I will take a 2x2 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results
Number Grid Coursework
My task is to investigate a 2x2 box on a 100 square
I will take a 2x2 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.
Test 1
54 55 54 x 65= 3510
64 65 55 x 64= 3520
3520-3510= 10coff ff" . "r se" . ff . "ff" . "w or". ff . " " . ff . "k inff foff " . ff . "!
DIFFERENCE = 10
Test 2
5 6 5 x 16= 80
5 16 6 x 15= 90
90-80= 10
DIFFERENCE = 10
Test 3
8 19 18 x 29= 522
28 29 19 x 28= 532
532-522= 10
DIFFERENCE = 10
Prediction
I predict that in a two by two square the difference will always be 10
Proof
83 84 83 x 94= 7802coaf af" . "r se" . af . "af" . "w or". af . " " . af . "k inaf foaf " . af . ".
93 94 84 x 93= 7812
7812-7802= 10
DIFFERENCE = 10
Algebraic Explanation
I will assign a letter to the first number in the 2 x 2 square, n.
The next number to the right will therefore be n+1
The number directly below it will then be n+10
The number diagonally across from it will be n+11
I will then times the corners together, like In did on the above examples.
Top Left hand corner x bottom right hand corner = n(n+11) = n² + 11n
Top right hand corner x bottom left hand corner = n² +1n+10n+10
n² +11n+10
(n² +11n+10) - (n² + 11n) = 10
Therefore the difference between the corners multiplied together will always be 10.
Expanding the Task
I now feel it will be interesting to look at a 3x3 number square on a 100 grid. I will take a 3x3 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.
Test 1
37 38 39 37 x 59= 2183
47 48 49 39 x 57= 2223
57 58 59 2223 - 2183 = 40
DIFFERENCE 40
Test 2
72 73 74 72 x 94= 6768
82 83 84 74 x 92= 6808
92 93 94 6808 - 6768 = 40
DIFFERENCE = 40
Test 3
2 3 1 x 23 = 23
1 12 13 3 x 21 = 63
21 22 23 63 - 23 = 40
DIFFERENCE = 40
Prediction
I predict that in a 3 x 3 square the difference will always be 40
Proof
26 27 28 26 x 48 = 1248
36 37 38 28 x 46 = 1288
46 47 48 1288 - 1248 = 40
DIFFERENCE = 40
Algebra
I will assign a letter to the first number in the 3x3square, n.
The right hand top corner will therefore be n+2
The left hand bottom corner will then be n+20 Carstens theorised sabkaurb's functionalism .
The corner diagonally across from it will be n+22
I will then times the corners together, like I did on the above examples.
Top Left hand corner x bottom right hand corner = n(n+22) = n² + 22n
Top right hand corner x bottom left hand corner = (n+20)(n+2) = n²+40+22n
(n²+40+22n) - (n² + 22n) = 40
Therefore the difference between the corners multiplied together will always be 40.
Expanding the Task Further
I now feel it will be interesting to look at a 4x4 number square on a 100 grid. I will take a 4x4 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.
57 58 59 60 57 x 90 = 5130cofd fd" . "r se" . fd . "fd" . "w or". fd . " " . fd . "k infd fofd " . fd . ".
67 68 69 70 60 x 87 = 5220
77 78 79 80 5220 - 5130 = 90
87 88 89 90 DIFFERENCE = 90
22 23 24 25 22 x 55 = 1210
32 33 34 35 25 x 52 = 1300
42 43 44 45 1300 - 1210 = 90
52 53 54 55 DIFFERENCE = 90
4 5 6 7 4 x 37 = 148
14 15 16 17 7 x 34 = 238
24 25 26 27 238 - 148 = 90
34 35 36 37 DIFFRENCE = 90
Prediction
I predict that in a 4 x 4 square the difference will always be 90
Proof
33 34 35 36 33 x 66 = 2178
43 44 45 46 36 x 63 = 2268
53 54 55 56 2268 - 2178 = 90
63 64 65 66 DIFFREMCE = 90
Algebra
I will assign a letter to the first number in the 4x4 square, n.
The right hand top corner will therefore be n+3
The left hand bottom corner will then be n+30
The corner diagonally across from it will be n+33
I will then times the corners together, like I did on the above examples.
