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• Level: GCSE
• Subject: Maths
• Word count: 1645

# Number Grid

Extracts from this document...

Introduction

Maths Coursework - Number Grid

Friday 13th July 2007

Number Grid Coursework

For this piece of coursework, I will investigate the difference when 2x2, 3x3, 4x4, 5x5 and rectangle snapshots are taken from a 10x10 number grid and have their corners multiplied and the difference worked out. For the first part, I will use 2x2 snapshots.

2x2 Boxes

Box 1

 2 3 12 13

2x13=26

36-26=10

3x12=36

Box 2

 32 33 42 43

32x43=1376

1386-1376=10

33x42=1386

Box 3

 6 7 16 17

6x17=102

112-102=10

7x16=112

Box 4

 5 6 15 16

5x16=80

90-80=10

6x15=90

I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 10. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 2x2 snapshot, the difference I will find will be 10.

Box 5

 25 26 35 36

25x36=900

910-900=10

26x35=910

My predictions that I made earlier about ‘Box 5’ have turned out to be correct as when I multiplied the numbers in the 2x2 box and worked out the difference, I was left with 10.

Middle

59

60

67

68

69

70

77

78

79

80

47x80=3760

3850-3760=90

50x77=3850

My predictions that I made earlier about ‘Box 4’ have turned out to be correct as when I multiplied the numbers in the 4x4 box and worked out the difference, I was left with 90.

I should now try the same method but with boxes of 5x5 dimensions.

5x5 Boxes

Box 1

 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

1x45=45

205-45=160

5x41=160

Box 2

 6 7 8 9 10 16 17 18 19 20 26 27 28 29 30 36 37 38 39 40 46 47 48 49 50

6x50=300

460-300=160

10x46=460

I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 160. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 5x5 snapshot, the difference I will find will be 160.

Box 3

 23 24 25 26 27 33 34 35 36 37 43 44 45 46 47 53 54 55 56 57 63 64 65 66 67

23x67=1541

1701-1541=160

27x63=1701

My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 5x5 box and worked out the difference, I was left with 160.

I should now try the same method but with rectangles.

Rectangles

Rectangle 1

 1 2 3 4 11 12 13 14 21 22 23 24

1x24=24

84-24=60

4x21=84

Rectangle 2

 7 8 9 10 17 18 19 20 27 28 29 30

7x30=210

270-210=60

10x17=270

I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 60. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 4x3 snapshot, the difference I will find will be 60.

Rectangle 3

 22 23 24 25 32 33 34 35 42 43 44 45

22x45=990

1050-990=60

25x42=1050

My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 4x3 box and worked out the difference, I was left with 60.

Algebra Variables

For this specific investigation, I have decided to use algebra to represent and then make formulae for each snapshot I have taken from the 1-100 grid. I will use the letter H to represent the height of each snapshot and use the letter W for the width of each snapshot, no matter of size and dimensions. In this, S is the top left corner in each snapshot. I will now prove how 2x2 boxes always leave you with a difference of 10 through algebra.

Equation

 S S+1 S+10 S+10+1

Conclusion

2

= S2-20W-11S+11WS+10W2+10

Next, I will show how to find the formula for any Rectangle.

Finding the Formula Which Can Be Used On Any Rectangle

S+10L-10xS+(W-1)=

S2+10LS-10S(W-1)S+10(L1)(W-1)

SxS+10(L-1)+(W-1)=

S2+10LS-10S(W-1)S+10(L-1)(W-1)-(S2+10LS-10S+(W-1)S)

=10(L-1)(W-1)

I have now found the general rule to work out the formula for any rectangle on a 10x10 grid. The rule is 10(L-1)(W-1).

Expanding the Brackets

s(s+11)

 S +11 S S2 +11

s2+11s

(s+10)(s+1)

 s +10 s s2 +10s +1 +s +10

s2+11s+10-(s2+11s)

s2+11s+10-(s2-11s)

=+10

Algebra

2x2 Box

 s s +1 s+10 s+10+1

s(s+11)(s+1)-s(s+11)

3x3 Box

 s s+1 S+1+1 S+10 s+10+1 S+10+1+1 s+10+10 s+10+10+1 S+10+10+1+1

s(s+10+10+1+1)-(s+1+1)

(s+10+10)

s+20

4x4 Box

 S S+1 S+1+1 S+1+1+1 S+10 S+10+1 S+10+1+1 S+10+1+1+1 S+10+10 S+10+10+1 S+10+10+1+1 S+10+10+1+1+1 s+10+10+10 s+10+10+10+1 s+10+10+10+1+1 S+10+10+10+1+1+1

s(s+33)-(s+30)(s+3)

5x5 Box

 s s+1 s+1+1 s+1+1+1 s+1+1+1+1 s+10 S+10+1 S+10+1+1 S+10+1+1+1 S+10+1+1+1+1 S+10+10 S+10+10+1 S+10+10+1+1 S+10+10+1+1+1 S+10+10+1+1+1+1 S+10+10+10 S+10+10+10+1 S+10+10+10+1+1 S+10+10+10+1+1+1 S+10+10+10+1+1+1+1 s+10+10+10+10 s+10+10+10+10+1 s+10+10+10+10+1+1 s+10+10+10+10+1+1+1 s+10+10+10+10+1+1+1+1

s(s+44)-(s+40)(s+4)

DIFFERENCE IS 10 WHEN YOU GO DOWN A BOX.

Shiva Dhunna

11S/Ma2

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