Conclusion
From the diagram above I am going to form a table and try to identify an overall formula:
My first part of my investigation is complete. The table on the previous page shows my results found throughout chapter one. From drawing this table up I was able to see a trend of multiples of 10, which allowed me to continue on and make a perfect square.
This table as a whole allowed me to meet my aim of identifying a formula. My result for this chapter on squares in a 10x10 number grid was:
10(r - 1)
Chapter Two
My initial investigation was looking at various squares randomly selected from the provided 10x10 grid. To make my investigation more interesting I am going to repeat my previous process except this time I will be using rectangles.
I predict with rectangles that I will get a different result than with squares, but my aim is still the same, I want to be able to establish a trend and an overall formula.
I shall now repeat my process using randomly selected rectangles from the 10x10 grid, starting with a 3x2 rectangle.
14x22 – 12x24
308 – 288
Difference = 20
48x56 – 46x58
2688 - 2668
Difference = 20
I am now going to try the same size of rectangle except turning it vertically to see if the outcome differs.
13x32 – 12x33
416 - 396
Difference = 20
47x66 – 46x67
3102 - 3082
Difference = 20
As can be seen, whether the rectangle is horizontal or vertical the outcome is the same. Because the outcome is the same I am going to concentrate on horizontal rectangles only.
I am now going to use algebra to prove my defined difference of 20 is correct when using 3x2 rectangles.
s s+1 s+2
s+10 s+11 s+12
(s+2) (s+10)-s (s+12)
= s(s+10) +2 (s+10)- s -12s
= s +10s+2s+20- s -12s
=20
Furthering my investigation
I am going to further my investigation by increasing the size of my rectangle to 5x3. These will be randomly selected from the 10x10 number grid, my aim being to find a pattern in my results.
I feel unable to make a prediction of a defined difference at this point; I hope to make able to predict some type of trend or pattern further on.
9x25 – 5x29
225 –145
Difference = 80
56x72 – 52x76
4032 –3952
Difference = 80
As can be seen my defined difference for any 5x3 rectangle gives me an answer of 80. I am going to use algebra to ensure my answer is accurate.
(s+4)(s+20) – s(s+24)
s(s+20)+4(s+20) – s -24s
s +20s +4s+80 – s -24s
=80
I have now calculated the answer for my 3x2 rectangles and came to a difference of 20 and an answer for my 5x3 rectangles and came to a defined difference of 80. Still I do not see any major patterns forming and the only trend I can establish is that my answers are still multiples of 10. I will continue my investigation by increasing to a larger rectangle.
Furthering my investigation
I am now going to look at 6x4 rectangles, I would hope after this I will be able to establish some type of pattern to help me reach my aim of finding an overall formula for any rectangle that could be investigated.
30x55 – 25x60
1650 –1500
Difference =150
66x91 – 61x96
6006 – 5856
Difference = 150
By looking at the above calculations it can be assumed that they are correct, I will use algebra to prove this.
(s+5)(s+30) – s(s+35)
s(s+30) +5(s+30) – s - 35s
s +30s+5s+150 – s -35s
=150
I find it very difficult to see any major trend, this is because I am randomly selecting various sizes of rectangles and therefore they do not come in any particular order and the sizes do not carry on from the one before.
Furthering my investigation
At this stage of my investigation of rectangles I had hoped to find a major trend, unfortunately this has not happened so I will again increase the size of my rectangles and aim to come up with a major pattern.
I will increase the size of my rectangles to 7x4, and with any luck this will help me reach my aim.
8x32 – 2x38
f 256 –76
Difference = 180
58x82 – 52x88
4756 – 4576
Difference =180
Below is the algebra to prove that my defined difference for 7x4 rectangles is correct.
