# Number grids

Extracts from this document...

Introduction

Number grids On a grid number 1 to 100 I have a drawn a rectangle on the grid around the numbers 29, 30, 39 and 40. Then I have multiplied diagonally the opposite numbers in each corner of the rectangle. 1-100 number 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 30 31 32 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 29 30 39 40 On the grid I have located a 2 by 2 rectangle in a different area and I will see if there is any kind of pattern in the answers. I will place ma rectangle around the following numbers 45, 46, 55 and 56. 45 46 55 56 I have noticed a pattern between both of the answers as when I place the 2 by 2 rectangle on the number grid. ...read more.

Middle

22 23 24 25 32 33 34 35 800-770=30 As I predicted the difference would be 30, which it has turned out to be, as before I will check it using algebra. n n+1 n+2 n+3 n+10 n+11 n+12 n+13 I know that the difference will be 30 for any 2 by 4 rectangle drawn anywhere on the number grid. Now I am going to look at a 2 by 5 grid and see if the pattern carries as I increase the rectangle. If I'm correct the difference will be 40. 61 62 63 64 65 71 72 73 74 75 n n+1 n+2 n+3 n+4 n+10 n+11 n+12 n+13 n+14 As before I'm correct the pattern dose continue to carry on as it goes up n 10's each time. If I was to change the shape of the rectangle would it make a difference? I am going to look at the following grid sizes and see what happens: 3 by 2, 4 by 2 and 5 by 2 rectangles. 45 46 55 56 65 66 So far I have noticed that the difference is the same as the 2 by 3 rectangle. I will use algebra to check. n n+1 n+10 n+11 n+20 n+21 The difference between the 2 values is 20 again. ...read more.

Conclusion

55 56 57 58 59 65 66 67 68 69 75 76 77 78 79 85 86 87 88 89 n n+1 n+2 n+3 n+4 n+10 n+11 n+12 n+13 n+14 n+20 n+21 n+22 n+23 n+24 n+30 n+31 n+32 n+33 n+34 We have got a difference of 120 between the two values so I now have rule I will clearly show in how this rule works in a table by bring all my data together as I have looked at the different kinds of rectangles and worked out a pattern for each sequence. Size of Rectangle 1st differences 2nd differences (differences of differences ) Solution 2 x 2 10 +10 1x1x10=10 2 x 3 20 +10 1x2x10=20 2 x 4 30 +10 1x3x10=30 2 x 5 40 +10 1x4x10=40 +10 3 x 3 40 +20 2x2x10=40 3 x 4 60 +20 2x3x10=60 3 x 5 80 +20 2x4x10=80 +10 4 x 4 90 +30 3x3x10=90 4 x 5 120 +30 3x4x10=120 4 x 6 150 +30 3x5x10=150 I have showed how, the pattern seems to build up as we increase the rectangle by 1 row. I have added an extra column on the end which is the solution of the value of each difference found using a different multiplication as I have worked out in the solution column. The way I have worked out this is by taking away 1 less from each number of the size of each rectangle Sandeep Patel ...read more.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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