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Introduction

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

Middle

2 + 10(b-1)2 + 11n(b-1)

## Find the difference between the two multiplied corners

[n2 + 10(b-1)2 + 11n(b-1)] – [n2 + 11n(b-1)] =

10(b-1)2

Which is the same answer as I got when finding the formula before.

This means I have proved beyond any doubt that the formula for finding the difference of the corners multiplied together in a square on a 100 grid is 10(b-1)2.

Investigating rectangle boxes in a 100 square

I will take a 2x3 rectangle 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. 13  14                12 x 24 = 288

22  23  24                14 x 22 = 308

308 – 288 = 20

DIFFERENCE = 20

## Test 2

1. 43  44                42 x 54 = 2268
1. 53  54                44 x 52 = 2288

2288 – 2268 = 20

DIFFERENCE = 20

## Test 3

1. 5    6                4 x 16 = 64

14  15 16                6 x 14 = 84

84 – 64 = 20

DIFFERENCE = 20

## Prediction

I predict that in a 2 x 3 rectangle the difference will always be 20

## Proof

1. 28  29                27 x 39 = 1053

37  38  39                29 x 37 = 1073

1073 – 1053 = 20

DIFFERENCE = 20

## Algebra I will assign a letter to the first number in the 2x3 rectangle, n.

The right hand top corner will therefore be n+2

The left hand bottom corner will then be n+10

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+12) = n² + 12n

Top right hand corner x bottom left hand corner = (n+2)(n+10) = n²+2n+10n+20

= n²+12n+20

n²+12n+20 - n² + 12n = 20

Therefore the difference between the corners multiplied together will always be 20.

## Expanding the Task

I now feel it will be interesting to look at a 3x4 number rectangle on a 100 grid. I will take a 3x4 rectangle 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. 54  55  56        53 x 76 = 4028
1. 64  65  66        56 x 73 = 4088

73  74  75  76        4088 – 4028 = 60

DIFFERENCE = 60

## Test 2

1. 68  69  70        67 x 90 = 6030
1. 78  79  80        70 x 87 = 6090
1. 88  89  90        6090 – 6030 = 60

DIFFERENCE = 60

## Test 3

1. 32  33  34        31 x 54 = 1674
1. 42  43  44        34 x 51 = 1734
1. 52  53  54        1734 – 1674 = 60

DIFFERENCE = 60

## Prediction

I predict that in a 3 x 4 rectangle the difference will always be 60 Conclusion

## Find the difference between the two multiplied corners

[n2 + 9n(w-1) + n(L-1) + 9(L-1)(w-1)] – [n2 + 7n(w-1) + n(L-1)] =

9(L-1)(w-1)

I will now try this formula out with 3 examples on a 49 square, each with different sized rectangles, to prove whether it is correct

2x3 sized rectangle

Formula

9(L-1)(w-1) = 9(3-1)(2-1)

= 9 x 2 x 1

= 18

Therefore in theory the difference between the two corners multiplied together in a     2 x 3 rectangle on a 81 square grid will be 18.

Example

1. 38  39                37 x 48 = 1776

46  47  48                39 x 46 = 1794

1794 – 1776 = 18

DIFFERENCE = 18

The formula worked for this example

3x4 sized rectangle

Formula

9(L-1)(w-1) = 9(3-1)(4-1)

= 9 x 2 x 3

= 54

Therefore in theory the difference between the two corners multiplied together in a     3 x 4 rectangle on a 81 square grid will be 54.

Example

1.  69  70  71        68 x 89 = 6052
1. 78  79  80        71 x 86 = 6106

86  87  88  89        6106 – 6052 = 54

DIFFERENCE = 54

The formula worked for this example

4x5 sized rectangle

Formula

9(L-1)(w-1) = 9(4-1)(5-1)

= 9 x 3 x 4

= 108

Therefore in theory the difference between the two corners multiplied together in a     4 x 5 rectangle on a 81 square grid will be 108.

Example

1. 3     4   5    6                2 x 33 = 66
1. 12  13  14  15                6 x 29 = 174
1. 21  22  23  24                174 – 66 = 108

29  30  31  32  33                DIFFERENCE = 108

The formula worked for this example

I am now going to compare the formula for different sized number grids

Size Of Grid                                Formula

10 x 10                                10(L-1)(w-1)

7 x 7                                                 7(L-1)(w-1)

9 x 9                                                    9(L-1)(w-1)

Prediction:       6 x 6                                          6(L-1)(w-1)

I predict this, as there is a relationship between the size of the grid and the formula. The (L-1)(w-1) part of the formula stays the same and the constant that multiplies it is the size of the grid. Therefore in a 6x 6 grid the formula should be 6(L-1)(w-1).

Proof – 2 x 3 rectangle

Formula

6(L-1)(w-1) = 6(3-1)(2-1)

= 6 x 2 x 1

= 12

Example

1. 26  27                25 x 33 = 825
1. 32  33                27 x 31 = 837

837 – 825 = 12

DIFFERENCE = 12

My prediction was correct.

This means I can create a formula that will tell the difference between the corners multiplied together of any rectangle in any size of number grid.

Therefore my target has been achieved. I have been able to investigate differences of multiplied corners in… squares, then leading to rectangles, and finally actually changing the size of the number grid.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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