# Number Grid Investigation.

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Introduction

Number Grid Investigation 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 Task. My task is to use the box above to do the following things: * Draw a box around 4 numbers. * Find the product of the top left number and the right number within this box. * Do the same with the top right and the bottom left numbers. * Calculate the difference between these products. * INVESTIGATE FURTHER... This assignment is concerned with product differences in different size matrices for different sized grids. Above is a 10 X 10 grid. We are trying to investigate product differences within this 10 X 10 grid. As you can see I have drawn a 2 X 2 grid inside the 10 X 10 matrix. To find the product difference I do: Top left number X bottom right - top right X bottom left. TL X BR - TR X BL TL = Top left BR = Bottom right TR = Top right BL = Bottom left. The product difference I found was 10. This was anywhere on the grid where a 2 X 2 grid could be drawn. To prove this I will take a 2 X 2 grid and use the above formula to work out the product difference. Below is the result of me picking a 2 X 2 grid from this 10 X 10 grid and calculating the product difference. ...read more.

Middle

( d - 1 ) n = 5 d = 7 (n - 1 = 4) (d - 1 = 6) 4 X 6 = 24 24 X 10 = 240. Using the above formula, the product difference is correct. What next? So far I have changed: * The size of the box: 2 X 2, 3 X 3, 4 X 4 etc and found a formula. * The width of the box: 3 X 2, 4 X 2, 5 X 2 etc and found a formula. * A combination of the two above and found a formula. Will my formulas work in different sized grids? So far, all my formulas have been gained from calculations within a 10 wide grid. I am now going to take a different approach in this investigation. I am going to investigate the effect on the formula when I change the grid size to say 12 X 12 or 8 X 8. I am going to take one number higher than ten and one lower to see what happens. Starting with: 8 X 8 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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 First I will see what the Product difference of a 3 X 3 square is. Here are 3 examples: 1 2 3 9 10 11 17 18 19 (TL X BR) - (TR X BL) (1 X 19) - (3 X 17) = 32. 6 7 8 14 15 16 22 23 24 (6 X 24) - (8 X 22) = 32 41 42 43 49 50 51 57 58 59 (41 X 59) ...read more.

Conclusion

I may also have varied the shape of the grid. I could have had an 'L' shaped one like the one below: 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 This approach especially could have gone into greater depth. I could have tried cross shape, triangles etc... I could have made my numbers run vertically rather than horizontal and used this approach in grid with multiples, grids of different shapes, grids of different sizes. The approaches are un-countable. I also think it would be hard for me to present all my findings and investigations in an interesting way without losing the interest of the reader. Final conclusion. In conclusion to the project as a whole I think that I handled the given task very well. I have followed my plan and brainstorm accurately gathering different calculations and formulas as I worked my way down. I have presented my results and formulas in an appropriate way so as to make sense to the reader. I have challenged my own thoughts by using mini predictions throughout the experiment and tested them using the phrase 'let's see...' I have used algebra where necessary to prove why the formula or calculation I have gathered has worked in the way that it has. Unfortunately I have not been able to show my results in any form chart but I have used number patterns to make my results more easily understandable. To gather a larger set of formulas and results and to improve this investigation I could have ventured into some of the approaches written above or gone into greater depth with my approaches carried out. This could prove difficult as presenting it in an interesting way would cause great problems. Stuart Small. 11 - O D. 1 Page 1 of 28 ...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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