I am to conduct an investigation involving a number grid.

you go up 20 is added onto the number i.e.

2 x 2 = 10 (+ 30)

3 x 3 = 40 (+ 50)

4 x 4 = 90 (+ 70) so the next box will be:

5 x 5 = 160

For example the 4 x 4 grid can be predicted by the 3 x 3 box used

before. 3 x 3 = 9, multiply it by 10 = 90. The 4 x 4 grid has the

difference of 90 between the number. This proves that the formula

works correctly.

To predict the 5 x 5 box I will use the 4 x 4 number. 4 x 4 = 16 x 10

= 160. I am now going to do the 5 x 5 box to see if my predictions

are true.

ü 5 x 5 Boxes

Box 1

6

7

8

9

10

X

X+1

X+2

X+3

X+4

16

17

18

19

20

X+10

X+11

X+12

X+13

X+4

26

27

28

29

30

X+20

X+21

X+22

X+23

X+24

36

37

38

39

40

X+30

X+31

X+32

X+33

X+34

46

47

48

49

50

X+40

X+41

X+42

X+43

X+44

[image015.gif] [image016.gif] 6 x 50 =

300 x (x + 44) x2 + 44

[image008.gif] 46 x 10 = 460 (x + 4) (x +

40) x2 + 40x + x + 40

=(x2 + 44x + 40) - (x2

+ 44x) = 160

460 - 300 = 160

The difference between the two numbers is 160

Box 2

22

23

24

25

26

32

33

34

35

36

42

43

44

45

46

52

53

54

55

56

62

63

64

65

66

[image017.gif] 22 x 66 = 1452

62 x 26 = 1612

1612 - 1452 = 160

The difference between the two numbers is 160

Box 3

51

52

53

54

55

61

62

63

64

65

71

72

73

74

75

81

82

83

84

85

91

92

93

94

95

[image018.gif] 51 x 95 = 4845

91 x 55 = 5005

5005 - 4845 = 160

The difference between the two numbers is 160

I have now calculated the 5 x 5 boxes using a 10 x 10 grid and have

learned that the number which I predicted was accurate and correct.

This determines that the formula that I chose to use was the proper

one to do this certain part of the investigation with.

I now intend to do a 5 x 5 grid to determine the formula. This will

be set out the same way as the 10 x 10 grid only the numbers will be

different this time around.

5 x 5 Grid

ü 2 x 2 Boxes

Box 1

4

5

X

X+1

9

10

X+5

X+6

[image008.gif] [image019.gif] 4 x 10 =

40 x (x + 6) x2 + 6x

[image003.gif] 9 x 5 = 45 (x + 1) (x +

5) x2 + 5x + x + 5

= (x2 + 6x + 5) - (x2

+ 6x) = 5

45 - 40 = 5

The difference between the two numbers is 5

Box 2

48

49

53

54

[image020.gif] 48 x 54 = 2592

53 x 49 = 2597

2597 - 2597 = 5

The difference between the two numbers is 5

Box 3

65

66

70

71

[image021.gif] 65 x 71 = 4615

70 x 66 = 4620

4620 - 4615 = 5

The difference between the two numbers is 5

ü 3 x 3 Boxes

Box 1

2

3

4

X

X+1

X+2

7

8

9

X+5

X+6

X+7

12

13

14

X+10

X+11

X+12

[image003.gif] [image022.gif] 2 x 14 =

28 x (x + 12) x2 + 12x

[image023.gif] 12 x 4 = 48 (x + 2) (x +

10) x2 + 10x + x + 10

= (x2 + 12x + 10) -

(x2 + 12x) = 20

48 - 28 = 20

The difference between the two numbers is 20

Box 2

24

25

26

29

30

31

34

35

36

[image024.gif] 24 x 36 = 864

34 x 26 = 884

884 - 864 = 20

The difference between the two numbers is 20

Box 3

56

57

58

61

62

63

66

67

68

[image025.gif] 56 x 68 = 3808

66 x 58 = 3828

3828 - 3808 = 20

The difference between the two numbers is 20

ü 4 x 4 Boxes

Box 1

5

6

7

8

X

X+1

X+2

X+3

10

11

12

13

X+5

X+6

X+7

X+8

15

16

17

18

X+

16

17

18

20

21

22

23

20

21

22

23

[image008.gif] [image026.gif] 5 x 23 =

115 x (x + 23) x2 + 23x

[image003.gif] 20 x 8 = 160 (x + 3) (x +

20) x2 + 20x + x + 20

= (x2 + 23x + 20) -

(x2 + 23x) = 45

160 - 115 = 45

The difference between the two numbers is 45

Box 2

32

33

34

35

37

38

39

40

42

43

44

45

47

48

49

50

[image027.gif] 32 x 50 = 1600

47 x 35 = 1645

1645 - 1600 = 45

The difference between the two numbers is 45

Box 3

44

45

46

47

49

50

51

52

54

55

56

57

59

60

61

62

[image028.gif] 44 x 62 = 2728

59 x 47 = 2773

2773 - 2728 = 45

The difference between the two numbers is 45

Evaluation

This formula is working with the same pattern as before using square

numbers only instead of multiplying by 10 it is now being multiplied

by 5 using the same size of grid.

