# I am going to investigate by taking a square shape of numbers from a grid, and then I multiply the opposite corners to find the difference of these two results. Firstly I am going to start with a 10x10 grid

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Introduction

Natalien nasir

Gcse Math's – number grid coursework

I am going to investigate by taking a square shape of numbers from a grid, and then I multiply the opposite corners to find the difference of these two results.

Firstly I am going to start with a 10x10 grid and pick up 4 different squares, I will start with the 2x2 square. Then I move on and use the 3x3, 4x4 and the 5x5 square.

number | Left corner x right corner | Right corner x left corner | Products difference |

1 | 13x24=312 | 14x23=322 | 10 |

2 | 14x25=350 | 15x24=360 | 10 |

3 | 25x36=900 | 26x35=910 | 10 |

I have noticed that the products difference of 2x2 squares in a 10x10 grid equal to 10. I predict if I move the 2x2 square to the right or down I will get the same answer.

4 | 34x45=1530 | 35x44=1540 | 10 |

My prediction is right. I am going to use algebra to test my results.

n | n+1 |

n+10 | n+11 |

(n+1)(n+10)=n²+10+11n

n(n+11)=n²+11n

Products difference is equal to (n²+10+11n) – (n²+11n) =10

In the same grid I will now work out a 3x3 square.

number | Left corner x right corner | Right corner x left corner | Products difference |

5 | 15x37=555 | 17x35=595 | 40 |

6 | 6x24=144 | 4x26=104 | 40 |

7 | 16x38=608 | 18x36=648 | 40 |

I have noticed that the products difference of 3x3 squares in a 10x10 grid equal to 40. I predict if I move the 3x3 square to the right or up I will get the same answer.

8 | 26x48=1248 | 28x46=1288 | 40 |

My prediction is right. I am going to use algebra to test my results.

n | n+2 |

n+20 | n+22 |

(n+2)(n+20)=n²+40+22n

n(n+22)=n²+22n

Products difference is equal to (n²+40+22n) – (n²+22n) =40

In the same grid I will now work out a 4x4 square.

number | Left corner x right corner | Right corner x left corner | Products difference |

9 | 61x94=5734 | 64x91=5824 | 90 |

10 | 62x95=5890 | 65x92=5980 | 90 |

11 | 51x84=4284 | 54x71=4374 | 90 |

I have noticed that the products difference of 4x4 squares in a 10x10 grid equal to 90. I predict if I move the 4x4 square up, I will get the same answer.

12 | 52x85=4420 | 55x82=4510 | 90 |

My prediction is right. I am going to use algebra to test my results.

n | n+3 |

n+30 | n+33 |

(n+3)(n+30)=n²+90+33n

n(n+33)=n²+33n

Products difference is equal to (n²+90+33n) – (n²+33n) =90

In the same grid I will now work out a 5x5 square.

number | Left corner x right corner | Right corner x left corner | Products difference |

13 | 6x50=300 | 10x46=460 | 160 |

14 | 16x60=960 | 20x56=1120 | 160 |

15 | 15x59=885 | 19x55=1045 | 160 |

I have noticed that the products difference of 5x5 squares in a 10x10 grid equal to 160. I predict if I move the 5x5 square to the left I will get the same answer.

16 | 5x49=245 | 9x45=405 | 160 |

My prediction is right. I am going to use algebra to test my results.

n | n+4 |

n+40 | n+44 |

(n+4)(n+40)=n²+160+44

n(n+44)=n²+44n

Products difference is equal to (n²+160+44n) – (n²+44n) =160

Box size | Results |

2x2 | 10 |

3x3 | 40 |

4x4 | 90 |

5x5 | 160 |

I have put my results in a table and I am now going to try to predict the 6x6 square in a 10x10 grid.

n | 1 | 2 | 3 | 4 |

Tn | 10 | 40 | 90 | 160 |

10n² | 10 | 40 | 90 | 160 |

10, 40, 90, 160, 250,

+30 +50 +70 +90

+20 +20 +20

nth term= 10n²

The n is not the box size because for example if I put the 2x2 square in a 10x10 grid I will get 40. Unfortunately, this formula does not work but if I minus the box size by one I will then get 10 which is the right answer.

