# Number Grid

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Introduction

Kim Williams St Austell College GCSE Maths

Number Grid

For this piece of coursework I have been asked to identify the connections between calculating the top numbers in a square sized grid with the bottom numbers. This is to identify what connections there are for the four sets of numbers.

This is the 10 x 10 number grid, which I will use to investigate the products of the top and bottoms numbers when multiplied.

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 |

This is achieved through the following rule:

- A box is drawn around four numbers
- The top left number is multiplied with the bottom right number to find the product
- Then the top right number is multiplied with the bottom left number to find the product
- Then through calculating these products the difference will be attained

For example

2 x2 Grid

12 13

22 23

12 x 23 = 276 The difference from the two products are 10.

13 x 22 = 286 This is how I worked the difference out: 286-276 = 10

From this example I can now form my hypothesis, which in this case is a statement. I will aim to either prove my hypothesis correct or incorrect through investigating further.

Hypothesis

## “Calculating the product of the top left and bottom right number will be less than the product of the top right and bottom left number when calculated”.

This is a general hypothesis that I will aim to prove throughout the investigation.

## Investigate further

I will start by developing the task and investigating different sized grids.

Middle

7 8 9 10 7x40 = 280 The difference for this 4x4 square grid is 90.

17 18 19 20 10x37 = 370 This calculation was worked out by: 370-280 = 90

27 28 29 30

37 38 39 40

### I will do another 4x4 square grid to prove that this is correct.

45 46 47 48 45x78 = 3510 The difference for this 4x4 square grid is 90.

55 56 57 58 48x75 = 3600 This calculation was worked out by: 3600-3510 = 90

65 66 67 68

75 76 77 78

### From the calculations of the 4x4 square grids above the difference of the products is 90.

The algebraic calculation for the 4x4 square grid is the following:

n | n + 3 |

n + 30 | n + 33 |

(n + 3)(n + 30) – n(n+33)

n² +30n+3n+90-n² -33n

n²+33n+90 - n²- 33n = 90

From this calculation it shows that the answer is the correct, hence the 4x4 square grid product difference is 90.

This algebraic formula proves that it can be used for any 4x4 square grid in a 10x10 master grid.

I will now calculate the products of the 5x5 square grid to find the difference.

## 35 36 37 38 39 35x79 = 2765 The difference for this 5x5 square grid is 160.

45 46 47 48 49 39x75 = 2925 This calculation was worked out by:

## 55 56 57 58 59 2925-2765 = 160

## 65 66 67 68 69

## 75 76 77 78 79

55 56 57 58 59

## 65 66 67 68 69 55x99 = 5445 The difference for this 5x5 square grid is 160.

75 56 77 78 79 59x95 = 5605 This calculation was worked out by:

85 86 87 88 89 5605-5445 = 160

95 96 97 98 99

## From the above two calculation for the 5x5 square grid the difference is 160.

## The algebraic calculation for the 5x5 square grid is the following:

n | n + 4 |

n + 40 | n + 44 |

(n + 4)(n + 40) – n(n + 44)

n² +40n+4n+160-n² -44n

n² + 44n+160 - n² - 44n =160

Conclusion

I decided to carry out two examples for each individual size grid in a 10x10 master gird, as I wanted to ensure that I was conducting the calculations correctly. It therefore gave accurate and reliable results. Although I did not do two examples for the individual sized grids in the 6x6 master grid or for the rectangle size grids. This was because my method procedure was correct throughout the calculations for the squares in the 10x10 master grids so I did not feel it was necessary.

From the results/findings in the 6x6 master grid table it shows that the product differences of the 2x2, 3x3 and 5x5 square grids are multiples of six but not the 4x4 square grid product difference. The calculation for the 4x4 product difference is correct and as noted doesn’t follow the same pattern as the other square grid sizes.

The rectangles product differences increases by 40 each time as the grid size goes up, which proves my prediction correct. I stated in my prediction that as ‘the rectangle sizes goes up i.e. 2x5, 3x5 and 4x5 the product difference will increase. This would be because the rectangle grid sizes are increasing and therefore would have a higher number in the bottom left and right corner of the rectangle to multiply by when calculating’ From the prediction I showed an examples of an 2x5 and a 4x5 grid to justify my prediction.

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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