(s+6)(s+30) – s (s-36)
s(s+30) +6(s+30) – s -36s
s +30s+6s+180 – s -36s
=180
At this point I feel unable to establish a sequence. This has happened because I have jumped in sizes of rectangles and have not kept to a pattern. I have shown my answers and difference between each below:
60 70 30
20 80 150 180
Furthering my investigation
Still with the results I have I am unable to see any major trend forming, I am going to increase the size of my rectangles to and 8x5.
30x63 – 23x70
1890 – 1610
Difference = 280
I am confident that the defined difference of 280 for any 8x5 is correct. I will use algebra to ensure this is true.
(s+7)(s+40) – s (s+47)
s(s+40) +7 (s+40) – s - 47s
s +40s +7s +280 – s - 47s
=280
Conclusion
From the results I have I am going to draw up a table and hope that I can meet my aim of establishing an overall formula.
By completing the second part of my investigation and drawing up the table above, showing my results calculated in Chapter Two I have been able top establish an accurate formula for rectangles in a 10x10 number grid, my result here was:
10(s - 1)(r - 1)
Chapter Three
So I have now investigated squares and rectangles in a 10x10 grid and have discovered two formulas.
My formula for squares was 10(r –1)
My formula for rectangles was 10(s –1)(r – 1).
I notice that the number 10 appears in both formulas and the reason I think this has happened might be because the squares and rectangles I used were selected from the provided 10x10 grid.
I am now going to repeat my process using a different grid to see what affect, if anything, there will be to the figure 10.
I am going to be using a 12x12 square grid and will repeat my processes from chapter one.
(12x12 grid)
I am again going to start with 2x2 squares randomly selected from my new 12x12 grid.
30x41 – 29x42
1230 - 1218
Difference =12
77x88 – 76x89
6776 – 6764
Difference = 12
I am now confident that any 2x2 square selected from my new 12x12 number grid will give me a defined difference of 12.
I feel I have got this defined difference of 12 because I am using a 12x12 grid.
I will use algebra to prove that my found result is correct:
(r+1)(r+12) – r (r+13)
r(r+12) +1 (r+12) – r -13r
r +12r +r +12 – r -13r
= 12
From the algebra I now feel my answer is accurate and will move on to further investigate.
Furthering my investigation
I am now going to investigate 3x3 squares using my 12x12 grid. I will repeat my same process used previously.
I predict I will get an answer of 48 I came to this prediction from using the results I found in chapter one.
67x89 – 65x91
5963 – 5915
Difference = 48
17x39 – 15x41
663 – 615
Difference = 48
My prediction was correct. When investigating a 3x3 square from the 12x12 number grid the defined difference is 48.
I will again use algebra to prove my defined difference of 48 is accurate:
(r+2)(r+24) – r(r+26)
r(r+24)+2(r+24) – r -26r
r +24r+2r+48 – r -26r
=48
I have now calculated a trend for my 2x2 squares and came to a difference of 12 and a trend for my 3x3 squares and came to a difference of 48. I am now going to continue my investigation by increasing again to a larger square.
Furthering my investigation
I will now continue on and increase my square size to 4x4 as I did in chapter one, this is to help me come to an overall formula. The 4x4 squares will be randomly selected from my new 12x12 number grid.
- 4x37 – 1x40
148 – 40
Difference = 108
59x92 – 56x95
5428 – 5320
Difference = 108
As can be seen I am getting a defined difference of 108 when using 4x4 squares randomly selected from the 12x12 grid.
I will use algebra to prove my defined difference of my 4x4 squares is correct.
(r+3)(r+36) – r(r+39)
r(r+36)+3(r+36) – r – 39r
r +36r+3r+108 – r - 39r
=108
It is already possible for me to see some type of trend forming, I can see that all my results are all multiples of 12 and I suspect this is so because I have used a new 12x12 number grid.
