4648 – 4558 = 90
Data calculations for 5x5 squares
51 x 95 = 4845 55 x 91 = 5005
5005 – 4845 = 160
6 x 50 = 300 46 x 10 = 460
460 – 300 = 160
1 x 45 = 45 41 x 5 = 205
205 – 45 = 160
56 x 100 = 5600 96 x 60 = 5760
5760 – 5600 = 160
14 x 58 = 812 54 x 18 = 927
927 – 812 = 160
34 x 78 = 2652 38 x 74 = 2812
2812 – 2652 = 160
Rectangular calculation
For this part of my investigation I will find out the patterns and differences in rectangular boxes. These rectangular boxes include 2x3, 2x4, 2x5, 3x2, 3x4, 3x5 and 4x2, 4x3, 4x5, 5x2, 5x3, 5x4.
Statement for 2x rectangles
By working out my rectangular calculation I have created a table of differences for 2x3, 2x4, 2x5
This table on the left is the difference table I created for the rectangular boxes.
By looking at the results in the table above it shows us that all the differences are all multiples of 10 because it all ends in 0. I worked these differences out by doing a number of calculations of each term, which you can find on the next page.
The algebraic formulae that I found for the rectangles is
D = 10(r-1)(c-1)
D = difference
r= rows
c= columns
Rectangle calculations for 2x rectangles
4 x 25 = 100 24 x 5 = 120
120 – 100 = 20
49 x 70 = 3430 69 x 50 = 3450
3450 – 3430 = 20
71 x 92 = 6532 91 x 72 = 6552
6552 – 6532 = 20
1 x 32 = 32 31 x 2 = 62
62 – 32 = 30
56 x 87 = 4872 86 x 57 = 4902
4902 – 4872 = 30
9 x 40 = 360 39 x 10 = 390
390 – 360 = 30
3 x 44 = 132 43 x 4 = 172
172 – 132 = 40
29 x 70 = 2030 69 x 30 = 2070
2070 – 2030 = 40
51 x 92 = 4692 91 x 52 = 4732
4732 – 4692 = 40
Prediction
My prediction for the 2x6 will be a multiple of 10. I think it will be 50. Below are my results for my 2x6 rectangles.
1 x 52 = 52
51 x 2 = 102
102 – 52 = 50
35 x 86 = 3010
85 x 36 = 3060
3060 – 3010 = 50
9 x 60 = 540
59 x 10 = 590
590 – 540 = 50
I predict that my 2x7 rectangles would be a multiple of ten again and I think it will be 60. Below are the results of my 2x7 rectangles.
2 x 63 = 126 16 x 77 = 1232
62 x 3 = 186 76 x 17 = 1292
186 – 126 = 60 1292 – 1232 = 60
38 x 99 = 3762
98 x 39 = 3822
3822 – 3762 = 60
The table above on the top right is the difference table but as you can see I have added the 2x3, 2x4, 2x5, 2x6 and 2x7 rectangle differences in.
Statement 3x rectangles
By working out my rectangular calculation I have created a table of differences for 3x2, 3x4 and 3x5.
This table on the left is the difference table I created for the 3x rectangular boxes.
By looking at the results in the table above it shows us that all the differences are all multiples of 10 because it all ends in 0. I worked these differences out by doing some calculations of each term, which you can find on the next page.
The algebraic formulae that I found for the rectangles is
D = 10(r-1)(c-1)
D = difference
r= rows
c= columns
3x rectangles
6 x 18 = 108 16 x 8 = 128
128 – 108 = 20
18 x 30 = 540 28 x 20 = 560
560 – 540 = 20
53 x 65 = 3445 63 x 55 = 3465
3465 – 3445 = 20
1 x 33 = 33 31 x 3 = 93
93 x 33 = 60
66 x 98 = 6468 96 x 68 = 6528
6528 – 6468 = 60
4 x 36 = 144 34 x 6 = 204
204 – 144 = 60
1 x 43 = 43 41 x 3 = 123
123 – 43 = 80
22 x 64 = 1408 62 x 24 = 1488
1488 – 1408 = 80
58 x 100 = 5800 98 x 60 = 5880
5880 – 5800 = 80
Prediction
My prediction for my 3x6 rectangle is that it will be a multiple of ten. I am guessing that it will be 100. Below are my results of finding the differences for my 3x6 rectangles.
