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  • Level: GCSE
  • Subject: Maths
  • Word count: 3671

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

Extracts from this document...

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

...read more.

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:

...read more.

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.

...read more.

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Here's what a teacher thought of this essay

4 star(s)

****
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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