46 x 64 = 2944
44 x 66 = 2904
2944 x 2904 = 40
Again the result is 40 this does seem to follow a pattern, but maybe not the same as the grid before.
77 x 99 = 7623
79 x 97 = 7663
7623 – 7663 = 40
There is definitely a relationship between opposite corners in this grid, as again my result was 40.
I will now see if I can make an algebraic grid to help me work out my results quicker.
I will now test this by picking a random set of numbers and working out the other numbers in the box.
X = 71
= 71 + 1 = 72
= 71 + 2 = 73
= 71 + 10 = 81
= 71 + 11 = 82
= 71 + 12 = 83
= 71 + 20 = 91
= 71 + 21 = 92
= 71 + 22 = 93
This system will give me results on a 3 by 3 square on a ten by ten grid.
Ten by Ten (4x4)
I am now looking for a relationship in a 4 by 4 square; I will use the same system as above.
1 x 34 = 34
4 x 31 = 124
34 – 124 = 90
The final answer for this square is 90; still I have no idea of an equation for my work as a whole but I can predict that for this part of my investigation the results should be 90.
27 x 60 = 1620
30 x 57 = 1710
1620 – 1710
Again the final answer is 90 so not only was prediction correct but I now know that I could predict my next result (the answer will be 90).
61 x 94 = 5734
64 x 91 = 5824
5734 – 5824 = 90
The result was predicted correct. I still can’t see an overall pattern but I am going to try and draw an algebraic grid.
I will now test this by picking a random box and working out the other numbers in the box.
X = 67
= 67 + 1 = 68
= 67 + 2 = 69
= 67 + 3 = 70
= 67 + 10 = 77
= 67 + 11 = 78
= 67 + 12 = 79
= 67 + 13 = 80
= 67 + 20 = 87
= 67 + 21 = 88
= 67 + 22 = 89
= 67 + 23 = 90
= 67 + 30 = 97
= 67 + 31 = 98
= 67 + 32 = 99
= 67 + 33 = 100
This box works as you can see my results are the same as those in the green box.
Ten by Ten (5x5)
This is the last box I am going to do in the ten by ten grid. I hope that it will show a pattern.
I have had to use numerous colours to show the squares that I am using as they are so close together and could not be seen clearly. I will now carry on with my investigation following the same instructions as before.
1 x 45 = 45
5 x 41 = 205
45 – 205 = 160
The answer to this square is 160 so I predict that the answer to the next square will be 160.
51 x 95 = 4845
55 x 91 = 5005
4845 –5005 = 160
My prediction was correct so I will try again with this square to see if I my results where accurate. I still can’t see a constant pattern.
56 x 100 = 5600
60 x 96 = 5760
5600 – 5760 = 160
I still can’t see a common pattern but maybe something will arise in the nine by nine grid.
Nine by Nine (2x2)
I am now going to start my nine by grid. I will use exactly the same processes as in the ten by ten grid
Now I will try and find a relationship between a 2x2 squares on a nine by nine grid.
5 x 15 = 75
6 x 14 = 84
75 – 84 = 9
I can now see a pattern between my grids but I will have to repeat this process to see if anything changes or if the pattern is consistent.
28 x 38 = 1064
29 x 37 = 1073
1064 – 1073 = 9
There is definitive pattern between the result here and the result in grid 10 by 10-square 2x2. I can make prediction about an eight by eight grid when investigate the 2x2 square I predict that the answer would be 8.
49 x 59 = 2891
50 x 58 = 2900
2891 – 2900 = 9
Instead of producing an algebraic grid I am just going to produce an equation that is specific to my work (an equation specific to nine by nine grid 2x2 square.)
My equation =
=X (X+10)
I will now test my equation:
X = 1
= 1(1+10) = 11
Nine by Nine (3x3)
I am now going to investigate a square that is 3x3 and see if there is a pattern i.e. if I have to subtract one from the answer in Ten by Ten 3x3.
I am now going to tests on the boxes marked in red.
1 x 21 = 21
3 x 19 = 57
21 – 57 = 36
I can’t see a pattern between this square and any others, but I am still going to check that these results are correct, and will still observe for patterns.
34 x 54 = 1836
36 x 52 = 1872
1836 – 1872 = 36
My original result was correct but I still feel that could be a pattern so I am going to carry on so that I can find a relationship between the boxes.
38 x 58 = 2204
40 x 56 = 2240
2204 – 2240 = 36
I can’t find a pattern within this section of my work; but I am still going to work out an equation.
