STAIRCASE NUMBERS

1
STAIRCASE NUMBERS

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

Staircase number is 3
Staircase number is 7
INITIAL FEELINGS I did not know what to
investigate! I did not understand the concept at
all and ended up reading the question several
times. After several readings of the question and
consulting with my partner I began to understand
what was being asked.
2
STEPS I TOOK

• I firstly had to read the question several times
  so that I could understand what we had to
  investigate. Included in the kit was the
  investigation question, blocks which could be
  fitted together and toothpicks. We quickly
  identified that the toothpicks were distracters.
• Next we used the blocks contained in the kit to
  further understand what we had to do and to see
  what happens to the staircase number when we
  increased the height of the staircase.
• We discovered a pattern which showed that the
  staircase number went up by the number of steps.
  Eg, for 2 steps the staircase number goes up by
  2s. The staircase numbers would be 3, 5, 7, 9,
  11, 13 etc. For 3 steps the staircase number goes
  up by 3s. 6, 9, 12, 15, 18 etc. And so on
• This can be used to determine what a staircase
  number will be without having to physically build
  the staircase with blocks. For each number of
  steps you add the same number of blocks onto the
  previous staircase.

3
Steps continued

• Next we put our findings into spreadsheet and
  this was the result.

By using the spreadsheet we were able to see
other patterns which emerged from the
investigation. These patterns are shown on the
next slide.
4
Patterns from the spreadsheet

• When looking down each column it can be seen that
  the staircase numbers go down consecutively as
  one more step is added. Consecutive numbers are
  when numbers go in order. Eg, 1, 2, 3, 4 etc.
  Three rows only have been included.

5
6
7
8
5
Patterns from the Spreadsheet

• We also noted that there was a pattern going down
  the rows of staircase numbers. It was odd,
  even/odd, even, odd/even then it repeated. It can
  be seen in the following spreadsheet where odd is
  yellow, even/odd is green, even is blue and
  odd/even is red.

6
Other patterns discovered

• We also discovered some other patterns when we
  were colouring in the staircase numbers of 2
  steps, 3 steps and 4 steps on a numbered grid.
  They can be seen below. They start on the
  smallest staircase number for that amount of
  steps.

Pattern for 3 Steps
Pattern for 2 Steps
Pattern for 4 Steps
7
Changing the width of the stairs

• We looked at what would happen if we altered the
  width of the staircase instead of the normal one
  cube thick. We discovered that the number of
  times you extended the width that was how many
  times you multiplied the staircase number by. For
  example, 2 steps has the staircase number of 3,
  1 cube wide 3 cubes. 2 steps with the staircase
  number of 3, 2 cubes wide 6 cubes. 2 steps with
  the staircase number of 3, 3 cubes 9 cubes. We
  discovered that you multiply the staircase number
  by the width (cubes) of the staircase and you can
  establish the new staircase number. Here is an
  example below showing the different staircase
  numbers for different cube widths.

8
Our Cheat Sheet

• When we were handed our cheat or hint sheet for
  our investigation we thought it would give us
  help and show us what we needed to know.
• It stated we needed to find staircase numbers
  and non-staircase numbers and find a recipe for
  writing a number as a sum of consecutive numbers.
  
• In all honesty I thought that the cheat sheet
  confused me more than I already was about the
  investigation.
• We were able to discover non-staircase numbers.
  We started with 2, 4, 8, 16, 32, 64 etc. We then
  realised that each non-staircase number was
  simply doubling the previous number.

9
E-PORTFOLIO

• I would include this PowerPoint in the Curriculum
  and Knowledge component of my E-Portfolio. This
  is because problem solving and investigations are
  part of the Mathematics curriculum.
• The theme its the process that matters not the
  answer is also a key aspect of the Mathematics
  curriculum and I believe that this investigation
  supports this mathematics theme.

10
WHAT I LEARNT

• Whilst investigating Staircase numbers I learnt
  how to look for and use patterns, work
  systematically and record what I was doing. I
  also learnt techniques for explaining what I have
  done to someone else so they can understand.
• I dont know if I solved the investigation or not
  but the things I discovered were
• a pattern which showed that the staircase number
  went up by the number of steps in the staircase.
• When looking down each column on the spreadsheet
  it can be seen that the staircase numbers go down
  consecutively as one more step is added.
• There were certain patterns of odd and even when
  the spreadsheet was shaded in.
• The staircase numbers make specific patterns when
  coloured in on a checkerboard/ numbered grid.
• The number of times you extended the width of the
  staircase that was how many times you multiplied
  the staircase number by.
• Each non-staircase number was simply doubling the
  previous number Eg, 2, 4, 8 etc.