number grid

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12800/w2                                                                                                   Joanne Barton

Number grid

Task

      My coursework task is to look at the number grid

  • A box is drawn round four numbers.
  • Find the product of the top left number and the bottom right number in this box.
  • Do the same with the top right and bottom left numbers.
  • Calculate the difference between these products.

Investigate further.

In this investigation I will look at the different number grids and different sized rectangles within these grids, I will explain the patterns and give algebraic equations for the results found during this investigation. I will also try to find a formula and prove my findings.

To find out the product difference I need to:-

Multiply the top left number with the bottom right and multiply the top right with the bottom left and then - the products from each other.  

I will first try with a 2 x 2 square.

 

a)

12 x 23 = 276   The difference of the two products are 10.

13 x 22 = 286   I worked the difference out by doing, 286-276 = 10

b)      

38 x 49 = 1862

39 x 48 = 1872, I worked the difference out by 1872 - 1862 = 10

The product is again 10.

My theory is that all 2 x 2 squares will have the product of 10, I will show one more 2 x 2 square to prove the theory.

c)      

86 x 97 = 8342

87 x 96 = 8352, I worked the difference out by 8352 – 8342 = 10

The product is 10 also, so the theory works all 2 x2 squares = D 10

D = Difference.

I have drawn a table to show this clearer.

For all of the 2x2 squares you can see that the difference is 10, so anywhere you put a 2x2 square on a 10x10 grid you get the difference of 10. This can also be shown algebraically.

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For example,

For the 2x2 square.

n = any number

(n+1)(n+10) - n(n+11)

   n²+10n+1n +10-n² -11n

     n²+11n+10 - n²-11n = 10

This proves that for any number in a 2x2 square the grid difference will always be 10.

I would now like to look at the difference in a 3x3 square on a 10x10 grid.

My prediction is that once I find out the distance for the first 3x3 square then the rest will be the same I will give 3 examples.

a)    

If I multiply the top ...

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