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  • Level: GCSE
  • Subject: Maths
  • Word count: 2571

My coursework task is to investigate why, in a number grid square of 1-100, when a section of two by two squares is extracted and the two opposite squares are multiplied and then subtracted the result is always 10.

Extracts from this document...

Introduction

Math Coursework

My coursework task is to investigate why, in a number grid square of 1-100, when a section of two by two squares is extracted and the two opposite squares are multiplied and then subtracted the result is always 10.

I will also be testing and studying whether it is true for three by three, four by four, five by five e.t.c number squares. I shall also be studying what will happen if I change the size of the grid square upon which I am extracting the numbers from.

E.g.

image49.png

image00.pngimage01.png

image10.png

image50.png

image20.png

                     2x2 squareimage29.png

image68.pngimage37.png

image43.png

                                           3x3 square        

2 by 2 Analyses

image79.png

                                                     (2x11) - (1x12) = 10image44.png

image89.png

                (35x44) - (34x45) = 10image44.png

image100.png

                                  (3x12) - (2x13) = 10image02.png

image111.png

                (99x90) - (89x100) = 10image02.png

As we can see the results clearly show that no matter what selection of 2x2 square we use the result will always  be 10.

We can show how and why the result is always 10 by using Algebra, (representing numbers by using letters).

image132.pngimage122.png

   =

This is how we can express numbers using letters. As with numbers we can also put letters into a formula. This is how it would look:

(P+10)(P+1)- P (P+11) = 10

image03.png

Testing the Formula

Now we can put the formula to the test by using it with numbers:

(23x32)- (22x33) =10

image51.pngimage140.png

           =

As we can see the formula has proven to work well and the result is once again 10 as the formula suggests.

...read more.

Middle

²X10 is 360.

image90.png

(8x62)(2x68) = 360

image91.png

                                                                             (P+60) (P+6) – P (P+66)

image19.png

I will now try an 8x8 grid square … I predict using the table that the result will be 490, because 7²x10, is 490.

image92.png

        (29x92)(22x99) = 490

image21.png

As the other results have proven the rule that the number, -1, multiplied by ten and squared gives the result. I can safety say that I will try a 9x9 and 10x10 and I shall predict the result.

9x9 the result will be 640, as 10²x8= 640

image22.png

image93.png

(10x82)-(2x90)= 640

image23.png

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This is the ten by ten grid square, I predict that when the opposite corners are multiplied and then subtracted from each other that the result will be 810, because judging by my results table so far I have formed a theory that it will be 10x9², is 810.

I can now test the theory that states:     P²+99P+810 -P² - 99P = 810          

(10x91)- (1x100) = 810

The result clearly shows us that the formula expressing the theory is correct.

image24.pngimage94.png

image25.png

image27.pngimage26.pngimage27.pngimage26.png

Now that I have tried and tested the results within a 1-100 grid square or 10x10, I am now going to try it with a 9x9 grid square like this:

image95.png

image28.png

image30.png

image96.png

image97.png

I will be extracting the square of 2x2, 3x3, 4x4 and so on.  Like shown above and subtracting the product of the two numbers multiplied together.

2 by 2 analyses

                                                                                 (2x10)-(1x11)=9image98.png

image31.png

image99.png

                                                  (81x89)-(80-89)=9image31.png

image101.png

         (74x82)-(73x83)=9

image31.png

        (43x51)-(42x52)=9image31.pngimage102.png

...read more.

Conclusion

always  be 8. Even if it is extracted from the bottom right or left or top left or right hand corners.

We can show how and why the result is always 8 by using Algebra, (representing numbers by using letters).

image149.pngimage150.png

             =

This is how we can express numbers using letters. As with numbers we can also put letters into a formula. This is how it would look:

(P+9)(P+1)- P (P+8) = 8

image45.png

image52.png

        (3x17)-(1x19)= 32

image53.png

        (23x37)-(21x39)= 32

image54.png

           (21x35)-(19x37) = 32

image55.png

             (4x18)-(2x20)=32

As we can see the results clearly show that no matter what selection of 3x3 square we use the result will always  be 32. Even if it is extracted from the bottom right or left or top left or right hand corners.

We can show how and why the result is always 32 by using Algebra, (representing numbers by using letters).

image56.pngimage57.png

             =

This is how we can express numbers using letters. As with numbers we can also put letters into a formula. This is how it would look:

(P+18)(P+2)- P (P+16) = 32

image46.png

As we can see there is a formula beginning to evolve as there was with the squares. So now we can draw a results table explaining and displaying our results.

image58.png

image47.pngimage48.png

 I have now proved, using algebra, that when the two corner numbers of any size square or rectangle are extracted from a number grid of any size, the results will always be, the size of the square (n-1)².

...read more.

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