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• Level: GCSE
• Subject: Maths
• Word count: 2333

# 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

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

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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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