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.

ImplementationI planned an investigative lesson on fractionsby introducing the ideaof equivalent fraction through fraction bars whole group.Afterwards, the students used their understanding offractions to create a fraction book.

Note: This would be a lesson that a third grade student coulddo 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 thebasics. I had them use tiles to draw and compareFraction equivalences. After having them create a fraction book, Studentsobserved:1. That all pieces were equal in length.2. When put together, some smaller pieceswould equal to larger wholes.

Once the students had completed this exploratory investigation, theycreated a fraction book.

Implementation

Students then wrote in their journals theirresults and drew picture models to make aconcrete 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 slices2 Pepperoni slices3 Ham slices6 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-3-5-Mathematics-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 andImproper Fractions. Teaching Children Mathematics, 13, (9), 488 – 492.

Norton, A., & McCloskey, A. (2008). Modeling Students’ Mathematics Using Steffe’s FractionSchemes. 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 throughPatterns 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.