# 10x10 number grid

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Introduction

Stephanie Evans – 20002303

GCSE Maths Coursework

Ma 1 Number Grids

Aim:

My task is to look at a 10x10 number grid and take a box of numbers from within that grid. I will then find the products of the numbers in opposite corners of the box and calculate the difference between the products.

I will start with a box of 4 numbers from the grid, and then change the size of the box to see what the variation may be and record my results as I progress.

Hypothesis:

Based on the example given in the original question (12x23= 276 13x22=286 difference = 10) I predict that the results for other number groups with a square or rectangle of the same size will have the same difference. i.e. a group of 4 numbers where the products of the numbers in opposite corners are subtracted to find the difference will always have a difference of 10.

I predict that if I investigate this theory further using varying sizes of square that providing the same method is used the difference between products will remain the same for that size square. E.g. a group of 4 numbers the difference is always 10. A group of 9 numbers will always have a difference that is the same as another group of 9 numbers and so on.

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 |

12 x 23=276

13 x 22=286

Difference = 10

36 x 47=1692

Middle

A 4x4 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 | 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 |

11 x 44=484

14 x 41=574

A product difference of 90

Stephanie Evans – 20002303

16 x 49=784

19 x 46=874

Again product difference of 90

61 x 94=5734

64 x 91=5824

Difference=90

66 x 99=6534

69 x 96=6624

Product difference = 90

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 |

1 x 45=45

5 x 41=205

Product difference =160

6 x 50=300

10 x 46=460

Product difference= 160

51 x 95=4845

55 x 91= 5005

Product difference = 160

56 x 100=5600

60 x 96=5760

Product difference= 160

Stephanie Evans – 20002303

Looking at square grids my theory is proved. Now can we find a rule? A simple algebra equation which would allow us to find out what the product difference is providing we know the smaller grid sizes without working it all out.

Length | Height | Area | Product Difference |

2 | 2 | 4 | 10 |

3 | 3 | 9 | 40 |

4 | 4 | 16 | 90 |

5 | 5 | 25 | 160 |

The first thing I notice is that the area of the previous grid multiplied by 10 is equal to the difference of the following grid. I.e. the area of the 2x2 grid =4 x 10 = 40 and the difference of the product difference of the 3x3 grid is 40. This is the case with the other grids I tested.

So a possible rule could be the length/ height of the grid minus one squared multiplied by ten. The rule looks like this:

(L-1) x (H-1) squared x 10.

Length | Height | Area | Product Difference |

2 | 2 | 4 | 10 |

3 | 3 | 9 | 40 |

4 | 4 | 16 | 90 |

5 | 5 | 25 | 160 |

So in the first grid 2-1=1 and 1 squared is 1 x 10 =10,

3-1 = 2 squared is 4 x 10 = 40,

4-1= 3 squared =9 x 10 = 90,

5-1=4 squared = 16 x 10 = 160.

The rule works so far.

Conclusion

By looking at my first small grid in this 5x10 grid the difference is 5.

Could my rule then be (L-1) x (H-1) x grid width (G)?

The length of the grid is irrelevant as the order in which the

Numbers appear would not change. If the new rule works a 3x3 square in this grid will have a difference of 20 because 3-1 = 2 and 2x2x5 = 20, and a 4x4 square will have a difference of 45.

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 |

1 x 13= 13

3 x 11= 33

Difference = 20

21 x 39=819

24 x 36=864

Difference = 45

Stephanie Evans – 20002303

A 3x2 or 2x3 rectangle should have a difference of 10, and a 3x4 or 4x3 a difference of 45.

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 |

1 x 8= 8

3 x 6= 18

Difference = 10

31 x 49= 1519

34 x 46= 1564

Difference= 45

Summary

In conclusion my rule is correct and will work for and sized square or rectangle within any grid.

H = Height of the square/ rectangle

L= Length of square/ rectangle

G= Width of the number grid

In algebra terms the equation can then be written as (L-1) x (H-1) x G.

My investigation has progressed in such a way that I have found a rule which I am convinced will apply to any square or rectangle in any given grid width. I am happy that I have extended the investigation in such a way that has proved my theory, and that my rule works.

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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