Number Grid Coursework
The grid I will using is a 10 x 10 grid, numbers ranging from 1 to 100.
Grid Size = 10 x 10
Box size = 2 x 2
Task 1:
I will take a 2 x 2 box on the 100 numbered grid and I will find the product of the top left number and the bottom right number of this box, and will do the same with the top right and bottom left numbers of this box. I will then look at the relationship between the two products, by finding the difference.
2
3
22
23
2 x 23 = 276
3 x 22 = 286
286-276 = 10
Difference = 10
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
78
79
88
89
78 x 89 = 6942
79 x 88 = 6952
6952-6942 = 10
Difference = 10
I want to further test out the experiment:
55
56
65
66
55 x 66 = 3630
56 x 65 = 3640
3640-3630 = 10
Difference = 10
I will now represent all the numbers in the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
Table of results:
a
ad
bc
bc-ad
2
276
286
0
78
6942
6952
0
55
3630
3640
0
39
950
960
0
7
26
36
0
From the results of this experiment, I have come to realize that there is a distinct pattern in the results. From the results, this suggests that the difference between the products of ad and bc is always 10.
One can also see that if a is increased then so does ad and bc even though the difference between the two is not affected.
I will now assign the first number in the box with x, thus meaning that the number next to it will be x+1 and the one directly below x will be x+10 and then one next to that will be x+11.
x
x+1
x+10
x+11
This box shows how the numbers layout in each 2 x 2 box.
I will also represent the difference of ad and bc with the letter y.
Therefore, the two variables will be x and y:
x = Independent variable
y = Dependant Variable
I know:
y = bc - ad
Result:
y = 10
Proof:
y = (x+1) (x+10) - x(x+11)
= x2 + 11x + 10 - x2 - 11x
= 10
Therefore this proves that y = 10
Therefore proving that in a 2 x 2 box the difference between ad and bc will always be 10.
However, just to be sure, I want to check whether this is true by replacing x with a random number.
Check:
x= 45
y = (45+1) (45+10) - 45(45+11)
= 452 + 495 + 10 - 452 - 495
= 10
This further suggests that y = 10
However, I want to check once more whether y = 10:
x = 94
y = (94+1) (94+10) - 94(94+11)
= 942 + 1034+ 10 - 942 - 1034
= 10
These examples conform to the formula y = 10
This checks that y = 10
I believed that I have now proved that y = 10 in a 2 x 2 box on a grid with 100 numbers starting from 1 going to 100.
Task 2:
In task 2, I have decided to change the box size, by expanding it to a 3 x 3 box.
Grid size = 10 x 10
Box size = 3 x 3
I will now take a 3 x 3 box on the 100 numbered grid and I will find the product of the top left number and the bottom right number of this box, and will do the same with the top right and bottom left numbers of this box. I will then look at the relationship between the two products, by finding the difference. I want to find out whether there is any pattern in a 3 x 3 box, in the same way there was in a 2 x 2 box.
2
3
4
22
23
24
32
33
34
2 x 34 = 408
4 x 32 = 448
448-408= 40
Difference = 40
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
78
79
80
88
89
90
98
99
00
78 x 100 = 7800
80 x 98 = 7840
7800-7840 = 40
Difference = 40
I want to further test out the experiment:
55
56
57
65
66
67
75
76
77
55 x 77 = 4235
57 x 75 = 4275
4235-4275 = 40
Difference = 40
I will now represent all the corners of the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
Table of results:
a
ad
bc
bc-ad
2
408
448
40
78
7800
7840
40
55
4235
4275
40
38
2280
2320
40
7
203
243
40
From the results of this experiment, I have come to realize that there is a distinct pattern in the results. From the results, this suggests that the difference between the products of ad and bc is always 40, suggesting that a 3 x 3 box also has a pattern within it, as did the 2 x 2 box.
Prediction:
I predict.....