Furthering my investigation
As I have previously stated, it is possible to see some type of trend forming, I am going to increase my square size to 5x5 and from this I hope to see a trend more clearly.
hjhjhjhjhjh 24x68 – 20x72
1632 –1440
Difference = 192
8x52 – 4x56
416 – 224
Difference = 192
By seeing the calculations above it is possible to see that they are correct as they give the same defined difference of 192. To ensure my calculations are right I will use algebra:
(r+4)(r+48) – r(r+52)
r(r+48) +4(r+48) – r - 52r
r +48r +4r +192 – r - 52r
=192
As can be seen my calculations are correct, I will now continue to further my investigation.
By investigating this 5x5 square and identifying a defined difference of 192 and by using chapter one and looking at my previous answers from this new grid I have defined the following trend:
36 60 84
12 48 108 192
To confirm that this pattern I have identified is accurate I will predict that in an 8x8 square the defined difference will be 588.
I am going to continue past 6x6 and 7x7 squares as if my prediction is correct it will be unnecessary to carry out further investigation. If my prediction for my 8x8 is correct I will simply use the pattern I have identified to find the defined differences of any 6x6 or 7x7 squares.
11x88 – 4x95
968 –380
Difference = 588
I will again use algebra to prove my defined difference of 588 for any 8x8 square selected from the 12x12 grid is correct:
(r+7)(r+84) – r (r+91)
r(r+84) +7(r+84) – r - 91r
r +84r +7r +588 – r - 91r
=588
Again the algebra has proved my calculations to be accurate.
I now feel that I have identified and proven a major trend within my investigation of my new 12x12 square number grid.
I am confident with my result and will continue on to show the complete pattern trend identified starting from a 2x2 square to a 12x12 square.
36 60 84 108 132 156 180 204 228 252
12 48 108 192 300 432 588 768 972 1200 1452
2x2 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10 11x11 12x12
Looking at the diagram, the top number indicates the difference between each defined difference found and the numbers along the bottom indicate the defined differences of each square investigated.
Conclusion
From the sequence on the previous page I will draw up a table to try and establish an overall formula when investigating the 12x12 number grid or any other size of grid.
I furthered my investigation by looking at squares in a 12x12 number grid. In Chapter Three I looked at the exact same size of squares as in Chapter One, my aim of this part of my investigation was again to establish a formula and see how my results differed.
The table of results on the previous page shows my answers calculated and predicted within Chapter Three, I was able to see a trend of multiples of 12 and was able to continue on and calculate perfect squares. By revising this table I was able to meet my aim of identifying a formula of any size squares from a 12x12 number grid, my formula here was:
12(r - 1)
Chapter Four
So I have now investigated squares and rectangles in a 10x10 number grid and squares in a new 12x12 number grid.
With my new 12x12 grid I have noticed my results are all multiples of 12. I am now going to repeat my processes from chapter two, except this time I will be randomly selecting the rectangles from my new 12x12 number grid.
I am going to start by investigating 3x2 rectangles.
8x18 – 6x20
144 – 120
Difference = 24
52x62 – 50x64
3224 – 3200
Difference = 24
I am again going to try a 2x3 rectangle to see if the outcome differs.
7x30 – 6x31
210 –186
Difference = 24
51x74 – 50x75
3774 -3750
Difference = 24
As can be seen whether the rectangle is horizontal (3x2) or vertical (2x3) my answer is still the same. Because the outcome does not differ I am going to concentrate on horizontal shaped rectangles only.
The algebra below will prove that my defined difference of 24 is correct when investigating any 3x2 rectangles from a 12x12 number grid.
(s+2)(s+12) – s(s+14)
s(s+12)+2(s+12) – s - 14s
s +12s+2s+24 – s - 14s
=24
Furthering my Investigation
I am going to further my investigation by increasing the size of my rectangles to 5x3. These will be randomly selected from the new 12x12 number grid, my general aim being to establish an overall formula.
I feel unable to make any predictions when investigating various rectangles.
48x68 – 44x72
3264 –3168
Difference = 96
77x97 –73x101
7469 –7373
Difference = 96
As can be seen above I get a defined difference of 96 when looking at 5x3 rectangles, I will use algebra to prove this is correct.
