# number grid

Extracts from this document...

Introduction

12800/w2 Joanne Barton

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 |

Number grid

Task

My coursework task is to look at the number grid

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

Investigate further.

In this investigation I will look at the different number grids and different sized rectangles within these grids, I will explain the patterns and give algebraic equations for the results found during this investigation. I will also try to find a formula and prove my findings.

To find out the product difference I need to:-

Multiply the top left number with the bottom right and multiply the top right with the bottom left and then - the products from each other.

I will first try with a 2 x 2 square.

12 | 13 |

22 | 23 |

a)

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

13 x 22 = 286 I worked the difference out by doing, 286-276 = 10

38 | 39 |

48 | 49 |

b)

38 x 49 = 1862

39 x 48 = 1872, I worked the difference out by 1872 - 1862 = 10

The product is again 10.

My theory is that all 2 x 2 squares will have the product of 10, I will show one more 2 x 2 square to prove the theory.

86 | 87 |

96 | 97 |

c)

86 x 97 = 8342

87 x 96 = 8352, I worked the difference out by 8352 – 8342 = 10

The product is 10 also, so the theory works all 2 x2 squares = D 10

D = Difference.

I have drawn a table to show this clearer.

First sum answer | Second sum answer | difference | |

a) | 8352 | 8342 | 10 |

b) | 1862 | 1872 | 10 |

c) | 276 | 286 | 10 |

Middle

7104

7144

40

c)

1560

1600

40

I will again use algebra to prove this.

## For a 3x3 square:

n = any number

n | n +2 |

n + 20 | n +22 |

(n + 20) (n + 2) – n(n + 22)

n² +2n+20n+40-n² -22n

n²+22n+40 - n²- 22n = 40

From this algebraic calculation it shows that the answer is 40, which is the difference of the 3x3 square.

This formula could be used for any 3x3 square in a 10x10 grid.

I will now try with a 4x4 square in a 10 x 10 grid and my prediction is the same as above, the difference will be the same for all 3 4x4 squares.

1 | 2 | 3 | 4 |

11 | 12 | 13 | 14 |

21 | 22 | 23 | 24 |

31 | 32 | 33 | 34 |

a)

I will multiply the top right number and the bottom left and the top left and bottom right.

1x34 = 34

4 x31 =124

To find the difference I will subtract the numbers.

124 – 34 = 90

The difference is 90.

42 | 43 | 44 | 45 |

52 | 53 | 54 | 45 |

62 | 63 | 64 | 65 |

72 | 73 | 74 | 75 |

b)

42 x 75 = 3150

45 x 72 = 3240

3240 – 3150 = 90

The difference is 90.

24 | 25 | 26 | 27 |

34 | 35 | 36 | 37 |

44 | 45 | 46 | 47 |

54 | 55 | 56 | 57 |

c)

24 x 57 = 1368

27 x 54 = 1458

1458 – 1368 = 90

The difference is again 90.

The table below shows this.

1st sum answer | 2nd sum answer | difference | |

a) | 34 | 124 | 90 |

b) | 3240 | 3150 | 90 |

c) | 1368 | 1458 | 90 |

I will again prove this with algebraically.

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

My prediction was right any number in a 4x4 square within a 10x10 grid has a difference of 90.

I drew a table to show my results so far.

Square size | Difference |

2x2 | 10 |

3x3 | 40 |

4x4 | 90 |

From the product differences I noticed a pattern emerging so I took the noughts away from the numbers, I would be left with square numbers i.e. 1, 4, and 9.

The next square number would be 16 so if I added the naught I would get 160, so I predict that a 5x5 square will have the difference of 160.

I will now test my prediction.

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 |

a)

.

1 x 45 = 45

5 x 41 = 205

205 – 45 = 160

The difference is 160 as predicted.

50 | 51 | 52 | 53 | 54 |

60 | 61 | 62 | 63 | 64 |

70 | 71 | 72 | 73 | 74 |

80 | 81 | 82 | 83 | 84 |

90 | 91 | 92 | 93 | 94 |

b)

50 x 94 = 4700

54 x 90 = 4860

4860 – 4700 = 160

My prediction was right I will now prove this will algebra.

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

With this you can see the difference for any number in a 5x5 square on a 10x10 grid is 160, which proves my prediction.

Using the same method I predict that the 6x6 square on a 10x10 grid difference will be 250, because if I took the nought away it would be the next square number in the sequence, 25.

1, 4, 9, 16, and 25

I will test my theory using the same method I have used throughout.

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 |

Conclusion

1 | 2 | 3 | 4 | 5 |

11 | 32 | 13 | 14 | 15 |

a)

1 x 15 = 15

5 x 11 = 55

55 – 15 = 40

The difference is 40.

b)

65 | 66 | 67 | 68 | 69 |

75 | 76 | 77 | 78 | 79 |

65 x 79 = 5135

69 x 75 = 5175

5175 – 5135 = 40

My prediction was correct the difference was 40, I will prove this algebraically.

n | n + 4 |

n + 10 | n + 14 |

n(n + 14) – (n + 4)(n + 10) =

(n² + 14n) – (n² +14n + 40) = 40.

This proves the difference is 40 and the number has gone up by 10 again so you can then predict 2x6 difference of 60 and 2x7 difference of 70.

I am now going to find a formula.

I tried

(n – 1) X 10

n =the width of the box.

22 | 23 | 24 | 25 | 26 | 27 | 28 |

32 | 33 | 34 | 35 | 36 | 37 | 38 |

7 x 2

7 – 1 = 6

6 x 10 = 60

I predict using my formula that the difference of this 7 x 2 square will be 60.

22 x 38 = 836

28 x 32 = 896

896 – 836 = 60

My prediction was correct.

I could use this formula to work out any size of rectangle.

I.e. 18 X 2 = 170

Conclusion

I found 3 things out about rectangles in a 10x10 grid.

- The rectangles I used differences increased by 10 every time.
- The 2nd difference is always 10.
- The formula for any rectangle is (n – 1) X 10

Evaluation

By doing lots of experiments and recording my results I found different patterns in a square and a rectangle by finding the difference of the corner numbers, I also used algebra to prove the differences. I found 2 formulas which can be used in any size square and any size rectangle in a 10x10 grid.

I could have expanded my coursework if I had more time and looked at different grid sizes and maybe different shapes like triangles or t shape patterns.

This coursework has helped me to understand number patterns and has developed my ability to introduce algebra when trying to spot patterns and I am happy with my results.

This student written piece of work is one of many that can be found in our GCSE Miscellaneous section.

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