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

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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

- 55 54 x 65= 3510

64 65 55 x 64= 3520

3520-3510= 10

DIFFERENCE = 10

## Test 2

- 6 5 x 16= 80

15 16 6 x 15= 90

90-80= 10

DIFFERENCE = 10

## Test 3

- 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

- 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

- 38 39 37 x 59= 2183

47 48 49 39 x 57= 2223

57 58 59 2223 – 2183 = 40

DIFFERENCE 40

## Test 2

- 73 74 72 x 94= 6768

- 83 84 74 x 92= 6808

92 93 94 6808 – 6768 = 40

DIFFERENCE = 40

## Test 3

- 2 3 1 x 23 = 23

- 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

- 27 28 26 x 48 = 1248

- 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

## 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

- 13 14 12 x 24 = 288

22 23 24 14 x 22 = 308

308 – 288 = 20

DIFFERENCE = 20

## Test 2

- 43 44 42 x 54 = 2268

- 53 54 44 x 52 = 2288

2288 – 2268 = 20

DIFFERENCE = 20

## Test 3

- 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

- 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

- 54 55 56 53 x 76 = 4028

- 64 65 66 56 x 73 = 4088

73 74 75 76 4088 – 4028 = 60

DIFFERENCE = 60

## Test 2

- 68 69 70 67 x 90 = 6030

- 78 79 80 70 x 87 = 6090

- 88 89 90 6090 – 6030 = 60

DIFFERENCE = 60

## Test 3

- 32 33 34 31 x 54 = 1674

- 42 43 44 34 x 51 = 1734

- 52 53 54 1734 – 1674 = 60

DIFFERENCE = 60

## Prediction

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

## Algebra

Conclusion

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

## 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

- 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

- 69 70 71 68 x 89 = 6052

- 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

- 3 4 5 6 2 x 33 = 66

- 12 13 14 15 6 x 29 = 174

- 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

- 26 27 25 x 33 = 825

- 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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