French Holidays
4 star(s)
Opposite corner
Extracts from this document...
Introduction
Opposite Corner
Candidate Sheet
Opposite corner
Firstly i will investigate the difference between the products of the numbers in oppsite corners of any rectangle that can be drawn in a 100-sqaure grid. After that i will choose randomly some rectangles from the 100-sqaure grids. After that i will mutiply the top right corner with bottom left corner (which will be coloured in blue) then i will multiply the bottom right corner with the top left corner (which will be coloured in yellow). After that i will find the answer then i will subtract the smallest number with biggest number, which will give me the answer of the opposite corner.
This is a diagram of a 100-sqaure grid.
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 |
1) (A) From the 100 sqaure i will choose a rectangle sided 2 by 3 horizontally
4 | 5 | 6 |
14 | 15 | 16 |
The products of the number in the opposite corner of this rectangle are:
4x16 = 64
14x6 = 84
The difference between these products is
84-64=20
The difference is 20
8 | 9 | 10 |
18 | 19 | 20 |
The product of the number in the opposite corner of this rectangle is:
8x20=160
18x10=180
The differences between these products are:
160-180=20
Middle
5
6
7
14
15
16
17
The product of the number in the opposite corner of this rectangle is:
4x17=68
14x7=98
The difference between these products is:
68-98=30
The difference is 30
24 | 25 | 26 | 27 |
34 | 35 | 36 | 37 |
The products of the number in the opposite corner of this rectangle are:
24x37=888
34x27=918
The differences between these products are:
888-918=30
The difference is 30
Now i have found out that a rectangle sized 2 by 4 is on anywhere in a 100 sqaure grid the difference is always going to be 30
I am going to choose a rectangle from the 100-sqaure grids sided 2 by 5 horizontally
1 | 2 | 3 | 4 | 5 |
11 | 12 | 13 | 14 | 15 |
The products of the number in the opposite corner of this rectangle are:
1x15=15
11x5=55
The differences between these products are:
55-15=40
The difference is 40
21 | 22 | 23 | 24 | 25 |
31 | 32 | 33 | 34 | 35 |
The products of the number in the opposite corner of this rectangle are:
21x35=735
31x25=775
The difference between these products is:
735-775=40
The difference is 40
6 | 7 | 8 | 9 | 10 |
16 | 17 | 18 | 19 | 20 |
The products of the number in the opposite corner of this rectangle are:
6x20=120
16x10=160
The difference between these products is:
120-160=40
The difference is 40
Now i have found out that a rectangle sized 2 by 5 is on anywhere in a 100 sqaure grid it is always going to be 40
Now i am going to choose a rectangle from the 100-sqaure grids sided 2 by 6 horizontally
25 | 26 | 27 | 28 | 29 | 30 |
35 | 36 | 37 | 38 | 39 | 40 |
The product of the number in the opposite corner of this rectangle is:
25x40=1000
35x30=1050
The difference between these products is:
1000-1050=50
The difference is 50
85 | 86 | 87 | 88 | 89 | 90 |
95 | 96 | 97 | 98 | 99 | 100 |
The product of the number in the opposite corner of this rectangle is:
85x100=8500
95x90=8550
The difference between these products is:
8500-8550=50
The difference is 50
62 | 63 | 64 | 65 | 66 | 67 |
72 | 73 | 74 | 75 | 76 | 77 |
The product of the number in the opposite corner of this rectangle is:
62x77=4774
72x67=4824
The difference between these products is:
4824-4774=50
The difference is 50
Now i have found out that a rectangle sized 2 by 6 is on anywhere in the 100 sqaure grid the difference will always going to be 50
Now i have found out that if i go up by 1 horizontally (e.g. 2 by 7, 2 by 8, 2 by 9...) the answer will go up by 10.
From the 100-sqaure grids I will now choose a rectangle 2 by 3 vertically
1 | 2 |
11 | 12 |
21 | 22 |
The product of the number in the opposite corner of this rectangle is:
1x22=22
21x2=42
The difference of this product is:
42-22=20
The difference 20
9 | 10 |
19 | 20 |
29 | 30 |
Conclusion
82-42=40
The difference is 40
4 | 5 |
14 | 15 |
24 | 25 |
34 | 35 |
44 | 45 |
The product of the number in the opposite corner of this rectangle is:
4x45=180
44x5=220
The difference between these products is:
220-180=40
Now i have found out that a rectangle sized 2 by 5 vertically is on anywhere in the 100 sqaure grid the difference is always going to be 40
From the 100-sqaure grids I will now choose a rectangle 2 by 6 vertically
1 | 2 |
11 | 12 |
21 | 22 |
31 | 32 |
41 | 42 |
51 | 52 |
The product of the number in the opposite corner of this rectangle is:
1x52=52
51x2=102
The difference between these products is:
102-52=50
The difference is 50
41 | 42 |
51 | 52 |
61 | 62 |
71 | 72 |
81 | 82 |
91 | 92 |
The product of the number in the opposite corner of this rectangle is:
41x92=3772
91x42=3822
The difference between these products is:
3822-3772=50
The difference is 50
4 | 5 |
14 | 15 |
24 | 25 |
34 | 35 |
44 | 45 |
54 | 55 |
The product of the number in the opposite corner of this rectangle is:
4x55=220
54x5=270
The difference between these products is:
270-220=50
The difference is 50
Now I have found out that a rectangle sized 2 by 6 vertically is on anywhere in the 100 sqaure grid the difference is always going to be 50
Now I have found out that if i go up by 1 vertically (e.g. 2 by 7, 2 by 8, 2 by 9...) the answer will go up by 10.
Omar Ferdous 10a - -
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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