5 x 57 = 285
55 x 7 = 385
385 – 285 = 100
47 x 99 = 4653
97 x 49 = 4753
4753 – 4653 = 100
1 x 53 = 53
51 x 3 = 153
153 – 53 = 100
My prediction for my 3x7 rectangle is that it will also be a multiple of 10 and also I am guessing that it will be 120. Below are the results of my 3x7 rectangles.
33 x 95 = 3135
93 x 35 = 3255
3255 – 3135 = 120
26 x 88 = 2288
86 x 28 = 2408
2408 – 2288 = 120
31 x 93 = 2883
91 x 33 = 3003
3003 – 2883 = 120
The table on the right is the
difference table but as you
can see I have filled in the 3x2,
3x4, 3x5, 3x6 and 3x7 boxes.
Statement 4x rectangles
By working out my rectangular calculation I have created a table of differences for 4x2, 4x3 and 4x5.
This table on the left is the difference table I created for the 4x rectangular boxes.
By looking at the results in the table above it shows us that all the differences are all multiples of 10 because it all ends in 0. I worked these differences out by doing some calculations of each term e.g. 3x2.
The algebraic formulae that I found for the rectangles is
D = 10-1)(c-1)
D = difference
r= rows
c= columns
4x rectangles
1 x 14 = 14 11 x 4 = 44
44 – 14 = 30
24 x 37 = 888 34 x 27 = 918
918 – 888 = 30
77 x 90 = 6930 87 x 80 = 6960
6960 – 6930 = 30
4 x 27 = 108 24 x 7 = 168
168 – 108 = 60
57 x 80 = 4560 77 x 60 = 4620
4620 – 4560 = 60
11 x 34 = 374 31 x 14 = 434
434 – 374 = 60
1 x 44 = 44 22 x 65 = 1430
41 x 4 = 164 62 x 25 = 1550
164 – 44 = 120 1550 – 1430 – 120
17 x 60 = 1020
57 x 20 = 1140
1140 – 1020 = 120
Prediction
My prediction for my 4x6 is that it will also be a multiple of 10 because I feel that the last digit on the differences will be 0. I am guessing that the 3x6 rectangles will be 150 I am guessing this because I am following the algebraic formula.
1 x 54 = 54 34 x 87 = 2958
51 x 4 = 204 84 x 37 = 3108
204 – 54 = 150 3108 – 2958 - 150
12 x 65 = 780
62 x 15 = 930
930 – 780 = 150
My prediction for my 4x7 rectangle is that it will be 180 because following the difference from the 4x6 rectangle that was 150 I just add 30 and then that is the result for my 4x7 rectangle. Below are some results for my 4x7 rectangle.
2 x 65 = 130
62 x 5 = 310
310 – 130 = 180
36 x 99 = 3564
96 x 39 = 3744
3744 – 3564 = 180
17 x 80 = 1360
77 x 20 = 1540
1540 – 1360 = 180
The difference table below is now filled with all the 4x rectangles. It concerns me that all the differences are all multiples of 10 and in patterns.
Statement for 5x rectangles and 6x7 and 7x6
By working out my rectangular calculation I have created a table of differences for 5x2, 5x3, 5x4 and to complete my table fully I will do 6x7 and 7x6 rectangles to finish off the difference table.
This table on the left is the difference table I created for the 5x rectangular boxes and 6x7 and 7x6 rectangles.
By looking at the results in the table above it shows us that all the differences are all multiples of 10 because it all ends in 0. I worked these differences out by doing some calculations of each term e.g. 5x2.
