• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month   number grid

Extracts from this document...

Introduction

NUMBER GRID

COURSE WORK

 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

To investigate the patterns generated from using rules in a square grid

The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

In order to find the difference of n x n square first step is to take nxn square and then multiply the corners diagonally. For example take 2x2 square and multiply its corners diagonally.

 25 26 35 36

25 x 36= 900             26 x 35= 910                                   910 - 900= 10

 77 78 87 88

77 x 88 = 6776              78 x 87= 6786                     6786 - 6776= 10

After a few results I observed that

Middle

61

62

63

64

31 x 64 = 1984              34 x 61=2074       2074 – 1984 = 90

 n th term SIZED DIFFERENCE 1 2 x 2 10 2 3 x 3 40 3 4 x 4 90

I put my results into table and observed that the difference is square number because the box and the grid are square shaped.

I predict that for a 5 by 5 square the difference between the products will be 160.

 46 47 48 49 50 56 57 58 59 60 66 67 68 69 70 76 77 78 79 80 86 87 88 89 90

46 x 90 = 4140        50 x 86 = 4300              4300 - 4140 = 160

I will now attempt to prove my results algebraically.

After doing this I will check my answer using the nxn formula

2x2 square

 23 24 33 34 n            n+1

n+10       n+11                                             (24x33)- (23x34)

792-782=10

(n+1) (n+10)-n (n+11)

n2 +10n + n+10 - n2 +11n                                                                    n2 +11n +10 - n2 +11n =10

In both ways the difference is

5x5 square

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 n                 n+4

n+40         n+44

n (n+44) - [(n+4) (n+40)]                                                   (5x41)-(1x45)

n2+44n - n2+40n + 4n +160                                                   205-45= 160

n2+44n +160 - n2+44n =160

In both ways the difference is 160

After doing this I will check if the nxn formula will work in a bigger grid size like 11x11

 1 2 12 13

Conclusion

S-1 x G = D

Sized = 2 x 3   G = 10   D = 20

2-1 =1

3-1 =2

1 x 2 =2

2 x 10 =20

 37 38 39 40 47 48 49 50 57 58 59 60

37 x 60 =2220    40 x 57 = 2280 2280 - 2220 = 60

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

SIZED = 3 x 4       G = 10   D = 60

3-1 = 2

4-1 = 3

2 x 3 = 6

6 x 10 = 60

I could prove that this formula can work for both rectangle and square.

 53 54 63 64

53 x 64 = 3392       54 x 63 = 3402      3402 - 3392 = 10

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 2 x 2       G = 10    D = 10

2-1 = 1

2-1 = 1

1 x 1 = 1

1 x 10 = 10

FINALLY

In this project I found that not all formula can work for rectangle and square.

But I found a formula can work for rectangle and square it does not matter the size of the box and I will proof that it does not matter for grid sizes too.

 28 29 30 40 41 42 52 53 54

28 x 54 = 1512    30 x 52 = 1560 1560 - 1512 = 48

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 3 x 3 G = 12    D = 48

3-1 = 2

3-1 = 2

2 x 2 = 4

4 x 12 = 48

If I were to extend this project further I would try and do cube in three dimensions.

ISRAA AL-RIFAEE                                                    -  -

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

Related GCSE Number Stairs, Grids and Sequences essays

1. Algebra Investigation - Grid Square and Cube Relationships

It is possible to use the 2x2, 3 and 4 rectangles above to find the overall, constant formula for any 2xw rectangle on a 10x10 grid, by implementing the formulae below to find a general calculation and grid rectangle. It is additionally possible to see that the numbers that are added to n (mainly in the corners of the grids)

2. Number grid Investigation

x 201 = 33969 34029 - 33969 = 60 Then I found the difference of 60. I repeated this process four times with other numbers from the grid to see if the difference would change. 39 40 41 54 55 56 69 70 71 69 x 41 = 2829 39

1. number grid

top right number and the bottom left number for any rectangle were I change the length but keep the width the same. Results Length Of Rectangle (L) Difference (d) 3 20 4 30 5 40 6 50 After looking at my table I have found out that the difference is 10 subtracted off 10 multiplied the length of the rectangle.

2. number grid investigation]

For example, if one of the previously tested 4x4 boxes is examined: This means that if the width (w) of 4 is inserted into the formula, the difference of 90 should be returned. Difference (d) = 10(4-10� Difference (d) = 10 x 3� Difference (d)

1. Number Grid

Check: x= 74 y = (74+3) (74+30) - 74(74+33) = 742 + 2442 + 90 - 742 - 2442 = 90 This further suggests that y = 90 However, I want to check once more whether y = 90: x = 63 y = (63+3) (63+30) - 63(63+33)

2. Maths - number grid

Again I am going to use algebra to prove that the defined difference of my 3x3 squares is correct. r (r+2) (r+20)(r+22) (r+2)(r+20)-(r+22)r =r (r+20)+2(r+20) - r -22r =r +20r+2r+40-r -22r =40 I have now calculated a trend for my 2x2 squares and came to a difference of 10 and

1. number grid

it is necessary to use summary tables. Using these summary tables, it will be possible to establish and calculate the algebraic steps needed to gain an overall formula. 2 x 2 Grid Prediction I believe that this difference of 10 should be expected to remain constant for all 2x2 number boxes.

2. I am doing an investigation to look at borders made up after a square ...

3 16 4 20 5 24 Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to
improve your own work 