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  • Level: GCSE
  • Subject: Maths
  • Word count: 2183

Opposite Corners

Extracts from this document...

Introduction

Opposite corners

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100

On a 10*10 square grid, choose any 2*2 square, multiply the corners in that grid and then find the difference between the two corners investigate.

image00.pngimage01.png

12  13           12       13                    16  17             16        17

22  23        * 23    * 22                    26  27          * 27     * 26

276 284332342

284-276=10                                     342-332=10image03.pngimage09.png

                          Difference=10

The two answers are the same. I think it would be the same for any 2*2 square. To prove this I will use algebra to show that in any 2*2 square the difference will 10.

z- number in the top left cornerimage27.png

  z     z+1                                                                              z(z+11)=z²+11z

z+10z+11                (z+1)(z+10)=z²+11z+10      (z²+11z+10)-(z²+11z)=10

 Difference = 10

This proves that with any 2*2 square the corners multiplied then subtracted always = 10

To further my investigations I am now going to use a 3*3 square and do the same as I did with the 2*2 square.

   3     4      5               3          5               1      2      3              3          1image35.pngimage35.png

   13   14    15        * 25* 23               11    12    13       * 21     * 23

   23   24    25           75      115               21    22    23          63        23

115-75=40                    Difference = 4063-23=40image62.pngimage51.png

Opposite corners

These answers are the same; just as the answer for the 2*2 squares are the same. I think that any 3*3 square would have a difference of 40. To prove this I will use algebra.

image66.png

  z     z+1   z+2                                                                  z(z+22)=z²+22zz

z+10 z+11z+12                                                      (z+2)(z=20)=z²+22z+40                      

z+20 z+21z+22                                                    (z²+22z+40)-(z²+22z)=40

This proves that with any 3*3 square the corners multiplied the subtracted always = 40

Now I am going to further my investigations again. I am now going to use a 4*4 square and do the same, as I did with the 2*2 and 3*3 square.

image02.pngimage02.png

1      2      3     4             1           4           7      8     9     10          7         10

  11    12    13   14       * 34      * 31          17    18   19    20     * 40      * 37

  21    22    23   24          34       124          27    28   29    30      280       370

  31    32    33   34       124-34=90            37    38    39   40     370-280=90image03.pngimage04.png

Difference = 90

The answer for the 4*4 squares are the same, just as the 2*2

...read more.

Middle

4*4 squares had the same difference. I think that any 5*5 square would have a difference of 160. To prove this I will use algebra.

   z     z+1   z+2  z+3   z+4                                                  z(z+44)=z²+44zimage07.png

z+10 z+11 z+12z+13 z+14                                 (z+4)(z+40)= z²+44z+160

z+20 z+21z+22 z+23 z+24(z²+44z+160)-(z²+44z)=160image10.png

z+30 z+31z+32 z+33 z+34                               Difference = 160

z+40 z+41z+42 z+43 z+44

This proves that with any 5*5 square the opposite corners multiplied then subtracted from each other = 160

I will now put the results from my investigations in a table, and comment on them.

image12.pngimage11.png

 Square Size               Difference of the corners multiplied then subtracted

           2*2                                                            10

           3*3                                                        40

           4*4                                                     90

           5*5                                                    160

I have noticed that my results are:

  • Multiples of 2,5 and 10.
  • They are square numbers.
  • They are one less than the size of the square.

From my results I can predict that the difference for a 6*6 square will be,

            6*6            (6-1)²*10=250image13.png

Opposite corners

1      2     3     4      5     6                1            6image14.png

11   12   13   14    15   16          * 56       * 51

21   22   23   24    25   26             56        306

31   32   33   34    35   36             306-56=250image15.png

41   42   43   44    45   46                                        Difference

51   52   53   54    55   56

This proves my prediction is correct. Through my investigations I can say that the general rule for finding the difference is,

10(m-1)²

m-size of the square                          

image16.png

image17.png

m                                10

m

                                              Now I am going to prove my general

                                              formula.

10

z-number in the top left corner.       m-length of the squareimage18.png

z                         z+(m+1)image19.png

                                                  z(z+M)=z²+12z

       (m+1)=M                  m     (z+M)(z+10M)=y²+11Mz+10m²image21.pngimage20.png

                                           (z²+11Mz+10m²)-(z²+11My)=10m²image22.png

z+10(m+1)     z+10(m+1)         10(m-1)²

...read more.

Conclusion

m-length of the square

                    9image49.png

                    m

   z                       z+(m-1)                                            z(z+10M)=z²+10Mzimage50.png

                                                              (z +M)(z+9M)=z²+9Mz+Mz+9M²image52.png

              (m-1)=M            m     9           (z²+10Mz+9M²)-(z²+10Mz)=9M²                    image53.pngimage54.pngimage55.png

                                                                                   9(m-1)²

   z+9(m-1)     z+10(m-1)

 This proves that the general formula for finding the difference of any square in a 9*9 grid is 9(m-1)².

Now I am going to further my investigations by finding the general formula for finding the difference of any rectangle in a 9*9 grid.

m-length of the rectangle.   n-width of the rectangle.   z-number in the top

left corner

                        m                                                        image56.png

z                                       z+(m-1)

                                                                             z(z+9N+M)=z²+9Nz+Mzimage57.png

                                                   (z+M)(z+9N)=z²+9Mz+Mz+9MN

(n-1)=N                                         nimage58.pngimage59.png

                                      (m-1)=M       (z²+10Mz+9MN)-(z²+10Mz)=9MNimage60.pngimage61.png

                                                                                9((m-1)(n-1)

z+9(n-1)             z+9(n-1)+(m-1)        

This proves that the general formula for finding the difference of any rectangle in a 9*9 is 9((m-1)(n-1)).

Now I am going to further my investigation further by finding the general formula for any square, rectangle or square grid.

Opposite corners

m-length of square or rectangle.

n-width of square or rectangle.

y-size of the grid.

z-number in the top left corner.

mimage49.png

z                                 z+(m-1)                            z(z+yN+M)=z²+yNz+Mzimage63.png

(z+M)(z+yN)=z²+yNz+Mz+yNM

(z²+yNz+Mz+yNM)-(z²+yNz+Mz)=yNMimage64.png

(n-1)=N                                                                     y((n-1)(m-1)image59.pngimage65.png

(m-1)=Mnimage61.png

z+y(n-1)        z+y(n-1)+(m-1)

This proves that the general formula for finding the difference for any square, rectangle and square grid.

...read more.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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4 star(s)

This is an excellent pieces of mathematical investigation. It is well structured moving from the concrete to the algebraic easily. The are a few small mathematical errors which limit the piece to four stars. There are specific strengths and improvements suggested throughout.

Marked by teacher Cornelia Bruce 18/04/2013

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