The algebraic formulae that I found for the rectangles is
D = 10(r-1)(c-1)
D = difference
r= rows
c= columns
5x rectangles
5 x 19 = 95 15 x 9 = 135
135 – 95 = 40
34 x 48 = 1632 4444 x 38 = 1672
1672 – 1632 = 40
1 x 15 = 15 11 x 5 = 55
55 – 15 = 40
76 x 100 = 7600 96 x 80 = 7680
7680 – 7600 = 80
22 x 46 = 1012 42 x 26 = 1092
1092 – 1012 = 80
54 x 78 = 4212
74 x 58 = 4292
4292 – 4212 = 80
1 x 35 = 35
31 x 5 = 155
155 – 35 = 120
35 x 69 = 2415
65 x 39 = 2535
2535 – 2380 = 120
63 x 97 = 6111
93 x 67 = 6231
6231 – 6111 = 120
Prediction
My prediction for my 5x6 is that it will be 200 and a multiple of 10. I think this because the pattern that has caught my attention was to add 40 so add 40 to the difference of 5x5 then you get 200. Below are my results of my 5x6.
1 x 55 = 55
51 x 5 = 255
255 – 55 = 200
34 x 88 = 2992
84 x 38 = 3192
3192 – 2992 = 200
16 x 70 =1120
66 x 20 = 1320
1320 – 1120 = 200
My prediction for my 5x7 rectangles is that it will be a multiple of 10. I am predicting that the difference would be 240.
3 x 67 = 201
63 x 7 = 441
441 – 201 - 240
31 x 95 = 2945
91 x 35 = 3185
3185 – 2945 = 240
26 x 90 = 2340
86 x 30 = 2580
2580 – 2340 = 240
my prediction for my 6x7 and 7x6 will both be 300. I predict this because I just added 60 to the 6x5 and 5x6.
34 x 99 = 3366
94 x 39 = 3666
3666 – 3366 = 300
11 x 67 = 737
61 x 17 = 1037
1037 – 737 = 300
The difference table on the left is now full with the entire differences filled in. as you can see I have added the 5x rectangles and the 6x7 and 7x6 to complete the table.
The formula for this difference table is D = 10(r-1)(c-1)
D = difference
R = rows
C = columns
If you check the formula with my calculation you can see that the formula was right for the rectangles.
Other number grids
To further my investigation more I have chosen to do squares with a different number grid. The number grid that I will use now is shown below.
This number grid above is 15x15 am going to see what differences there are in this number grid and I am going to compare it with the 10x10 number grid, which was the last number grid I just did.
Statement for squares
I have found out the differences for the 2x2, 3x3, 4x4 and 5x5 squares for my 15x15 grid and they are shown below. Following in this statement are my prediction and my results for my 6x6 squares and 7x7 squares. I did 2 examples for each of my boxes
Difference
2x2 3x3 4x4 5x5
15 60 135 240
One thing that stood out to me straight away was to see all my differences were a multiple of 5. Like the 10x10 grid you can tell this because all the differences end in 5. The algebraic formula that I found for the squares on the 15x15 number grid was D = 15(n-1) 2. D stands for difference.
5 x 21 = 105 20 x 6 = 120
120 – 105 = 15
1 x 17 = 17 16 x 2 = 32
32 – 17 = 15
4 x 36 = 144 34 x 6 = 204
204 – 144 = 60
106 x 138 = 14628 136 x 108 = 14688
14688 – 14628 = 60
12 x 60 = 720 57 x 15 = 855
855 – 720 = 135
80 x 128 = 10240 125 x 83 = 10375
10375 – 10240 = 135
4 x 68 = 272 64 x 8 = 512
512 – 272 = 240
86 x 150 = 12900 146 x 90 = 13140
13140 – 12900 = 240
Prediction
My prediction for my 6x6 squares is that it will be a multiple of 5. I predict that the 6x6 square for the 15x10 number grid will be 375. I predict this because I worked it out from my algebraic formula. Below are my results for my 6x6 squares.
4 x 84 = 336
79 x 9 = 711
711 – 336 = 375
66 x 146 = 9636
141 x 71 = 10011
10011 – 9636 = 375
my prediction for 7x7 squares is that it will also be a multiple of 5 as well as the 6x6 squares. My results for my 7x7 are shown below.
I predict that my 7x7 square will be 540 from looking at my formula.
9 x 105 =945
99 x 15 = 1485
1485 – 945 = 540
112 x 208 = 23296
202 x 118 = 23836
23836 – 23296 = 540
This difference table is the difference table that I created for my 15x15 number grid. As you can see I have filled in the 2x2, 3x3, 4x4, 5x5, 6x6 and 7x7 squares already.