(s+4)(s+24) – s(s+28)
s(s+24) +4 (s+24) – s – 28s
s +24s +4s+96 – s - 28s
=96
I cannot see a trend emerging and therefore I am going to again increase my rectangles to 6x4 as I did in chapter two.
Furthering my investigation
As can be seen the only trend I have found so far in this part of my investigation is that my defined differences are all multiples of 12, the reason I would say I’m getting this result is because the grid I am investigating now is a 12x12 grid.
I am now going to investigate 6x4 rectangles randomly selected from the new 12x12 number grid.
6x37 – 1x42
222 – 42
Difference =180
57x88 – 52x93
5016 - 4836
Difference = 180
By looking at the calculations above it can be assumed they are correct, the algebra below will prove this.
(s+5)(s+36) – s (s+41)
s(s+36) +5(s+36) – s - 41s
s +36s +5s +180 – s – 41s
=180
I have now calculated the defined differences for any 3x2, 5x3 and 6x4 rectangle. Still I am unable to predict any sequences emerging or see any major trends, the reason I give for this is the same problem that I came across when investigating rectangles in the 10x10 number grid, I have used exactly the same size rectangles from both grids, but my rectangles are not in any particular order, I feel to establish a major trend I would have needed to start with a 3x2 and move up one each time, yet I feel this would be very time consuming and that through more investigating I will still be able to reach my aim of establishing an overall formula for any number grid and any square or rectangle that may be investigated.
Furthering my investigation
Because I do not yet feel confident that I have enough information to reach my aim I am going to continue my investigation by increasing the size of my rectangles to 7x4. These will be randomly selected from the 12x12 number grid.
12x42 – 6x48
f 504 –288
Difference = 216
59x89 – 53x95
5251 – 5035
Difference =216
I feel my defined difference of 216 for any 7x4 rectangle investigated in a 12x12 number grid is accurate, I will use algebra to prove this is true.
(s+6) (s+42) – s(s+52)
s(s+42)+6 (s+42) – s - 52s
s +42s +6s +216 – s - 52s
=216
72 64 36
24 96 180 216
As can be seen from the pattern above, showing my results and the differences between each there is no major trend. The results along the bottom are the defined differences to each rectangle I have investigated in the new 12x12 number grid.
Furthering my investigation
As there is no major trend emerging this means I must continue on with this part of the investigation. I will randomly select an 8x5 rectangle from the new grid and hope that from this I can draw up some type of trend.
96x137 – 89x144
13152 –12816
Difference = 336
Below is the algebra to prove my defined difference of 336 is correct.
(s+7)(s+48) – s(s+55)
s(s+48) +7 (s+48) – s - 55s
s +48s +7s +336 – s - 55s
=336
As can be seen the only trend within this part of my investigation is that all my results for each rectangle is a multiple of 12. I feel that by bringing my results together I may now be able to establish an overall formula.
Conclusion
The table below shows all my findings and trends I may spot for me to complete this part and my investigation as a whole.
My final part of my investigation was looking at the exact same size of rectangles as in Chapter Two except using my new 12x12 number grid. My aim still remaining the same, to identify a formula.
By drawing up the table above formed from my results in Chapter Four I was able to see a trend of multiples of 12 and was successful in establishing a formula, which was:
12(s - 1)(r - 1)
Overall Summary of chapter 1- 4
I was set the task of investigating a provided 10x10 number grid. The only instructions I was given was to find the product of the top left number and the bottom right number within the highlighted box and to repeat this process with the top right and bottom left numbers with the aim being to calculate the difference between these products and investigate further.
I started my investigation by finding the difference of the highlighted box, I then furthered my investigation by increasing the size of the square and finding a trend of multiples of 10 and successfully identifying a sequence, this allowed me to draw up a table and establish a formula 10(r – 1).