I will now assign the first number in the box with x, thus meaning that the number next to it will be x+1 and the one directly below x will be x+10 and then one next to that will be x+11 and so on.
x
x+1
x+2
x+10
x+11
x+12
x+20
x+21
x+22
This box shows how the numbers layout in each section of the 3 x 3 box
.
I will also represent the difference of ad and bc with the letter y.
Therefore, the two variables will be x and y:
x = Independent variable
y = Dependant Variable
I know:
y = bc - ad
Result:
y = 40
Proof:
y = (x+2) (x+20) - x(x+22)
= x2 + 22x + 40 - x2 - 22x
= 40
Therefore this proves that y = 40
Therefore proving that in a 3 x 3 box the difference between ad and bc will always be 40.
However, just to be sure, I want to check whether this is true by replacing x with a random number.
Check:
x= 53
y = (53+2) (53+20) - 53(53+22)
= 532 + 1166 + 40 - 532 - 1166
= 40
This further suggests that y = 40
However, I want to check once more whether y = 40:
x = 32
y = (32+2) (32+20) - 32(32+22)
= 322 + 704+ 40 - 322 - 704
= 40
These examples conform to the formula y = 40
This checks that y = 40
I believed that I have now proved that y = 40 in a 3 x 3 box on a grid with 100 numbers starting from 1 going to 100.
Task 3:
In task 3, I have decided to change the box size, by expanding it to a 4 x 4 box.
Grid size = 10 x 10
Box size = 4 x 4
I will now take a 4 x 4 box on the 100 numbered grid and I will find the product of the top left number and the bottom right number of this box, and will do the same with the top right and bottom left numbers of this box. I will then look at the relationship between the two products, by finding the difference. I want to find out whether there is any pattern in a 4 x 4 box, in the same way there was in a 2 x 2 box and 3 x 3 boxes. I have wanted to continue this approach of expanding the box size on the 10 x 10 grid so that I can see if there are patterns concerning with the change in box size, so then I can see if there is a correlation and compare all of them later.
4
5
6
7
24
25
26
27
34
35
36
37
44
45
46
47
4 x 47 = 658
7 x 44 = 748
658-748= 90
Difference = 90
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
61
62
63
64
71
72
73
74
...
This is a preview of the whole essay
4
5
6
7
24
25
26
27
34
35
36
37
44
45
46
47
4 x 47 = 658
7 x 44 = 748
658-748= 90
Difference = 90
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
61
62
63
64
71
72
73
74
81
82
83
84
91
92
93
94
61 x 94 = 5734
64 x 91 = 5824
5824-5734 = 90
Difference = 90
I want to further test out the experiment:
37
38
39
40
47
48
49
50
57
58
59
60
67
68
69
70
37 x 70 = 2590
40 x 67 = 2680
2680-2590 = 90
Difference = 90
I will now represent all the corners of the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
Table of results:
a
ad
bc
bc-ad
4
5734
5824
90
61
658
748
90
37
2590
2680
90
24
34
90
27
620
710
90
From the results of this experiment, I have come to realize that there is a clear pattern in the results. From the results this suggests that the difference between ad and bc is always 90, therefore y = bc-ad, suggesting y=90 in a 4 x 4 box, suggesting that a 4 x 4 box also has a pattern within it, as did the 2 x 2 box and the 3 x 3 box.
Prediction:
I predict.....
I will now assign the first number in the box with x, thus meaning that the number next to it will be x+1 and the one directly below x will be x+10 and then one next to that will be x+11 and so on.
x
x+1
x+2
x+3
x+10
x+11
x+12
x+13
x+20
x+21
x+22
x+23
x+30
x+31
x+32
x+33
This box shows how the numbers layout in each section of the 4 x 4 box.
I will also represent the difference of ad and bc with the letter y.
