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					                       SNP Workshop 5




                     Fraction Concepts




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                                    Learning intentions




1) To develop teacher understanding of
   the strategies and knowledge required
   in the fraction, ratio and proportion
   domain, in particular, fraction concepts.




2) To enable teachers to plan a lesson
   sequence for fraction concepts based
   on existing student knowledge




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                              Workshop Overview
1)      Reflection Time
        (a)   Where are people up to at the moment?       [individual report back]
        (b)   What issues/problems do people currently have?
        (c)   What has gone well? Any good activities to share?

2)      Learning intentions

3)      Starters
        (a)   Uncover one at a time, answer “question” at bottom of page
        (b)   ½ of 56
               students may treat halving as an operation (like doubling)
               introduce ideas of fraction constructs - halving as the operator
                 construct – start list on right hand side of board
               may wish to integrate this work with / teaching
        (c)   ¼ of 36
               halve and halve again
               relationship between times tables and division and fractional
                 strategies
               Draw a picture to show what ¼ of 36 looks like. {Ask: What is the
                 whole?]
               (36  4 = 9 by sharing out counters. Extending this to asking “what is
                 one quarter of 36”)
               Issue of what happens when a student is eating a block of chocolate
                 with 7 pieces in it.
                 If eat 4 pieces how much is left? – Students saying “3 pieces” are
                 reunitising – changing the unit to the piece rather than maintaining
                 the block as the whole. ie repackaged
                 Getting students to do the problem with sevenths – actually doing 1 –
                 4/7
        (d)   2/3 of 21
               built on the idea that 1/3 of 21 is 7
               students need to be multiplicative thinkers to be able to answer this

4)      (a)      What does two thirds mean?
                  Two out of three equal sized pieces
                   So this is not cut into thirds (though some families may share pizzas
                   like this…)




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                  Does P(X) = 2/3 fit here?… [add this to the list of constructs]
        (b)      draw a picture to show two thirds




                    discuss this as the part-whole construct of fractions (add it to the
                     list). Research suggests that understanding the part-whole construct
                     is essential to understanding fractions.
                     introduce the concept of discrete and continuous fraction models,
                     how they differ and to where they lead [OHT]
                    Which is the bigger fraction?




                   Many miss the comparative nature of fractions – the relationship
                    between the numerator and denominator. (Some research suggests
                    that the failure to understand this is the reason why students have
                    difficulty with fractions)
        (c)      so what do you “understand” when you say you understand what 2/3
                 means?
                  In 2001 42% of Y7 & 8 students could not name the fractions ½, 1/3
                    and ¼
                  Issues of representation – verbal/mental – materials – symbol
                    translations     [diagram on board]        add in written words.
                  handout

5)      Line-ups
        (a)   Fraction activities – fraction line-ups [cards]
              [get teachers to do these as if they are students]
              (i)    ½ to 1/6
                      why are you where you are?
                      Example of the need to tie up the material pieces with the
                        symbol and a language based understanding. (Put pieces on
                        OHT to show and hand to those with the cards)
              (ii)   harder ones
                      why are you where you are?
                      Discuss strategies to sort out which is bigger – equivalent,
                        half, bigger or smaller than half, knowing size of piece, etc
              (iii) what curriculum levels do you think these activities are at?

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                          [level 3 – but split into i and ii]
                        What stages do they relate to?
                          [ stage 5 and 8]
                        Wave the maths exemplar with these questions
        (b)      Sorting fractions from Number Sense 4-6 [handout]
                  close to zero, close to one etc building the idea of coordinating the
                    numerator and denominator

6)      A progression for fractions
        (a)   Outline of a fraction progression         [OHT and handout]

7)      More starter questions… [OHT]
        (a)   Discuss the problems in pairs.
        (b)   Choose volunteers to feed back to the group

