# 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

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Introduction

Page 1

Michelle Solley

AQA – Set Coursework Task

Module 4 Investigation

We were asked to 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

Eg:

1 | 2 |

11 | 12 |

1 x 12 = 12

We did the same with the top right and bottom left numbers.

Eg:

1 | 2 |

11 | 12 |

2 x 11 = 22

We were then asked to find the difference between the two answers. I am going to try different 2 x 2 boxes in a 10 x 10 grid to try and find a pattern.

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 |

2 x 2 Box

1 x 12 = 12 15 x 26 = 390

> 10 > 10

2 x 11 = 22 16 x 25 = 400

81 x 92 = 7452 89 x 100 = 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.

Middle

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

5 x 5 Box

1 x 45 = 45 56 x 100 = 5600

> 160 > 160

5 x 41 = 205 60 x 96 = 5760

Page 3

Michelle Solley

I am going to put this information into a table to show my findings.

Column A shows the difference between the answers to each sum.

Column B shows the increase from the answers to the sums in column A.

BOX SIZE | a | b |

2 X 2 | 10 | 30 |

3 X 3 | 40 | 50 |

4 X 4 | 90 | 70 |

5 X 5 | 160 | 90 |

6 X 6 | 250 | 110 |

As you can see I have predicted that the 6 x 6 box in a 10 x 10 grid has a difference of 250 between each answer. The increase in column B rises by 20 each time.

For the purposes of writing a rule in algebra I am calling the box B. For example:

1 | 2 |

11 | 12 |

This 2 x 2 box in a 10 x 10 grid will be called B.

1 | 2 | 3 |

11 | 12 | 13 |

21 | 22 | 23 |

This 3 x 3 box in a 10 x 10 grid will also be called B.

I am now going to find a rule for getting the difference without having to draw a 10 x 10 grid to check.

Using the Above table to refer to I have worked out the following:

2 x 2 Box

(2 – 1) x 10 = 10

3 x 3 Box

(3 – 1) x 10 = 40

4 x 4 Box

(4 – 1) x 10 = 90

Page 4

Michelle Solley

This rule works for all equal sided boxes in a 10 x 10 grid, the rule is:

(B-1) X 10

I am now going to try the same thing on a 9 x 9 grid to see if the rule will work for all equal sided boxes whatever the grid size.

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 |

2 x 2 Box

1 x 11 = 11 4 x 14 = 56

> 9 > 9

2 x 10 = 20 5 x 13 = 65

3 x 3 Box

7 x 27 = 189 55 x 75 = 4125

> 36 >

9 x 25 = 225 57 x 73 = 4161

4 x4 Box

19 x 49 = 931 42 x 72 = 3024

> 81 > 81

22 x 46 = 1012 45 x 69 = 3105

I will now continue by drawing a 5 x 5 box in a 9 x 9 grid and predict what the differences will be for a 6 x 6 box in a 9 x 9 grid.

Page 5

Michelle Solley

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 |

5 x 5 Box

1 x 41 = 41 41 x 81 = 3321

> 144 > 144

5 x 37 = 185 45 x 77 = 3465

As I have done before I will place the information I have gathered into a table, and as above Column A shows the difference between the answers to each sum. Column B shows the increase from the answers to the sums in column A.

BOX SIZE | a | b |

2 X 2 | 9 | 27 |

3 X 3 | 36 | 45 |

4 X 4 | 81 | 63 |

5 X 5 | 144 | 81 |

6 X 6 | 225 | 99 |

Conclusion

Michelle Solley

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 |

2 x 3 Rectangle

1 x 20 = 20 5 x 24 = 120

> 18 > 18

2 x 19 = 38 6 x 23 = 138

2 x 4 Rectangle

3 x 31 = 93 7 x 35 = 245

> 27 > 27

4 x 30 = 120 8 x 34 = 272

2 x 5 Rectangle

39 x 76 = 2964 44 x 81 = 3564

> 36 > 36

40 x 75 = 3000 45 x 80 = 3600

2 x 6 Rectangle

28 x 74 = 2072 32 x 78 = 2496

> 45 > 45

29 x 73 = 2117 33 x 77 = 2541

This investigation of rectangles in a 9 x 9 grid has shown me that the answers to the sums increase in 9’s. As with the 10 x 10 grid I will show my findings in a table, Column A shows the difference between the answers to each sum. Column B shows the increase from the answers to the sums in column A.

Page 10

Michelle Solley

Rectangle size | a | b |

2 X 3 | 18 | 9 |

2 X 4 | 27 | 9 |

2 X 5 | 36 | 9 |

2 X 6 | 45 | 9 |

2 X 7 | 54 | 9 |

My findings have proved that my prediction for the rule for rectangles is correct. As I did with the 10 x 10 grid I will show my workings that prove the rule.

(2 – 1) (3 – 1) x 9 = 18

(2 – 1) (4 – 1) x 9 = 27

As you can see I have predicted the answer to the 2 x 7 rectangle

(2 – 1) (7 – 1) x 9 = 54

My investigation has proved that there is a rule for finding the answer to a sum created by any equal sided box in any size grid. Once again, the rule is:

(B – 1) G

The investigation also proved that there is a rule for finding the answer to a sum created by any rectangle. The rule is:

(W – 1) (L – 1) G

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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