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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

Extracts from this document...

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:

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2

11

12

1 x 12 = 12

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

Eg:

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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.

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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.

...read more.

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:

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11

12

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

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21

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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.

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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

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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

...read more.

Conclusion

Michelle Solley

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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

...read more.

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