Let’s check if my prediction was correct.
The product of the top left number and the bottom right number in the square will be:
The product of the top right and bottom left numbers will be:
Difference between these products will be:
Conclusion:
My prediction was correct. There was a difference of 10. This may mean that it doesn't matter which 2 by 2 square you select within a 10 by 10 number grid the difference will still be 10.
As I see, where ever we move the square there is certain tendency which makes me be able to call any top left number (n), any top right number (n + 1), any bottom left number (n + 10) and bottom right number (n + 11) in any 2 by 2 square on the 10 by 10 grid.
This lets me check my conclusion using algebra.
The product of the top left number and the bottom right number in the square is:
The product of the top right and bottom left numbers is:
Difference between these products is:
This proves that all 2 by 2 squares taken from a 10 by 10 number grid
result in the answer 10.
I am going to investigate 3 by 3 squares in a 10 by 10 number grid.
The product of the top left number and the bottom right number in the square is:
The product of the top right and bottom left numbers is:
The difference is:
I will now move the square and do same calculations.
1)*
2)
3)
* The calculations that are being done every time (i.e. multiplying the top left and bottom
right corners and the top right and bottom left corners and finding
the difference) will only be shown by numbers (i.e. 1), 2), 3)) in the further content of the coursework.
I predict that my next 3 by 3 square will result in the answer 40.
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My prediction was correct. There was a difference of 40. This may mean that it doesn't matter which 3 by 3 square you select within a 10 by 10 number grid the difference will still be 40.
I will now prove my conclusion using algebra.
Any top left number will be , any top right number will be , any bottom left number will be , any bottom right number will be in any 3 by 3 square on a 10 by 10 number grid.
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This proves my conclusion and means that all 3 by 3 squares taken from a 10 by 10 number grid result in the answer 40.
I will now investigate 4 by 4 squares in a 10 by 10 number grid.
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I predict that my next 4 by 4 square will result in the answer 90.
Let’s check my prediction.
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My prediction was correct. There was a difference of 90. This means that it doesn't matter which 4 by 4 square you select within a 10 by 10 number grid the difference will still be 90.
I will now try to prove my conclusion using algebra.
Any top left number will be , any top right number will be , any bottom left number will be , any bottom right number will be in any 4 by 4 square on a 10 by 10 number grid.
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This proves that all 4 by 4 squares taken from a 10 by 10 number grid result in the answer 90.
I will now make a table of all the results that I had.
Using th term I can tell that the formula is
Example (2 by 2 square)
I predict that a 5 by 5 square would result in the following:
I will now test my prediction.
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The same result keeps occurring. It is the result that I predicted.
Using algebra, I will now prove that this is the result for any 5 by 5
square taken from a 10 by 10 number grid.
Any top left number will be , any top right number will be , any bottom left number will be , any bottom right number will be in any 5 by 5 square on a 10 by 10 number grid.
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All 5 by 5 squares on a 10 by 10 number grid result in the answer 160.
Now, using algebra I will prove that an x by x square will result in
Any top left number will be , any top right number will be , any bottom left number will be , any bottom right number will be in any x by x square on a 10 by 10 number grid.
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I can see that repeats in all parts of equation. To simple equation I will rename. It will now be called T.
So
Now I will put instead of T.
So it is:
This means that an x by x square would result in. This means my
prediction was correct. This is the result for any x by x grid taken from
a 10 by 10 number grid.
In this investigation I found a formula that can be used to find difference in any x by x square on a 10 by 10 number grid. This was the main purpose of my investigation, so I can say that the investigation passed successfully.
If I were to extend this investigation further, I would experiment with different sized master grids