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  • Level: GCSE
  • Subject: Maths
  • Word count: 2909

Number grid.

Extracts from this document...

Introduction

NUMBER

GRID

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Project by: Clare Bray

Date completed: May 2003

For my coursework on investigation I have chosen to use a number grid. From the grid I will draw a box around a selected amount of numbers I will then find the product of the top left number and the bottom right number this will be repeated with the top right number and the bottom left number. When this is completed I will then find the difference between the two numbers.

During this process I will be looking for patterns in and relationships between the differences that I collect.

The grid that I have started with is numbered from 1 to 100 with ten in each row and ten in each column.

10 x 10 Grid

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I drew a box around four numbers and then found the product of the top left number:

12 x 23 = 276

I then repeated this with the top right number and the bottom left number:

13 x 22 =286

On completion of this I found the difference:

286 – 276 = 10

I repeated this process and recorded the results in a table:

Box size: 2 x 2

Box

numbers

difference

1

12 x 23 = 276

13 x 22 = 286

286 – 276 = 10

2

27 x 38 = 1026

28 x 37 = 1036

1036 – 1026 = 10

 Box

numbers

difference

3

44 x 55 = 2420

45 x 54 = 2430

2430 – 2420 =10

4

73 x 84 = 6132

74 x 83 = 6142

6142 – 6132 = 10

When using the 3 x 3 box obviously I was only using the numbers in the four corners of the box thereby omitting five numbers in between i.e.

6image07.png

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28

6 x 28 = 168                      208 – 168 = 40

8 x 26 = 208

...read more.

Middle

11 x 23 = 253

13 x 21 = 273

273 – 253 = 20

2

15 x 27 = 405

17 x 25 = 425

425 – 405 = 20

3

42 x 54 = 2268

44 x 52 = 2288

2288 – 2268 = 20

4

64 x 76 = 4864

66 x 74 = 4884

4884 – 4864 = 20

I therefore chose to repeat this exercise again with a larger rectangle to try to prove my prediction. Following the same action as previously I drew a 4 x 3 rectangle around the numbers and again omitting the numbers in the middle.

14image03.png

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Box size: 4 x 3

Box

numbers

Difference

1

14 x 37 = 518

17 x 34 = 578

578 – 518 = 60

2

41 x 64 = 2624

44 x 61 = 2684

2684 – 2624 = 60

3

46 x 69 = 3174

49 x 66 = 3234

3234 – 3174 = 60

4

75 x 98 = 7350

78 x 95 = 7410

7410 – 7350 = 60

Box numberimage04.png

Box size

Difference

1

3 x 2

20

2

4 x 3

60

3

5 x 4

120

4

6 x 5

200

To find the difference for box 7 x 6 use box 6 x 5

                                       6 x 5 = 30

                                       30 x 10 = 300

Pattern:

Box size

3 x 2

4 x 3

5 x 4

6 x 5

7 x 6

=

6

12

20

30

42

image05.pngimage06.pngimage05.pngimage05.png

                                            x 10                 x 10                  x 10                  x 10                                    

Difference

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60

120

200

300

On closer examination, the differences for both box shapes display another pattern. For the square box the difference is: add 30 to the 10 to make the difference 40, then add 20 to the 30 to find the next difference and so on.

Box size

2 x 2

3 x 3

4 x 4

5 x 5

6 x 6

7 x 7

Difference

10

40

90

160

250

360

image09.pngimage10.pngimage09.pngimage08.pngimage10.pngimage11.pngimage11.pngimage10.pngimage12.pngimage10.png

image14.pngimage15.pngimage14.pngimage15.pngimage16.pngimage16.pngimage17.pngimage14.pngimage14.png

                   +30                  +50                  +70                   +90                   +110                                      

                               +20                  +20                  +20                     +20

The process to find the difference for the rectangular box is, again, almost the same as for the square box:

Box size

3 x 2

4 x 3

5 x 4

6 x 5

7 x 6

Difference

20image12.pngimage10.pngimage11.pngimage10.pngimage11.pngimage10.pngimage10.pngimage09.png

60

120

200

300

image14.pngimage16.pngimage14.pngimage17.pngimage16.pngimage14.pngimage15.png

                               +40                  +60                  +80                    +100

                                           +20                  +20                  +20                                                                                                                    

I believe that the patterns are so similar because the grid used was the same for both shaped boxes, I also believe that the number sequences of the differences are multiples of ten because the grid has ten numbers in each row.

Because of this belief I chose to try another number grid, this time I would be using a grid that was 8 x 8.

I will be using the same process as before of finding the product of the top left number and bottom right number, then the same with the top right number and bottom left number. I will then find the difference by subtracting the smaller of the two products from the larger.

8 x 8 Grid

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Box size 2 x 2

image21.png

Box

numbers

Difference

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9 x 18 = 162

10 x 17 = 170

170 – 162 = 8

2

21 x 30 = 630

22 x 29 = 638

638 – 630 = 8

3

34 x 43 = 1462      

35 x 42 = 1470

1470 – 1462 = 8

4

53 x 62 = 3286

54 x 61 = 3294

3294 – 3286 = 8

12image22.png

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12 x 30 = 360                               392 – 360 = 32

14 x 28  = 392                        

Box size 3 x 3

Box

numbers

Difference

1image23.png

12 x 30 = 360

14 x 28 = 392

392 – 360 = 32

2

25 x 43 = 1075

27 = 41 = 1107

1107 – 1075 = 32

3

30 x 48 = 1440

32 x 46 = 1472

1472 – 1440 = 32

4

43 x 61 = 2623

45 x 59 = 2655

2655 – 2623 = 32

image24.png

Box number

Box size

Difference

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2 x 2

8

2

3 x 3

32

3

4 x 4

72

4

5 x 5

128

2 x 47 = 94                         294 – 94 = 200

7 x 42 = 294

Pattern:  

Box size

2 x 2

3 x 3

4 x 4

5 x 5

6 x 6

=

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9

16

25

36

image05.pngimage06.pngimage05.pngimage05.png

                                             x 8                   x 8                    x 8                    x 8                                      

Difference

8

32

72

128

200

...read more.

Conclusion

On closer inspection of the patterns, I was able to see an easier way to explain how to work out the difference using algebra. By using the tables showing the number patterns, I could draw a simpler version that also enabled me to find the nth term:

Number pattern for square box, 10 x 10 grid

Box number (n)

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5

6

Difference (d)

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40

90

160

250

360

Therefore n = box number and d = difference

d = 10n2

Example:    d  = 10 x 22  

                       =  10 x 4

                       = 40

Therefore the next term of the sequence would be:

d = 10 x 32

   = 10 x 9

   = 90

This way to find the way the number pattern works also works for the 8 x 8 grid:

Number pattern for square box, 8 x 8 grid

Box number (n)

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2

3

4

5

Difference (d)

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32

72

128

200

d = 8n2  

Therefore: d = 8 x 22  

                     = 8 x 4

                     = 32

And the next term would be:

d = 8 x 32

   = 8 x 9

   = 72

As before this same, process applies to the 12 x 12 grid:

Number pattern for square box, 12 x 12 grid

Box number (n)

1

2

3

4

Difference (d)

12

48

108

192

d = 12n2  

Therefore: d = 12 x 22

                     = 12 x 4

                     = 48

And the next term in the sequence:

d = 12 x 32  

   = 12 x 9

   = 108

The same rule for finding the nth term applies for all number patterns with regards to the square boxes, and, although the patterns for the rectangular boxes are very similar to that of the square boxes, I was unable to find the nth term. I believe that this is because the patterns created by the rectangular boxes are not linear sequences.

...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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