Mathematical Investigation: Staircase Numbers

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Mathematical InvestigationStaircase Numbers

• By
• Lisa Stephens
• Megan Lombo

2
Initial Reaction

• Negative attitude
• - Limited understanding
• - Lack of information given
• Persistence
• - Approached investigation as best we knew how,
  using an array of strategies
• Basic outline of process
• - Tried out formulas
• - Tested formulas
• - Made conjectures
• - Tested conjectures

3
Our topic

• A staircase number is the number of cubes needed
  to make a staircase that has at least two steps
  with each step (other than the first) being one
  cube high. Investigate.

Staircase number is 6
Staircase number is 9
4
But what is a staircase number?

• To our knowledge, a staircase number is one that
  can be expressed as a sum of consecutive numbers.
  For example, 12 is a staircase number because
  34512. When staircases are drawn, it is clear
  to see that we must add consecutive numbers to
  find the staircase number.

See here, 1236
5
Our aim for the investigation- Use a combination
of the following strategies to explain what we
did and how we did it.
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The Simultaneous Approach

• The first formula that we came up with was
• (l x h) 1
• Where l no. of cubes in length of block
• h no. of cubes in height of block

Testing conjectures
Analysing conjectures
7
Testing conjectures
4
1
5
2
4
2
2
1
3
6
5
3
3
1
(l x h) 1 (2 x 3) 1 6 1 5
(l x h) 1 (2 x 2) 1 4 1 3
(l x h) 1 (3 x 3) 1 9 1 8 6
Back
8
Analysing conjectures

• But it doesnt work for more than two steps!
• Maybe we need a new approach
• How about we try a different formula for each
  number of steps!

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Lisas Formula

• Lisa devised the following formula
• 2 steps (l x h) 1
• 3 steps (l x h) 3
• 4 steps (l x h) 6
• Number deducted from block increases by 3.

Testing conjectures
Analysing conjectures
10
Testing Lisas Formula
7
10
1
4
8
11
6
2
4
9
5
2
3
7
12
3
5
6
1
3
6
10
13
8
4
1
3 steps (l x h) 3 (3 x 3) 3 9 3 6
14
9
5
2
4 steps (l x h) 6 (4 x 4) 6 16 6 10
4 steps (l x h) 6 (4 x 5) 6 20 6 14
However
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However

• Further testing of this formula indicated that,
  in fact, the numbers were not increasing by 3 at
  all!

1
5 steps (l x h) 9 (5 x 5) 9 25 9 16
15
Back to the drawing board!
6
2
10
7
3
13
11
8
4
14
15
12
9
5
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Analysing Lisas Formula

• Upon further evaluation of Lisas formula, we
  continued to find new formulas for each step
  staircases in an attempt to find a regular
  pattern in the number of cubes that needed to be
  extracted from the block to form the staircase
  number.
• And then

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HINT!

• TRIANGULAR NUMBERS

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Megans Formula

• This can be shown by the formula
• (l x h) (l 1)
• Where (l 1) becomes the nth term of the
  triangular number sequence.
• Number deducted from the block forms the
  triangular sequence with each increasing step.

Analysing conjectures
Testing conjectures
15
Testing Megans Formula
7
1
10
4
8
2
5
11
6
9
5
2
3
6
8
3
7
12
1
3
6
10
9
7
4
13
8
4
1
(l x h) (l 1) (3 x 4) (3 1) 12 2nd
triangular no. 12 3 9
14
9
5
2
(l x h) (l 1) (4 x 4) (4 1) 16 3rd
triangular no. 16 6 10
(l x h) (l 1) (4 x 5) (4 1) 20 3rd
triangular no. 20 6 14
Back
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Analysing Megans Formula

• For larger staircases, finding the triangular
  number to subtract from the block may become too
  difficult.
• To overcome this problem, we needed to find a
  formula that would enable us to easily work out
  triangular numbers.

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Sum of an arithmetic sequence

• Triangular numbers when placed in sequence form
  an arithmetic progression.
• Therefore, to find the sum of triangular numbers,
  we can use the formula
• Sn n/2 (a l)
• (l 1) in Megans formula is equivalent to the n
  in the above formula.
• a is the first term in the series (1 in these
  cases), and l is the number of terms you are
  looking for.

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Final testing- 2 step formula
Firstly, find triangular no.- Sn n/2 (a l)
5/2 (1 5 ) 2.5 (6) 15 Secondly,
substitute this into Megans formula- (l x h)
15 (6 x 6) 15 36 15 21
5 is found by using (l 1) in Megans formula,
as mentioned on the previous slide
1
2
7
3
8
12
4
9
13
16
Where (l x h) is equivalent to Sn.
5
10
14
17
19
6
11
21
20
18
15
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More testing
Firstly, find triangular no.- Sn n/2 (a l)
3/2 (1 3) 1.5 (4) 6 Secondly,
substitute this into Megans formula- (l x h)
6 (4 x 5) 6 20 6 14
10
11
6
3
7
12
13
8
4
1
14
9
5
2
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Conclusion

• This is where we chose to end our investigation
  of staircase numbers.
• If time permitted we would have further
  investigated other aspects, such as whether or
  not you can work out if any given number is a
  staircase number by using a set formula.

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E-portfolio

• This investigation PowerPoint presentation can
  be viewed in the Curriculum Knowledge section
  of both of our E-portfolios.