Planning
What is your task about? What do you have to do?
My task is to take a number grid, draw a square around four numbers, and work out the product of the top left corner and bottom right corner. I am to do the same for the top right and bottom left corner. I must then work out the formula for my calculations so it can apply to any four numbers in a square on that size grid.
Have you got any questions you would like to add to the original tasks?
After each section of results I will do an Analysis of my results to look for any kind of relationship forming.
What will you do first?
To begin with I will plan my results table and then begin work on my task analysis.
What information or data will you need to collect in order to complete the task?
The information I need to collect is the product of the top right and bottom left numbers in a two-by-two square on a ten-by-ten number grid. I will also need to find the product of the top right and bottom left numbers. I will then use these numbers to calculate an nth term.
How much information do you think you will need to collect?
I will need to take 3, 4 or 5 readings (depending on grid size) and an nth term to prove that my calculations are correct.
How will you know when you have got enough information?
I will know I have got enough data when I am able to successfully and easily calculate an nth term.
How accurate does your data have to be?
My data will all be in whole numbers so it does not need to be accurate to any decimal places.
How will you check that your data is accurate?
I will use a calculator to verify my results so I know that they are accurate.
What will you do with the data in order to investigate your original task?
I will place the data into a table, and from there I will work out an equation with three variables that will show how to work out any sized square on any sized grid.
Working Through
What sort of calculations will you need to get your solutions?
To get my initial results I will need only to use simple and long multiplication. When it comes to working out my nth terms and overall equation, I will need to use algebra.
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This is a preview of the whole essay
What will you do with the data in order to investigate your original task?
I will place the data into a table, and from there I will work out an equation with three variables that will show how to work out any sized square on any sized grid.
Working Through
What sort of calculations will you need to get your solutions?
To get my initial results I will need only to use simple and long multiplication. When it comes to working out my nth terms and overall equation, I will need to use algebra.
How will you record your results?
I will record my results in a table to make them easy to view and access.
Are there any special Mathematical techniques you will need to use?
The only special Mathematical technique I will be using will be algebra and that will be used at the end of the task to calculate my final equation.
Can you give reasons for your choice of calculations/techniques?
I have chosen to use simple and long multiplication for my task because these are the methods used to work out the product of two numbers. I have chosen to use algebra because it is needed to work out nth terms and an equation with three variables.
Are you going to change anything to make the task different?
In collecting my results I will be changing the grid and square sizes, to see if I can come up with any relationships and hopefully find the equation with three variables that will allow me to calculate any size square on any size grid.
Is there more then one way of tackling this task?
No.
Have you thought of trying a different method?
No methods other then the one I have chosen are useable for this task.
What methods will you use to present your results? Why will you use this particular method? What advantages does it have?
I will be presenting my results in a table form. I am using this form because it has the advantage of being easy to use and read, and can include space for my calculations.
Results: First Group
.1 - Ten-by Ten Grid, Two-by-Two Square
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
52
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
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57
58
59
60
61
62
63
64
65
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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
00
Numbers
Top Left x Bottom Right
Top Right x Bottom Left
Difference
, 2, 11, 12
x 12 = 12
2 x 11 = 22
0
26, 27, 36, 37
26 x 37 = 962
27 x 36 = 972
0
58, 59, 68, 69
58 x 69 = 4002
59 x 68 = 4012
0
73, 74, 83, 84
73 x 84 = 6132
74 x 83 = 6142
0
89, 90, 99, 100
89 x 100 = 8900
90 x 99 = 8910
0
n, n + 1, n + 10, n + 11
n(n + 11) = n2 + 11n
(n + 1)(n + 10) = n2 + 11n + 10
0
.2 - Seven-by-Seven Grid, Two-by-Two Square
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
Numbers
Top Left x Bottom Right
Top Right x Bottom Left
Difference
, 2, 8, 9
x 9 = 9
2 x 8 = 16
7
2, 13, 19, 20
2 x 20 = 240
3 x 19 = 247
7
25, 26, 32, 33
25 x 33 = 825
26 x 32 = 832
7
30, 31, 37, 38
30 x 38 = 1140
31 x 37 = 1147
7
41, 42, 48, 49
41 x 49 = 2009
42 x 48 = 2016
7
n, n + 1, n + 7, n + 8
n(n + 8) = n2 + 8n
(n + 1)(n + 7) = n2 + 8n + 7
7
.3 - Five-by-Five Grid, Two-by-Two Square
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
Numbers
Top Left x Bottom Right
Top Right x Bottom Left
Difference
, 2, 6, 7
x 7 = 7
2 x 6 = 12
5
9, 10, 14, 15
9 x 15 = 135
0 x 14 = 140
5
6, 17, 21, 22
6 x 22 = 352
7 x 21 = 357
5
9, 20, 24, 25
9 x 25 = 475
20 x 24 = 480
5
n, n + 1, n + 5, n + 6
n(n + 6) = n2 + 6n
(n + 1)(n + 5) = n2 + 6n + 5
5
Results Analysis One
There appears to be some sort of relationship between the sizes of the grid (5 x 5, 7 x 7, etc) and the difference in-between the product of the top left/bottom right and the top right/bottom left. As you can see on the above 5 x 5 grid, there is a difference of 5 between each reading. I will investigate further using different sized boxes and different
Results: Second Group
2.1 - Ten-by Ten Grid, Three-by-Three Square
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
52
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
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46
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51
52
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63
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65
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92
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98
99
00
Numbers
Top Left x Bottom Right
Top Right x Bottom Left
Difference
, 3, 21, 23
x 23 = 23
3 x 21 = 63
40
5, 17, 35, 37
5 x 37 = 555
7 x 35 = 595
40
47, 49, 67, 69
47 x 69 = 3243
49 x 67 = 3283
40
63, 65, 83, 85
63 x 85 = 5355
65 x 83 = 5395
40
78, 80, 98, 100
78 x 100 = 7800
80 x 98 = 7840
40
n, n + 2, n + 20, n + 22
n(n + 22) = n2 + 22n
(n + 2)(n + 20) = n2 + 20n + 40
40
2.2 - Seven-by-Seven Grid, Three-by-Three Square
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
Numbers
Top Left x Bottom Right
Top Right x Bottom Left
Difference
, 3, 15, 17
x 17 = 17
3 x 15 = 45
28
4, 6, 18, 20
4 x 20 = 80
6 x 18 = 108
28
23, 25, 37, 39
23 x 39 = 897
25 x 37 = 925
28
33, 35, 47, 49
33 x 49 = 1617
35 x 47 = 1645
28
n, n + 2, n + 14, n + 16
n(n + 16) = n2 + 16n
(n + 2)(n + 14) = n2 + 16n + 28
28
2.3 - Five-by-Five Grid, Three-by-Three Square
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
Numbers
Top Left x Bottom Right
Top Right x Bottom Left
Difference
, 3, 11, 13
x 13 = 13
3 x 11 = 33
20
3, 15, 23, 25
3 x 25 = 325
5 x 23 = 345
20
n, n + 2, n + 10, n + 12
n(n + 12) = n2 + 12n
(n + 2)(n + 10) = n2 + 12n + 20
20
Results Analysis Two
In this set of results I have realised that the difference between the two products is 4 times the grid size, i.e. on a 7x7 grid, with a 3x3 square, the difference will be 4x7. On a 5x5 grid with a 3x3 square, the difference will be 4x5.