Rectangular differences
To further my investigation more I will now find the differences in rectangular boxes for my 15x15 number grid. On my calculation the differences are underlined and in bold font.
Statement for 2x rectangles
By working out my rectangular calculation I have created a table of differences for 2x3, 2x4, 2x5
This table on the left is the difference table I created for the 2x rectangular boxes.
By looking at the results in the table above it shows us that all the differences are all multiples of 15 because they can all be divided into 15. I worked these differences out by doing a couple of calculations of each term, which you can find on the next page. Followed after my calculations are my prediction for my 2x6 and 2x7 rectangles.
I also have found an algebraic formula to fit the method of this term. The formulae was D = 15(r-1)(c-1)
D meaning difference, r standing for rows and c standing for columns.
2x rectangle calculations
2x3 rectangles
17 x 48 = 816 47 x 18 = 8446
846 – 816 = 30
85 x 116 = 9860 115 x 86 = 9890
9890 – 9860 = 30
2x4 rectangles
94 x 140 = 13160
139 x 95 = 13205
13205 – 13160 = 45
143 x 189 = 27027
188 x 144 = 27072
27072 – 27027 = 45
2x5 rectangles
42 x 103 = 4326
102 x 43 = 4386
4386 – 4326 = 60
1 x 62 = 62
61 x 2 = 122
1222 – 62 = 60
Prediction
My prediction for my 2x6 rectangle is that it will be a multiple of 15 and I predict that it will be around 75. My reason for predicting this number is because I have noticed a pattern occurring in the 2x rectangles; this pattern is that it starts from 30 then goes up in 15. Add 15 to the 2x5 difference and then you get your 2x6 difference. Below are my results of my 2x6 rectangles.
21 x 97 = 2037
96 x 22 = 2112
2112 x 2037 = 75
55 x 131 = 7205
130 x 56 = 7280
7280 – 7205 = 75
My prediction for my 2x7 is that it will be a multiple of 15 and following the pattern that I found I predict that it will be 90. Below are my results for my 2x7 rectangles.
125 x 216 = 27000
215 x 126 = 27090
27090 – 27000 = 90
14 x 105 = 1470
104 x 15 = 1560
1560 – 1470 = 90
Above on the top right is my difference table for my 15x15 number grid. Again as same as the 10x10 number grid I will gradually fill this difference table in as I go along. Also you can see that I have already added the 2x rectangle differences in.
Statement for 3x rectangles
By working out my rectangular calculation I have created a table of differences for 3x2, 3x4 and 3x5.
This table on the left is the difference table I created for the 3x rectangular boxes.
By looking at the results in the table above it shows us that all the differences are all multiples of 15 because they can all be divided into 15. I worked these differences out by doing some calculations of each term, which you can find on the next page.
The algebraic formulae that I found for the 3x rectangles is
D = 15(r-1)(c-1)
D = difference
r = rows
c = columns
3x2 rectangles
51 x 68 = 3468 66 x 53 = 3498
3498 – 3468 = 30
133 x 150 = 19950 148 x 135 = 19980
19980 – 19950 = 30
3x4 rectangles
142 x 189 = 26838
187 x 144 = 26928
26928 – 26838 = 90
7 x 54 = 378
52 x 9 = 468
468 – 378 = 90
3x5 rectangles
94 x 156 = 14664
134 x 96 = 14784
14784 – 14664 = 120
13 x 75 = 975
73 x 15 = 1095
1095 – 975 = 120
Prediction
My prediction for my 3x6 rectangle is that it will be a multiple of 15 because there is another pattern I found for my 3x rectangles it is add 30, so I add 30 to my 3x5 difference and then I get my 3x6 difference. I predict that my 3x6 difference will be 150. I predict this because I am following the pattern that I found. On the next page are the results for my 3x6 rectangles.
65 x 142 = 9230
140 x 67 = 9380
9380 – 9230 = 150
91 x 168 = 15288
166 x 93 = 15438
15438 – 15288 = 150
I predict that my 3x7 rectangle will be a multiple of 10 and 15 and it will be 180. I predict this because following my pattern by just adding 30 to my 3x6 rectangle.