My equation is: (X+20)
I will now test my equation:
X=7
= (7+20)= 27
Nine by Nine (4x4)
I am about to start my nine by nine 4x4 grid: I have not yet spotted an overall pattern.
5 x 35 = 175
8 x 32 = 256
175 – 256 = 81
37 x 67 = 2479
40 x 64 = 2560
2479 – 2560 = 81
51 x 81 = 4131
54 x 78 = 4212
4131 – 4212 = 81
I can’t see a pattern in my overall work but maybe a complete set of results will give me an idea.
I am still going to find an equation for this section of work.
My equation: X (X+30)
I will now test my equation:
X = 1
= 1(1+ 30) =31
Nine by Nine (4x4)
I will now work towards getting a complete set of results I will do this by carrying on with my investigation.
1 x 41 = 41
5 x 37 = 185
41 – 185 = 144
38 x 78 = 2964
42 x 74 = 3108
2964 – 3108 = 144
41 x 81 = 3321
45 x 47 = 3165
3331 – 3165 = 144
I think that I have found a pattern, when analysing my results as a whole I noticed:
That depending on the numbers in the square you can multiply the grid number by a particular number and you can predict the subtracted number.
E.g. On any grid when looking at the square 2x2, you can work out the subtracted number by multiplying the grid number by 1.
On any grid when looking at the square 3x3, you can work out the subtracted number by multiplying the grid number by 4.
On any grid when looking at the square 4x4, you can work out the subtracted number by multiplying the grid by 9
On any grid when looking at the square 5x5, you can work out the subtracted number by multiplying the grid by 16.
I will now test these results:
(2x2) 1x10 =10
(3x3) 4x10 = 40
(4x4) 9x10 = 90
(5x5) 16x10 = 160
The subtracted number is a multiple of the grid number.
I am now going to predict the subtracted numbers for grid eight.
(2x2) 1x8= 8
(3x3) 4x8=32
(4x4) 9x8 =72
(5x5) 16x8 = 128
This is what I predict my subtracted numbers to be in my eight by eight grid
Eight by Eight (2x2)
I am now Going to see if my predictions where correct.
3 x12 = 36
4 x 11 = 44
36 – 44 = 8
My prediction was correct; so I have found a pattern. I will now see if the rest of prediction was correct.
29 x 38 = 1102
30 x 37 = 1110
1102 – 1110 = 8
55 x 64 = 3520
56 x 63 = 3528
3520 – 3528 = 8
My first prediction was definitely correct. I will now look for a general equation: but first I will find a specific equation.
My equation: (X+9)
X= 3
=(3+9)=12
My equation is correct.
Eight by Eight (3x3)
I have found a pattern so now I’ m going to carry on and see if I can find a common formula.
6 x 24 = 144
8 x 2 = 176
144 – 176 = 32
The second part of my prediction was correct I will now carry on trying to find out more evidence to support this.
19 x 37 = 703
21 x 35 = 735
703 – 735 = 32
46 x 64 = 2944
48 x 62 = 2976
2944 – 2976 = 32
This part is correct maybe I can find more relationships and final formula if I carry on. I will now look at an algebraic equation for this section of work.
My equation: (X+18)
X = 6
= (6 + 18) =24
My equation works.
Eight by Eight (4x4)
I have to use multiple colours so that my squares can be clearly seen
I will now see if the third part of my prediction is correct.
5 x 32 = 160
8 x 29 = 232
160 – 232 = 72
The third part of my prediction was correct. I am going to find more evidence to prove this.
33 x 60 = 1980
36 x 57 = 2052
1980 – 2052 = 72
37 x 64 = 2368
40 x 61 = 2440
2368 – 2440 = 72
The third part of my prediction was definitely correct so I think that the fourth part of this section of investigation will be correct. I am now going to produce an algebraic equation for this section of work.
My equation:
(X+27)
I will now test my equation:
X= 33
= (33+27)= 60
My equation works.
Eight by Eight (5x5)
This is the final part of my numeric work and hopefully I can find an equation that works of all opposite corners.
For the end part of my investigation I am only going to take two examples from my grid. As I think that it will be too hard to recognise what squares I have used.
1 x 37= 37
5 x 33 = 165
37 – 165 = 128
27 x 63 = 1701
31 x 59 = 1829
1701 – 1829 = 128
My predictions where all correct as I’ve just completed the numeric section of work and it was I predicted. Now I am going to analysise all work to see if there is an overall section of work.
