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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

²+72+27n

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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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

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