8)      Teaching demonstration using fractions materials
         Unifix or multilink cubes
           Chewing gum packs with 5 and 4 pieces
           Sums to one
           4 reds to six yellows as a ratio, as fractions and percentages
         Fraction circles
           Naming fractions and size of the pieces
           Improper fractions and mixed numbers
           Equivalent fractions
         Number stick
           Avalon results – 1/77 naming ¾ and 1¼ correct. Hardest construct for
           students
           Measure construct for fractions (add to list)
           Need to use whole numbers first, and teach conventions of this construct
           Need to get students to draw a number stick and discuss the key
           conventions
           Locating fractions with spaces left, and without marked divisions
           Starter activity is to use a piece of paper and get kids to fold it in half
              [progression of student strategies]
         Fraction dominoes        [handout]

9)      Some    more fraction constructs…
        (a)     quotient construct 2/3 as 2  3
        (b)     ratio construct with 2/3 as the ratio 2:1
        (c)     probability construct – how does this utilise the concept of 2/3?
        (d)     handout of fraction constructs

10)     Looking higher up this domain…

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        (a)      Teaching percentage, ratio, fraction manipulation.
                 [Note that some may decide to do their decimal teaching here too]

11)     Identifying resources
        (a)   Rangi’s square pizzas    [important resource to support idea that
              fractions must have equally sized pieces]
               what curriculum level is the activity at? [2?]
               What numeracy skills are needed for the activity?      [fractions,   ]
               What stage is the activity at?       [3-4]
        (b)   Fraction games – set of cards [handout]

12)     Group planning for fractions       [if time…]   [OHT]

13)     For next time       [OHT]
        Read articles – fractions
                        Beginning to learn fractions




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                                     Resources needed

Beforehand remind people to bring along
 copies of/materials for one good activity used in the past
  to teach fractions, along with any equipment needed for
  the activity and any resources prepared for practice or
  independent groups relating to the activity.
 “Number Framework” and Getting Started Booklets.
 Set of FIO


On site
   Board, pens and duster
   OHP
 Sets of “Figure it out” are needed for the planning session


To Take
   Newsprint, blue-tac and vivids
   Box of numeracy equipment
       Multilink or unifix
       Fraction circles
       Number stick
   Fractions dominoes

Handouts
   Issues of representation
   Sorting fractions (from Number Sense 4-6)
   Progression for fractions
   Fraction constructs
   Set of fraction cards and games to use them




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                                           Starters



1) Half of 56




2) Quarter of 36




3) Two thirds of 21




4) Half an apple



     At which stage do you think it would be reasonable to pose
      each of these problems?

     What underlying knowledge is needed?




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            Discrete and continuous fraction models



Continuous fraction model

Fractions are met as cutting up an entire object

         This single object soon extends to multiple
          objects to be shared
            For example:sharing two apples between 5 people

         Leads to the concept that fractions are
          divisions

Discrete fraction model

Fractions are met through sharing a collection of
objects

         Problems initially lead to discrete, whole
          number answers

         Later problems merge with the continuous as
          “remainder” objects are cut


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                A teaching progression for fractions


                 Discrete                              Continuous
Stage 2                                    Stage 2

Grouping [and sharing]                     Understanding terms like
                                           half an apple, quarter of
                                           an orange

                                           Understand ordinal
                                           numbers (second, third,
                                           fourth…)

                                           Stage 3

                                           Recognise the words for
                                           ordinal numbers

Stage 3/4                                  Stage 3/4

Skip counting [and skip                    Idea that fractions are
sharing and repeated                       of “equal size and shape”
subtraction]                               Recognise symbols for
Doubling and Tens                          ordinal numbers
[Halving and Fives]                        Symbols for halves,
Solves sharing problems                    thirds, quarters, fifths …
involving whole numbers
by dealing out the                         Identifying what fraction
amount                                     is shaded
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Stage 4/5                                  Stage 4/5

Quarters and thirds.    Symbols for all fractions
                        with denom 2, 3, 4, 5, 10
Fraction of a number by [at least] – inc improper
addition                fractions

Solves sharing problems Ordering fractions with
by dealing out the        the same denominator
amount, dividing up left-
overs by dividing and     Order unit fractions
sharing halves then
qaurters…

Stage 5/6                                  Stage 5/6

Fraction of a number by Placing fractions on and
multiplication [simple  reading fractions from
fractions]              number lines

