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  • Level: GCSE
  • Subject: Maths
  • Word count: 1847

Number Grids Investigation.

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Firstly, I will test a 2x2 grid from a 10x10 master grid. I will then find the sum of the top-left and bottom right numbers multiplied together and do the same for the top right and bottom-left numbers.

 15 x 26 = 390

 16 x 25  = 400 the difference between the top number and bottom number = 10

I will repeat this method again with another 2x2 grid.                                  

1 x 12 = 12

2 x 11 = 22 the difference between the top number and bottom number = 10

I will again follow the same method with another 2x2 grid.

85 x 96 = 8160

86 x 95 = 8170 the difference between the top number and bottom number = 10

I will now show my working in algebraic terms.

n

n+1

n+10

n+11

  (n+1)(n+10) - n(n+11)

    n²+11n+10 -  n²11n          = 10
By working using algebra, I can see that I will always get an answer of 10 on a 2x2 grid.

I will now test a 3x3 grid and carry out the same methods as before for my investigation.

 1 x 23 = 23

 3 x 21 = 63 the difference between the top number and the bottom number = 40

I will again do the same for another 3x3 grid.

 31 x 53 = 1643

 33 x 51 = 1683 the difference between the top number and bottom number = 40

I will again do the same for another 3x3 grid.

 76 x 98 = 7800

 78 x 96  = 7840 the difference between the top number and bottom number = 40

I will now show my working in algebraic terms.

n

n+2

n+20

n+22

(n+20)(n+2) - n(n+22)                                                    n²+22n+40 - n²- 22n       = 40

...read more.

Middle

I will now show my working in algebraic terms.

n

n+3

n+30

n+33

     (n+3)(n+30) - n(n+33)

       n²+33n+90 - n²- 33n   = 90                              

By working using algebra, I can see that I always get a difference of 90 on a 4x4 grid.

I will now test out a 5x5 grid and carry out the same methods as before for my investigation.

1 x 45 = 45

41 x 5 = 205 the difference between the top number and bottom number = 160

I will do the same for another 5x5 grid.

45 x 89 = 4005

49 x 85 = 4165 the difference between the top number and bottom number = 160

I will again do the same for another 5x5 grid.

56 x 100 = 5600

60 x 96 = 5760 the difference between the top number and bottom number = 160

I will now show my working in algebraic terms.

n

n+4

n+40

n+44

(n+40)(n+4) - n(n+44)

 n²+44n+160 - n² - 44n =160

By working using algebra, I can see I always get a difference of 160 on a 5x5 grid.

By using my information from the other grids, I have made a prediction for a 6x6 grid. The difference should be 250.

I will check this by using the same technique as before (using the 4 corners of a square) for a 6x6 grid in algebra.

n

n+5

n+50

n+55

 (n+5)(n+50) - n(n+55)

   n²+ 55n+250 - n² -55n = 250

By working using algebra, I can that I will always get a difference of 250 on a 6x6 grid.

...read more.

Conclusion

n+12

(n+10)(n+2) – n(n+12)

n² +12n+20- n²-12n   = 20

By using algebra I can see I will always get a difference of 20 on a 2x3 rectangle

n

n+3

n+10

n+13

(n+10)(n+3) – n(n+13)

  n² +13n+30- n²-13n   = 30

By using algebra I can see I will always get a difference of 30 on a 2x4 rectangle.

I will investigate the algebra for 4x3 grid.

n

n+3

n+20

n+23

(n+20)(n+3)-n(n+23)

  n²+23n+60 - n²-23n  = 60

By using algebra, I can see that I will always get a difference of 60.

I will now investigate the algebra for 5x3 grid.

n

n+4

n+20

n+24

 (n+20)(n+4)-n(n+24)

 n²+24n+80 - n²-24n =80

By using algebra, I can see I will always get a difference of 80.


Here are my overall results for the rectangles:

Difference of 2x3 = 20     =10 x 2

Difference of 2x4 = 30     =10 x 3

Difference of 3x4 = 60     =10 x 6

Difference of 3x5 = 80     =10 x 8

I can see that the formula will definitely have a 10 involved. I need to find out how to relate the size of the rectangle with the difference between the top and bottom numbers:

  (2x3)                                    2x10 = 1x2x10

  (2x4)                                    3x10 = 1x3x10image00.png

  (3x4)                                    6x10 = 3x2x10image00.png

  (3x5)                                    8x10 = 4x2x10
I can see from the grids with the width of 2 that I need to take 1 from the front number and 1 from the back number.
image00.png

For an nxn rectangle the difference will be 10(m-1)(n-1).

This is the same result that I got with the nxn squares but with rectangles the length and width variey.

...read more.

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