For example:

When multiplying by 10 the pattern was going up by 20

[image029.gif] 2 x 2 = 10 (+ 30)

[image030.gif] [image031.gif] 3 x 3 = 40 (+ 50)

4 x 4 = 90 (+ 70)

5 x 5 = 160

When multiplying by 5 the pattern was going up by 10

[image032.gif] 2 x 2 = 5 (+15)

[image033.gif] [image034.gif] 3 x 3 = 20 (+25)

4 x 4 = 45 (+35)

So if I was to do a 5 x 5 table the result would be:

5 x 5 = 80 (45 + 35)

Part 2

For the second part of this investigation I plan to predict the

difference between numbers in a different sized number grid.

When doing the grid before which was a 10 x 10 grid the numbers turned

out:

2 x 2 = 10 (+ 30)

3 x 3 = 40 (+ 50)

4 x 4 = 90 (+ 70)

5 x 5 = 160

The formula given from these numbers turned out to be:

Square number x 10 i.e. 3 x 3 = 9 x 10 = 90 which was the 4 x 4 box

difference between numbers.

When I am doing a different size number grid I am now predicting that

the formula for the numbers will stay the same and that only the

actual numbers will change.

I have decided to predict a 7 x 7 grid and I believe the formula will

be:

Square number x 7 which means that each sized box will be:

2 x 2 = 1 x 7 = 7 (+ 21)

3 x 3 = 4 x 7 = 28 (+ 35)

4 x 4 = 9 x 7 = 63 (+ 49)

5 x 5 = 16 x 7 = 112

The pattern for each number is rising up by 14 each time.

I am now going to do the 7 x 7 to determine whether my prediction is

correct or not.

7 x 7 Grid

ü 2 x 2 boxes

Box 1

6

7

X

X+1

13

14

X+7

X+8

[image008.gif] [image035.gif] 6 x 14 =

84 x (x + 8) x2 + 8x

[image003.gif] 13 x 7 = 91 (x + 1) (x +

7) x2 + 7x + x + 7

= (x2 + 8x + 7) - (x2

+ 8x) = 7

91 - 84 = 7

The difference between the two numbers is 7

Box 2

21

22

28

29

[image036.gif] 21 x 29 = 609

28 x 22 = 616

616 - 609 = 7

The difference between the two numbers is 7

Box 3

78

79

85

86

[image037.gif] 78 x 86 = 6708

85 x 79 = 6715

6715 - 6708 = 7

The difference between the two numbers is 7

ü 3 x 3 Boxes

Box 1

24

25

26

X

X+1

X+2

31

32

33

X+7

X+8

X+9

38

39

40

X+14

X+15

X+16

[image008.gif] [image038.gif] 24 x 40 = 960

x (x + 16) x2 + 16x

[image008.gif] 38 x 26 = 988 (x + 2) (x +

14) x2 + 14x + x + 14

= (x2 + 16x + 14) -

(x2 + 16x) = 28

988 - 960 = 28

The difference between the two numbers is 28

Box 2

45

46

47

52

53

54

59

60

61

[image039.gif] 45 x 61 = 2745

59 x 47 = 2773

2773 - 2745 = 28

The difference between the two numbers is 28

Box 3

78

79

80

85

86

87

92

93

94

[image040.gif] 78 x 94 = 7332

92 x 80 = 7360

7360 - 7332 = 28

The difference between the two numbers is 28

ü 4 x 4 Boxes

Box 1

7

8

9

10

X

X+1

X+2

X+3

14

15

16

17

X+7

X+8

X+9

X+10

21

22

23

24

X+14

X+15

X+16

X+17

28

29

30

31

X+21

X+22

X+23

X+24

[image001.gif] [image041.gif] 7 x 31 =

217 x (x + 24) x2 + 24x

[image003.gif] 28 x 10 = 280 (x + 4) (x +

21) x2 + 21x + x + 21

= (x2 + 24x + 21) -

(x2 + 24x) = 63

280 - 217 = 63

The difference between the two numbers is 63

Box 2

22

23

24

25

29

30

31

32

36

37

38

39

43

44

45

46

[image042.gif] 22 x 46 = 1012

43 x 25 = 1075

1075 - 1012 = 63

The difference between the two numbers is 63

Box 3

56

57

58

59

63

64

65

66

70

71

72

73

77

78

79

80

[image043.gif] 56 x 80 = 4480

77 x 59 = 4543

4543 - 4480 = 63

The difference between the two numbers is 63

Evaluation for overall investigation

By completing this investigation I have discovered that all the

working out that I have used has been correct. My statement about the

formula has been proved to be correct. The formula for any grid is

the square number multiplied by the size of the grid used i.e. 7 x 7

is the square number multiplied by 7.

