# Find relationships between the stair total and the position of the stair shape on the grid for a three step stair.

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Introduction

Introduction

In this investigation I am going to see if I can find relationships between the stair total and the position of the stair shape on the grid for a three step stair. I will then try to work out the relationship between the stair total and the position of the stair shape on the grid for 2 step stairs to 6 step stairs. Furthermore I will try to find a formula for working out any stair size which is placed anywhere on a 10X10 grid.

For this investigation I will need:

- Calculator
- Pencil
- Pen
- Grid Tables

To carry out the investigation I will need to use algebra and substitution. If I succeed in this part of the investigation I will then try to extend my investigation further by working out the relationships between stairs and where they are placed on different sized grids. From that I should be able to work out a formula to work out any step stairs placed anywhere on any sized grid.

3 Step Stair on A 10X10 Grid

To work out this part of the investigation I used algebra and gathered the similar terms to find a formula.

21

11 12

1 2 3 I took this stair and added the numbers together. It totaled 50

I then made my key number 1 and let x = 1

Middle

c = -1 5/6

I then substituted a b and c into equation 1

8(1 5/6) + 4(0) + 2(-1 5/6) + d = 11

14 2/3 – 3 2/3 + d = 11

11+ d = 11 (11 cancels out on both sides)

d = 0

So : a = 1 5/6 : b = 0 : c = -1 5/6 : d = 0

I can say that c = -a

Having found this I then took the coefficients of the formulas for the steps and placed them in a table. I noticed they were triangle numbers from Pascal’s triangle.

M >> 3

3

M >> 6 1

4

M >> 10 1

5

M >> 15 1

6

M >> 21

Because there were only 2 steps to find a similarity I knew I needed this equation

ax² + bx +c

I then began to substitute.

x = 2 >> a2² + b2 + c = 3

x = 3 >> a3² + b3 + c = 6

x = 4 >> a4² + b4 + c = 10

Rearranging them and simplifying them I found these equations.

4a + 2b + c = 3 || equation 1

9a + 3b + c = 6 || equation 2

16a + 4b + c = 10 || equation 3

I had to then subtract equation 1 from 3 and equation 2 from 1

12a + 2b = 7 || Equation 4

5a + b = 3 || Equation 5

I then had to multiply equation 5 by 2 to make b = 2b

12a + 2b = 6 || Equation 6

Then subtract

2a = 1

a = ½

I then had to substitute a into equation 5

5(½) +b = 3

2.5 + b = 3

b = ½

2 + 1 + c = 3

c = 0

I gathered the terms and found y = (1/2x² + 1/2x) x Reference Number + c

Gathering c I formed this equation.

y = (1/2x² + 1/2x) x Reference Number + (1 5/6x³ - 1 5/6 x)

Conclusion

(1/2x² + 1/2x) x Reference Number + (G+1 x³ – G+1 n)

6 6

Where x is step size and G is grid size

Having found this I then decided to test it on a 9 x 9 grid with 2 step stair.

Using my formula I calculated :

(1/2(2²) + ½(2) 1 + (9+1 (2³) – 9 +1 (2)

6 6

(2 + 1) 1 + ( (10 x 8) – 10 x 2) )

6 6

(3) 1 + (13 1/3 – 3 1/3)

3 + (10) = 13

To check that this is correct

10

1 2

10 + 1 + 2 = 13

This proves that all my formulas are correct.

Conclusion

In conclusion I have answered all the questions I needed to find out in my investigation and I was able to successfully extend the problem further and work out a general formula for any stair size placed anywhere on any sized grid.

I was also able to use my algebraic knowledge to carry out the investigation as well as substitution.

I could also extend the investigation further by working out a formula for stair sizes on 3 dimensional grids.

My results were justified by solving other problems using them and checking them by working it out the longer way.

Contents

Page 1: Introduction

Page 2: 3 step stair on a 10 x 10 grid and other step stairs on a 10 x 10 grid

Page 3: Other step stairs on a 10 x 10 Grid

Page 4: To Work Out General Formula for a 10x10 Grid

Page 5: To Work Out General Formula for a 10x10 Grid

Page 6: To Work Out General Formula for a 10x10 Grid

Page 7: To Work Out General Formula for a 10x10 Grid

Page 8: 8 x 8 Grid

Page 9: 8 x 8 Grid

Page 10: General Formula for an 8 x 8 Grid

Page 11: General Formula for an 8 x 8 Grid and Ultimate Formula

Page 12: Ultimate Formula

Page 13: Conclusion

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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