• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 2389

Investigation into the patterns of mutiplication sqaures

Extracts from this document...

Introduction

Maths Coursework         1st DRAFT!!!!                                   Mr Gazey

Introduction

I am going to investigate and observe what happens when I find the product of the top left and bottom right hand side of a square, and vice versa find t he product of the top right and the bottom left hand side of a square. I will then calculate the difference between the two squares and try to see if there is a genuine or pattern with the numbers.  

I am first going to use a two by two square and see what happens.

12

13

22

23

12 X 23 = 276          Here we can see that the difference is equal to ten.

22 X 13 = 286

54

55

64

65

55 X 64 = 3520         The difference equals ten.

54 X 65 = 3510

68

69

78

79

68 X 79 = 5372          The difference equals ten.

69 X 78 = 5382

I have observed that the difference is always ten between the two totals. To prove that my results were accurate I did another example.

81

82

91

92

81 X 92 = 7452    I was correct as my concluding example also had a difference of                  

91 X 82 = 7462    ten.

To further my investigation I am now going to observe what happens with a three by three square.

1

2

3

11

12

13

21

22

23

1 X 23 = 23           Here we can see that the difference is forty.

3 X 21 = 63

35

36

37

45

46

47

55

56

57

35 X 57 = 1995      The difference is forty

37 X 55 = 2035  

71

72

73

81

82

83

91

92

93

71 X 93 = 6603        The difference is forty.

73 X 91 = 6643

This time the difference is always forty. To prove my results were correct I did another example.

78

79

80

88

89

90

98

99

100

78 X 100 = 7800     My results were correct.

98 X 80 = 7840

I then decided to do the same thing with a four by four square.

1

2

3

4

11

12

13

14

21

22

23

24

31

32

33

34

...read more.

Middle

20

25

26

27

28

29

30

35

36

37

38

39

40

45

46

47

48

49

50

55

56

57

58

59

60

5 x 60 = 300              The difference is two hundred and fifty.

10 x 55 = 550

45

46

47

48

49

50

55

56

57

58

59

60

65

66

67

68

69

70

75

76

77

78

79

80

85

86

87

88

89

90

95

96

97

98

99

100

50 x 95 = 4750           The difference is two hundred and fifty.

100 x 45 = 4500

23

24

25

26

27

28

33

34

35

36

37

38

43

44

45

46

47

48

53

54

55

56

57

58

63

64

65

66

67

68

73

74

75

76

77

78

23 x 78 = 1794           The difference is two hundred and fifty.

28 x 73 = 2044

This is clearly what I predicted as the difference is always equal to two hundred and fifty. I attempted another example to back up my results.

1

2

3

4

5

6

11

12

13

14

15

16

21

22

23

24

25

26

31

32

33

34

35

36

41

42

43

44

45

46

51

52

53

54

55

56

1 x 56 = 56                The difference again is always two hundred and fifty.

51 x 6 = 306

To make further investigations quicker. I worked out a formula from my table of results above. With the formula it will enable to see what the differences are for any square without having to work out the products of the top right and bottom left hand number, and the top left and bottom right hand numbers. I could see that the differences of each square is always the size of square penultimate. Therefore I worked out the formula to be this;

                                          (n-1) ² x 10

To ensure that my formula worked I tested it on a 2 x 32 square of which the difference was 10.

(n-1) ² x 10   =     (2-1)² x 10   =

                            1² x 10   = 10

This shows me that my formula was correct, as it gave me the correct difference.

To further my investigation I am going to prove my results algebraically for a 2 x 2 square.

x

x+1

x+10

x+11

Therefore : x(x+11) =                   (x+1)(x+10) =

                   x² + 11x                      x² + 10x + x + 10 =

                                                      x² + 11x + 10

Here we can see that no matter what the value of x may be, the difference will always be ten.

I am now going to prove my results algebraically for a three by three square.

x

x+2

x+20

x+22

Therefore : x(x+20)                        (x+1)(x+10) =

                   x² + 20x                        x² + 20x + 2x + 40 =

                                                         x² + 22x + 40

Again this proves my results, as the difference will always be forty between the two numbers.

I am now going to prove my results algebraically for a four by four square.

x

x+3

x+30

x+33

Therefore : x(x+33)                       (x+3)(x+30) =

                   x² +33x                        x² + 30x + 3x + 90 =  

                                                        x² + 33x + 90

Here the difference between the numbers is + 90, which proves and supports my results in the table.

