Number Stairs

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Number Stairs

Mathematics GCSE Coursework

Part 1

Introduction

I am going to investigate relationships between 3 step stairs on a ten by ten grid. I also intend to find algebraic expressions to link the 3 step stairs and there positions and totals.

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

The total of the 3 the step stair is the sum of all the numbers included in it.

Excluded

The numbers in the grid with a lighter colour do not have totals because a 3 step stair can not be formed with them.

Expressions

I am going to use these letters to show certain numbers:

N= bottom left hand number on the step stair, the position (the yellow number above)

T= total of the step stair

T(t-1)= total of step stair before

G= the grid size

R= horizontal row total

R(r-1)= total of horizontal row before

C= vertical column total

C(c-1)= total of column before

 

The Task

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

Data Collection

I am going to display some totals and positions in the grid.

The value before the equals sign is the bottom left hand position in the step stair, also known as N.

1=50, 2=56, 3=62, 4=68, 5=74, 6=80, 7=86 and 8=92. I can’t do 9 or 10 because there are not enough stairs to complete the step stair grid.

I noticed that the difference between all the totals is 6:

1=50

        6

2=56

        6

3=62

        6        

4=68

        6

5=74

        6

6=80        

        6

7=86

        6

8=92

        

I am now going to investigate the second row,

11=110, 12=116, 13=122, 14=128, 15=134, 16=140, 17=146, 18=152. These also have the same difference between the numbers i.e. 6.

This would mean the formula T=T(t-1)+6 would work for the second row as well as the top row, apart from N=19 and 20.

This shows on the first two rows the difference between the totals is 6.

I am now going to investigate the third row,

21=170, 22=176, 23=182, 24=188, 25=194, 26=200, 27=206, 28=212

I am now going to investigate the fourth row,

31=230, 32=236, 33=242, 34=248, 35=254, 36=260, 37=266, 38=272

I am now going to investigate the fifth row,

41=290, 42=296, 43=302, 44=308, 45=314, 46=320, 47=326, 48=332

I am now going to investigate the sixth row,

51=350, 52=356, 53=362, 54=368, 55=374, 56=380, 57=386, 58=392

I am now going to investigate the seventh row,

61=410, 62=416, 63=422, 64=428, 65=434, 66=440, 67=446, 68=452

I am now going to investigate the eighth,

71=470, 72=476, 73=482, 74=488, 75=494, 76=500, 77=506, 78=512

This formula summarises my findings above:

T=T(t-1)+6

I arrived at this formula because I saw that the total (T) of the step stair is the total of the step stair before it, T(T-1), plus 6. This works for the entire bottom row except N=1 because it doesn’t have a total before it, and N=9 and 10, because they don’t have totals. You can also see that the  step stair consists of 6 numbers each of these numbers moves 1 position to the right, e.g. 25 becomes 26 and 26 becomes 27 and 27 becomes 28 and 35 becomes 36 and 36 becomes 37 and 45 becomes 46, this means the step stair before becomes greater by 6 because each of the 6 positions in the step stair move 1 place to the right, 1x6=6 which means the step stair becomes greater by 6.

Join now!

An example of the formula T=T(t-1)+6 working in practice is the calculation for N=54,

T(t-1)=The total for the number N=53 i.e. 362.

T=362+6

T=368 where N=54

I checked my answer was correct by adding up the step stair N=54, 54+55+56+64+65+74=368.

This shows that the formula works for N=54.

To check the formula I am also going to use N=78 as an example,

T(t-1)=The total for the number N=77 i.e. 506

T=506+6

T=512 where N=78

I checked my answer was correct by adding up the step stair N=78,

78+79+80+88+89+98=512

This shows the formula works ...

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