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Mathematics Coursework; Number Grid

I have been asked to investigate a mathematical problem, in this piece of work I will explore the problem investigating a number of variables, using algebra to prove that the results hold for any combination of number.

12 x 23 = 276

13 x 23 = 286

The difference between the above two products is equal to 10.

I will repeat this using different numbers to see if there is a recurring pattern.

35 x 46 = 1610

36 x 45 = 1620

The difference between the above two products is equal to 10.

I predict that if I work out the following, the difference will be equal to 10.

38 x 49 =

39 x 48 =

I will now check to see if I am correct.

38 x 49 = 1862

39 x 48 = 1872

I was correct there is a difference of 10.

I am now going to try to prove that this holds for any 2 by 2 square using algebra.

Let n be the top left number and n + 11 be the bottom right number.

n(n+11) = n²+11n

Let n + 1 be the top right number and n + 10 be the bottom left number.

(n+1)(n+10) = n²+n+10n+10

              = n²+11n+10

I will now subtract the two algebraic products to get an overall product.

n²+11n+10

n²+11n       -

         

        +10

This proves that the difference is always 10



I have proved that the above results hold for any 2 by 2 square.

This shows that for any 2 by 2 square the difference will always be equal to 10.

To extend this investigation I am going to investigate the effect of changing the shape of the box. This variable will look at the effect of changing the length of the rectangle by 1 square each time.

I am going to change the square to a 3 by 2 rectangle.

15 x 27 = 405

17 x 25 = 425

The difference between the above two products is equal to 20.

I am now going to try to prove that this holds for any 3 by 2 rectangle using algebra.

Let n be the top left number and n + 12 be the bottom right number.

n(n+12) = n²+12n

Let n + 2 be the top right number and n + 10 be the bottom left number.

(n+2)(n+10) = n²+2n+10n+20

              = n²+12n+20

I will now subtract the two algebraic products to get an overall product.

n²+12n+20

n²+12n       -

         

        +20

This proves that the difference is always 20

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I have proved that the above results hold for any 3 by 2 rectangle.

This shows that for any 3 by 2 rectangle the difference will always be equal to 20.

I will now change the size of the rectangle to 4 by 2.

66 x 79 = 5214

69 x 76 = 5244

The difference between the above two products is equal to 30.

I am now going to try to prove that this holds for any 4 by 2 rectangle using algebra.





Let n be the top left number and ...

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