Helping Students to Grasp
Fractions: Concrete to Abstract
By Stephanie S. Hardy
Fractions Standards in
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
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
• 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
• Allow students to draw picture models, if
needed, but guide students toward just using
symbols and algorithms.
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.
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.
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
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.
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
• 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.?
• The students then drew sample models and
most concluded they would receive:
1 Sausage slices
2 Pepperoni slices
3 Ham slices
6 Cheese slices
• 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.
(2006). 3-5 Mathematics Georgia Performance Standards. Retrieved April 20, 2009, from GADOE Web
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.

Final Project PowerPoint