6 x 98 = 588
96 x 8 = 768
768 – 588 = 180
26 x 118 = 3068
116 x 28 = 3248
3248 – 3068 = 180
The difference table on the right has now been filled in with the 3x rectangle differences.
Statement for 4x rectangles
By working out my rectangular calculation I have created a table of differences for 4x2, 4x3 and 4x5
This table on the left is the difference table I created for the 4x rectangular boxes.
By looking at the results in the table above it shows us that all the differences are all multiples of 15 because it all ends in 0 and 5 and all the differences can be divided in to 15. I worked these differences out by doing some calculations of each term, which you can find on the next page.
The algebraic formula that I found for the 4x rectangles is same as the other rectangles.
D = 15(r-1)(c-1)
D = difference
r = rows
c = columns
4x2 rectangles
36 x 54 = 1944 51 x 39 = 1989
1989 – 1944 = 45
205 x 223 = 45715 220 x 208 = 45760
45760 – 45715 = 45
4x3 rectangles
1 x 34 = 34 31 x 4 = 124
124 – 34 = 90
39 x 72 = 2808 69 x 42 = 2898
2898 – 2808 = 90
4x5 rectangles
94 x 157 = 14758
154 x 97 = 14938
14938 – 14758 = 180
151 x 214 = 32314
211 x 154 = 32494
32494 – 32314 = 180
Prediction
My prediction for my 4x6 rectangle is that it will be a multiple of 15. I predict it will be 225. I predict this because I am following the pattern I found for the 4x rectangles which is add 45 from the last difference. I will add 45 to my 4x5 rectangle and then that will give me my 4x6 rectangle difference. Below are some results for my 4x6 rectangles.
7 x 85 = 595
82 x 10 = 820
820 – 595 = 225
41 x 119 = 4879
116 x 44 = 5104
5104 – 4879 = 225
My prediction for my 4x7 rectangles is that it will also be a multiple of 15 and I am guessing it will be 270. I predict this because I am adding 45 to my 4x6 difference and that should give me the result for my 4x7 rectangles. Below are my results for my 4x7 rectangles.
3 x 96 = 288
93 x 6 = 558
558 – 288 = 270
128 x 221 = 28288
218 x 131 = 28558
28558 – 28288 = 270
The difference table below has now got all the differences for the 4x rectangles filled in.
Statement for 5x rectangles
By working out my rectangular calculation I have created a table of differences for 5x2, 5x3, 5x4.
This table on the left is the difference table I created for the 5x rectangular boxes.
By looking at the results in the table above it shows us that all the differences are all multiples of 15 because they can all be divided into 15. I worked these differences out by doing some calculations of each term 5x2, 5x3 and 5x4, which you is below and on the next page.
The algebraic formulae that I found for the rectangles is
D = 15(r-1)(c-1)
D = difference
r= rows
c= columns
5x2 rectangles
4 x 23 = 92 19 x 8 = 152
152 – 92 = 60
70 x 89 = 6230 85 x 74 = 6290
6290 – 6230 = 60
5x3 rectangles
10 x 44 = 440 40 x 14 = 560
560 – 440 = 120
50 x 84 = 4200 80 x 54 = 4320
4320 – 4200 = 120
5x4 rectangles
1 x 50 = 50
46 x 5 = 230
230 – 50 = 180
82 x 131 = 10742
127 x 86 = 10922
10922 – 10742 = 180
Prediction
I predict that my 5x6 rectangle will be a multiple of 15 as well as all the other rectangles. I predict it will be 300. My reason for this prediction is because o found the pattern for the 5x rectangles that was to add 60 to the last rectangle. If I add 60 to my 5x5 box then it should give me the result for my 5x6 rectangle. Below are some of my results for my 5x6 rectangle.
20 x 99 = 1980
95 x 24 = 2280
2280 – 1980 = 300
92 x 171 = 15732
167 x 96 = 16032
16032 – 15732 = 300
My prediction for my 5x7 rectangles is that it will be a multiple of 15 and I am guessing it will be 360. My reason for this prediction is because if I follow my pattern and add 60 to my 5x6 rectangle it should give me my difference for my 5x7 rectangle. On the next page are some of my results for my 5x7 rectangle.
