Concept Maps for Fractions

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Concept Maps for Fractions Algebraic
thinking on the ARBsAlex Neill and Jonathan
Fisher NZAMT
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What are the Assessment Resource Banks?

• On-line
• Three curriculum areas
• Levels 2 5
• Free
• Trialled in New Zealand schools
• Research based
• Over 3600 resources for assessment

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Fractional Thinking

• more than just pizza?

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What are Fractions?

• Fractions can be
• about a relationship between a part and a
  whole (part-whole relationship) 
• a result of a division (quotient)
• compared and ordered (equivalence)
• a measurement and
• like a function (operation) performed on a
  quantity.

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Fractions assessment resources

• Our goal
• Developing assessment resources that support and
  elicit deeper understandings of fractions for all
  of these personalities.
• gt Fractional thinking concept map and linked
  resources

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Some important concepts

• Partitioning
• Extending the part-whole
• A fraction as a number
• Comparing fractions
• Fractions as operators
• Using representations to explore understandings.

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Partitioning

• Partitioning involves dividing an object or set
  of objects into parts.  It can involve the
  breaking up of quantities into equal or non-equal
  parts. 

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Partitioning example
Show how to share 2 cakes equally amongst 5 people
How much cake does each person get?
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Partitioning example
Show how to share 5 cakes equally amongst 9 people
How much cake does each person get?
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Partitioning example
Show how to share 7 cakes equally amongst 4 people
How much cake does each person get?
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Part-whole fractions

• The concept of a fraction as a part-whole
  relationship is about comparing the part to the
  whole.

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Part-whole fractions example

• What fraction is shaded?

Out of the 12 members of a cricket club, 4 have
their own gear. What fraction have their own
gear?
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Part-whole fractions examples

• What fraction is shaded?

Continuous Discrete or both
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Part-whole fractions example
How much of the square is shaded?
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Part-whole fractions example
How much of the shape is shaded?
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Part-whole fractions example

• If is 2/3 of all the counters, how much
  is 1/3? A whole?
• If 1/2 is worth 12, what will 3/4 be worth?

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Beyond Part-whole fractions
How can you multiply two pieces of pizza?
(Hart, 2000)
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Fractions as numbers

• (Students) need to be clearly told what a
  fraction is. A fraction has to be a number, and
  so the definition of a fraction as
  parts-of-a-whole simply doesnt cut it.
  Students have to be shown that fractions are the
  natural extension of whole numbers.
• Wu (2005)

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Fractions as numbers
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Fractions as numbers
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Fractions as numbers
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Comparing fractions

• Looking at equivalence involves recognising that
  fractions have a size, and can be compared.
• Comparing fractions involves a common referent
  whole.

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Comparing fractions examples

• Show or explain which fraction is larger.

• 1/9 or 1/8?
• 3/5 or 4/9?
• 7/6 or 8/9?
• 2/3 or 3/5?

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Fractions as operators

• Fractions can act as operators upon quantities,
  regions or setsFor example
• What is 1/4 of 160?
• Show what 1/3 of is.
• What is 2/5 of ?

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Representing understanding

• Using representations to explore understandings
  addition subtraction multiplication division.
• then apply the fractional notation

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Representing understanding

• Addition1/2 1/43/4 7/8

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Representing understanding

• Subtraction3 1/2 1 3/4Even 5 1/6 2 3/8

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Representing understanding

• MultiplicationVisualising what multiplying
  fractions actually looks like.1/2 x 1/3?3/4 x
  8/9?

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Representing understanding

• DivisionVisualising what division of fractions
  is actually doing.1 1/2 1/4?1/2 1/3?

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Representing understanding

• From these explorations students can develop
  conjecturesdiscuss them, test them, and write
  the rules they have uncovered.

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Concept map (Fractions)
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Assessment resource Banks

• www.nzcer.org.nz/arb
• Username arb
• Password guide