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• Level: GCSE
• Subject: Maths
• Word count: 1364

# To investigate the relationship between a 10 by 10-number grid and various stair shapes (as in the example below).

Extracts from this document...

Introduction

Aim: To investigate the relationship between a 10 by 10-number grid and various stair shapes (as in the example below).

 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

On posed with this investigation I began to look at the relativity of the stair shape on the grid and its relation to the numbers it coincides with. I then began to imagine the grid as an axis whereby the x and y lay on the grid.

## Part 1

For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shapes on the grid.

Firstly I had to find the relationship of the stair shape where it lay on the grid to the stair total. I began to look at different ways, I found that by working systematically on the grid i.e. starting by looking at horizontal, vertical and diagonal movements I would be able to find the common factors involved.

In the example we are given the numbers:

25, 26, 27, 35, 36, 45

They total to 194

To work out the placement of the stair shape I used the smallest number of the stair to translate the shape from, in this case 25.

Middle

182

33

34

23

24

25

 44 188 34 35 24 25 26
 45 194 35 36 25 26 27
 46 200 36 37 26 27 28

Although there is no common number here at the end of the total for each stair shape there is a common number fond when the difference is calculated.

200-194=6

194-188=6

188-182=6

This means that for every horizontal movement either left or right on the grid the total difference between each stair shape for a 3-step stair would be 6.

Positive diagonal movement on the grid:

 23 62 13 14 3 4 5
 34 128 24 25 14 15 16
 45 194 35 36 25 26 27
 56 260 46 47 36 37 38

There is no common number found here at the end of each total. The common difference is:

260-194=66

194-128=66

128-62=66

This means that when the stair shape is moved diagonally in the “y=x” direction the difference increase is 66. The total stair shape total for the stair shape will increase by 66 for every increase or decrease by 1.

Negative diagonal movement on the grid:

 36 140 26 27 16 17 18
 45 194 35 36 25 26 27
 54 248 44 45 34 35 36
 63 302 53 54 43 44 45

302-248=54

248-194=54

194-140=54

The negative diagonal difference when translating the shape in a “y=-x” manner is 54. So for every movement up 1 on the grid in that direction you can add 54 to the total.

After working out the vertical, diagonal and horizontal difference in the entire 3-step stair shapes I found that there was a common difference with all the stair shapes and that the difference found related to the position of the stair shape on the grid.

Part 2

Investigate further the relationship between the stair totals and other step stairs on other number grids.

To further my investigation I decided to observe the relationship between larger stair shapes and their position on the grid, also their relationship with each other.

2-step stair

Using the same process as the 3-step stair I began to look at the horizontal, vertical and diagonal differences. I first started with looking at the horizontal difference

 35 86 25 26 36 89 26 27 34 83 24 25

All the last numbers of the individual stair totals all have a multiple of three at the end.

89-86=3

86-83=3

The total difference overall is three and the multiple of the last number is also three. The horizontal difference is equalled to three.

Vertical difference:

 45 116 35 36 35 86 25 26 25 56 15 16

Conclusion

n>3

14

15

 35 86 25 26
 46 119 36 37

The numbers 3, 6 and 9 can be seen; they all are multiples of three. The total overall difference for the diagonal 2-step stair:

119-86=33

86-53=33

As can be seen the number 33 is also a multiple of 3.

Negative diagonal 2-step stair:

 26 59 16 17
 35 86 25 26
 44 113 34 35

As can be seen the numbers shown at the end are 3, 6 and 9, they are all multiples of three.

Working out the common difference:

113-86=27

86-59=27

Twenty-seven is seen to be the common difference tis however is also a multiple of three.

4 step-stair

Horizontal difference:

 54 350 44 45 34 35 36 24 25 26 27 55 360 45 46 35 36 37 25 26 27 28
 56 370 46 47 36 37 38 26 27 28 29

They all are multiples of 10, so already you can see a common difference.

Working out the overall difference:

370-360=10

360-350=10

Vertical difference:

 55 360 45 46 35 36 37 25 26 27 28
 45 260 35 36 25 26 27 15 16 17 18 35 160 25 26 15 16 17 5 6 7 8

60 is a common number in this, the total difference worked out:

360-260=100

260-160=100

There is a difference of 100 for each progressional movement vertically up or down on the 4-step star grid.

Positive diagonal movement:

 55 360 45 46 35 36 37 25 26 27 28
 44 250 34 35 24 25 26 14 15 16 17
 33 140 23 24 13 14 15 3 4 5 6

360-250=110

250-140=110

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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