To conclude I am going to prove my results algebraically for a five by five square.

x

x+4

x+40

x+54

Therefore : x(x+54) =                            (x+4)(x+40) =

                   x² + 54x                               x² + 40x + 4x + 160 =

                                                                x² + 44x + 160

Again the difference will always be +160. This example supports and proves my results.

To further my investigation, I am now going to find out what happens in a rectangle, starting with a three by two rectangle, and observe any patters or trends.

1

2

3

11

12

13

1 X 13 = 13                      The difference between them equals twenty.

3 X 11 = 33                

4

5

6

14

15

16

4 X 16 = 64                      The difference between them equals twenty.

6 X 14 = 84

88

89

90

98

99

100

90 X 98 = 8820                The difference between them equals twenty.

88 X 100 = 8800

I have noticed that the difference is always equal to twenty in a three by two rectangle. To prove my results are correct, I am going to conclude by doing another example.

81

82

83

91

92

93

81 X 93 = 7533                The difference between them equals twenty

83 X 91 = 7553

I have now chosen to enlarge my rectangle to three by four.

41

42

43

44

51

52

53

54

61

62

63

64

...read more.

Conclusion

5

6

7

8

9

10

15

16

17

18

19

20

25

26

27

28

29

30

35

36

37

38

39

40

45

46

47

48

49

50

5 X 50 = 250                   Again we see that the difference between the two numbers

10 X 45 = 450                 will always be two hundred.

I put my results in a table in order to enable me to discover a pattern or trend, and also to find a formula with the size of my rectangles.

Size Of Rectangle

Difference between the products

2 X 3

20

3 X 4

60

4 X 5

120

5 X 6

200

My results have shown me exactly the same pattern as to what happened with using squares. The size of the rectangle equals the next rectangle’s product after it. For example the 2 X 3 rectangle multiply together to give the product of the next rectangle 3 X 4, which is 200 (multiplied by ten). From my table of results I am going to work out a formula, making it easier and quicker to work out differences of any size rectangle. This is what I came up with.

(L-1)(W-1) X 10

To ensure that my formula was correct and worked I tested it with an example. I used a 5 X 4 rectangle of which the difference is one hundred and twenty.

(L-1)(W-1) X 10 =

(5-1)(4-1) X 10 =

4 X 3 = 12                 Therefore : 12 X 10 = 120

As you can see my formula clearly worked, and gave me the correct difference of one hundred and twenty.

I am now going to prove my results algebraically for a 2 X 3 rectangle.

x

x+2

x+10

x+12

Therefore : x(x+12) =                     (x+2)(x+10) =

                   x² + 12x                        x² + 20x + 2x +20 =  

                                                         x² + 22x + 20

This clearly shows that no matter what the value of x maybe in a 2 X 3 rectangle the difference is always going to be twenty.

Moses Ilori 11P             Miss Lucas

...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. Number Grids Investigation Coursework

    in my original calculation for the difference between the products of the opposite corners, I can prove my formula: (top right x bottom left) - (top left x bottom right) = (a + pm - p) (a + wpn - pw)

  2. Investigation of diagonal difference.

    51 52 53 54 55 56 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

  1. What the 'L' - L shape investigation.

    This suggests that -3 is to be added to the formula. From these two tables so far I have the formula 3g - 3. I will use this formula to prove its correctness and to look further for additional differences.

  2. Algebra Investigation - Grid Square and Cube Relationships

    Instead, in order to ensure the grid could be used to calculate a cube or cuboid of any height, width and depth, with any step size, the original terms are used. n: The top left number of the front face grid.

  1. Step-stair Investigation.

    28 15 16 17 18 19 20 21 8 9 10 11 12 13 14 1 2 3 4 5 6 7 By using the formula 10X+10g+10=S, I worked out the total of the numbers inside the blue area of the 4-step stair.

  2. Mathematical Coursework: 3-step stairs

    Therefore I will take the pattern number of the total: > 46-6=40 > b= 40 To conclusion my new formula would be: > 6n+40 8cm by 8cm grid 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46

  1. For my investigation I will be finding out patterns and differences in a number ...

    After I have completed all of my working out and results I should then do a conclusion to sum up what I have found out from my investigation. Statement I have found out the differences for the 2x2, 3x3, 4x4 and 5x5 squares and they are shown below.

  2. Algebra - Date Patterns.

    First I found separate formulas for each individual sum: n x (n + 8) or n2 + 7 x n + n This gives me the result of the root times it's diagonal, eg: 4 5 11 12 4 x 12 = 48 4 x (4 + 8)

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