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  • Level: GCSE
  • Subject: Maths
  • Word count: 1974

The problem is to investigate the differences of corner numbers on a multiplication grid.

Extracts from this document...

Introduction

Number Grids

The Problem

The problem is to investigate the differences of corner numbers on a multiplication grid.

Introduction

To solve this problem I will have to choose several examples of squares on a grid:

E.g. 2x2, 3x3, 4x4

To work this out I will need to take the opposite corners of the square and subtract them from the other sum of the two corners. This is easier seen in the diagram shown below:

*The RED numbers are always multiplied then subtracted from the sum of the Blue numbers.

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

This is an example of a 2x2 square. I will then make an equation with this:

Difference = (2x11)-(1x12)

= 10

The difference for this example is ten.

2x2 Squares

I will use on grid to use several example of 2x2 squares by placing them randomly on the grid.

12          3          4          5          6          7          8          9        10

1112        13        14        15        16        17        18        19        20

21        22        23        2425        26        27        28        29        30

31        32        33        3435        36        37        38        39        40

41        42        43        44        45        46        47        48        49        50

51        52        53        54        55        56        5758        59        60

61        6263        64        65        66        6768        69        70

71        7273        74        75        76        77        78        79        80

81        82        83        84        85        86        87        88        8990

91        92        93        94        95        96        97        98        99100

From the squares I can find the difference for a 2x2 grid.

(2x11)-(1x12) = 10

(25x34)-(24x35) = 10

(63x72)-(62x73) = 10

(58x67)-(57x68) = 10

(90x99)-(89x100)=10

For all of these you can that the difference is a constant 10, so anywhere you put a 2x2 square on a 10x10 grid you will always get the same difference of ten.

...read more.

Middle

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

(3x21)-(1x23) =40

(8x26)-(6x28) =40

(44x62)-(42x64)=40

(58x76)-(56x78)=40

For the 3x3 grids the constant difference is 40. Therefore if you put a 3x3 square anywhere on a 10x10 grid the difference will be equal to 40. We can again show the algebraic method of working out the difference.

Let ‘N’ equal the top left number in the square.

N

 N+2

N+20

 N+22

We then change this into the equation

(N+2)(N+20) – N (N+22)

Multiply out the brackets

N²+ 20N +2N+ 40-N²-22N

This can be simplified to so that you are only left with 40, which is the constant difference for a 3x3 grid.

4x4 Squares

I will now investigate for the last time with square boxes using a larger 4x4 square. I will place them randomly on the 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        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

(4x31)-(34x1) =90

(9x36)-(6x39) =90

(67x94)-(64x97)=90

The difference for a 4x4 grid is always going to be no matter where you place the square on a 10x10 grid. We can again prove that the difference will always be 90 using algebra.

Let ‘N’ be the number at the top left of the square.

N

 N+3

N+30

 N+30

...read more.

Conclusion

D=20(N-1)

From this we begin to see that the first difference is going up in tens, and so is the formula, so we can predict that the formula for a 4xN grid will be 30(N-1), but we will have to test this to see if it is true.

4xN Grid

I am doing this rectangle grid to see if my predictions are correct, and if they are I will be able to construct a formula for any grid where I know what ‘M’ is.

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

These rectangles and their differences are:

M N Difference First Difference

4x3 60 30

4x4 90  30

4x5 120 30

4x6 150  30

My prediction was correct. We need to multiply the nth term by 30 this time then subtract the first difference which is 30. So we can construct the formula D=30N-30 which can be reduced to the formula D=30(N-1)

Now that my predictions have been proven correct we can now work out any rectangle as long as we know the Mth term. For example if we want a rectangle that had the constant M as 7 then we could make the formula D=60(N-1).

But now we want a formula so that we can be given an Nth and an Mth term and from that we can work out the difference of the opposing corners.

...read more.

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