I then did a three by three box…
I did three different three by three boxes…
I then did a four by four box…
I then did a five by five box…
I have noticed that all these numbers are divisible by ten. I’m going to divide by 10 to see what answers I get…
I have notice that when I divide by 10 I get all square numbers as my answers. These answers are all less than the box size chosen. Eg 2×2 2-1=1 1²
So from this I can make an equation to test any box…
10 (n-1)(n-1)=10 (n-1)²
Now I will test rectangle boxes to further my investigation. I predicted that this formula would not work on the rectangles. I will have to alter the formula later.
A three by two box…
12×24=288 308-288=20
22×14=308
I tested the three by two box twice more in different places to see if it would be the same answer…
Now I will test the box as a two by three and see if this will alter my answer…
I will test this three times…
Looking at my results so far, I realise that when I test three times I always get the same answer so from now I will only test once…
Here I have put my results in to a table…
With the column staying the same
With the rows staying the same
Here I noticed that my answers go up in tens. This means they are also divisible by 10. I will use my formula to check that these answers are correct.
I have to alter my formula slightly so it works for squares and rectangles…
This is my old formula
10 (n-1)(n-1)=10 (n-1)²
This is my new formula
10(c-1)(r-1)
Testing…
10(2-1)(4-1)=30 10(2-1)(5-1)=40
I am now going to change the size of my grid to further my investigation. I predict that I will have to alter my formula by changing the starting number. Eg 10 in my original formula I’m using 8 grids so my differences should be divisible by 8.
First ill test a two by two boxes. I will test these each twice if I get the same answer I will not test them further. If I don’t ill test them more…
Three by three boxes …
Four by four boxes…
Five by five boxes…
I can use the same formula on this grid except I have to change it slightly… this was my formula for the 10 grid
10(c-1)(r-1)
as this is a 8 grid and all my difference are divisible by 8 I will change my formula to
8(c-1)(r-1)
I can check my answers using this formula
8(5-1)(5-1)=128 8(2-1)(2-1)=8
I now know that I can use my formula to find my difference of rectangles in this grid. Eg.
A two by three grid…
11x21=231 247-231=16
13x19=247
8(2-1)(3-1)=16
The answers are the same so I think it is more efficient to use the formula to find the answers…
8(2-1)(4-1)=24 8(4-1)(2-1)=24
8(2-1)(5-1)=32 8(5-1)(2-1)=32
I will now further my investigation using algebra…
If I know that it’s a two by grid…
2
2
TL=X TR=X+1
BL=X+10 BR=X+11
TL X BR TRXBL
(X)(X+11) (X+1)(X+10)
X²+11X X²+10+X+10X X²+10+11X
This helps to p
