Fraction Progressions

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Fraction Progressions

• Presented by
• Jenny Ray, Mathematics Specialist
• Kentucky Department of Education
• Northern KY Cooperative for Educational Services

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Childrens Ideas about
Fractions

• Show me where ½ could be on the number line
  below

Why do students sometimes choose this part of the
number line?
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Childrens Ideas about Whole Numbers

• 3 gt 2 ALWAYS.
• 1 1 ALWAYS.
• Sohow can it be that 1/3 gt ½ ?

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When students cant remember a procedure, they
resort to performing any operation they know they
can do

• Estimate the answer 12/13 7/8
• A) 1
• B) 2
• C) 19
• D) 21
• E) I dont know.

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instead of making sense of the numbers they are
attempting to add.
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National Assessment of Educational Progress
(NAEP) results show an apparent lack of
understanding of fractions by 9, 13, and 17 yr
olds.Estimate the answer 12/13 7/8

• Only 24 of the 13-yr-olds responding chose the
  correct answer, 2.
• 55 selected 19 or 21
• These students seem to be operation on the
  fractions without any mental referents to aid
  their reasoning.

Results from the 2nd Mathematical Assessment of
the National Assessment of Educational Progress
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Perhaps youve seen this reasoning

• 1/2 1/3 2/5
• If students have an understanding of the value of
  the fractions on a number line, or as parts of a
  whole, then they can argue the unreasonableness
  of this answer.

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How can students learn to think quantitatively
about fractions?

• Based on research
• students should know something about the
  relative size of fractions.
• They should be able to order fractions with the
  same denominators or same numerators as well as
• to judge if a fraction is greater than or less
  than 1/2.
• They should know the equivalents of 1/2 and other
  familiar fractions.
• The acquisition of a quantitative understanding
  of fractions is based on students' experiences
  with physical models and on instruction that
  emphasizes meaning rather than procedures.
  (Bezuk Cramer, 1989)

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Hands on experiences help students develop a
conceptual understanding of fractions numerical
values.

• FRACTION MANIPULATIVES

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Learning Activity Fraction Circles

• The white circle is 1. What is the value of each
  of these pieces?
• 1 yellow
• 3 reds
• 1 purple
• 3 greens    

Nowchange the unit The yellow piece is 1. What
is the value of those pieces?
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Learning Activity Using Counters

• Eight counters equal 1, or 1 whole.
• What is the value of each set of counters?
• 1 counter
• 2 counters
• 4 counters
• 6 counters
• 12 counters   

Now, change the unit Four counters equal 1.
What is the value of each set of counters?
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Learning Activity Cuisinaire Rods

• The green Cuisenaire rod equals 1.
• What is the value of each of these rods?
• red
• black
• white
• dark green    

Change the unit The dark green rod is 1. Now
what is the value of those rods?
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Learning Activity Number Lines
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A new way of thinking/teaching

• Many pairs of fractions can be compared without
  using a formal algorithm, such as finding a
  common denominator or changing each fraction to a
  decimal.

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Comparing without an algorithm

• Pairs of fractions with like denominators  
• 1/4 and 3/4 3/5 and 4/5
  
• Pairs of fractions with like numerators  
• 1/3 and 1/2 2/5 and
  2/3
• Pairs of fractions that are on opposite sides of
  1/2 or 1  
• 3/7 and 5/9 3/11 and
  11/3
• Pairs of fractions that have the same number of
  pieces less than one whole  
• 2/3 and 3/4 3/5
  and 6/8

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Comparing 3/7 and 5/9a students response

• The fractions in the third category are on
  "opposite sides" of a comparison point.
• One fourth-grade student compared 3/7 and 5/9 in
  the following manner (Roberts 1985)
• "Three-sevenths is less. It doesn't cover half
  the unit. Five-ninths covers over half."

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Comparing 6/8 and 3/5 A students response

• A fourth-grade student compared 6/8 and 3/5 in
  this way (Roberts 1985)
• "Six-eighths is greater. When you look at it,
  then you have six of them, and there'd be only
  two pieces left.
• And then if they're smaller pieces like, it
  wouldn't have very much space left in it, and it
  would cover up a lot more.
• Now here 3/5 the pieces are bigger, and if you
  have three of them you would still have two big
  ones left. So it would be less."

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Conceptual Understanding

• Notice that each child's reasoning from the
  previous two examples is based on an internal
  image constructed for fractions.
• Hands-on experiences with fractional parts, both
  smaller than and greater than one, helps to
  create this conceptual knowledge, so that
  procedures that they develop make sense.

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Exploring fractions with the same denominators

• Use circular pieces. The whole circle is the
  unit.
• A. Show 1/4
• B. Show 3/4
• Are the pieces the same size?
• How many pieces did you use to show 1/4?
• How many pieces did you use to show 3/4?
• Which fraction is larger? How do you know?

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Comparing fractions to ½ or 1

• Use circular pieces. The whole circle is the
  unit.
• A. Show 2/3 B. Show
  1/4
• Which fraction covers more than one-half of the
  circle?
• Which fraction covers less than one-half of the
  circle?
• Which fraction is larger? How do you know?
• Compare these fraction pairs in the same way.
• 2/8 and 3/5
• 1/3 and 5/6
• 3/4 and 2/3

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Resources for Activities

• Illuminations (NCTM)
• Rational Number Project
• nzmaths
• Mars/Shell Centre
• Teaching Channel
• www.jennyray.net

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