Therefore, the two variables will be x and y:
x = Independent variable
y = Dependant Variable
I know:
y = bc - ad
Result:
y = 90
Proof:
y = (x+3) (x+30) - x(x+33)
= x2 + 33x + 90 - x2 - 33x
= 90
Therefore this proves that y = 90
Therefore proving that in a 4 x 4 box the difference between ad and bc will always be 90.
However, just to be sure, I want to check whether this is true by replacing x with a random number.
Check:
x= 74
y = (74+3) (74+30) - 74(74+33)
= 742 + 2442 + 90 - 742 - 2442
= 90
This further suggests that y = 90
However, I want to check once more whether y = 90:
x = 63
y = (63+3) (63+30) - 63(63+33)
= 632 + 2079+ 90 - 632 - 2079
= 90
These examples conform to the formula y = 90
This checks that y = 90
I believed that I have now proved that y = 90 in a 4 x 4 box on a grid with 100 numbers starting from 1 going to 100.
Task 4:
In task 4, I have decided to change the box size, by expanding it to a 5 x 5 box.
Grid size = 10 x 10
Box size = 5 x 5
I will now take a 5 x 5 box on the 100 numbered grid and I will find the product of the top left number and the bottom right number of this box, and will do the same with the top right and bottom left numbers of this box. I will then look at the relationship between the two products, by finding the difference. I want to find out whether there is any pattern in a 5 x 5 box, in the same way there was in a 2 x 2 box and 3 x 3 boxes and the 4 x 4 box. I have wanted to continue this approach of expanding the box size on the 10 x 10 grid so that I can see if there are patterns concerning with the change in box size, so then I can see if there is a correlation and compare all of them later.
1
2
3
4
5
21
22
23
24
25
31
32
33
34
35
41
42
43
44
45
51
52
53
54
55
1 x 55 = 605
5 x 51 = 765
605-765= 160
Difference = 160
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
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
5005-4845 = 160
Difference = 160
I want to further test out the experiment:
6
7
8
9
0
6
7
8
9
20
26
27
28
29
30
36
37
38
39
40
46
47
48
49
50
6 x 50 = 300
0 x 46 = 460
460-300 = 160
Difference = 160
I will now represent all the corners of the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
Table of results:
a
ad
bc
bc-ad
1
605
765
60
51
4845
5005
60
6
300
460
60
56
5600
5760
60
33
2541
2701
60
From the results of this experiment, I have come to realize that there is a clear pattern in the results. From the results this suggests that the difference between ad and bc is always 160, therefore y = bc-ad, suggesting y=160 in a 5 x 5 box, suggesting that a 5 x 5 box also has a pattern within it, as did the 2 x 2 box , 3 x 3 box and a 4 x 4 box.
Prediction:
I predict.....
I will now assign the first number in the box with x, thus meaning that the number next to it will be x+1 and the one directly below x will be x+10 and then one next to that will be x+11 and so on.
x
x+1
x+2
x+3
x+4
x+10
x+11
x+12
x+13
x+14
x+20
x+21
x+22
x+23
x+24
x+30
x+31
x+32
x+33
x+34
x+40
x+41
x+42
x+43
x+44
This box shows how the numbers layout in each section of the 5 x 5 box.
I will also represent the difference of ad and bc with the letter y.
Therefore, the two variables will be x and y:
x = Independent variable
y = Dependant Variable
I know:
y = bc - ad
Result:
y = 160
Proof:
y = (x+4) (x+40) - x(x+44)
= x2 + 44x + 160 - x2 - 44x
= 160
Therefore this proves that y = 160
Therefore proving that in a 5 x 5 box the difference between ad and bc will always be 160.
However, just to be sure, I want to check whether this is true by replacing x with a random number:
Check:
x= 3
y = (3+4) (3+40) - 3(3+44)
= 32 + 132 + 160 - 32 - 132
= 160
This further suggests that y = 160
However, I want to check once more whether y = 160:
x = 24
y = (24+4) (24+40) - 24(24+44)
= 242 + 1056+ 160 - 242 - 1056
= 160
These examples conform to the formula y = 160
This checks that y = 160
I believed that I have now proved that y = 160 in a 5 x 5 box on a grid with 100 numbers starting from 1 going to 100.
