• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
• Level: GCSE
• Subject: Maths
• Word count: 2296

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

Extracts from this document...

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.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Number Stairs, Grids and Sequences essays

1. ## Number Grid Aim: The aim of this investigation is to formulate an algebraic equation ...

3 star(s)

= n2 + ng + n + g ? [n2 + ng + n +g] - [n2 + ng + n ] = g I will then replace g by 11 for this exercise: n(n + 11 + 1) = n2 + 11n + n n + 1(n + 11)

2. ## What the 'L' - L shape investigation.

By looking at the results in my table I have noticed that each formula is 5L minus a value. Using the plus and minus rule I have applied this rule to the formula. Therefore, -3g + 3 is equal to 3g - 3.

1. ## Number stairs

a 3-stepped, 4-stepped, or 5-stepped etc or where its position is in a numbered grid, such as 10x10, or 23x23 etc the general formula will always give the total of the grid numbers in that step stair.

2. ## Number Grid Investigation.

(d - 1) (N = 3) ... (D = 6) N - 1 = 2 D - 1 = 5 5 X 2 = 10 10 X 10 = 100. This formula is correct. I will now try it again but in the 3 X 4 calculation.

1. ## Maths-Number Grid

5 � 5 Grid:- (A + 4) (A + 8) (A + 44) (A + 48) Above, I have drawn out a 5 � 5 square grid showing the algebra proving to me that my all my 5 � 5 grids were correct as they all came out with a product difference of 160, which this algebraic method shows.

2. ## For other 3-step stairs, investigate the relationship between the stair total and the position ...

used to calculate the equation the total value of the squares can be found. This theory has been put to the test using a 10x10, 11x11 and 12x12 gird squares. The results are conclusive and consistent, proving the theory to be accurate and reliable.

1. ## Number Grid Investigation

2 x 2 difference to get the 3 x 3 difference so I feel another 30 will be added to get the 4 x 4 difference. 3 4 5 6 13 14 15 16 23 24 25 26 33 34 35 36 3 x 36 = 108 Difference = 90

2. ## Number Stairs

This is for the 8x8 grid with the stair number 1. Stair number= 2 Whereas the stair total= 2+3+4+10+11+18 = 48 = T Stair number=3 Whereas the stair total= 3+4+5+11+12+19= 54 = T The following table shows the stair total (T)

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to