Solves sharing problems Converting improper
by dealing and          fractions and mixed
appropriate division (& numbers
sharing) of remainders

Stage 6/7                                  Stage 6/7

Fraction of a number by Equivalent fractions (&
multiplication [more    simplifying fractions)

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complex fractions]                         [simple denominators]

What fraction? Items                       Fraction – decimal –
with same units)                           percentage conversions
                                           [simple cases]
Fractions as divisions
                                           Stage 7

                                           +/- (same denominator)

                                           

Stage 7/8                                  Stage 7/8

What fraction?                             Ordering fractions with
(involving unit                            different denominators
conversion)
                                           Given fraction
                                           conversions – eg eighths
                                           to decimals and
                                           percentages

                                           Reciprocals

                                           Simple 

                                           +/- (different
                                           denominators)

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NOTE:
For a more detailed explanation of what some of these
statements mean, refer to the fraction questions in the
diagnostic interview and examples in the Number Framework




c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc
                                 A teaching progression for fractions



                     Discrete                                      Continuous
Stage 2                                           Stage 2
Grouping [and sharing]                            Understanding terms like half an apple,
                                                  quarter of an orange

                                                  Understand ordinal numbers (second,
                                                  third, fourth…)

                                                  Stage 3
                                                  Recongnise the words for the ordinal
                                                  numbers

Stage 3/4                                         Stage 3/4
Skip counting [and skip sharing and               Idea that fractions are of “equal size and
repeated subtraction]                             shape”
                                                  Recognise the symbols for th ordinal
Doubling and Tens [Halving and Fives]             numbers
                                                  Symbols for halves, thirds, quarters,
Solves sharing problems involving whole           fifths …
numbers by dealing out the amount
                                                  Identifying what fraction is shaded

Stage 4/5                                         Stage 4/5
Quarters and thirds.                              Symbols for all fractions with denom 2, 3,
                                                  4, 5, 10 [at least] – inc improper fractions
Fraction of a number by addition
                                                  Ordering fractions with the same
Solves sharing problems by dealing out            denominator
the amount, dividing up left-overs by
dividing and sharing halves then                  Order unit fractions
qaurters…

Stage 5/6                                         Stage 5/6
Fraction of a number by multiplication            Placing fractions on and reading fractions
[simple fractions]                                from number lines

Solves sharing problems by dealing and            Converting improper fractions and mixed
appropriate division (& sharing) of               numbers
reaminders

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Stage 6/7                                  Stage 6/7
Fraction of a number by multiplication     Equivalent fractions (& simplifying
[more complex fractions]                   fractions)
                                           [simple denominators]
What fraction? Items with same units)
                                           Fraction – decimal – percentage
Fractions as divisions                     conversions
                                           [simple cases]

                                           Stage 7
                                           +/- (same denominator)

                                           

Stage 7/8                                  Stage 7/8
What fraction? (involving unit             Ordering fractions with different
conversion)                                denominators

                                           Given fraction conversions – eg eighths to
                                           decimals and percentages

                                           Reciprocals

                                           Simple 

                                           +/- (different denominators)




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                             More starter questions…



                                  2
1) Find 4 fractions equivalent to 7




2) Simplify the following
                             36
                         (a) 72


                                   28
                         (b)       35


3) In groups of 2 or 3, create a brief outline for a
   lesson that gets students to learn how to
             1    4
           1 to
   convert 3      3 (and vice versa) without being
   taught an algorithm




c2be0b5d-7e18-4977-862e-8caa9b33ac61.doc
                          Group planning for fractions



In pairs or threes

1) Pick one stage of the progression



2) Think of as many relevant fraction based
   activities as possible for that stage
    Consult the Getting Started book
    Consult the activity books
    Consult Figure It Out
    Think of good activities that were used in the
     past – and still fit
    Think them up
   Record your activities on newsprint



3) Reference everything – so someone doing the
   later planning can find them




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                                           For next time…




1) Read through the fraction booklet, and the
   readings provided




1) Continue planning the fraction lesson series




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