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• Level: GCSE
• Subject: Maths
• Word count: 4251

Number Grid

Extracts from this document...

Introduction

Number Grid

Introduction

I am going to be looking at a 10x10 square and drawing a square around a 2x2 square around 4 numbers and multiplying the top left number by the bottom right number and multiplying the top right number by the bottom left. I will find out the difference and do 2 more and see if there is a pattern. I will then investigate this further. I will first be looking at a 10x10 square.

2x2

Example 1

Top Left x Bottom Right-12 x 23= 276

Top Right x Bottom Left-13 x 22= 286

286-276=10

The difference is 10. I will now look at 2 more examples and see if this continues

Example 2

Top Left x Bottom Right-35 x 46= 1610

Top Right x Bottom Left- 36 x 45= 1620

1620-1610=10

Example 3

Top Left x Bottom Right- 81 x 92=7452

Top Right x Bottom Left- 82 x 91=7462

7462-7452= 10

The difference of all 3 examples was 10 so it must be the same for any 2 x 2 in that particular number grid.

General Rule For 2x2

If we look at the 2x2 algebraically then it would look like this:-

Top Left x Bottom Right- (X)(X+11)=X2+11X

Top Right x Bottom Left- (X+1)(X+10)=X2+11X+10

Prove

X=2

Top Left x Bottom Right-(2)2+11(2)=4+22=26

Top Right x Bottom Left-(2)2+11(2)+10=4+22+10=36

36-26=10    QED

This proves the difference is always 10

3x3

I will now be looking at a 3x3 square and seeing if it follows a rule as well.

Example 1

Top Left x Bottom Right- 36 x 58= 2088

Top Right x Bottom Left- 38 x 56= 2128

2128-2088= 40

The difference is 40. I will now do 2 further examples to see if this is the same for those 2

Example 2

Top Left x Bottom Right- 72 x 94= 6768

Top Right x Bottom Left- 74 x 92= 6808

6808-6768= 40

Example 3

Top Left x Bottom Right- 6 x 28= 168

Top Right x Bottom Left- 8 x 26= 208

208-168

General Rule For 3x3

Middle

2+10X(N-1)+X(M-1)+10(N-1)(M-1)

T.R. X B.L. – T.L. x B.R.=X2+10X(N-1)+X(M-1)+10(N-1)(M-1)-X2+10X(N-1)+X(M-1)

=10(N-1)(M-1)

Prove

M=3 N=2

10(2-1)(3-1)=10(1)(2)=20 QED

This is correct as the difference for a 3x2 rectangle is 20 so this proves this equation can be used for any MxN square

I will now be looking at the same shapes in a different number grid. One which has row’s of 11 instead of 10, basically a 11x10 number grid.

2x2

I will be looking at a 2x2 square in this new grid and seeing if it follows any patterns.

Example 1

Top Left x Bottom Right- 27 x 39=1053

Top Right x Bottom Left- 28 x 38= 1064

1064-1053= 11

The difference is 11. I will do 2 more examples to see if this is the pattern

Example 2

Top Left x Bottom Right- 54 x 66= 3564

Top Right x Bottom Left- 55 x 65= 3575

3575-3564=11

Example 3

Top Left x Bottom Right- 79 x 91= 7189

Top Right x Bottom Left- 80 x 90= 7200

General Rule For 2x2

If we look at the 2x2 square algebraically it would look like this:-

Top Left x Bottom Right- (X)(X+12)= X2+12X

Top Right x Bottom Left- (X+1)(X+11)=X2+12X+11

Prove

X=2

Top Left x Bottom Right- (2)2 +12(2)=4+24=28

Top Right x Bottom Left- (2)2+12(2)+11=4+24+11=39

39-28=11 QED

This proves the difference is always 11

3x3

I will now be looking at a 3x3 square in this number grid and seeing if it follows a pattern.

Example 1

Top Left x Bottom Right- 1 x 25= 25

Top Right x Bottom Left- 3 x 23= 69

69-25= 44

The difference is 44. I will do further examples to see if there is a pattern.

Example 2

Top Left x Bottom Right- 31 x 55= 1705

Top Right x Bottom Left- 33 x 53= 1749

1749-1705= 44

Example 3

Top Left x Bottom Right- 71 x 95= 6745

Top Right x Bottom Left- 73 x 93= 6789

6789-6745= 44

General Rule For 3x3

Conclusion

Top Left x Bottom Right- (X)(X+65)=X2+65X

Top Right x Bottom Left- (X+15)(X+50)= X2+65X+750

Prove

X=2

Top Left x Bottom Right- (2)2+65(2)=4+130=134

Top Right x Bottom Left- (2)2+65(2)+750=4+130+750= 884

884-134=750 QED

This proves the difference is always 750

5x2

I will now be looking at a 5x2 rectangle in this particular grid.

Example 1

Top Left x Bottom Right- 50 x 120= 6000

Top Right x Bottom Left- 70 x 100= 7000

7000-6000=1000

The difference is 1000. I will do 2 further examples to see if this is the same for those as well.

Example 2

Top Left x Bottom Right- 25 x 95= 2375

Top Right x Bottom Left- 45 x 75= 3375

3375-2375=1000

Example 3

If we look at a 4x2 rectangle algebraically it would look like this:-

Top Left x Bottom Right- 405 x 475= 192375

Top Right x Bottom Left- 425 x 455= 193375

193375-192375=1000

General Rule For 5x2

If we look at a 4x2 rectangle algebraically it would look like this:-

Top Left x Bottom Right- (X)(X+70)=X2+70X

Top Right x Bottom Left- (X+20)(X+50)= X2+70X+1000

Prove

X=2

Top Left x Bottom Right- (2)2+70(2)=4+140=144

Top Right x Bottom Left- (2)2+70(2)+1000=4+140+1000=1144

1144-144=1000 QED

This proves the difference is always 1000

General Rule For MxN

An M x N rectangle can also be written like this. I have only included the four numbers that are used in the multiplication as the others are not necessary in this equation.

Top Left x Bottom Right- (X)(X+50N-50+5M-5)=X2+50NX-50X+5MX-5X

Top Right x Bottom Left- (X+5M-5)(X+50N-50)= X2+50NX-5X+5MX-5X+(50N-50)(5M-5)

T.R.xB.L.-T.L.xB.R.=X2+50NX-5X+5MX-5X+(50N-50)(5M-5)- X2+50NX-50X+5MX-5X

= (50N-50)(5M-5)= 250MN-250N-250M+250

Prove

M=3 X=2

250MN-250N-250M+250=250(3)(2)-250(2)-250(3)+250=1500-500-750+250=500

*the difference for 3x2 is 500

Conclusion

In this piece I have proved that in any number grid, as long as the number sequences follow a regular pattern, there is a NxN and MxN pattern which will work and have separate rules and results for separate grids.

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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