I decided to continue on and make a Chapter Two, where I investigated rectangles in a 10x10 number grid and repeated my process from my initial piece of work. IN this chapter I found it difficult to find a sequence, I drew up a table and was able to see all the results were multiples of 10 and from revising this table I identified a formula 10(s –1)(r –1).
From Chapter Two I decided to make the investigation more interesting and I increased the provided number grid to a 12x12 grid. In this chapter I decided to simply repeat my processes from Chapter One. I again seen an accurate sequence and from this I drew up a table and seen that this time my results were multiples of 12, I was then able to work out perfect squares and establish the formula 12(r – 1).
Finally I continued on to Chapter Four, where I investigated the new 12x12 number grid but repeated my processes from Chapter Two. I was able to see a trend of multiples of 12 but again I was unable to find a major sequence for the rectangles. From drawing up a table I was able to find a formula
12 (s–1)(r –1).
From reviewing all my Chapters and formulas within this investigation I am able to establish an overall formula suitable for any sized number grid and any size of square or rectangle investigated.
Formula for Squares
p (n – 1)
This is true for any square (nxn) taken from any square grid (pxp).
Formula for Rectangles
p (m – 1) (n – 1)
This is true for any rectangle (mxn) taken from any square grid (pxp).
Chapter five
I will now investigate a serious of squares on a 10x12 number grid to distinguish if there is a trend that will lead to an overall formula. As I have gained a working knowledge of how to obtain a formula I believe that I will find it easier to distinguish possible patterns
I shall now examine a 2x2 square.
34x43 – 33x44
1462 – 1452
Difference = 10
I will now look at a 3x3 square in the 10x12 number grid as there is no point in me continuing to repeat the investigation of 2x2 as from previous experience I know that I am always going to find that the difference is 10, I need to investigate further to distinguish any patterns.
76x94 – 74x96
7144 – 7104
Difference = 40
I will use algebra to prove the defined difference of this 3x3 square within the 10x12 number grid is correct.
r (r+2)
(r+20)(r+22)
(r+2)(r+20)-(r+22)r
=r (r+20)+2(r+20)-r –22r
=r +20r+2r+40-r –22r
= 40
I will now examine a 4x4 square within the 10x12 number grid
8x35 – 5x38
280 – 190
Difference = 90
I can see that so far all the answers that I have obtained within this section of chapter five are all multiples of 10. In reflecting on my past investigation into the sequence of squares on a 10x10 number grid I have discovered that my answers are so far working out the same. I will now predict that the next answer will be 160 based on my previous investigation.
69x105 – 65x109
7245 – 7085
Difference = 160
I will now draw my findings into a table to help identify and overall formula. In identifying that the answers of chapter correspond with the answers in chapter one whereby I investigated squares within the 10x10 number grid I can now predict that the formula will be the same.
If I was to continue my investigation by examining bigger squares in the grid I will continue to get the same answers to that of what I found in chapter on. I can now conclude that in my 10x12 number grid the formula is 10(r-1) , and will continue to be this, as long as the number grid remains 10 long, as I have found that the width can alternate without affecting the formula.
Chapter 6
As an addition to my chapter five investigation I will now repeat my process using a 12x10 number grid to distinguish how this will affect the formula.
5x16 – 4x17
80 – 68
Difference = 12
Is this a coincidence that this first answer is a multiple of 12? I will continue to investigate by examining a 3x3 square, as I now know that no matter how many times I repeat this process of investigating a 2x2 square in this 12x10 number grid I will continue to get the difference of 12.
55x77 – 53x79
4235 – 4187
Difference = 48
Again this difference is a multiple of 12. I will now look at a 4x4 square in this number grid to see if I can distinguish a sequence.
11x44 – 8x47
484 – 376
Difference = 108
Like my prior investigation in chapter five I have discovered that the answers I have obtained are the same as in chapter three where I previously investigated a 12x12 number grid. I will now predict that a 5x5 square has a difference of 192.
