# Number Grids Investigation.

Extracts from this document...

Introduction

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 |

Firstly, I will test a 2x2 grid from a 10x10 master grid. I will then find the sum of the top-left and bottom right numbers multiplied together and do the same for the top right and bottom-left numbers.

15 x 26 = 390

16 x 25 = 400 the difference between the top number and bottom number = 10

I will repeat this method again with another 2x2 grid.

1 x 12 = 12

2 x 11 = 22 the difference between the top number and bottom number = 10

I will again follow the same method with another 2x2 grid.

85 x 96 = 8160

86 x 95 = 8170 the difference between the top number and bottom number = 10

I will now show my working in algebraic terms.

n | n+1 |

n+10 | n+11 |

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

n²+11n+10 - n²11n = 10

By working using algebra, I can see that I will always get an answer of 10 on a 2x2 grid.

I will now test a 3x3 grid and carry out the same methods as before for my investigation.

1 x 23 = 23

3 x 21 = 63 the difference between the top number and the bottom number = 40

I will again do the same for another 3x3 grid.

31 x 53 = 1643

33 x 51 = 1683 the difference between the top number and bottom number = 40

I will again do the same for another 3x3 grid.

76 x 98 = 7800

78 x 96 = 7840 the difference between the top number and bottom number = 40

I will now show my working in algebraic terms.

n | n+2 | |

n+20 | n+22 |

(n+20)(n+2) - n(n+22) n²+22n+40 - n²- 22n = 40

Middle

I will now show my working in algebraic terms.

n | n+3 | ||

n+30 | n+33 |

(n+3)(n+30) - n(n+33)

n²+33n+90 - n²- 33n = 90

By working using algebra, I can see that I always get a difference of 90 on a 4x4 grid.

I will now test out a 5x5 grid and carry out the same methods as before for my investigation.

1 x 45 = 45

41 x 5 = 205 the difference between the top number and bottom number = 160

I will do the same for another 5x5 grid.

45 x 89 = 4005

49 x 85 = 4165 the difference between the top number and bottom number = 160

I will again do the same for another 5x5 grid.

56 x 100 = 5600

60 x 96 = 5760 the difference between the top number and bottom number = 160

I will now show my working in algebraic terms.

n | n+4 | |||

n+40 | n+44 |

(n+40)(n+4) - n(n+44)

n²+44n+160 - n² - 44n =160

By working using algebra, I can see I always get a difference of 160 on a 5x5 grid.

By using my information from the other grids, I have made a prediction for a 6x6 grid. The difference should be 250.

I will check this by using the same technique as before (using the 4 corners of a square) for a 6x6 grid in algebra.

n | n+5 |

n+50 | n+55 |

(n+5)(n+50) - n(n+55)

n²+ 55n+250 - n² -55n = 250

By working using algebra, I can that I will always get a difference of 250 on a 6x6 grid.

Conclusion

## n+12

## (n+10)(n+2) – n(n+12)

n² +12n+20- n²-12n = 20

By using algebra I can see I will always get a difference of 20 on a 2x3 rectangle

n | n+3 | ||

n+10 | n+13 |

(n+10)(n+3) – n(n+13)

n² +13n+30- n²-13n = 30

By using algebra I can see I will always get a difference of 30 on a 2x4 rectangle.

I will investigate the algebra for 4x3 grid.

n | n+3 | ||

n+20 | n+23 |

(n+20)(n+3)-n(n+23)

n²+23n+60 - n²-23n = 60

By using algebra, I can see that I will always get a difference of 60.

I will now investigate the algebra for 5x3 grid.

n | n+4 | |||

n+20 | n+24 |

(n+20)(n+4)-n(n+24)

n²+24n+80 - n²-24n =80

By using algebra, I can see I will always get a difference of 80.

Here are my overall results for the rectangles:

Difference of 2x3 = 20 =10 x 2

Difference of 2x4 = 30 =10 x 3

Difference of 3x4 = 60 =10 x 6

Difference of 3x5 = 80 =10 x 8

I can see that the formula will definitely have a 10 involved. I need to find out how to relate the size of the rectangle with the difference between the top and bottom numbers:

(2x3) 2x10 = 1x2x10

(2x4) 3x10 = 1x3x10

(3x4) 6x10 = 3x2x10

(3x5) 8x10 = 4x2x10

I can see from the grids with the width of 2 that I need to take 1 from the front number and 1 from the back number.

For an nxn rectangle the difference will be 10(m-1)(n-1).

This is the same result that I got with the nxn squares but with rectangles the length and width variey.

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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