Top Left hand corner x bottom right hand corner = n(n+33) = n² + 33n
Top right hand corner x bottom left hand corner = (n+30)(n+3) = n²+33n + 90
(n²+33n + 90) - (n² + 33n) = 90
Therefore the difference between the corners multiplied together will always be 90
Looking at the relationship between the differences of the corners multiplied together in different size squares - using a table to compare the findings
Size of Square Difference Difference of the Differences
2 x 2 10 30
3 x 3 ...
This is a preview of the whole essay
Top Left hand corner x bottom right hand corner = n(n+33) = n² + 33n
Top right hand corner x bottom left hand corner = (n+30)(n+3) = n²+33n + 90
(n²+33n + 90) - (n² + 33n) = 90
Therefore the difference between the corners multiplied together will always be 90
Looking at the relationship between the differences of the corners multiplied together in different size squares - using a table to compare the findings
Size of Square Difference Difference of the Differences
2 x 2 10 30
3 x 3 40 50
4 x 4 90 70???
Prediction: 5 x 5 160
I predict this because the difference of the differences go up by 20 each time e.g from 30 to 50. Therefore it would make sense for 50 to go up to 70, add this onto the 90 which makes 160, which is my prediction for the 5 x 5 square.
Proof
21 22 23 24 25 21 x 65 = 1365
31 32 33 34 35 25 x 61 = 1525
41 42 43 44 45 1365 - 1525 = 160
51 52 53 54 55 DIFFERENCE = 160
61 62 63 64 65
Working out a general equation for the difference between the two corners multiplied together with any number of sides on the square
As all of the numbers in the table were in the 10 times table (10, 40, 90, 160) then I will assume the equation has a x10 in it, and I will therefore take off the 0 from each of the numbers. This leaves 1, 4, 9, and 16. These are all square numbers. I notice that the numbers are the square of the side of square minus 1(see table to the right). This would mean that b-12 would make sense. I will encompass this with the x10 so it becomes:
0(b-1)2
Trying the equation out
I am going to use the example of a 4 x 4 square; I already know that the difference is 90.
0(b-1)2 = 10(4-1)2
10 x 32
10 x 9 = 90
THIS IS THE CORRECT ANSWER, my equation is right.
Working out the same equation through looking at general cases.
I am going to look at the numbers in the corners and how they relate to the width of a side of the number square. When n is the number in the top left hand corner and b is the length of the square.
The top left hand corner is n, so therefore the top right hand corner will be n + the width of the box minus 1 (as shown), this is because you move along the width of the box, but as the top left hand corner (n) takes up one column you minus 1. The bottom right hand corner of the box will have plus n + 10 because as you move down vertically the number increases by 10 each time. This will be multiplied by the side of the square minus 1, n+10(b-1). This is because you move down the width of the square, however the top left hand corner (n) takes up 1 row so you minus 1. To find the bottom right hand corner we have to keep the n+10(b-1) of the left hand bottom corner and then add b-1. I will then multiply the corners out, as I do on my investigation.
Left hand top corner x bottom right corner
n x [n +10 (b-1) + (b-1)] = n2 + 10n(b-1) + n(b-1)
= n2 + 11n(b-1)
Right hand top corner x bottom left corner
b-1 = a
[n+(b-1)] [n+10(b-1)]
As both formula have b-1 in I will replace this with an a
(n+a)(n+10a) = n2 + 10a2 + an + 10an
= n2 + 10a2 + 11an
I will now substitute the b-1 back in, instead of the a
= n2 + 10(b-1)2 + 11n(b-1)
Find the difference between the two multiplied corners
[n2 + 10(b-1)2 + 11n(b-1)] - [n2 + 11n(b-1)] =
0(b-1)2
Which is the same answer as I got when finding the formula before.
This means I have proved beyond any doubt that the formula for finding the difference of the corners multiplied together in a square on a 100 grid is 10(b-1)2.
Investigating rectangle boxes in a 100 square
I will take a 2x3 rectangle on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.
