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.

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