After experimenting with the size of the box, starting with 2 x 2 and ending with 5 x 5 sized boxes, I want to compare the formulas that I got from each of them, as well as the results, in order for me to perhaps work out a formula relating with the box size:
Length of Box (columns)
Width of Box (rows)
Difference/ y = bc - ad
2
2
0
3
3
40
4
4
90
5
5
60
From this table, one can see that as both the length and width of the box is increased by one each time, the difference increases. It can be seen that the difference is increasing as you go down the table, I could also see that if all the differences are divided by ten then they all are squared numbers of 1, 2, 3 and 4.
Hence, I wanted to find a formula, which could relate all of these results together:
x(n + 10Sm +2 ) = n? + 10Smn + 2
(n + Sm)( n + 10Sm) = n? + 10Smn + Smn + 10Sm? = n ? + 10Smn +10Sm?
y =(x +mb) (x +10) - x(x+mb+10)
(x +mb) (x +10) = x² +10x + mbx +10mb
-x(x+mb+10) = x² -mbx +10x
Therefore:
y = x² + 10x + mbx + 10mb - x² - mbx - 10x
y = 10mb
I discovered that if the difference of one of the boxes is doubled and ten is added on to it then one gets the difference of the next box. The reason 10 was added was because it was the size of the grid, 10 x 10, and also I thought that it would make sense to add it onto the part of the formula with the brackets as all the differences are in the 10 times table.
Hence, due to this I could make a formula:
w = width
h = length
0(h-1)
Therefore, I decided to test the formula by substituting w with 3 from the 3 x 3 box:
y= difference
0(3-1) = y
y= 20
However, for some reason this formula did not work for the 3 x 3 box, because the difference was supposed to be 40, but I decided to try it with a 4 x 4 anyway:
0(4-1) = y
Y = 30
This showed that the formula was incorrect and I needed to change something in the formula, as I needed to get difference to equal 90 not 30 for a 4 x 4 box.
I realized that for the 3 x 3 box, I had to multiply the 10 by 4 in order to get 40 and in the 4 x 4 box, I needed to multiply the 10 by a 9 to get 90.
I noticed that the numbers are the square of the side of square minus 1. I discovered that if I squared the brackets then both 3 x 3 and 4 x 4 boxes would get their differences correct, as 40 and 90.
3 x 3 box:
0(3-1)²= y
0 x 2²= y
y= 40
4 x 4 box:
0(4-1)²= y
0 x 3²= y
y=90
Therefore, I proved that the general formula for the difference of bc and ad in a 10 x 10 squared grid was:
0(h-1)²
Task 5:
I wanted to see what change would occur if I changed the grid size.
The formula for the 10 x 10 grid was 10(h-1)², I predict that the grid size for a 5 x 5 grid will be 5(h-1)² and for a 8 x 8 grid, it will be 8(h-1)²:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
Task 5:
I now will investigate how changing the shape of the box will affect the patterns of the differences and whether it has any affect at all on the results, I will change the shape of the squared box into a rectangle:
Grid Size: 10 x 10
Box Size: 2 x 3
I will now take a 2 x 3 box on the 100 numbered grid and I will find the product of the top left number and the bottom right number of this box, and will do the same with the top right and bottom left numbers of this box. I will then look at the relationship between the two products, by finding the difference.
2
3
4
22
23
24
2 x 24 = 288
4 x 22 = 308
308-288= 20
Difference = 20
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
78
79
80
88
89
90
78 x 90 = 7020
80 x 88 = 7040
7040-7020 = 20
Difference = 20
I want to further test out the experiment so that I can make sure that the results above are not anomalies:
55
56
57
65
66
67
55 x 67 = 3685
57 x 65 = 3705
3685-3705 = 20
Difference = 20
I will now represent all the corners of the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
Table of results:
a
ad
bc
bc-ad
2
288
308
20
78
7020
7040
20
55
3685
3705
20
38
900
920
20
7
33
53
20
From the results of this experiment, I have come to realize that there is a distinct pattern in the results. From the results, this suggests that the difference between the products of ad and bc is always 20, suggesting that a 2 x 3 box also has a pattern within it.
