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  • Level: GCSE
  • Subject: Maths
  • Word count: 2421

number stairs

Extracts from this document...

Introduction

GCSE Maths        

Number Stairs

Part 1:For the other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid.

Here we have a 10 by 10 grid. In the grid, there is a shape called the 3-step stair

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The number at the bottom left of the 3-step stair is 25. This is called the Step-number.

The sum of all the numbers in the 3-step stair is 194.

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

This is called the Step-total.

I am going to investigate the relationship between the stair total and the position of the stair shape on the grid using the first step-number on the grid.

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The first 3-step stair is made up of

1 + 2 + 3 + 11 + 12 + 21 = 50

If I move the 3-step stair 1 unit to the right, the 3-step stair would be made up of

2 + 3 + 4 + 12 + 13 + 22 = 56

If I move the 3-step stair another unit to the right, the 3-step stair would be made up of

3 + 4 + 5 + 13 + 14 + 23 = 62

I have created a table of a few results of the 3-step stairs

Numbers in 3-step stair

Step-number

Step-total

1, 2, 3, 11, 12, 21

1

50

2, 3, 4, 12, 13, 22

2

56

3, 4, 5, 13, 14, 23

3

62

4, 5, 6, 14, 15, 24

4

68

5, 6, 7, 15, 16, 25

5

74

6, 7, 8, 16, 17, 26

6

80

From this information, you can see that;

  1. Each 3-step stair can be divided evenly by 2
  2. Each step-total increases by 6 starting from 50
  3. The step-number goes up by 1 and the step-total goes up by 6.
...read more.

Middle

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n + 20

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n + 10

n + 11

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Since all the numbers in the stair-shape adds up to the step-total, if I simplify all the values in the 3-step stair, I should get a formula for the step-total

n + (n + 1) + (n + 2) + (n + 10) + (n + 11) + (n + 20)        =        6n + 44

Therefore, the step-total should equal to 6n + 44

I called step-total t

For stair-step 1, n = 1

t = 6n + 44

  = 6(1) + 44

  = 6 + 44

  = 50

I have proven that the formula works for the first 3-step stair. To verify that this formula works for every other 3-step stair, I will test the first example given, stair-step 25. The step-total should equal 194.

t = 6n + 44

  = 6(25) + 44

  = 150 + 44

  = 194

The formula works for all the step-numbers on the 10 by 10 grid. The relationship between the stair-totals and the position of the stair-shape on the grid is that if you multiply the step-number by 6 and the add 44, you should find the step-total.

Part 2:Investigate further the relationship between the stair totals and other step stairs on other number grids.

I have worked out the formula for the 10 by 10 grid to be t = 6n +44 but this formula only works for a 10 by 10 grid. I will investigate the relationship between the step-total, step-number and the grid size.

In the 10 by 10 grid I noticed that the numbers 1, 11 & 21 etc increases by 10. If you subtracted n from the number above n, you would get the grid size number Therefore, the step-total and the step-number relate with the grid size.

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In this example, n is 25 and the number above n is 35. If I subtract n from 35, I would get 10. This is the grid size number.

The relationship between the step-number and the rest of the numbers are that;

The number 35 has a difference of 10 from the step-number. The number 45 has a different of 20 from the step-number, which is 2 multiplied by 10. The number 26 has a difference of 1 from the step number, which is 1 multiplied by 10, minus 9.The number 27 has a difference of 2 from the step number, which is 1 multiplied by 10, minus 8.

The number 36 has a difference of 11 from the step number, which is 1 multiplied by 10, plus 1

With this information, I will create a formula for each of the numbers in the 3 step-stair shape. The Grid size will be called g.

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n+2g

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= 25

n + ( g – 9)

= 25 + (10 – 9)

= 25 + 1

=26

n + ( g – 8 )

= 25 + (10 – 8)

= 25 + 2

=27

n + g

= 25 + 10

= 35

n + ( g + 1 )

= 25 + (10 + 1)

= 25 + 11

=36

n + 2g

= 25 + (10 x2)

= 25 + 20

=45

...read more.

Conclusion

n+6g – 14

From this information, I have come up with another equation that will work with a grid of any size;

6n + 6g – ( 16 ± 2d )

Where d is the difference of the grid from a 10 by 10 grid

I will test this new formula with the 10 by 10 grid and the 9 by 9 grid.

Step-number = 1

Grid size = 10 by 10

t = 6n + 6g – ( 16 ± 2d )

  = 6(1) + (6 x 10) – [16  ± (2 x 0)]

  = 66 – 16

  = 50

The new equation has worked for the 10 by 10 grid

Step-number = 1

Grid size = 9 by 9

t = 6n + 6g – ( 16 ± 2d )

  = 6(1) + (6 x 9) – [16 – (2 x 1)]

  = 60 – 14

  = 46

The new equation has also worked for the 9 by 9 grid

This formula should even work for a smaller scale, such as a 5 by 5 scale

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The answer we should get is;

1 + 2 + 3 + 6 + 7 + 11 = 30

t = 6n + 6g – ( 16 ± 2d )

  = 6(1) + (6 x 5) – [16 – (2 x 5)]

  = 36 – 6

  = 30

I have proven that my formula works with any grid

Conclusion:

Here I have put all the formula I have come up with. This formula will apply to a grid of any size;

t = 6n + 6g – ( 16 ± 2d )

Where;

t   : step- total

n  : step-number

g  : grid size

d  : difference of grid size from a 10 by 10 grid

In this project I have found out many ways in which to solve the problem I have with the stair-shape being in various positions with different sizes of grids. The way I have made the calculations less difficult is by creating a main formula that works for all the different circumstances.

...read more.

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