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• Level: GCSE
• Subject: Maths
• Word count: 2212

Number Stairs Investigation

Extracts from this document...

Introduction

Beth Connolly        Maths Coursework         2004

Number Stairs Investigation

Introduction:

This is a 3-step stair taken from a 10 x 10 grid. The total of the numbers inside the stair shape is:

25+26+27+35+36+45 = 194

We can write this as T    = 194

Part 1 Working:

For other 3-step stairs investigate the relationship between the stair total and the position of the stair shape on the 10 x 10 grid (given by the label x in Tx)

Results Table:

X                        T (total)

1                                T     = 50

25                        T     = 194

26                        T     = 200

27                        T     = 206

28                        T     = 212

78                        T     = 512

As the x number rises by one, the total number rises by six. This is because there are six numbers in the 3-step stair, and if x increases by one then so do the rest of the numbers.

E.g.

1. 46

35     36                        36     35

25     26     27                26     27     28

As we can see, each of the six numbers has increased by one.  Therefore;

1 x 6 = 6.

So far the equation stands like so; X = 6X???

We can discover the remainder of the equation by using examples.

X                6X                                Total        (Tx)

1                6                +44                50

25                150                +44                194

26                156                +44                200

27                162                +44                206

28                168                +44                212

78                468                +44                512

or we can find it algebraically;

x + 20

x + 10        x + 11

x                x + 1                x + 2

The formula for the 3-step stair on a 10 x 10 grid is    Tx = 6x + 44

The letter “g” represents the grid number, in this case 10.

Using this formula I will predict that T     = (6 x 4) + 44

T     = 24 + 44

T     = 68

24

14    15

4      5      6                           24+14+15+4+5+6 = 68

I will also predict that   T     = (6 x 72) + 44

T     = 432 + 44

T     = 476

92

82     83

72     73     74                        92+82+83+72+73+74 = 476

My prediction was correct consequently the formula is also correct.

Middle

Results Table:

X                        T (total)

1                        T    = 235

23                        T    = 340

24                        T    = 355

25                        T    = 370

26                        T    = 385

As the x number rises by one the total number rises by 15. This is because there are 15 numbers in a 5-step stair, and as x increases by one so do the rest of the 15 numbers.

E.g.

1. 42

31     32                                32     33

21     22     23                        22     23     24

11     12     13     14                12     13     14     15

1        2        3        4     5                    2         3         4         5        6

As we can see each of the 15 numbers has increased by one. Therefore;

1 x 15 = 15

So far the equation stands like so; X = 15X???

We can find out the remaining of the equation by using example.

X                15X                                        Total (Tx)

1                15                  +220                235

3                45                  +220                265

4                60                  +220                280

23                345                  +220                565

24                360                  +220                580

25                375                  +220                595

26                390                  +220                610

56                840                  +220                1060

The formula for the 5-step stair on a 10 x 10 grid is   Tx = 15x + 220

Using this formula I will predict that T    = (15 x 2) + 220

T    = 30 + 220

T    = 250

42

32    33

22     23     24

12     13     14     15

2         3        4        5      6

42+32+33+22+23+24+12+13+14+15+2+3+4+5+6  = 250

My prediction and formula was correct. I can now produce a graph showing the relationship between x and Tx with 5-step stairs.

I am now going to investigate further by finding a formula for 3, 4 and 5 step stairs on an 8 x 8 grid.

3-step stair:

Results Table:

X                        T (total)

1. 42
2. 48
3. 54
4. 60

45                        306

46                        312

So far the equation stands like so; x= 6X???

We can find the remaining of the equation in the same way as before:

X                6X                                Total (Tx)

1                6                +36                T    = 42

2                12                +36                T    = 48

3                18                +36                T    = 54

4                24                +36                T    = 60

46                276                +36                T    = 312

The formula for a 3-step stair on any 8 x 8 grid is        Tx = 6x + 36

I am going to predict the totals of 3-step stairs on an 8x8 grid

I predict that T    = (6 x 6) + 36

Conclusion

1. 215
2. 230
3. 245
4. 260
5. 275

So the equation stands like so; X = 15X???

X                15X                                        Total (Tx)

1                15                +200                        T    = 215

2                30                +200                        T    = 230

3                45                +200                        T    = 245

4                60                +200                        T    = 260

5                75                +200                        T    = 275

The formula for any 5-step stair on a 9 x 9 grid is

Tx = 15x + 200

I am going to predict that T    = (15 x 30) + 200

T    = 1500 + 200

T    = 1700

66

57    58

48    49    50

39    40    41    42

30    31    32    33    34

66+57+58+48+49+50+39+40+41+42+30+31+32+33+34 = 1700

My prediction was correct, I can now draw up another graph, similar to the others showing the relationship between x and Tx with any 5-step stair on a 9 x 9 grid.

By drawing all my formulas together I can see a pattern occurring. The first set of numbers are in a CUBIC SEQUENCE; 6, 10, 15 etc.

3-step                4-step                5-step

8 x 8                        6x + 36                10x + 90                15x + 220

9 x 9                        6x + 40                10x + 100                15x + 200

10 x 10                6x + 44                10x + 110                15x + 180

I can predict that for a 3-step stair on an 11 x 11 grid that the formula will be 6x + 48.

T    = (6 x 1) + 48

T    = 6 + 48

T    = 54

23

12     13

1             2        3                23+12+13+1+2+3 = 54

My prediction was correct.

I will now predict that the next formula on an 11 x 11 grid for a 4-step stair will be   Tx = 10x + 120

T    = (10 x 3) + 120

T    = 30 + 120

T    = 150

36

25    26

14    15    16

3        4        5        6        36+25+26+14+15+16+3+4+5+6 = 150

My prediction was correct.

I now predict the next formula for a 5-step stair on an 11 x 11 grid will be Tx = 15x + 160

T    = (15 x 1) + 160

T    = 15 + 160

T    = 175

45

34    35

23    24    25

12    13    14    15

1        2      3     4             5

45+34+35+23+24+25+12+13+14+15+1+2+3+4+5 = 255

Unfortunately, my prediction was not correct. This is something that maybe if I extended my investigation further I could find out.

In conclusion to my investigating I have discovered the formulas for any 3, 4 and 5- step stairs on 8 x 8, 9 x 9, and 10 x 10 grids. I have also made a prediction about 11 x 11 formulas too. I think of this to be a success.

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