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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

image00.png

image01.png

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

image12.png

Top Left x Bottom Right-35 x 46= 1610

                          Top Right x Bottom Left- 36 x 45= 1620

                           1620-1610=10

Example 3

image23.png

                        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+11Ximage34.png

                             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= 2088image45.png

                                          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

image56.png

                              Top Left x Bottom Right- 72 x 94= 6768image67.pngimage67.png

                              Top Right x Bottom Left- 74 x 92= 6808

                              6808-6768= 40

Example 3

image81.png

                           Top Left x Bottom Right- 6 x 28= 168

                           Top Right x Bottom Left- 8 x 26= 208

                           208-168

General Rule For 3x3

...read more.

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.

image22.png

2x2

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

Example 1

image24.png

               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

image25.png

                Top Left x Bottom Right- 54 x 66= 3564

                Top Right x Bottom Left- 55 x 65= 3575

3575-3564=11

Example 3

image26.png

                         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:-

image27.png

                       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

image28.png

                             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

image29.png

                             Top Left x Bottom Right- 31 x 55= 1705

                             Top Right x Bottom Left- 33 x 53= 1749

                             1749-1705= 44

Example 3

image30.png

                                 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

...read more.

Conclusion

image75.png

                                                       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= 6000image76.png

                                                          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

image77.png

                                                          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:-

image78.png

                                                        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:-

image79.png

                                                                        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.

image80.png

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.

...read more.

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