10(b-1) ² this is my new formula.

I will test this formula on two box sizes I already have the results for:

E.g. 3x3 and 4x4 box size (see page 2).

10(3-1) ² = 40

10(4-1) ² =90 my new formula works.

number | Left corner x right corner | Right corner x left corner | Products difference |

17 | 43x98=4214 | 48x93=4464 | 250 |

18 | 33x88=2904 | 38x83=3154 | 250 |

I predict that the 6x6 square in a 10x10 grid will be 250 by using this formula:

- 10(b-1) ²
- =10(6-1) ²
- =10x25
- =250

My prediction is right.

number | Left corner x right corner | Right corner x left corner | Products difference |

19 | 11x88=968 | 18x81=1458 | 490 |

20 | 1x78=78 | 8x71=568 | 490 |

I predict that the result for an 8x8 square in a 10x10 grid will be 490 by using this formula:

- 10(b-1) ²
- =10(8-1) ²
- =10x49
- =490

My prediction is right.

I predict that the result for a 10x10 square in a 10x10 grid will be 810 by using this formula:

- 10(b-1) ²
- =10(10-1) ²
- =10x81
- =810

number | Left corner x right corner | Right corner x left corner | Products difference |

21 | 1x100=100 | 10x91=910 | 810 |

22 | 10x91=910 | 1x100=100 | 810 |

My prediction is right.

Conclusion

I found a new formula which will find the difference of the two opposing corners on a 10x10 grid for any square shape. Now that I have worked out a quadratic formula for the squares on a 10x10 grid, I can investigate further to see if I can work out a formula for a different sized number grid. I will have to use the same process as before.

Firstly I am going to start with an 8x8 grid and pick up 4 different squares and I will start with the 2x2 square. Then I move on and use the 3x3, 4x4 and the 5x5.

number | Left corner x right corner | Right corner x left corner | Products difference |

1 | 28x37=1036 | 29x36=1044 | 8 |

2 | 29x38=1102 | 30x37=1110 | 8 |

3 | 27x36=972 | 28x35=980 | 8 |

I have noticed that the products difference of 2x2 squares in an 8x8 grid equal to 8. I predict if I move the 2x2 square down I will get the same answer.

4 | 52x61=3172 | 53x60=3180 | 8 |

My prediction is right. I am going to use algebra to test my results.

n | n+1 |

n+8 | n+9 |

(n+1)(n+8)=n²+8+9n

n(n+9)=n²+9n

Products difference is equal to (n²+8+9n) – (n²+9n) =8

In the same grid I will now work out a 3x3 square.

number | Left corner x right corner | Right corner x left corner | Products difference |

5 | 3x21=63 | 5x19=95 | 32 |

6 | 11x29=319 | 13x27=351 | 32 |

7 | 4x22=88 | 6x20=120 | 32 |

I have noticed that the products difference of 3x3 squares in an 8x8 grid equal to 32. I predict if I move the 3x3 square down I will get the same answer.

8 | 44x62=2728 | 46x60=2760 | 32 |

My prediction is right. I am going to use algebra to test my results.

n | n+2 |

n+16 | n+18 |

(n+2)(n+16)=n²+32+18n

n(n+18)=n²+18n

Products difference is equal to (n²+32+18n) – (n²+18n) =32

In the same grid I will now work out a 4x4 square.

number | Left corner x right corner | Right corner x left corner | Products difference |

9 | 5x32=160 | 8x29=232 | 72 |

10 | 4x31=124 | 7x28=196 | 72 |

11 | 3x30=90 | 6x27=162 | 72 |

I have noticed that the products difference of 4x4 squares in an 8x8 grid equal to 72. I predict if I move the 4x4 square down I will get the same answer.

