• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  • Level: GCSE
  • Subject: Maths
  • Word count: 1729

Math Grid work

Extracts from this document...

Introduction

Number Grids Coursework – Maths – Mr Danes

Maths Coursework

Introduction

We are looking at number grids and are using the numbers in the edges of any square or rectangle. I am multiplying the opposite edged numbers and subtracting the smaller number from the bigger one.

This is a 10 x 10 square

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

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

I pick any square from this grid and do the multiplying and subtracting and see what I get. I multiply the top right hand number to the bottom left and number and then subtract the number from the other two numbers multiplied from it.

2 x 2 square

24

image00.png

25

34

35

46

image01.png

47

56

57

image12.png

89

90

99

100

I notice that I get the same number for any same size square so to prove this I will use X and prove why the answer is always 10. I can use a square in which the boxes are the edges of the square on the grid.

X

image27.pngimage23.png

X + 1

image32.png

X + 10

X + 11image33.png

image35.pngimage34.png

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+1)(X+10)…which is = X2+11X+10…and…

X(X+11)…which is = X2 +11X

[X2+11X+10]- [X2 +11X] = 10

X can be any number and will always equal 10!

3 x 3 square

36

37

image36.png

38

 46

47

48

56

57

58

image02.png

X

X + 2

X + 20

X + 22

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+2)(X+20)…which is = X2+22X+40…and…

X(X+22)…which is = X2 +22X

[X2+22X+40]-[X2 +22X] = 40

X can be any number and will always equal 40!

4 x 4 square

4

5

6

7

14

15

16

image03.png

17

24

25

26

27

34

35

36

37

image04.png

X

X + 3

X + 30

X + 33

...read more.

Middle

54

55

61

62

63

64

65image05.png

71

72

73

74

75

81

82

83

84

85

91

92

93

94

95

X

image06.png

X + 4

X + 40

X + 44

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+4)(X+40)…which is = X2+44X+160…and…

X(X+44)…which is = X2 +44X

[X2+44X+160]-[X2 +44X] = 160

X can be any number and will always equal 160!

Working out a formula for any square size on a 10 by 10 number grid

image07.png

The sequence I get is 10, 40, 90 and then 160.

So I can do the use a square to find out a formula that works out N for any square size on a 10 by 10 number grid.

image08.png

I do what I do with these figures if they were numbers:

1. [X+(S-1)][X+10(S-1)

    I get X2+11X(S-1)+10(S-1)2

And…

2. X[X+11(S-1)]

    I get X2+ 11X(S-1)

I then subtract 2 from 1…

[X2+11X(S-1)+10(S-1) 2] -  [X2+ 11X(S-1)]

And whatever that is left over must be the formula, so therefore the formula to work out N for any square size in a 10 by 10 grid square is…

N= 10(S-1)2

Test

When the square size is 2…

89

image09.png

90

99

100

Using the formula, (2-1) 2 = 1.. x 10 = 10  

When the square size is 5…

2

image10.png

6

42

46

Using the formula, (5-1) 2 = 16… x 10 = 160  

The formula works!

What happens if I change the grid size?

I think that I have come to the stage where I can use algebra to work out what the number each time will be. So, I don’t have to show examples with numbers all the time.

A 6 by 6 grid square on a 2x2 square

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

image11.png

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+1)(X+6)…which is = X2+7X+6…and…

X(X+7)…which is = X2 +7X

[X2+7X+6]-[X2 +7X]= 6

X can be any number and will always equal 6 on a 2 by 2 square on a 6 by 6 grid.

An 8 by 8 grid square on a 2x2 square

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

51

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

...read more.

Conclusion

N= g(S-1)2

Test

When the square size is 2 and the grid size is 10

89

image09.png

90

99

100

Using the formula, (2-1) 2 = 1.. x 10 = 10  

When the square size is 3 and the grid size is 6

2

image29.png

4

14

16

Using the formula, (3-1) 2 = 4… x 6 = 24  

The formula works!

Final formula: for any size square/rectangle in any grid square

Common sense tells me that, this is basically replacing the S by the length and width of any square/rectangle.

image30.png

I do what I do with these figures if they were numbers:

1. [X+(w-1)][X+g(L-1)]

    I get X2+gX(L-1)+X(w-1)+g(w-1)(L-1)

And…

2. X[X+g(L-1)+(w-1)]

    I get X2+ gX(L-1)+X(w-1)

I then subtract 2 from 1…

[X2+gX(L-1)+X(w-1)+g(w-1)(L-1)]  - [X2+ gX(L-1)+X(w-1)]

The only thing left over is g(w-1)(L-1)  

And whatever that is left over must be the formula, so therefore the formula to work out N for any square/rectangle in a g by g grid square is…

N= g(w-1)(L-1)

Test

When the grid size is 10 and the length and width are 2

89

image09.png

90

99

100

Using the formula, 10(2-1)(2-1) = 10x1x1= 10

When the grid size is 5, length is 2 and width is 4

2

image31.png

5

7

10

Using the formula, 5(4-1)(2-1) = 5x3x1 = 15

The formula works!

***

By: Haroon Motara

...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.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Number Stairs, Grids and Sequences essays

  1. Algebra Investigation - Grid Square and Cube Relationships

    Difference in Cube: Stage A: TF Top Left x BF Bottom Right Stage B: TF Bottom Left x BF Top Right Stage B - Stage A: Difference Difference in Cube: Stage A: = 1 x 1000 = 1000 Stage B: = 91 x 910 = 82810 Difference in Cube: Stage

  2. Number Grid Coursework

    the location of the box upon the grid. 2) Method Varying values of p will be tested to give different lengths of sides for the boxes. The lengths will range from 3 to 7. With these boxes, in 5 different random locations on the width 10 grid, the differences of the two products will be calculated.

  1. Staircase Coursework

    A 2-step stair on 6x6 stair case 19 20 21 22 23 24 13 14 15 16 17 18 7 8 9 10 11 12 1 2 3 4 5 6 So again because n = 1 and g = 2 for stair 1 I get the formula: n+(n+1)+(n+gx3)

  2. number grid investigation]

    3 x Width Rectangles 3x2 Rectangles Firstly, a rectangle with applicable numbers from the grid will be selected as a baseline model for testing. n n+1 n+10 n+11 n+20 n+21 Stage A: Top left number x Bottom right number = n(n+21)

  1. number grid

    Because the second answer has +10 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of 10 will always be present. 3 x 3 Grid Prediction I believe that this difference of 40 should be expected to remain constant for all 3x3 number boxes.

  2. I will take a 2x2 square on a 100 square grid and multiply the ...

    148 = 90 34 35 36 37 DIFFRENCE = 90 Prediction I predict that in a 4 x 4 square the difference will always be 90 Proof 33 34 35 36 33 x 66 = 2178 43 44 45 46 36 x 63 = 2268 53 54 55 56 2268

  1. number grid

    2 X 2 Grid I have chosen the top left number of the square randomly. I have done this by using the random number function on my calculator. In my investigation I am going to find the product of the top left number and the bottom right number.

  2. Mathematical Coursework: 3-step stairs

    As I would have to follow the grid side ways from term 1 towards e.g. term 20. Thus making it's time consuming. After analysing the grids, the table of results and my prediction I have summed up an algebraic equation which would allow me to find out the total of any 3-step stair shape in a matter of minutes.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work