• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
• Level: GCSE
• Subject: Maths
• Word count: 2237

# Number Stairs Coursework

Extracts from this document...

Introduction

Number Stairs Coursework

Mathematics

Number stairs coursework

Aim:

I have been given a 10 by 10 number grid (as shown below) with a stair shape drawn on it. The stair shape is a 3–step stair and the total of the numbers inside it is 194.

 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

I am required to carry out the followings:

• Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid for other 3-step stairs.
• Part 2: To investigate further the relationship between the stair totals and other step stairs on other number grids.

Plan for Part 1

I shall now investigate part 1 where I will show and explain the relationship between the stair total and the position of the stair shape on the grid for other 3- step stairs. The stair shape on the diagram is a 3 step stair because both the length and the width of the stair are 3 steps. Part 1 is basically all about drawing patterns and conclusions from the relationships, and constructing formulas based on them. To do this, I will change the position of the 3-step stair from the bottom left hand corner (see blue on diagram) from positions 1-5 (see green), vertically across the grid and calculate the total of the numbers inside it.

Middle

Total = an + b                           Therefore a= 6

6+b = 50

b = 50-6

b = 44

Therefore, 6n+44 can be used as a formula for every position possible on the number grid.

Testing my Predictions

To prove that my formula works, I can test my predictions by firstly manually calculating the total of the 3-step stair, and then algebraically calculating the total using my formula finally I will randomly select three positions to test. If my formula works I should have the same total for both of the stairs;

• Position 58 = 58 + 59 + 60 + 68 + 69 + 78 = 392

By using my formula (6n + 44), I should get the same total, thus if n = 58,

6 x 58 + 44 = 392

• Position 16 = 16 + 17 + 18 + 26 + 27+ 36 = 140

By using my formula (6n + 44), I should get the same total, thus if n = 16,

6 x 16 + 44 = 140

• Position 32 = 32+ 33 + 34 +  + 27+ 36 = 140

By using my formula (6n + 44), I should get the same total, thus if n = 16,

6 x 16 + 44 = 140

Plan for Part 2

For part 2 of this task I shall investigate further the relationship between the stair totals and other step stairs on other number grids. To do this, I will change the size of the number stair (from a 1-step stair to a 5-step stair) and calculate the total of the algebraic letters inside it. This can indicate what the relationship between the totals of the number stairs are compared to their sizes.

Conclusion

Extending the Investigation

There are a number of possible ways to extend this investigation. For example, it is possible to determine formulas for any step stair on say an 8 by 8 grid and then comparing them with those from a 10 by 10 grid. This can be further extended to finding an overall formula for any step stair on say an 8 by 8 number grid as shown below:

I shall first investigate what the formulas are for other step stairs in order to find an overall formula for any step stair possible on an 8 by 8 grid. I then compare this to other step stairs.

1-Step stair

 n

Total: n

Formula: 1n

2-Step stair

 n+9 n n +1

Total: n + n + 1 + n + 9

Formula: 3n + 10

3-Step stair

 n + 17 n +9 n + 10 n n +1 n + 2

Total: n + n + 1+ n + 2 + n + 9 + n + 10

n + 17

Formula: 6n + 39

4-Step stair

 n + 25 n +17 n +18 n +9 n + 10 n + 11 n n +1 n + 2 n + 3

Total: n + n + 1+ n + 2+ n + 3 + n + 9 + n + 10 + n + 11+ n + 17 + n + 18 + n + 25

Formula: 10n + 96

5-Step stair

 n +33 n +25 n+ 26 n +17 n +18 n + 19 n +9 n + 10 n + 11 n + 12 n n +1 n + 2 n + 3 n + 4

Total: n + n + 1+ n + 2+ n + 3 + n + 4 + n + 9 + n + 10+ n + 11+ n + 12 + n + 17 + n + 18+ n + 19 + n + 25 + n + 26 + n + 33

Formula: 15n + 190

By looking at the number stairs I can see another pattern similar to the numbers on a 10 by 10 grid.

10 by 10 grid        8 by 8 grid

 Step Stair (s) Formulas 1 n 2 3n  + 11 3 6n  +  44 4 10n + 110 5 15n + 220 Step Stair (s) Formulas 1 n 2 3n  + 10 3 6n  +  39 4 10n + 96 5 15n + 190

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. ## This is a 3-step stair. The total of the numbers inside the step square ...

So we put it as. N 6 But as we put a number into the formula we notice that the number still isn't right 3 6=4.6 4 6=10.67 5 6=20.83 As we look at the results we can tell that there is a point something after the number we want.

2. ## Number Stairs

I went on to see if the 3rd difference for the y values on a different sized grid was the same as the number they were all multiples of. I checked if the 3rd difference for the y values on a 9�9 grid was 10 as the y values for

1. ## Number stairs

3-step stair I can start to establish if there is a pattern I need to find a pattern so that I can find an algebra formula to represent this pattern and use the formula for the 12 by 12 Number Grid By looking at the 3 step stair diagram we

2. ## Number Stairs.

Now that I have found these equations I can see three sets of sequences forming that will help to form the final equation for any size stair on any size grid. It is clear that these three sets of sequences are the co-efficient in front of x sequence, the coefficient in front of g sequence and the constant sequence.

1. ## Number Grids Investigation Coursework

2 3 4 5 6 11 12 13 14 15 20 21 22 23 24 29 30 31 32 33 38 39 40 41 42 (top right x bottom left) - (top left x bottom right) = 6 x 38 - 2 x 42 = 228 - 84 = 144 So my formula works.

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

- 44 = 226 45 46 47 2: 21+33+34+45+46+47=226 53 FORMULA = 6x - 44 65 66 1: (6 x 77) - 44 = 418 77 78 79 2: 53+65+66+77+78+79=418 From the above 6 examples, it clearly indicates that the algebra equation works and the theory is accurate.

1. ## Number Stairs

This will lead to the fact that the nth term has to have 6n in the formula, the extra is calculated by working out what is left over this will be +36.Therefore, the formula has to be T =6n + 36.

2. ## GCSE Maths Sequences Coursework

of a pattern in the 1st difference, but when I calculate the 2nd difference I can see that it goes up in 3's, therefore this is a quadratic sequence and has an Nth term. The second difference is 3 therefore the coefficient of N� must be half of 3 i.e.

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