12 | 37x64=2368 | 40x61=2440 | 72 |

Middle

n(n+27)=n²+27n

Products difference is equal to (n²+72+27n) – (n²+27n) =72

Box size | Results |

2x2 | 8 |

3x3 | 32 |

4x4 | 72 |

I have put my results in a table and I am now going to try to predict the 5x5 square in an 8x8 grid.

n | 1 | 2 | 3 |

Tn | 8 | 32 | 72 |

8n² | 8 | 32 | 72 |

8, 32, 72, 128,

+24 +40 +56

+16 +16

nth term= 8n²

The n is not the box size because for example if I put the 3x3 square in an 8x8 grid I will get 72. Unfortunately, this formula does not work but if I minus the box size by one I will then get 32 which is the right answer.

8(b-1) ² this is my new formula.

I will test this formula on two box sizes I already have the results for:

8(2-1)2 = 8

8(3-1)2 =32 my new formula works.

I predict that the 5x5 square in an 8x8 grid will be 128 by using this formula:

- 8(b-1) ²
- =8(5-1) ²
- =8x16
- =128

number | Left corner x right corner | Right corner x left corner | Products difference |

13 | 10x46=460 | 14x42=588 | 128 |

14 | 25x61=1525 | 29x57=1653 | 128 |

My prediction is right.

I predict that the result for a 7x7 square in an 8x8 grid will be 288 by using this formula:

- 8(b-1) ²
- =8(7-1) ²
- =8x36
- =288

number | Left corner x right corner | Right corner x left corner | Products difference |

15 | 2x56=112 | 8x50=400 | 288 |

My prediction is right.

I predict that the result for an 8x8 square in an 8x8 grid will be 392 by using this formula:

- 8(b-1) ²
- =8(8-1) ²
- =8x49
- =392

number | Left corner x right corner | Right corner x left corner | Products difference |

16 | 1x64=64 | 8x57=456 | 392 |

My prediction is right.

I found a new formula which will find the difference of the two opposing corners on an 8x8 grid for any box size. I also found a new formula which will find any box sizein any number grid which is:

Conclusion

Right corner x left corner

Products difference

13

20x64=1280

24x60=1440

160

14

19x63=1197

23x59=1357

160

15

18x62=1116

22x58=1276

160

I have noticed that the products difference of 5x6 rectangles in an 8x8 grid equal to 160. I predict if I move the 5x6 rectangle up I will get the same answer.

16 | 9x53=477 | 13x49=637 | 160 |

My prediction is right. I am going to use algebra to test my results.

n | n+4 |

n+40 | n+44 |

(n+4)(n+40)= n²+160+44n

n(n+44)=n²+44n

Products difference is equal to (n²+160+44n) – (n²+44n) =160

Box size | Results | |

2x3 | 16 | 1x2x8 |

3x4 | 48 | x3x82 |

4x5 | 96 | 3x4x8 |

5x6 | 160 | 4x5x8 |

I found a new formula which will find the difference of the two opposing corners in an 8x8 grid for any rectangle shape which is:

(c-1) x (d-1) x8

I also found a new formula which will find any rectangle shape in any number grid which is:

(c-1) x (d-1) x g

g = the grid size so if I want to find the formula of:

- 10x10 grid = (c-1) x (d-1) x10
- 8x8 grid = (c-1) x (d-1) x8

I will choose at random from previous work and apply it to this formula:

E.g. 5x6 box size in a 10x10 grid (see page 14).

- 4x5x10
- =200

This new formula works.

E.g. 3x4 box sizes in an 8x8 grid (see page 17).

- 2x3x8
- =48

The new formula works.

Now I am going to try this formula on a 7x7 grid and see if it works.

E.g. 5x6 box size in a 7x7 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 |

- 4x5x7
- =140

number | Left corner x right corner | Right corner x left corner | Products difference |

17 | 3x42=126 | 7x38=266 | 140 |

This proves that the new formula works.

Conclusion

I found a new formulawhich will find any rectangle shape in any number grid.

## In this project I hoped to extend the investigation further in order to find more interesting patterns, which can be relevant to the task.

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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## Here's what a teacher thought of this essay

****

This is a very well structured investigation. All mathematical working is correct and appropriately tested throughout. Specific strengths and improvements are suggested throughout. This is a good example of this coursework task.

Marked by teacher Cornelia Bruce 18/07/2013