ODD NUMBERS

1
9
1
5
ODD NUMBERS
7
3
By Ellen Lawler and Michelle Ross
2
Choose any odd number
Square it
Divide it by 2
Choose neighbouring whole numbers
What do you find
3
Michelle's Investigation of Odd Numbers

• I came up with the idea that there had to be a
  way to investigate a series of odd numbers. I
  created an excel formular for the investigating
  question on odd numbers.
• Once I constructed the excel spreadsheet I could
  investigate the patterns of odd numbers both
  vertically and horizontally
• When investigating vertically I found numerous
  patterns that led to no findings.

4
Excel Spreadsheet
IF(MOD(C35,1)0,C351,TRUNC(D35,0)1)
IF(MOD(C3,1)0,C3-1,TRUNC(C3,0))
A3A3
B3/2
5
Ellen's Investigation of Odd Numbers

• On Michelles excel spreadsheet I initially noted
• Column B ( square it) was always odd
• Column C( divided by 2) always ended in a decimal
  of .5
• Column D (lower neighbouring number) was always
  even
• Column E (higher neighbouring number) was always
  odd
• Column D and E always had a variance of one

• My second idea was to investigation columns in
  excel looking for patterns running vertically
• Columns C,D and E all increase in a pattern
  which were multiples of four
• e.g. 4,8,12,16,20

6
Our Findings of Odd Numbers

• After sometime we decided to explore other
  areas. This drew us back to the original question
  and we paid special attention to neighbouring
  numbers.
• We discovered that adding column D and E and
  finding the square root of the answer gives you
  the original odd number.
• e.g. 40 41 81 v81 9

7
Unsuccessful Findings of Odd Numbers

• We thought we had discovered the formula to
  always give a result of an odd number
• By adding an even and an odd number ( that have
  the difference of one) and taking the square
  root, you will always end with an odd number
• e.g. x (x 1) v of odd number
• When applying an even and an odd number (that
  have the difference of one) and taking the square
  root the result was not an odd number
• e.g. 6 (61) v133.6055513

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Odd Numbers What If?

• What if you choose any odd number. Square it.
  Divide it by 4. Choose neighbouring whole
  numbers.
• What do you find?
• What we discovered when dividing odd numbers by
  two, doesnt apply when dividing by four

9
10
6
2
EVEN NUMBERS
4
8
10
Choose any even number
Square it
Divide it by 4
Choose neighbouring whole numbers
What do you find
11
Michelle's Investigation of Even Numbers

• I came up with the idea that there had to be a
  way to investigate a series of even numbers. I
  created an excel formula for the investigating
  question on even numbers.
• Once I constructed the excel spreadsheet I could
  investigate patterns both vertically and
  horizontally
• When investigating vertically I found numerous
  patterns that led to no findings.

12
Excel Spreadsheet
IF(MOD(C40,1)0,C40-1,TRUNC(C40,0))
IF(MOD(C40,1)0,C401,TRUNC(D40,0)1)
A40A40
B40/4
13
Ellen's Investigation of Even Numbers

• On Michelles excel spreadsheet I initially noted
• Column B ( square it) was always even
• Column C( divided by 4) were whole numbers
• Column D (lower neighbouring number) had a
  repetitive pattern of even then odd
• Column E (higher neighbouring number) had a
  repetitive pattern of even then odd
• Column D and E always had a variance of two

• My second idea was to investigate the columns
  created in excel looking for pattern running
  vertically
• Columns C,D and E all increased by two and were
  odd
• e.g. 3,5,7,9,11..

14
Our Findings of Even Numbers

• After some time we decided to explore other
  areas. This drew us back to the original question
  and we paid special attention to neighbouring
  numbers
• We discovered that by adding column D and E,
  multiplying the answer by two and taking the
  square root, it will take you back to the
  original even number
• e.g. 2 (80 82) 324 v324 18

15
Unsuccessful Findings of Even Numbers

• We thought we had discovered the formula to
  always give a result of an even number
• By adding any two numbers ( that have the
  difference of two) multiply by two and taking the
  square root, you will always end with an even
  number
• e.g. 2 ( x (x 2)) vof even number
• When applying any two numbers (that have the
  difference of two) multiplying by two and taking
  the square root, the result will always be an
  even number
• e.g. 2 ( 6 (62)) v20 4.472136

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Even Numbers What If?

• What if you choose any even number. Square it.
  Divide it by 2. Choose neighbouring whole numbers
  
• What do you find?
• The finding of dividing even numbers by four
  dont apply to dividing by two