Prediction:
I predict.....
I will now assign the first number in the box with x, thus meaning that the number next to it will be x+1 and the one directly below x will be x+10 and then one next to that will be x+11 and so on.
x
x+1
x+2
x+10
x+11
x+12
This box shows how the numbers layout in each section of the 2 x 3 box.
I will also represent the difference of ad and bc with the letter y.
Therefore, the two variables will be x and y:
x = Independent variable
y = Dependant Variable
I know:
y = bc - ad
Result:
y = 20
Proof:
y = (x+2) (x+10) - x(x+12)
= x2 + 12x + 20 - x2 - 12x
= 20
Therefore this proves that y = 20
Therefore proving that in a 2 x 3 box the difference between ad and bc will always be 20.
However, just to be sure, I want to check whether this is true by replacing x with a random number.
Check:
x= 25
y = (25+2) (25+10) - 25(25+12)
= 252 + 300 + 20 - 252 - 300
= 20
This further suggests that y = 20
This example conforms to the formula y = 20
This checks that y = 20
I believed that I have now proved that y = 20 in a 2 x 3 box on a grid with 100 numbers starting from 1 going to 100.
Task 6:
I now will investigate how changing the shape of the box will affect the patterns of the differences and whether it has any affect at all on the results, I will further extend the square to make it become a longer rectangle.
Grid Size: 10 x 10
Box Size: 2 x 4
I will now take a 2 x 4 box on the 100 numbered grid and I will find the product of the top left number and the bottom right number of this box, and will do the same with the top right and bottom left numbers of this box. I will then look at the relationship between the two products, by finding the difference.
44
45
46
47
54
55
56
57
44 x 57 = 2508
47 x 54 = 2538
2538-2508= 30
Difference = 30
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
4
5
6
7
4
5
6
7
4 x 17 = 68
7 x 14 = 98
98-68 = 30
Difference = 30
I want to further test out the experiment so that I can make sure that the results above are not anomalies:
77
78
79
80
87
88
89
90
77 x 90 = 6930
80 x 87 = 6960
6960-6930 = 30
Difference = 30
I will now represent all the corners of the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
Table of results:
a
ad
bc
bc-ad
4
68
98
30
44
2508
2538
30
77
6930
6960
30
82
7790
7820
30
63
4788
4818
30
From the results of this experiment, I have come to realize that there is a distinct pattern in the results. From the results, this suggests that the difference between the products of ad and bc is always 30, suggesting that a 2 x 4 box also has a pattern within it.
One can also see clearly that as a increased so did the products of the corners of the box.
Prediction:
I predict.....
I will now assign the first number in the box with x, thus meaning that the number next to it will be x+1 and the one directly below x will be x+10 and then one next to that will be x+11 and so on.
x
x+1
x+2
x+3
x+10
x+11
x+12
x+13
This box shows how the numbers layout in each section of the 2 x 4 box.
I will also represent the difference of ad and bc with the letter y.
Therefore, the two variables will be x and y:
x = Independent variable
y = Dependant Variable
I know:
y = bc - ad
Result:
y = 30
Proof:
y = (x+3) (x+10) - x(x+13)
= x2 + 13x + 30 - x2 - 13x
= 30
Therefore this proves that y = 30
Therefore proving that in a 2 x 4 box the difference between ad and bc will always be 30.
However, just to be sure, I want to check whether this is true by replacing x with a random number.
