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• Level: GCSE
• Subject: Maths
• Word count: 3089

To find any relationships and patterns in the total of all the numbers in 'Number Stairs', that might occur if this stair was in a different place on the grid.

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Introduction

Number Stairs Course Work

Aim:

To find any relationships and patterns in the total of all the numbers in ‘Number Stairs’, that might occur if this stair was in a different place on the grid.

The 10 by 10 grid below is where I am to get all the information I need from to enable me to carry out this investigation.

 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

Part One

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

 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

By looking at this grid, the stair total is:

25 + (25+1) + (25+2) + (25+3) + (25+10) + (25+11) + (25+20)

If we call 25, the number in the corner of the stair, ‘x’ then we get:

X + (x + 1) + (x + 2) + (x + 3) + (x + 10) + (x + 11) + (x + 20) =

6x + 1 + 2 + 3 + 10 + 11 + 20 =

6x + 44

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

The stair total for this three-step stair is 194

This formula should work with every number stair that can fit onto a 10 by 10 grid. To support my judgement I will repeat the formula another two times on the same three-stair grid, but in a different position to the previous three stairs.

 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

By looking at this grid, the stair total is:

57 + (57+1) + (57+2) + (57+3) + (57+10) + (57+11) + (57+20)

If we call 57, the number in the corner of the stair, ‘x’ then we get:

X + (x + 1) + (x + 2) + (x + 3) + (x + 10) + (x + 11)

Middle

5

6

7

8

6x

6        56        62        68        74        80        86

12

18

24

30

36

42

48

6x doesn’t give me the correct answer so I add 44 to 6x and get the Stair Total:

 Stairs number (x) 1 2 3 4 5 6 7 8 Stairs total (tx) 50 56 62 68 74 80 86 92 6x 6 12 18 24 30 36 42 48 +44 44 44 44 44 44 44 44 44 6x+44 50 56 62 68 74 80 86 92

Testing the Formula

To ensure the formula works in every case, I tested it in five situations where

‘x’ was different each time.

Using the formula 6x + 44

If n = 1 then

·        6 x 1 + 44 = 6 + 44 = 50

If x = 12 then

·        6 x 12 + 44 = 72 + 44 = 116

If x = 23 then

·        6 x 23 + 44 = 138 + 44 = 182

If x = 34 then

·        6 x 45 + 44 = 270 + 44 = 314

All the results using the formula are correct, so I can come to the conclusion that the formula for the total of a stair:

·        On a 10 by 10 grid

·        Which travels downwards from left to right

·        With a height of 3 squares

Is: 6n + 44

In every case of a three step, the difference in position has a different total of 6.

Part Two (Extension)

Investigate further the relationship between the stair total and other step

stairs on the number grids.

For past of this investigation I am going to find out the totals for each step, starting from 2 step, to a 10step. As I found a formula for part one, I will continue to use that, in this part of this investigation, to see if there is any initial relationship.  To see if there are any relationships between the number of stairs and there total, I will then put the results into a table, to make it clearer.

Two and Three Step

 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

Two Step

‘x’ =43

X + (x + 1) + (x + 10) =

3x+11

=43+44+53=140

The stairs total for this two-step stair is 140

Three Step

‘X’=17

X + (X + 1) +  (X + 2) (X + 3) (X + 10) (X + 11) + (X + 20) =

6x+44

=17+18+19+27+28+37 = 146

The stairs total for this two-step stair is 146

Four and Five Step

 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

Four Step

‘X’=33

X + (X + 1) +  (X + 2) (X + 3) (X + 10) (X + 11) + (X + 12) + (X + 20) +        (X + 21) + (X + 30) =

10x+110

=33+34+35+36+43+44+45+53+54+63= 440

The stairs total for this two-step stair is 440

Step Five

‘X’=56

X + (X + 1) +  (X + 2) (X + 3) (X + 4) (X + 10) + (X + 11) + (X + 12) + (X + 20) + (X + 21) + (X + 22) + (X + 30) + (X + 31) + (X + 40) =

15x+220

=56+57+58+59+60+66+67+68+69+76+77+78+86+87+96= 1060

The stairs total for this two-step stair is 1060

Six Step

 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

Conclusion

=1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+21+22+23+24+25+26+27+28+31+32+33+34+35+36+36+37+41+42+43+44+45+46+51+52+53+54+55+61+62+63+64+71+72+73+81+82+91=1906

The stairs total for this two-step stair is 1906

Having found out the stairs totals and the formula’s for the steps of 2-10, I am then going to put my findings into a table, as I am not able to see any instant relationship between the group of results as they are scattered all about.

 Stairs Number Formulae Stairs Total 2 3x+11 140 3 6x+44 146 4 10x+110 440 5 15x+220 1060 6 21x+385 1330 7 28x+616 952 8 36x+924 1752 9 45x+1320 1814 10 55x+1815 1906

Having interpreted the results of the formulas and stairs totals into a table, I am now able to see a relationship between the formulas for working out each step. This being that the formulae’s for each step are triangular numbers.

The first 10 triangle numbers are: 3,6,10,15,21,28,36,45,55

As you can see, they all are in the formulas.

Starting from the basic two stairs, the formula (relationship) is the first triangular number, and continues through out the table, and beyond ten stairs.

As you carry on with the number stairs, for example working out the 11th stair formula, you will notice that it will be the 11th triangular number, this will be for all of the formulas.

The table below states this:

 Stairs Number Formulae Stairs Total 2 3x+11 140 3 6x+44 146 4 10x+110 440 5 15x+220 1060 6 21x+385 1330 7 28x+616 952 8 36x+924 1752 9 45x+1320 1814 10 55x+1815 1906 11 66x+{} 12 78x+{} 13 91x+{} 14 105x+{} 15 120x+{} 16 136x+{} 17 153x+{} 18 171x+{} 19 190x+{} 20 210x+{}

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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