Post, T', Behr, M',

1
Post, T., Behr, M., Lesh, R. (1982, April).
Interpretations of Rational Number Concepts. In
L. Silvey J. Smart (Eds.), Mathematics for
Grades 5-9, 1982 NCTM Yearbook (pp. 59-72).
Reston, Virginia NCTM.

• Rational number concepts are among the most
  important concepts children will experience
  during their pre-secondary years.
• From a practical perspective the ability to deal
  effectively with rational numbers vastly improves
  one's ability to understand and deal with
  situations and problems in the real world.
• From a psychological perspective an
  understanding of rational number provides a rich
  ground from which children can develop and expand
  the mental structures necessary for continued
  intellectual development.
• From a mathematical point of view rational
  number understandings are the foundation on which
  basic algebraic operations will later be based.
• Students have consistently experienced
  significant difficulty dealing with and applying
  these concepts. Perhaps one reason is that for
  the most part school programs tend to emphasize
  procedural skills and computational aspects
  rather than the development of important
  foundational understanding.

2
Fractions
George Anderberg  Mathematics Consultant Ph
0421151043 Fax [phone] Web
www.mathcon.com.au
3
(No Transcript)
4
AGENDA

• Fraction Constructs
• Fractions as a Quotient
• Fractions as a Rate or Ratio
• Activities to support Learning about Fractions

5
5 Fraction Constructs(Kieren, T. 1976)

• Fractions as
• Part/Whole,
• Measures,
• Quotients,
• Operators
• Rates Ratios

6
Part-Whole

• This is where students construct a fraction as
  part of a whole. For example to describe ¾
  students take a whole object, partition it into 4
  equal (hopefully) parts and then select 3 of
  those parts.
• There is less emphasis within part-whole on
  flexibility, eg. If this is ¾ what does ¼ look
  like and if this is ²/3 what would 1 whole look
  like.
• Students need to have access to a variety of
  wholes

7
Fractions as a measure

• Fractions as a number that can be placed on a
  number line in its appropriate position with
  whole numbers and decimals etc.

8
Fractions as an operator

• A fraction is an operator when it enlarges or
  reduces the size of something.
• eg determining of 28 metres where the
  fraction is
• operating on the 28.
• Dividing 4 by where 4 represents of a
  share
• Then a whole share would be 6, (dividing 4 by
  3 implies that 4 is 3 shares).
• Students need to be aware that the numerator and
  denominator are related through multiplication
  and division, not addition.

9
Fractions as a quotient

• The notion of a fraction as a division or a
  quotient is not a common construct in peoples
  minds (Clarke D. 2006).
• Yet if we look at the division symbol, , we see
  that it represents a fraction.
• is in fact 3 8.

Numerator Denominator
10
Fractions as a quotient

• Chocolate Activity APMC Vol 11 Number 3 2006.
  aamt

11
Fractions as a Rate or Ratio

• As a rate when something increases or decreases
  by a fixed amount. Eg radioactive decay, half
  life etc. Water is escaping from a tank at a rate
  of 3 litres per second.
• Ratios Linking quantities eg 5 cups of flour to
  2 cups of sugar.

12
Fractions as a Rate

• M Ms activity (also useful as fractions of a
  collection)
• Fibonacci Ratios

13
Fractions as a Ratio

• In order to compare ratios students need an
  understanding of equivalence.
• One way to demonstrate this is with an equivalent
  fraction chart, another is by using a
  multiplication grid.

Equivalent fractions
14

• Multiplication grids/equivalence table

15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
How many ratios?
How can you compare them?
20
Ratios The magic rectangle
Fibonacci 1,1,2,3,5,8,13,21,34, 55, 89, 144,
233, 377, 610, Find some ratios in the
pentagram that fit with the ratio of any two
consecutive Fibonacci Numbers. Try comparing
body measurements eg the ratio of head width to
head length should be approximately equal to the
ratio of two consecutive Fibonacci numbers How
can you compare them?
21
Putting them all together

• The ratio of land to water on Planet Earth
• What percentage of the Earth is water
• What percentage is land
• How else can we use this activity

22
Calculators
23

• Calculator Practice with the TI 15 Explorer.

24
WHERE TO NOW

• Planning for the future

25
2010 Locations in Sydney Metro, Hunter and the
Mid North Coast
These will run T1 and T2 2010
26
Mathematicians have all the fun
27
George AnderbergThe Mathematics Connection

• Thank you for your attendance
• Please complete the evaluation form

www.mathcon.com.au
28
(No Transcript)
29

• Understanding of what a fraction is
• Fractions relative to each other
• Equivalent fractions
• Fraction number lines
• Small strips
• Big Board