My task is to investigate a 2x2 box on a 100 square

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       Lizzy Gunstone

Number Grid Coursework

My task is to investigate a 2x2 box on a 100 square

I will take a 2x2 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.

Test 1

  1. 55                54 x 65= 3510

64    65                 55 x 64= 3520

                        3520-3510= 10

DIFFERENCE = 10

Test 2

  1. 6                5 x 16= 80        

15   16                6 x 15= 90

                        90-80= 10

DIFFERENCE = 10

Test 3

  1. 19                18 x 29= 522

28   29                19 x 28= 532

                        532-522= 10

DIFFERENCE = 10

Prediction

I predict that in a two by two square the difference will always be 10

Proof

  1. 84                83 x 94= 7802

93   94                84 x 93= 7812

                        7812-7802= 10

DIFFERENCE = 10

Algebraic Explanation

I will assign a letter to the first number in the 2 x 2 square, n.

The next number to the right will therefore be n+1

The number directly below it will then be n+10

The number diagonally across from it will be n+11

I will then times the corners together, like In did on the above examples.

Top Left hand corner x bottom right hand corner = n(n+11) = n² + 11n

Top right hand corner x bottom left hand corner = n² +1n+10n+10

                                                         n² +11n+10

 

(n² +11n+10) – (n² + 11n) = 10

Therefore the difference between the corners multiplied together will always be 10.

Expanding the Task

I now feel it will be interesting to look at a 3x3 number square on a 100 grid. I will take a 3x3 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.

Test 1

  1. 38  39                37 x 59= 2183

47  48  49                39 x 57= 2223

      57  58  59                2223 – 2183 = 40

DIFFERENCE  40

Test 2

  1. 73  74                72 x 94= 6768
  1. 83  84                74 x 92= 6808

92  93  94                6808 – 6768 = 40

DIFFERENCE = 40

Test 3

  1. 2    3                1 x 23 = 23
  1. 12  13                3 x 21 = 63

21  22  23                63 – 23 = 40

DIFFERENCE = 40

Prediction

I predict that in a 3 x 3 square the difference will always be 40

Proof

  1. 27  28                26 x 48 = 1248
  1. 37  38                28 x 46 = 1288

46  47  48                1288 – 1248 = 40

DIFFERENCE = 40

Algebra

I will assign a letter to the first number in the 3x3square, n.

The right hand top corner will therefore be n+2

The left hand bottom corner will then be n+20

The corner diagonally across from it will be n+22

I will then times the corners together, like I did on the above examples.

Top Left hand corner x bottom right hand corner = n(n+22) = n² + 22n

Top right hand corner x bottom left hand corner = (n+20)(n+2) = n²+40+22n

(n²+40+22n) – (n² + 22n) = 40

Therefore the difference between the corners multiplied together will always be 40.

Expanding the Task Further

I now feel it will be interesting to look at a 4x4 number square on a 100 grid. I will take a 4x4 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.

  1. 58  59  60        57 x 90 = 5130
  1. 68  69  70        60 x 87 = 5220
  1. 78  79  80        5220 – 5130 = 90

87  88  89  90        DIFFERENCE = 90

  1. 23  24  25        22 x 55 = 1210
  1. 33  34  35        25 x 52 = 1300
  1. 43  44  45        1300 – 1210 = 90

52  53  54  55        DIFFERENCE = 90

 4    5    6    7        4 x 37 = 148

14  15  16   17        7 x 34 = 238

24  25  26   27        238 – 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

  1. 34  35  36        33 x 66 = 2178
  1. 44  45  46        36 x 63 = 2268
  1. 54  55  56        2268 – 2178 = 90

63  64  65  66        DIFFREMCE = 90

Algebra

 I will assign a letter to the first number in the 4x4 square, n.

The right hand top corner will therefore be n+3

The left hand bottom corner will then be n+30

The corner diagonally across from it will be n+33

I will then times the corners together, like I did on the above examples.

Top Left hand corner x bottom right hand corner = n(n+33) = n² + 33n

Top right hand corner x bottom left hand corner = (n+30)(n+3) = n²+33n + 90

(n²+33n + 90) – (n² + 33n) = 90

Therefore the difference between the corners multiplied together will always be 90

Looking at the relationship between the differences of the corners multiplied together in different size squares – using a table to compare the findings

Size of Square                        Difference        Difference of the Differences

Join now!

2 x 2                                10                        30

3 x 3                                40                        50

4 x 4                                90                        70???

Prediction:        5 x 5                                160                        

I predict this because the difference of the differences go up by 20 each time e.g from 30 to 50. Therefore it would make sense for 50 to go up to 70, add this onto the 90 which makes 160, which is my prediction for the 5 x 5 square.

Proof

  1. 22  23  24  25                21 x 65 = 1365
  1. 32  33  34  35                25 x 61 = 1525
  1. 42  43  44  45                1365 – 1525 = 160
  1. 52  53  54  55                DIFFERENCE = 160

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