Test 1
2 13 14 12 x 24 = 288
22 23 24 14 x 22 = 308
308 - 288 = 20
DIFFERENCE = 20
Test 2
42 43 44 42 x 54 = 2268
52 53 54 44 x 52 = 2288
2288 - 2268 = 20
DIFFERENCE = 20
Test 3
4 5 6 4 x 16 = 64
14 15 16 6 x 14 = 84
84 - 64 = 20
DIFFERENCE = 20
Prediction
I predict that in a 2 x 3 rectangle the difference will always be 20
Proof
27 28 29 27 x 39 = 1053
37 38 39 29 x 37 = 1073
1073 - 1053 = 20
DIFFERENCE = 20
Algebra
I will assign a letter to the first number in the 2x3 rectangle, n.
The right hand top corner will therefore be n+2
The left hand bottom corner will then be n+10
The corner diagonally across from it will be n+22
I will then times the corners together, like I did on the above examples.
Top Left hand corner x bottom right hand corner = n(n+12) = n² + 12n
Top right hand corner x bottom left hand corner = (n+2)(n+10) = n²+2n+10n+20
= n²+12n+20
n²+12n+20 - n² + 12n = 20
Therefore the difference between the corners multiplied together will always be 20.
Expanding the Task
I now feel it will be interesting to look at a 3x4 number rectangle on a 100 grid. I will take a 3x4 rectangle on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.
Test 1
53 54 55 56 53 x 76 = 4028
63 64 65 66 56 x 73 = 4088
73 74 75 76 4088 - 4028 = 60
DIFFERENCE = 60
Test 2
67 68 69 70 67 x 90 = 6030
77 78 79 80 70 x 87 = 6090
87 88 89 90 6090 - 6030 = 60
DIFFERENCE = 60
Test 3
31 32 33 34 31 x 54 = 1674
41 42 43 44 34 x 51 = 1734
51 52 53 54 1734 - 1674 = 60
DIFFERENCE = 60
Prediction
I predict that in a 3 x 4 rectangle the difference will always be 60
Algebra
I will assign a letter to the first number in the 3x4 rectangle, n.
The right hand top corner will therefore be n+3
The left hand bottom corner will then be n+20
The corner diagonally across from it will be n+23
I will then times the corners together, like I did on the above examples.
Top Left hand corner x bottom right hand corner = n(n+23) = n² + 23n
Top right hand corner x bottom left hand corner = (n+20)(n+3) = n²+60+20n+3n
` = n2+60+23n
(n²+60+23n) - (n² + 23n) = 60
Therefore the difference between the corners multiplied together will always be 60.
Expanding the Task
I now feel it will be interesting to look at a 4x5 rectangle on a 100 grid. I will take a 4x5 rectangle on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference.
66 67 68 69 70 66 x 100 = 6660
76 77 78 79 80 70 x 96 = 6720
86 87 88 89 90 6720 - 6600 = 120
96 97 98 99 100 DIFFERENCE = 120
41 42 43 44 45 41 x 75 = 3075
51 52 53 54 55 45 x 71 = 3195
61 62 63 64 65 3195 - 3075 = 120
71 72 73 74 75 DIFFERENCE = 120
36 37 38 39 40 36 x 70 = 2520 sabkaurb, please do not redistribute this project. We work very hard to create this website, and we trust our visitors to respect it for the good of other students. Please, do not circulate this project elsewhere on the internet. Anybody found doing so will be permanently banned."
46 47 48 49 50 40 x 66 = 2640
56 57 58 59 60 2640 - 2520 = 120
66 67 68 69 70 DIFFERENCE = 120
Prediction
I predict that in a 4 x 5 rectangle the difference will always be 120
Proof
3 14 15 16 17 13 x 47 = 611
23 24 25 26 27 17 x 43 = 731
33 34 35 36 37 731 - 611 = 120
43 44 45 46 47 DIFFERENCE = 120
Therefore my prediction is right
Algebraic Explanation
I will assign a letter to the first number of the 4 x 5 rectangle, n
The right hand top corner will therefore be n+4
The left hand bottom corner will then be n+30
The corner diagonally across from it will be n+34
I will then multiply the corners together, like I did on the above examples.
Top Left hand corner x bottom right hand corner = n(n+34) = n2+34n
Top right hand corner x bottom left hand corner = (n+30)(n+4) = n2+120+30n+4n
= n2+120+34n
(n2+120+34n) - (n2+34n) = 120
Therefore the difference between the corners multiplied together will always be 120.
