By working using algebra, I can see I will always get an answer of 40 on a 3x3 grid.
I will now test out a 4x4 grid and carry out the same methods as before for my investigation.
1 x 34 = 34
4 x 31 = 124 the difference between the top number and bottom number = 90
I will again do the same for another 4x4 grid.
7 x 30 = 210
10 x 27 = 300 the difference between the top number and bottom number = 90
I will again do the same for another 4x4 grid.
62 x 95=5890
65 x 92=5980 the difference between the top number and bottom number = 90
I will now show my working in algebraic terms.
(n+3)(n+30) - n(n+33)
n²+33n+90 - n²- 33n = 90
By working using algebra, I can see that I always get a difference of 90 on a 4x4 grid.
I will now test out a 5x5 grid and carry out the same methods as before for my investigation.
1 x 45 = 45
41 x 5 = 205 the difference between the top number and bottom number = 160
I will do the same for another 5x5 grid.
45 x 89 = 4005
49 x 85 = 4165 the difference between the top number and bottom number = 160
I will again do the same for another 5x5 grid.
56 x 100 = 5600
60 x 96 = 5760 the difference between the top number and bottom number = 160
I will now show my working in algebraic terms.
(n+40)(n+4) - n(n+44)
n²+44n+160 - n² - 44n =160
By working using algebra, I can see I always get a difference of 160 on a 5x5 grid.
By using my information from the other grids, I have made a prediction for a 6x6 grid. The difference should be 250.
I will check this by using the same technique as before (using the 4 corners of a square) for a 6x6 grid in algebra.
(n+5)(n+50) - n(n+55)
n²+ 55n+250 - n² -55n = 250
By working using algebra, I can that I will always get a difference of 250 on a 6x6 grid.
I will now show all the results of difference from each nxn grid and try to distinguish a pattern:
Difference of 2x2 = 10 =10 x 1²
Difference of 3x3 = 40 =10 x 2²
Difference of 4x4 = 90 =10 x 3²
Difference of 5x5 = 160 =10 x 4²
Difference of 6x6 = 250 =10 x 5²
Therefore, he formula for a 10x10 master grid is 10(n-1)²
Also, from my results I can see that I always have to square a number by 1 less than the size of the grid.
I will now do the same experiment I conducted on a 10x10 master grid with an 8x8 master grid.
I will now investigate a 2x2 grid on this 8x8 master grid.
1 x 10 = 10
9 x 2 = 18 the difference between the top number and bottom number = 8
I will do the same with another 2x2 grid.
35 x 44 = 1540
36 x 43 = 1548 the difference between the top number and bottom number = 8
I will again do the same with another 2x2 grid.
50 x 59 = 2950
51 x 58 = 2958 the difference between the top number and bottom number = 8
I will now show my working in algebraic terms.
(n+1)(n+8) - n(n+9)
n²+9n+8 - n²-9n = 8
By using algebra, I can see that I will always get a difference of 8 on a 2x2 grid.
I will now test out a 3x3 grid and carry out the same method of investigation as before.
1 x 19 = 19
17 x 3 = 51 the difference between the top number and bottom number = 32
I will now do the same again for another 3x3 grid.
6 x 24 = 144
8 x 22 = 176 the difference between the top number and bottom number = 32
I will now do the same again for another 3x3 grid
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I will now show my working in algebraic terms.
(n+2)(n+16) - n(n+18)
n²+ 18n+32 - n²-18n = 32
By working using algebra, I can see I will always get a difference of 32 on a 3x3 grid.
I will now test out a 4x4 grid and carry out the same method of investigation as before.
1 x 28 = 28
4 x 25 = 100 the difference between the top number and bottom number = 72
I will now do the same for another 4x4 grid.
5 x 32 = 160
8 x 29 = 232 the difference between the top number and bottom number = 72
I will show my working in algebraic terms.
(n+24)(n+3) - n(n+27)
n²+ 27n+72-n²+27 = 72
By working using algebra, I can see that I will always get a difference of 72 on a 4x4 grid.
I will now test out a 5x5 grid and carry out the same method of investigation as before.
1x37 = 37
5x33 = 165 the difference between the top number and bottom number is 128.
I will now do the same for another 5x5 grid
2x38 = 76
6x34 = 204 the difference between the top number and bottom number is again 128.
I will now show my working using algebraic terms.
(n+32)(n+4) – n(+36)
n²+36n+128-n²+36n =128
By working using algebra, I can see I will always get a difference of 128 on a 5x5 grid.
I will now show all the results of difference from each nxn grid and try to distinguish a pattern:
Difference on a 2x2= 8 =1x8 = n²x8
Difference on a 3x3= 32 =4x8 = n²x8
Difference on a 4x4= 72 =9x8 = n²x8
Difference on a 5x5= 128 =16x8 = n²x8 8(n-1)²
From my result I can see that I always have to square a number 1 less than the size of the grid.
When I use a 10x10 grid the difference is 10(n-1), and when I use an 8x8 grid the difference is 8(n-1).
I am now going to investigate rectangles.
I will now investigate a 2x3 rectangle on a 10x10 master grid.
(n+10)(n+2) – n(n+12)
n² +12n+20- n²-12n = 20
By using algebra I can see I will always get a difference of 20 on a 2x3 rectangle
(n+10)(n+3) – n(n+13)
n² +13n+30- n²-13n = 30
By using algebra I can see I will always get a difference of 30 on a 2x4 rectangle.
I will investigate the algebra for 4x3 grid.
(n+20)(n+3)-n(n+23)
n²+23n+60 - n²-23n = 60
By using algebra, I can see that I will always get a difference of 60.
I will now investigate the algebra for 5x3 grid.
(n+20)(n+4)-n(n+24)
n²+24n+80 - n²-24n =80
By using algebra, I can see I will always get a difference of 80.
Here are my overall results for the rectangles:
Difference of 2x3 = 20 =10 x 2
Difference of 2x4 = 30 =10 x 3
Difference of 3x4 = 60 =10 x 6
Difference of 3x5 = 80 =10 x 8
I can see that the formula will definitely have a 10 involved. I need to find out how to relate the size of the rectangle with the difference between the top and bottom numbers:
(2x3) 2x10 = 1x2x10
(2x4) 3x10 = 1x3x10
(3x4) 6x10 = 3x2x10
(3x5) 8x10 = 4x2x10
I can see from the grids with the width of 2 that I need to take 1 from the front number and 1 from the back number.
For an nxn rectangle the difference will be 10(m-1)(n-1).
This is the same result that I got with the nxn squares but with rectangles the length and width variey.
