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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.

 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10
 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.

 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10
 Square      Total 1            50 11          110 21          170 31          230

Middle

33

34

35

36

37

38

39

40

21

22

23

24

25

26

27

28

29

30

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

 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.

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.

 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10
 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.

15n+220

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

 Corner No.

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

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

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

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

(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.

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