Looking at the relationship between the differences of the corners multiplied together in different size rectangles - using a table to compare the findings
Size of Rectangle Difference Difference of the Differences
2 x 3 20 40
3 x 4 60 60
4 x 5 120 80???
Prediction: 5 x 6 200
I predict this because the difference of the differences go up by 20 each time e.g from 40 to 60. Therefore it would make sense for 60 to go up to 80, add this onto the 120 which makes 200, which is my prediction for the 5 x 6 rectangle.
Proof
2 13 14 15 16 17 12 x 57 = 684
22 23 24 25 26 27 17 x 52 = 884
32 33 34 35 36 37 884 - 684 = 200
42 43 44 45 46 47 DIFFERENCE = 200
52 53 54 55 56 57
Working out a general equation for the difference between the two corners multiplied together with any size of rectangle
As all of the numbers in the table were in the 10 times table (20, 60, 120, 200) then I will assume the equation has a x10 in it. This makes sense, as the x10 is the factor that shows every time you go down a row you plus 10 every time. I will therefore take off the 0 from each of the numbers so I am able to discover the next part of the formula (see table to right). This leaves 2, 6, 12, and 20. I notice that the numbers are the length and width of the multiplied together minus 1(see table to the right). This would mean that (L -1)(w-1) would make sense. I will encompass this with the x10 so it becomes:
10(L-1)(w-1)
Trying the equation out
I am going to use the example of a 4 x 5 rectangle; I already know that the difference is 120.
0(L-1)(w-1) = 10(5-1)(4-1)
0 x 4 x 3
120
THIS IS THE CORRECT ANSWER, my equation is right.
Working out the same equation through looking at general cases.
I am going to look at the numbers in the corners and how they relate to the width and length of the rectangle. When n is the number in the top left hand corner, L is the length and w is the width.
The top left hand corner is n, so therefore the top right hand corner will be n + the length of the rectangle minus 1 (as shown), this is because you move along the length of the box, but as the top left hand corner (n) takes up one column you minus 1. The bottom right hand corner of the rectangle will have plus n + 10 because as you move down vertically the number increases by 10 each time. This will be multiplied by the width of the rectangle minus 1, n+10(w-1). This is because you move down the width of the rectangle, however the top left hand corner (n) takes up 1 row so you minus 1. To find the bottom right hand corner we have to keep the n+10(w-1) of the left hand bottom corner and then add L-1 (see diagram below for visual explanation of the algebra). I will then multiply the corners out, as I do on my investigation.
Left hand top corner x bottom right corner
n x [n +10 (w-1) + (L-1)] = n2 + 10n(w-1) + n(L-1)
Right hand top corner x bottom left corner
[n+(L-1)] [n+10(w-1)] = n2 + 10n(w-1) + n(L-1) + 10 (L-1)(w-1)
Find the difference between the two multiplied corners
[n2 + 10n(w-1) + n(L-1) + 10 (L-1)(w-1)] - [n2 + 10n(w-1) + n(L-1)] =
0(L-1)(w-1)
Which is the same answer as I got when finding the formula before.
This means I have proved beyond any doubt that the formula for finding the difference of the corners multiplied together in a rectangle on a 100 square grid is 10(L-1)(w-1).
Expanding the Task
I now feel it will be interesting to look at rectangles in different sized number grids. In doing this I hope to be able to create a general formula to find the difference between the corners of any rectangle on any sized number grid.
7x7 square grid. Working out a general equation for the difference between the two corners multiplied together with any size of rectangle.
I am going to look at the numbers in the corners and how they relate to the width and length of the rectangle. When n is the number in the top left hand corner, L is the length and w is the width.
The top left hand corner is n, so therefore the top right hand corner will be n + the length of the rectangle minus 1 (as shown), this is because you move along the length of the box, but as the top left hand corner (n) takes up one column you minus 1. The bottom right hand corner of the rectangle will have plus n + 7 because as you move down vertically the number increases by 7 each time. This will be multiplied by the width of the rectangle minus 1, n+7(w-1). This is because you move down the width of the rectangle, however the top left hand corner (n) takes up 1 row so you minus 1. To find the bottom right hand corner we have to keep the n+7(w-1) of the left hand bottom corner and then add L-1 (see diagram below for visual explanation of the algebra). I will then multiply the corners out, as I do on my investigation.
