Transcript Staircases - General Education @ Gymea

Staircases
A staircase number is the number of
cubes needed to make a staircase
which has at least two steps with each
step being one cube high.
INVESTIGATE!
Constraints and Ideas
Constraints:
 Each step being one cube high
 A staircase consists of at least 2 steps.
Ideas:
 Staircases with steps ranging from 2 steps,
altering the number of steps and the number
of cubes in the first step.
 3-D staircases
Staircases with 2 steps
Using blocks, constructed
staircase containing two
steps.
With a variable of the
number of cubes present
in the first step.
no. of cubes is
1st step
1
2
3
4
5
6
…
Total no. of
cubes
3
5
7
9
11
13
…
Staircases with 3 steps
The same process
was used though this
time with three steps
making up the
staircase.
no. of cubes is
1st step
1
2
3
4
5
6
…
Total no. of
cubes
6
9
12
15
18
21
…
Staircases with More than 3
steps
The same process was used up until there being 6
steps in the staircase and a clear pattern was
beginning to form.
No. of step in the staircase
2
3
4
5
6
1
3
6
10
15
21
2
5
9
14
20
27
3
7
12
18
25
33
4
9
15
22
30
39
5
11
18
26
35
45
6
13
21
30
40
51
7
15
24
34
45
57
No. of cubes in first step
Patterns
I was able to identify that each set of staircases (i.e. those with the
same number of steps) presented staircase numbers that formed an
arithmetic series.
An arithmetic series is when there is a common
difference between each number in the series.
For example, the series representing staircases of 3 steps;
6, 9, 12, 15, 18, 21, 24 ….
There is a common difference of 3 between each of the terms.
The same was noticed with the series of numbers representing
staircases of 6 steps;
21, 27, 33,39, 45, 51, 57 …
Where the common difference is six.
Patterns cont.
Used to find term in a
series once the first
term is known.
Arithmetic Series formulas:
1) Tn = a + (n – 1)d
2) Sn = n/2(a + Tn) = n/2[2a + (n – 1)d]
Where a = the first term in the series
d = the common difference
T = term
Used to find the first term in
series involving a large number
of steps, for example 15 steps.
n = number of term within the series
S = sum
In this instance this term stands for the
number of cubes in the first step on the
staircase
Using Arithmetic Series
Formulas.
For example:
Find the number of cubes required to form a staircase that contains
100 steps, with the first step being made up of 100 cubes.
Sn = n/2[2a + (n – 1)d]
Where a =1 , n = 100, and d = 1
Therefore,
S100 = 50[2+99]
S100 = 5050
Therefore if there are 100 steps in a staircase and the first step is made up
of 1 cube there are a total of 5050 cubes in the stair case.
Tn = a + (n – 1)d
Where a = 5050, n = 100, and d = 100
Therefore
T100 = 5050 + 99(100)
T100 = 14950
Therefore a staircase of 100 steps, with the first step containing 100 cubes,
contains a total of 14950 cubes.
3-D staircase
Do 3-D staircases present a different pattern?
Restricted to the area formed by a cube so that staircase are regular and
consistent in shape.
Therefore the number of base cubes in a 3-D staircase are the squares of odd
numbers.
2
2
2
1
2
3
2
1
2
2
2
1
1
1
1
2
1
1
Total no. of cubes = 10
Total no. of cubes = 19
This process was continued and for the 3 by 3 square the values of 10, 19, 28,
37, 46, 55, etc were calculated.
3-D Staircases cont.
The same process was used for 5 by 5 squares, 7 by 7
squares and 9 by 9 squares.
No. of step in the staircase
3x3
5x5
7x7
9x9
1
10
35
84
165
2
19
60
133
246
3
28
85
182
327
4
37
110
231
408
5
46
135
280
489
6
55
160
329
570
7
64
185
378
651
No. of cubes in first set of steps
3-D staircase patterns
An arithmetic series is formed for each sequence of
calculations, as there is a common difference. This
common difference is relative to the size of the square
base. i.e. 9 for 3 x 3, 25 for 5 x 5, 49 for 7 x 7, etc.
As set of arithmetic series, a value in the series can be
calculated using:
Sn = n/2(a + Tn) = n/2[2a + (n – 1)d].
Though, as the 3-D staircase is not linear the first
value cannot be calculated using the formula:
Tn = a + (n – 1)d.
Conclusion
Staircase numbers are numbers that can be arranged
in a number of arithmetic series, in which the
staircases contain the number of steps.
Though this is using the constraints stated at the
beginning of the investigation process.
Though I am sure with more time and persistence a
number of ideas, involving staircase numbers could
be investigated.