Helping Students to Grasp Fractions: Concrete to Abstract

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Helping Students to Grasp Fractions Concrete to
Abstract

• By Stephanie S. Hardy

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Fractions Standards in Elementary

• 3rd Grade
• Understand fractions as part of a whole
• Equivalent fractions
• Adding and subtracting fractions with like
  denominators
• 4th Grade
• Equivalent fractions
• Adding and subtracting fractions that are mixed
  with like denominators to 12.
• 5th Grade
• Equivalent fractions
• Simplifying fractions
• Multiply and dividing fractions
• Adding and subtracting fractions with unlike
  denominators and mixed fractions

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Strategies for Solving the Fraction Mystery

• Use concrete manipulatives for exploratory
  learning prior to introducing theory.
• Suggested types of manipulatives include
• Pattern Blocks Discover parts of a whole and
  equivalent fractions
• Unifex / Connector Cubes Adding fractions,
  discovering parts of a whole, and equivalent
  fractions
• Base Ten Blocks Use to discover tenths,
  hundredths, fraction to percents
• Number Lines Useful in comparing and ordering
  fraction sets
• Clocks Useful to teach adding and subtracting of
  fractions w/ unlike denominator association

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Strategies for Solving the Fraction Mystery

• Use manipulatives and picture models that can be
  associated with fraction concepts.
• Use picture models when solving word problems.
• Have students keep a journal of their fraction
  discoveries with explanations.
• Note The above three strategies guide students
  towards abstract thinking of math concepts
  because they begin to visualize the process.

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Strategies for Solving the Fraction Mystery

• As students begin to think more in the abstract,
  provide word problems without manipulatives.
• Allow students to draw picture models, if needed,
  but guide students toward just using symbols and
  algorithms.

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Implementation

• I planned an investigative lesson on fractions
• by introducing the idea
• of equivalent fraction
• through fraction bars
• whole group.
• Afterwards, the students used their understanding
  of
• fractions to create a fraction book.
• Note
• This would be a lesson that a third grade student
  could
• do to develop an understanding of equivalent
  fractions.
• For a third grade group, I would keep the
  fractions to 1 whole, 1 halves, and 4 ¼ pieces.
• If incorporated into fourth as an activating
  strategy, I would suggest bumping the
• fraction pieces up to 1/12s, 1/6, 1/3, ¼s,
  and ½s.
• If this lesson was used as an activating strategy
  for fifth grade, I would increase the book to
  1/16s, 1/8s, ¼s, ½s. Their may be a lesser
  amount of fractions to relate to, but the
  students can incorporate in visual images instead
  by drawing models.

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Implementation

• For my fifth grade group, I went back to the
• basics. I had them use tiles to draw and compare
• Fraction equivalences.
• After having them create a fraction book,
  Students
• observed
• That all pieces were
• equal in length.
• 2. When put together, some smaller pieces
• would equal to larger wholes.
• Once the students had completed this exploratory
  investigation, they
• created a fraction book.

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Implementation

• Students then wrote in their journals their
• results and drew picture models to make a
• concrete connection.

Sample of a concrete model of equivalent fractions
Student Work Sample Equivalent Fractions
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Implementation

• Finally, I removed the manipulatives and assessed
  the students by giving them a task to complete.
  It was as follows
• There are four equally sized pizzas pepperoni,
  ham, cheese, and sausage. The pepperoni is cut
  into 8 slices, the sausage is cut into 4 slices,
  the ham is cut into 12 slices and the cheese
  pizza is cut into 24 slices. Student a want an
  equal amount of pizza and got 1 slice of sausage
  pizza. How many slices of the other pizza would
  the student receive.?

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Implementation

• The students then drew sample models and most
  concluded they would receive
• 1 Sausage slices
• 2 Pepperoni slices
• 3 Ham slices
• 6 Cheese slices

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Results

• The exemplar on fractions was scored on a rubrics
  that identifies their knowledge of fractions
  concepts as a beginner (level 1), a practitioner
  (level 2), or Expert (level 3).
• Level one needed manipulatives to solve the
  problem, level 2 drew pictures only, and level
  three drew picture symbols and used algorithms to
  solve the task.
• About 80 percent of the students could use the
  strategies taught in order to solve the problems
  and score a level 2 or 3 on the exemplar.
• Some of the skills can be flawed.

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References

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  Standards. Retrieved April 20, 2009, from GADOE
  Web site
• https//www.georgiastandards.org/Standards/Georgi
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  -Standards.pdf.
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