Check:
x= 25
y = (25+3) (25+10) - 25(25+13)
= 252 + 325 + 30 - 252 - 325
= 30
This further suggests that y = 30
This example conforms to the formula y = 30
This checks that y = 30
I believed that I have now proved that y = 30 in a 2 x 4 box on a grid with 100 numbers starting from 1 going to 100.
Task 7:
I want to do one more task on the rectangle so I can get enough data to compare results and subsequently after this task, I will compare all these results in a table and try to work out a formula.
Grid size: 10 x 10
Box Size: 2 x 5
1
2
3
4
5
21
22
23
24
25
1 x 25 = 275
5 x 21 = 315
315-275= 40
Difference = 40
Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else:
51
52
53
54
55
61
62
63
64
65
51 x 65 = 3315
55 x 61 = 3355
3355-3315 = 40
Difference = 40
I want to further test out the experiment:
81
82
83
84
85
91
92
93
94
95
81 x 95 = 7695
85 x 91 = 7735
7735-7695 = 40
Difference = 40
I will now represent all the corners of the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
Table of results:
a
ad
bc
bc-ad
1
275
315
40
51
3315
3355
40
81
7695
7735
40
25
975
015
40
33
551
591
40
From the results of this experiment, I have come to realize that there is a clear pattern in the results. From the results this suggests that the difference between ad and bc is always 40, therefore, y = bc-ad, suggesting y=40 in a 2 x 5 box, suggesting that a 2 x 5 box also has a pattern within it.
Prediction:
I predict.....
I will now assign the first number in the box with x, thus meaning that the number next to it will be x+1 and the one directly below x will be x+10 and then one next to that will be x+11 and so on.
x
x+1
x+2
x+3
x+4
x+10
x+11
x+12
x+13
x+14
This box shows how the numbers layout in each section of the 2 x 5 box.
I will also represent the difference of ad and bc with the letter y.
Therefore, the two variables will be x and y:
x = Independent variable
y = Dependant Variable
I know:
y = bc - ad
Result:
y = 40
Proof:
y = (x+4) (x+10) - x(x+14)
= x2 + 14x + 40 - x2 - 14x
= 40
Therefore this proves that y = 40
Therefore proving that in a 2 x 5 box the difference between ad and bc will always be 40.
However, just to be sure, I want to check whether this is true by replacing x with a random number:
Check:
x= 32
y = (32+4) (32+40) - 32(32+44)
= 322 + 448 + 40- 322 - 448
= 40
This example conforms to the formula y = 40
This checks that y = 40
I believed that I have now proved that y = 40 in a 2 x 5 box on a grid with 100 numbers starting from 1 going to 100.
Now I will also introduce another variable z to further investigate the problem. I want to see what patterns, if any, come out of the new formula I will use.
This will be used for the rectangles produce before:
The equation I will be using is z= ab-cd
Meaning I will multiply the two top corners and I will also multiply the two bottom corners of the box and then take away the product of the bottom from the top corners in order to get a difference, represented by z.
Grid Size: 10 x 10
Box size: 2 x 2:
I will now represent all the corners of the box with a letter, therefore the first letter in the box (left- top corner) will be a:
a
b
c
d
x
x+1
x+10
x+11
z = x(x+1) - (x+10) (x+11)
= (x² + x) - (x² +2) (x+110)
z = 22x +110
Check:
00I will now compare the rectangles 2 x 3, 2 x 4 and 2 x 5, but I will also compare them with 2 x 2.
Size of Box
Result
2 x 2
y= 10
2 x 3
y= 20
2 x 4
y= 30
2 x 5
y= 40
2 x 6
y= 50
From the results one can see that there is a clear pattern in the results. As one increases the length of the box by one each time ,represented now with the letter h, one sees that y increases by ten. This is interesting firstly because the grid size is 10 x 10, but also the fact that they go up by 10 each time, resembling the 10 times table.
