding the formula by drawing a grid. This method would be good because it will show me the diagonal difference of any square of any size, because I would just have to insert the value of n in the formula to find
 Word count: 2442

Number grid.
5 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 Example One 12 13 22 23 12 13 DIFFERENCE x 23 x 22 286 = 276 = 286  276 = 10 I would like to show one more example of this to see if the difference will always be 10. 14 15 24 25 14 15 Difference x 25 x 24 360 350 360  350 10 I am now going to use
 Word count: 2697

Number Grid.
I am going to try one more: 17 18 27 28 17 x 28 = 476 18 x 27 = 486 The difference is 10 again! I am now going to try and work out a formula, and try another one using the formula: n n+1 n +10 n +11 If this formula is correct I should be able to pick any number from the grid and use the formula to get the right answer If n =77 then: (n) (N +11) = n2 + 11n. & (n + 1) (n + 10) = n2 + 11n +10. (77) (77+11)
 Word count: 2399