6 x 100 = 600
96 x 10 = 960
960 – 600 = 360
106 x 200 = 21200
196 x 110 = 21560
21560 – 21200 = 360
The difference table on the below is now filled in with all the 5x rectangles. To complete this table all I have to do is to fill in the 6x7 and 7x6 rectangles.
To complete my difference table for my 15x15 number grid I will now do some calculations for 6x7 and 7x6 rectangles.
My prediction for the 6x7 rectangle is that it will be a multiple of 15 and I am guessing that it will be 450. My reason for guessing this is because I found a pattern for the 6x rectangles which is to add 75 to the last rectangle before it, in this case I have to add 75 to my 6x6 square, which was 375, so that would give us 450 and that is our difference for the 6x7 rectangle.
Both these calculations are shown on the next page.
5 x 100 = 500
95 x 10 = 950
950 – 500 = 450
122 x 217 = 26474
212 x 127 = 26924
26924 – 26474 = 450
My prediction for my 7x6 rectangle would be exactly the same as my 6x7 rectangle that was 450. I predict this because it is exactly the same term except the row as been swooped around with the columns. My results for my 7x6 rectangle are shown below and on the following page.
37 x 118 = 4366
112 x 43 = 4816
4816 – 4366 = 450
144 x 225 = 32400
219 x 150 = 32580
32580 – 32400 = 450
The difference table below is now fully complete and now I can compare my 15x15 number grid difference table with my 10x10 number grid difference table. This will be done in my conclusion and evaluation.
Conclusion
Throughout this, confusing but helpful, investigation I have learnt new and educational things. When I first started this investigation I thought it would be fun to learn something new, it was new and I did learn something but it was not that fun. I began to be amazed and interested when I was working through and found new formulae and patterns. These things I have come across did not only interest me but also helped me further throughout my investigation. In the beginning of this investigation I found that I had to do a lot of calculations in order to find out the formula and any sort of patterns that occurred. Doing a lot of calculations made it easy and quicker to get a sort of pattern and formula for the number grids. The further I got through my investigation the less calculations I needed to do. In a way I found this investigation pretty boring because it was very repetitive, by drawing all the boxes and calculating them, this took quiet a bit of time.
The first number grid was the easiest to find out an algebraic formula for the squares. However as I moved on to finding out rectangular boxes it became a bit harder and more challenging, I think this is what got me further more interested in this investigation. It did take me a while to work out the formulae but in the end I got there. The formulae that I found in the 10x10 grid for the squares was 10 (n-1) 2. This formula means 10x number in the sequence minus 1 squared. The formula that I found for the rectangles on the 10x10 number grid was 10(r-1)(c-1) this formulae helped me to work out a lot of predictions and other formulas.
Evaluation
If I was to do this investigation again I think I would further my investigation more by doing much more calculations. Also I would have found more differences and formulas in different size number grids such as 20x5 or 30x10 etc. This would have made my view and points clearer and proved my opinion more. I would also like to see if my investigation were to change if I were to instead of multiply but add the top left number with the bottom right, perhaps this method might have changed my entire investigation. This piece of coursework was different from our last data handling coursework because the last coursework included using standard deviation and graphs. For this investigation on number grids I had to use a lot of tables and a lot of calculator calculations were needed. Not much equipment was used to do this investigation because I did do most of it on the computer. As you can see I filled out a difference table for both number grids shown on pages 23(10x10 number grid) and page 40(15x15 number grid). This gave me a clear look at what patterns I could notice in each table. I noticed that in my 10x10 number grid all the differences were a multiple of 10 and in my 15x15 number grid all my differences were multiples of 15. I think I have done quiet well for the presentation of my investigation but I do feel that it could have been extended more with further methods of investigation. I found this investigation very challenging and it did teach me a great deal on number girds and their differences. I never expected my investigation to be a great amount of pages but looking at some other pieces of coursework I found it to be normal, This just shows that I did a lot of calculations and methods of investigating formulas and patterns. In conclusion, this investigation has been very challenging and useful to my class work because it has taught me a lot on formulas and differences in number grids also it taught me a lot on how to investigate problem solving.
Sharish Rughoo 11L Candidate number 8146 Centre number 13428