32x76 – 28x80
2432 – 1120
Difference = 192
As you can see my prediction was correct. This means that the defined differences will also be the same as those identified within chapter three. I will now put these findings from this investigation into a table to help establish an overall formula. before doing this I imagine that the formula for this chapter will be reflected in that of what I found in chapter three:- 12(r-1) .
This trend demonstrates that any size of square within a 12x10 grid will have the formula of 12(r-1) .
Conclusion on chapter 5 & 6
Form conducting these investigations I have discovered that in a general PxQ grid it is only the first length that appears in the formula, ie P(r-1) .
Chapter 7
Throughout my investigations I have only looked at the affects of sizes of number grids and sizes of both squares and rectangles, so to enhance my investigation I will now examine the affects that a sequence of different sized rhombuses will have within a 10x10 number grid on the overall output formula.
I will start by calculating a 2x2 rhombus. All the rhombuses that I am investigating in this 10x10 grid will remain to be this shape:-
13x21 – 12x22
273 – 264
Difference = 9
I will calculate another 2x2 rhombus to ensure that the difference will always remain 9 within a 2x2 rhombus.
28x36 – 27x37
1008 – 999
Difference = 9
(r+1)(r+9)-r(r+10)
r(r+1)+1(r+9)-r –10r
r +8+1r+9-r –10r
= 9
I am now going to calculate a 3x3 rhombus within the 10x10 number grid.
36x52 – 34x54
1872-1836
Difference = 36
So far I can see that there is a sequence of multiples of 9’s being revealed. I predict that a 4x4 rhombus will too be a multiple of 9.
28x52 – 25x55
1456 – 1375
Difference = 81
81 = 9, as I predicted.
9
I am now going to try a 5x5 rhombus, again I predict that it will be a multiple of 9.
20x52 – 16x56
1040 – 896
Difference =144
Again another multiple of 9.
The algebra helps show that any given 5x5 rhombus within a 10x10 number grid will equal to that of 144.
(r+4)(r+36)-r(r+40)
r(r+36)+4(r+36)-r –40r
r +36+4r+144-r –40r
= 144
With the results that I have collected from this investigation into rhombuses in a 10x10 number grid, I will place into a table to help distinguish a sequence which will help me to identify an overall formula for this shape of rhombus
I can now draw upon the conclusion that the formula for this sequence of rhombuses is (10-1)(r-1) .
Chapter 8
To conduct a thorough investigation I felt it was important to examine the affects that may occur if I was to reflect he shape of the rhombus within the 10x10 number grid. To ensure validity I will examine the same sizes of rhombuses that I used in my prior investigation in chapter seven, only this time all the rhombuses that I use will be this shape:-
13x23 – 12x24
299 – 288
Difference = 11
55x75 – 53x77
4125 – 4081
Difference = 44
I will now show the algebra to this 3x3 rhombus to show that this calculation is correct.
(r+2)(r+22)-r(r+24)
r(r+22)+2(r+22)-r –24r
r +22r+2r+44-r –24r
= 44
I have distinguished that these answers are multiples of 11, therefore I will predict that the next answer will too be a multiple of 11.
24x54 – 21x57
1296 – 1197
Difference = 99
Again I can see that my prediction was correct.
36x76 – 32x80
2736 – 2560
Difference = 176
I will show the algebra for any 5x5 rhombus within a 10x10 number grid, this will show that the defined difference will always be 176.
(r+4)(r+44)-r(r+48)
r(r+44)+4(r+44)-r –48r
r +44r+4r+176-r –48r
=176
I will now place my findings from this investigation into a table to help find the end formula for any given rhombus within a 10x10 number grid.
In conclusion of this chapter I can now revel that the overall formula for any
shaped rhombus is (10+1)(r-1) .
Conclusion for chapter 7 & 8
Overall this investigation into the sizes and shapes of rhombuses has provided me with evidence to suggest that the determination on the formula will depend on the actual shape of the rhombus ie.
Will always be a –1 whereas Will always be a +1