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  • Level: GCSE
  • Subject: Maths
  • Word count: 2067

Number Grid and Stairs

Extracts from this document...

Introduction

Mark Burgess        10-1

Maths investigative coursework

For my G.C.S.E coursework I will be investigating the relationship between the position, the size of the number step, the size of the grids and the sum of these numbers. I will also attempt to find the ‘super formula’ or the formula that will find the sum of a number step no matter what size what position or what grid size.

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Square        Total

     1              50

     2              56

     3              62

     4              68

            The total of square 2 is 56

         The total of square 3 is 62        

The total of square 4 is 68

This shows my prediction was correct and that moving the number stair to the right increased the encased total by 6.

I will now look at the same 3x3 stair in the same 10x10 grid but this time I will try moving the stair upwards by one to see how this affects the total of the encased numbers. Each row of squares you move up the grid adds 10 on to the total so I predict that, as when you move up a row ten is added, then if all 6 of the squares in the stair moves up then this will add on 60 to the total of the encased numbers.

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   Square      Total

        1            50

       11          110

       21          170

       31          230

...read more.

Middle

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Corner no             Total

1                            120

11                          220

21                          320

31                          420

This table shows that my prediction was correct and that as you move one roe up the sum of the encased numbers increases by 100.

As I have worked my way through the 4x4 stair I have come to realise that there are some rules that this stair also abides by.

Rules

1        = +10        1        = -100

1         = -10         1        = +100

I have investigated the relationship between the position and the sum of the encased numbers in a 4x4 stair I will now find the algebraic rule to support my investigating.

image07.png

10n +110

I will now test the algebraic rule that I have come up with on 5 random numbers to see if the rule is correct.

Corner No.       Total Using Formula                      Working Out                            Total

       12                           230                    12+13+14+15+22+23+24+32+33+42        230

       27                           380                    27+28+29+30+37+38+39+47+48+57        380        

       31                           420                    31+32+33+34+41+42+43+51+52+61        420

       48                           590                    48+49+50+51+58+59+60+68+69+78        590

       65                           760                    65+66+67+68+75+76+77+85+86+95        760

This shows that my algebraic rule works, I have made a prediction, tested it, made a formula, tested it and have come to the conclusion that it works.

I will now investigate the relationship between the position of a 5x5 stair in a 10x10 grid I will try to find the formula and uncover some rules.

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Corner No.      Total

       1                235

       2                250

       3                265

       4                280

      11               385

      21               535

      31               685


I have tested the 5x5 stair and I am now going to find a formula to explain it.

image08.png    15n+220

I will now test my rule on 5 different and random square/corner numbers.

Corner No.

...read more.

Conclusion

Corner         Total             Increased

Number                                         By

     1                      130          -----

     2                      140                    10

     3                      150                    10

     4                      160                    10

     1                      130                   -----

    12                     240                   110

    23                     350                   110

    34        460                    110

Looking at the table I can see that the algebraic rule will be

10n+120

image01.png

1+2+3+11+12+13+22+23+33=120

I will now look at a 5x5 stair in an 11x11 grid, I predict that as I move the step to the right the sum will increase by 115, I also predict that as I move the stair up the sum will increase by 165 because 11x15=165

Corner         Total             Increased

Number                                         By

     1                   255          -----

     2                   270                       15

     3                   285                       15

     4                   300                       15

     1                   255                      -----

    12                  420                      165

    23                  585                      165

    34                  750                          165

I can tell from looking at the table that the algebraic rule will be

15n+240

image02.png

1+2+3+4+11+12+13+14+22+23+24+33+34+44=240

Here is a table that incorporates all the algebraic rules I have come across so far this may help point out any ways that I might come across the super rule.

Grid Size      Steps       Rule          Difference

11x11         3        6n+48

10x10              3        6n+44             4

9x9                  3        6n+40

11x11              4        10n+120

10x10              4         10n+110        10

9x9                  4         10n+100

11x11              5         15n+240

10x10              5         15n+220        20

9x9                  5         15n+200

If I piece together what I have found I can build up the super rule

image03.png

I need to test my super rule to make sure it is correct.

Imagine the corner number is 45 the step size is 3 and in a 11x11 grid

You would work out the total number in this way

image04.pngimage05.png

(3(3+1)                 +        (11+1)           x          

     2      n

 6     n              +          12        x                    1(1+1)  +   2(2+1)

                                                                                          2                2

6      n              +              12                   x                       1       +         3

6      n              +              12                  x                               4

6     n               +               48

6x45                +               48

270                 +                48                     =                 318

45+46+47+56+57+67=318

I have came up with a rule and tested it and it works. Woop

I hope you have enjoyed reading and marking my coursework as much as I have enjoyed doing it.

...read more.

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