ü Summarised table of results

Grid size

Box size

Difference

10 x 10

2 x 2

10

3 x 3

40

4 x 4

90

5 x 5

160

5 x 5

2 x 2

5

3 x 3

20

4 x 4

45

7 x 7

2 x 2

7

3 x 3

28

4 x 4

63

Part 3

I am now going to do a formula for a 10 x 10 grid using a rectangle

box instead of a square box.

ü 3 x 2 boxes

Box 1

7

8

9

X

X+1

X+2

17

18

19

X+10

X+11

X+12

[image015.gif] [image044.gif] 7 x 19 =

133 x (x + 12) x2 + 12x

[image008.gif] 17 x 9 = 153 (x + 2) (x +

10) x2 + 10x + x + 10

= (x2 + 12x + 20) -

(x2 + 12x) = 20

153 - 133 = 20

The difference between the two numbers is 20

Box 2

22

23

24

32

33

34

[image045.gif] 22 x 34 = 748

32 x 24 = 768

768 - 748 = 20

The difference between the two numbers is 20

Box 3

45

46

47

55

56

57

[image025.gif] 45 x 57 = 2565

55 x 47 = 2585

2585 - 2565 = 20

The difference between the two numbers is 20

4 x 3 boxes

16

17

18

19

X

X+1

X+2

X+3

26

27

28

29

X+10

X+11

X+12

X+13

36

37

38

39

X+20

X+21

X+22

X+23

[image001.gif] [image046.gif] 16 x 39 = 624

x (x + 23) x2 + 23x

[image012.gif] 36 x 19 = 684 (x + 3) (x +

20) x2 + 20x + x + 20

= (x2 + 23x + 20) -

(x2 + 23x) = 60

684 - 624 = 60

The difference between the two numbers is 60

Box 2

31

32

33

34

41

42

43

44

51

52

53

54

[image047.gif] 31 x 54 = 1674

51 x 34 = 1734

1734 - 1674 = 60

The difference between the two numbers is 60

Box 3

63

64

65

66

73

74

75

76

83

84

85

86

[image048.gif] 63 x 86 = 5418

83 x 66 = 5478

5478 - 5418 = 60

The difference between the two numbers is 60

Prediction

From doing the previous two rectangle boxes I am now going to predict

the number for the next box (5 x 4). The pattern for the previous two

is as follows:

3 x 2 = 20 (+ 40)

4 x 3 = 60 (+ 60)

I predict the pattern is going up by 20 each time and that the 5 x 4

grid will be 5 x 4 = 120

ü 5 x 4 boxes

1

2

3

4

5

X

X+1

X+2

X+3

X+4

11

12

13

14

15

X+10

X+11

X+12

X+13

X+14

21

22

23

24

25

X+20

X+21

X+22

X+23

X+24

31

32

33

34

35

X+30

X+31

X+32

X+33

X+34

[image049.gif] [image015.gif] 1 x 35 =

35 x (x + 35) x2 + 35x

[image001.gif] 31 x 5 = 155 (x + 4) (x +

31) x2 + 30x + x + 31

= (x2 + 35x + 31) -

(x2 + 35x) = 120

155 - 35 = 120

The difference between the two numbers is 120

Box 2

13

14

15

16

17

23

24

25

26

27

33

34

35

36

37

43

44

45

46

47

[image050.gif] 13 x 47 = 611

43 x 17 = 731

731 - 611 = 120

The difference between the two numbers is 120

Box 3

35

36

37

38

39

45

46

47

48

49

55

56

57

58

59

65

66

67

68

69

[image051.gif] 35 x 69 = 2415

65 x 39 = 2535

2535 - 2415 = 120

The difference between the two numbers is 120

Now that I have completed the grid I can see that my prediction was

correct and that whether you do a square or rectangle box the formula

will always have a similar pattern.

For example:

Overall summarised table of results

Grid size

Box size

Difference

10 x 10

2 x 2

10

3 x 3

40

4 x 4

90

5 x 5

160

5 x 5

2 x 2

5

3 x 3

20

4 x 4

45

7 x 7

2 x 2

7

3 x 3

28

4 x 4

63

10 x 10

3 x 2

20

4 x 3

60

5 x 4

120