# Number grid algebraic course work

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Introduction

Number grid algebraic 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 |

Firstly I was given this number grid, the intrusions were…

- A box is drawn round four numbers.
- Find the product of the top left number and the bottom left in this box.
- Do the same with the top right and bottom left numbers.
- Calculate the difference between these products.
- Investigate further.

The first thing I did was follow these instructions. Then I changed the box size and looked for patterns.

A two by two box…

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 |

12×23=276 286-276= 10

13×22=286

I tested a two by two box two more times to ensure each time I got the answer of 10, also to ensure that the answer was the same in different areas of the grid. I will do this for each different size boxes.

78×89=6942 79×88= 6952 | 9×20=180 10×19= 190 |

6952-6942 =10 | 190-180=10 |

Middle

46

52

53

54

55

56

12×56=672 16×52=832 | 1×45=45 5×41=205 | 96×60=5760 100×56=5600 |

832-672=160 | 205-45=160 | 5760-5600=160 |

I have noticed that all these numbers are divisible by ten. I’m going to divide by 10 to see what answers I get…

box | Difference | Divide by 10 | answer |

2×2 | 10 | 10÷10= | 1 |

3×3 | 40 | 40÷10= | 4 |

4×4 | 90 | 90÷10= | 9 |

5×5 | 160 | 160÷10= | 16 |

I have notice that when I divide by 10 I get all square numbers as my answers. These answers are all less than the box size chosen. Eg 2×2 2-1=1 1²

box | Difference | Divide by 10 | answer | Square no |

2×2 | 10 | 10÷10= | 1 | 1² |

3×3 | 40 | 40÷10= | 4 | 2² |

4×4 | 90 | 90÷10= | 9 | 3² |

5×5 | 160 | 160÷10= | 16 | 4² |

So from this I can make an equation to test any box…

10 (n-1)(n-1)=10 (n-1)²

Now I will test rectangle boxes to further my investigation. I predicted that this formula would not work on the rectangles. I will have to alter the formula later.

A three by two box…

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 |

12×24=288 308-288=20

22×14=308

I tested the three by two box twice more in different places to see if it would be the same answer…

5×17=85 7×15=105 | 67×79=5293 69×77=5313 |

105-85=20 | 5313-5293=20 |

Now I will test the box as a two by three and see if this will alter my answer…

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 |

Conclusion

4293-4165=128

I can use the same formula on this grid except I have to change it slightly… this was my formula for the 10 grid

10(c-1)(r-1)

as this is a 8 grid and all my difference are divisible by 8 I will change my formula to

8(c-1)(r-1)

I can check my answers using this formula

8(5-1)(5-1)=128 8(2-1)(2-1)=8

I now know that I can use my formula to find my difference of rectangles in this grid. Eg.

A two by three grid…

11 | 12 | 13 |

19 | 20 | 21 |

11x21=231 247-231=16

13x19=247

8(2-1)(3-1)=16

The answers are the same so I think it is more efficient to use the formula to find the answers…

8(2-1)(4-1)=24 8(4-1)(2-1)=24

8(2-1)(5-1)=32 8(5-1)(2-1)=32

I will now further my investigation using algebra…

If I know that it’s a two by grid…

2

X | |

2

TL=X TR=X+1

BL=X+10 BR=X+11

TL X BR TRXBL

(X)(X+11) (X+1)(X+10)

X²+11X X²+10+X+10X X²+10+11X

## This helps to p

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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