Helping Students to Grasp
Fractions: Concrete to Abstract
By Stephanie S. Hardy
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
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
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.
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.
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 1/12’s, 1/6’, 1/3, ¼’s, and ½’s.
•
If this lesson was used as an activating strategy for fifth grade, I would increase the book to 1/16’s,
1/8’s, ¼’s, ½’s. Their may be a lesser amount of fractions to relate to, but the students can
incorporate in visual images instead by drawing models.
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:
1.
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.
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
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.?
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
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.
References
(2006). 3-5 Mathematics Georgia Performance Standards. Retrieved April 20, 2009, from GADOE Web
site:
https://www.georgiastandards.org/Standards/Georgia%20Performance%20Standards/Grades-3-5Mathematics-Standards.pdf.
Chick, C., Tierney, C., & Storeygard, J. (2007). Seeing Students’ Knowledge of Fractions:
Candace’s Inclusive Classroom. Teaching Children Mathematics, 14, (1), 52 – 57.
Meagher, M. (2002). Teaching Fractions: New Methods, New Resources. ERIC Digest.
Retrieved February 17, 2009, from ERIC Database.
Neumer, C. (2007). Mixed Numbers Made Easy: Building and Converting Mixed Numbers and
Improper Fractions. Teaching Children Mathematics, 13, (9), 488 – 492.
Norton, A., & McCloskey, A. (2008). Modeling Students’ Mathematics Using Steffe’s Fraction
Schemes. Teaching Children Mathematics, 15, (1), 48-54.
Ortiz, E. (2003). The Roll Out Fraction Game: Comparing Fractions. Teaching Children Mathematics,
13, (1), 56-62.
Phillip, R., & Vincent, C. (2003). Reflecting on Learning Fractions Without Understanding.
ON-Math, 2, (2). Retrieved February 17, 2009, from NCTM database.
Roddick, C., & Silvas-Centeno, C. (2007). Developing an Understanding of Fractions through
Patterns Blocks and Fair Trade. Teaching Children Mathematics, 14, (3), 140 – 145.
Tzur, R. (2002). From Theory to Practice: Explaining Successful and Unsuccessful Teaching Activities
(Case of Fractions). ERIC Digest. Retrieved February 17, 2009.