Left hand top corner x bottom right corner
n x [n + 7 (w-1) + (L-1)] = n2 + 7n(w-1) + n(L-1)
Right hand top corner x bottom left corner
[n+(L-1)] [n+7(w-1)] = n2 + 7n(w-1) + n(L-1) + 7 (L-1)(w-1)
Find the difference between the two multiplied corners
[n2 + 7n(w-1) + n(L-1) + 7 (L-1)(w-1)] - [n2 + 7n(w-1) + n(L-1)] =
7(L-1)(w-1)
I will now try this formula out with 3 examples on a 49 square, each with different sized rectangles, to prove whether it is correct
2x3 sized rectangle
Formula
7(L-1)(w-1) = 7(3-1)(2-1)
= 7 x 2 x 1
14
Therefore in theory the difference between the two corners multiplied together in a 2 x 3 rectangle on a 49 square grid will be 14
Example
6 17 18 16 x 25 = 400
23 24 25 18 x 23 = 414
414 - 400 = 14
DIFFERENCE = 14
The formula worked for this example
3x4 sized rectangle
Formula
7(L-1)(w-1) = 7(4-1)(3-1)
= 7 x 3 x 2
= 42
Therefore in theory the difference between the two corners multiplied together in a 3 x 4 rectangle on a 49 square grid will be 42.
Example
31 32 33 34 31 x 48 = 1488
38 39 40 41 34 x 45 = 1530
44 46 47 48 1530 - 1488 = 42
DIFFERENCE = 42
The formula worked for this example
4x5 sized rectangle
Formula
7(L-1)(w-1) = 7(4-1)(5-1)
= 7 x 3 x 4
= 84
Therefore in theory the difference between the two corners multiplied together in a 4 x 5 rectangle on a 49 square grid will be 84.
Example
9 10 11 12 13 9 x 34 = 306
6 17 18 19 20 13 x 30 = 390
23 24 25 26 27 390 - 306 = 84
30 31 32 33 34 DIFFERENCE = 84
The formula worked for this example
The formula has worked for every example; therefore I think it is right. I am now going to look at a different sized grid.
9x9 square grid. Working out a general equation for the difference between the two corners multiplied together with any size of rectangle.
I am going to look at the numbers in the corners and how they relate to the width and length of the rectangle. When n is the number in the top left hand corner, L is the length and w is the width.
The top left hand corner is n, so therefore the top right hand corner will be n + the length of the rectangle minus 1 (as shown), this is because you move along the length of the box, but as the top left hand corner (n) takes up one column you minus 1. The bottom right hand corner of the rectangle will have plus n + 9 because as you move down vertically the number increases by 9 each time. This will be multiplied by the width of the rectangle minus 1, n+7(w-1). This is because you move down the width of the rectangle, however the top left hand corner (n) takes up 1 row so you minus 1. To find the bottom right hand corner we have to keep the n+9(w-1) of the left hand bottom corner and then add L-1 (see diagram below for visual explanation of the algebra). I will then multiply the corners out, as I do on my investigation.
Left hand top corner x bottom right corner
n x [n + 9 (w-1) + (L-1)] = n2 + 9n(w-1) + n(L-1)
Right hand top corner x bottom left corner
[n+(L-1)] [n+7(w-1)] = n2 + 9n(w-1) + n(L-1) + 9(L-1)(w-1)
Find the difference between the two multiplied corners
[n2 + 9n(w-1) + n(L-1) + 9(L-1)(w-1)] - [n2 + 7n(w-1) + n(L-1)] =
9(L-1)(w-1)
I will now try this formula out with 3 examples on a 49 square, each with different sized rectangles, to prove whether it is correct
2x3 sized rectangle
Formula
9(L-1)(w-1) = 9(3-1)(2-1)
= 9 x 2 x 1
= 18
Therefore in theory the difference between the two corners multiplied together in a 2 x 3 rectangle on a 81 square grid will be 18.
Example
37 38 39 37 x 48 = 1776
46 47 48 39 x 46 = 1794
1794 - 1776 = 18
DIFFERENCE = 18
The formula worked for this example
3x4 sized rectangle
Formula
9(L-1)(w-1) = 9(3-1)(4-1)