I will now try to find out 2 x h:
x
x+h-1
x+10
x+h+9
The reason I have added h-1 is because h is the length of the box and -1 because the corner on the top right is the amount of columns away from x and therefore x 's column cannot be added, hence -1. I have made d x+h+9 because it would be x+h+10-1.
Therefore, to make the equation easier to understand I have represented h-1 as a letter on its own: hb
Therefore I have found the formula:
y =(x +hb) (x +10) - x(x+hb+10)
(x +hb) (x +10) = x² +10x + hbx +10hb
-x(x+hb+10) = -x² -hbx -10x
Therefore:
y = x² + 10x + hbx + 10hb - x² - hbx - 10x
y = 10hb
Check:
I want to check if this rule works, so therefore I will substitute h with a number.
h = 3
y= 10hb
y= 10 x 2
y= 20
This shows that the formula is correct because with a 2 x 3 box y = 20
I want to see if the formula works with other random numbers to:
h = 10
y = 10hb
y = 10 x 9
y = 90
This is correct as if the sequence continued then 10 x 10 would be 90.
I further backed up this formula:
x = 4
w = 4
y = x² + 10x + hbx + 10hb - x² - hbx - 10x
6 + 40 + 12 + 30 - 16 - 12 - 40 = 30
30 was the correct answer, because the difference between the two products of the opposite corners bc and ad in a 2 x 4 rectangle was 30, checking that the formula is correct.
I have proven that this formula is correct.
After experimenting and finding a formula with the change in h I wanted to see what change would be brought up if I changed the width of the box, and to see whether the formula would be the same as that of h.
I will label the width of the box as w.
After experimenting with the size of the box, starting with 2 x 2 and ending with 5 x 5 sized boxes, I want to compare the formulas that I got from each of them, as well as the results, in order for me to perhaps work out a formula relating with the box size:
Length of Box (columns)
Width of Box (rows)
Difference/ y = bc - ad
2
2
0
3
3
40
4
4
90
5
5
60
2 x n or something- write the formula below here!!!!!!!!!
From this table, one can see that as both the length and width of the box is increased by one each time, the difference increases. It can be seen that the difference is increasing as you go down the table, I could also see that if all the differences are divided by ten then they all are squared numbers of 1, 2, 3 and 4.
Hence, I wanted to find a formula, which could relate all of these results together:
I discovered that if the difference of one of the boxes is doubled and ten is added on to it then one gets the difference of the next box. The reason 10 was added was because it was the size of the grid, 10 x 10, and also I thought that it would make sense to add it onto the part of the formula with the brackets as all the differences are in the 10 times table.
Hence, due to this I could make a formula:
w = width
h = length
0(h-1)
Therefore, I decided to test the formula by substituting w with 3 from the 3 x 3 box:
y= difference
0(3-1) = y
y= 20
However, for some reason this formula did not work for the 3 x 3 box, because the difference was supposed to be 40, but I decided to try it with a 4 x 4 anyway:
0(4-1) = y
Y = 30
This showed that the formula was incorrect and I needed to change something in the formula, as I needed to get difference to equal 90 not 30 for a 4 x 4 box.
I realized that for the 3 x 3 box, I had to multiply the 10 by 4 in order to get 40 and in the 4 x 4 box, I needed to multiply the 10 by a 9 to get 90.
I noticed that the numbers are the square of the side of square minus 1. I discovered that if I squared the brackets then both 3 x 3 and 4 x 4 boxes would get their differences correct, as 40 and 90.
3 x 3 box:
0(3-1)²= y
0 x 2²= y
y= 40
4 x 4 box:
0(4-1)²= y
0 x 3²= y
y=90
Therefore, I proved that the general formula for the difference of bc and ad in a 10 x 10 squared grid was:
0(h-1)²
Task 5:
I wanted to see what change would occur if I changed the grid size.
The formula for the 10 x 10 grid was 10(h-1)², I predict that the grid size for a 5 x 5 grid will be 5(h-1)² and for a 8 x 8 grid, it will be 8(h-1)²:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
