# Investigate the products of 2x2 number squares within a large 10x10 number grid.

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Introduction

Coursework Plan

My coursework task is to investigate the products of 2x2 number squares within a large 10x10 number grid. My investigation will start by isolating a 2x2 square of numbers, I will then find the differences between the products of the top left & bottom right numbers, and the top right and bottom left numbers. I will then do this again with 3 more sets of numbers, and try to find a pattern. I will then go on to investigate 3x3, 4x4, 5x5, etc number squares within a 10x10 large number grid and see how the pattern, if there is one, changes and try and explain why it changes. After finding a suitable formula for number squares within a 10x10 large number grid, I will change the size of the large number grid to 9x9, 11x11, 20x20 etc, and repeat the process again, going through 2x2, 3x3, etc number squares, trying to find patterns. Once I have found a pattern for each of the large number grids, I will then try and work out a general formula for all number grids. Throughout my coursework I will be explaining all the things I am doing, and giving reasons for them.

Year 10 Maths Coursework

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 | 28 | 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 |

- I will start my investigation by isolating a 2x2 square of numbers.

57 | 58 |

67 | 68 |

- I will now find the products of the top left & bottom right numbers, and the top right & bottom left numbers.

57 x 68 = 3876

58 x 67 = 3886

- I will then write down what I notice

Middle

3 x 3

40

30 (3 x 10)

4 x 4

90

50 (5 x 10)

I will now investigate 5 x 5 number squares in a large 10 x 10 number grid, and try to see if there are any more patterns. I predict that the difference will be 160, as this will be 70 (7 x 10) more than the last difference

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 | 28 | 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 |

I will investigate the 5 x 5 grid like I have all the other grids, finding the products of the top left and bottom right numbers, and the top right and bottom left numbers.

1 | 2 | 3 | 4 | 5 |

11 | 12 | 13 | 14 | 15 |

21 | 22 | 23 | 24 | 25 |

31 | 32 | 33 | 34 | 35 |

41 | 42 | 43 | 44 | 45 |

1 x 45 = 45

5 x 41 = 205

The difference between these two values is 160, 70 more than last time.

51 | 52 | 53 | 54 | 55 |

61 | 62 | 63 | 64 | 65 |

71 | 72 | 73 | 74 | 75 |

81 | 82 | 83 | 84 | 85 |

91 | 92 | 93 | 94 | 95 |

51 x 95 = 4845

55 x 91 = 5005

The difference between these two values is again 160.

36 | 37 | 38 | 39 | 40 |

46 | 47 | 48 | 49 | 50 |

56 | 57 | 58 | 59 | 60 |

66 | 67 | 68 | 69 | 70 |

76 | 77 | 78 | 79 | 80 |

The difference is again 160. At this point in my investigation I have noticed another pattern, which I shall now write down. Also my previous prediction was correct.

I have noticed that the differences are going up in order of 10 multiplied by square numbers. For example, in the 2 x 2 number square, the difference is 10, which is 10 multiplied by 1 squared. In the 3 x 3 number square, the difference was 40, 10 multiplied by 2 squared. I will now show these findings in a table to show them more clearly. I will then try and put my findings into a formula, then test it in 6 x 6 number squares to see if it would work for all sizes of number squares in a 10 x 10 number grid.

Size of number square | Difference | 10 x ? squared = difference |

2 x 2 | 10 | 10 x 1 squared |

3 x 3 | 40 | 10 x 2 squared |

4 x 4 | 90 | 10 x 3 squared |

5 x 5 | 160 | 10 x 4 squared |

I have also noticed that the number squared and multiplied by 10 is always 1 less than the size of the grid. For example, 1 is 1 less than 2, 2 is 1 less than 3 and so on.

I will now investigate 6 x 6 number squares, to see if this pattern carries on. I predict that the difference will be 250, 10 multiplied by 5 squared. If this is true, so will be my prediction about the difference between the differences going up in order of odd numbers multiplied by 10.

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 | 28 | 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 |

1 | 2 | 3 | 4 | 5 | 6 |

11 | 12 | 13 | 14 | 15 | 16 |

21 | 22 | 23 | 24 | 25 | 26 |

31 | 32 | 33 | 34 | 35 | 36 |

41 | 42 | 43 | 44 | 45 | 46 |

51 | 52 | 53 | 54 | 55 | 56 |

1 x 56 = 56

6 x 51 = 306

The difference is, as I predicted, 250, which is 10 multiplied by 5 squared, I will see if this is the same for 2 more sets of 6 x 6 number squares.

5 | 6 | 7 | 8 | 9 | 10 |

15 | 16 | 17 | 18 | 19 | 20 |

25 | 26 | 27 | 28 | 29 | 30 |

35 | 36 | 37 | 38 | 39 | 40 |

45 | 46 | 47 | 48 | 49 | 50 |

55 | 56 | 57 | 58 | 59 | 60 |

5 x 60 = 300

10 x 55 = 550

So far my prediction has been correct. I will do one mores et just to be 100% certain.

42 | 43 | 44 | 45 | 46 | 47 |

52 | 53 | 54 | 55 | 56 | 57 |

62 | 63 | 64 | 65 | 66 | 67 |

72 | 73 | 74 | 75 | 76 | 77 |

82 | 83 | 84 | 85 | 86 | 87 |

92 | 93 | 94 | 95 | 96 | 97 |

Conclusion

Size of number square | Difference of products | ? squared = difference | 9 x ? squared = difference |

2 x 2 | 9 | 3 squared | 9 x 1 squared |

3 x 3 | 36 | 6 squared | 9 x 2 squared |

4 x 4 | 81 | 9 squared | 9 x 3 squared |

5 x 5 | 144 | 12 squared | 9 x 4 squared |

The formula for a 9 x 9 large number grid, where n = size of number square (e.g. 2 = 2 x 2) is –

From this I think that I can safely say that I have come up with a general formula, for any size large number grids, where n = size of number squares, and k = size of large number grid (e.g. 10 = 10 x 10)

I do not have time to complete any other examples for any other size large number grids, so I will make an assumption that this formula will definitely work for any other size large number grids. E.g. in a 20 x 20 large number grid, the formula for the differences in the different number squares would be

Conclusion

In this investigation I have managed to a) find a formula for the differences in a 10 x 10 large number grid, b) find a formula for differences in a 9 x 9 large number grid, c) found a general formula for any size large number grid and d) noticed other patterns in the numbers in the large number grids that I investigated. Had I not run out of time, I could have worked with different shape large number grids or number squares, and seen what patterns there were there. I conclude that the formula for finding differences between products of top left + bottom right, and top right and bottom left numbers in any size large number grids, when n = size of number square and k = size of large number grid, is -

Kieran Paul 10E

Mrs Johnson

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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