# 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.

- 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.

- 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)

- 42
- 48
- 54
- 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

- 215
- 230
- 245
- 260
- 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.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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