# Investigation - stair shape on a 10x10 numbergrid

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Introduction

Investigation-stair shape on a 10x10 number grid

Introduction

For this investigation I will be investigating the relationship between no. total and stair no. on a 10x10 number grid. I will also be comparing results when the stairs are moved around the grid, finding patterns that may occur.

Method

A diagram of the stair shape is shown below. There are three numbers on the bottom layer, two on the second and one on the top, starting from left. The stair no. will be identified by the number in the bottom left hand corner of the stair. The grid starts from one in the bottom left hand corner and ascends to 100 in the top right. This is done so the stairs are in an up-right position rather than up side down. As shown below.

I began by planning the first set of stairs…

I shaded in no.s 1,2, 3, 4 and 5 to make the outline of the stair case clear. The no.

Middle

31

41

Stair total

50

110

170

230

290

+60 + 60 + 60 +60

As you can see the pattern increases by 60 every time

Therefore we can conclude that the rule

6n + 44 definitely does not work

Although it seems to work for stair no. 1

The horizontal rule does cannot be applied to the vertical rule

60n -10

Instead I will test the formulae 60n – 10

I was lead to this formulae as the pattern increases by 60; therefore 60n seems correct

The second half of the equation is – 10 as

110 – 60 = 50

The first stair total works using this formulae!

I will now try 11

11 x 60 =660

660 – 10 = 650

This formulae definitely does not work

After much consideration , I have come to the conclusion that working out the formulae is beyond my mathematical knowledge.Although I can see a pattern

Re- ocurring.The totals increase by 60 every time.

To expand the investigation further I will now be using a 7 x 7 grid …

7 x 7 Grid

The 7 x 7 grid will look something like this…..

Horizontal

I will take the stair no.s 3, 4 , 5, 6, and 7

The totals are as follows…

Stair no. | 3 | 4 | 5 | 6 | 7 |

Stair total | 42 | 48 | 54 | 60 | 66 |

6 6 6 6

As the difference is 6 the formula will be 6n

Conclusion

T= 6n + 24

Now I will try to find the vertical rule

Vertical

I will take the sequence 5 , 12 , 19 , 26 , and 33

Stair no | 5 | 12 | 19 | 26 | 33 |

Stair total | 54 | 96 | 138 | 180 | 222 |

42 42 42 42

The pattern shows an increase of 42

The equation should use 42 to multiply the subject

T=42n

If 5 x 42 = 210

To get to 54 we need to subtract 54 from 210

210 – 54=156

The formulae could be T= 42 n – 156

I will test the formulae using stair no. 12

It should total 96

12 x 42 = 504

504 – 156= 348

The formulae does not work

I will try another formulae using another sequence…

Sequence 6 , 13 , 20, 27 and 34

I will take stair no.s 6,13,20,27 and 34.

I then add up the totals for each stair no. and record it in the table below.

These were the results I found…

Stair no. | 6 | 13 | 20 | 27 | 34 |

Stair total | 60 | 102 | 126 | 186 | 228 |

42 24 60 42

18 36 18

18 18

The difference is 18

I will check to see if the horizontal formulae applies to the vertical as well.

T=6n+24

Stair no. 6

6 x 6 +24= 60

The formulae works for stair no.6

I will now check, using the other stairs in the sequence.

13x 6 +24=102

The formulae works for stair no. 13

20 x 6 + 24 =144

The formulae does not work

Therefore it is not the formulae

Although I can identify the patterns I cannot see a formulae that works.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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