Mark W. Ellis, Ph.D., NBCT
A 2010 report, “Developing Effective Fractions Instruction for Kindergarten through 8th Grade,” found that half of eighth graders could not order three fractions from least to greatest. The authors identified several recommendations for improving fraction instruction.
- Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts.
- Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.
- Help students understand why procedures for computations with fractions make sense.
These recommendations help students develop “fraction sense” by understanding key concepts and how they relate to prior knowledge of number and operations, and how they are reflected in the design of the learning progressions for fraction sense and fraction operations in the Common Core math standards.
Early work in grades 1 and 2 with fraction concepts—although not with fractions written as numbers—involves visually identifying halves, thirds, and fourths using geometric figures such as rectangles and circles. Before students are asked to consider operations with fractions, they must understand what fractions represent conceptually and how fractions fit into and extend the set of whole numbers. When planning for instruction, knowledge of learning progressions lets teachers think about concepts, strategies, and models that will be used over several grades as students develop their understanding.
Essential to strong fraction sense is learning to think about unit fractions with a numerator of 1 as building blocks for other fractions, based on an understanding that “1/n” measures of one part of a whole that is cut into “n” equal-sized parts. This extends students’ understanding that whole numbers are made up of units—ones, tens, and hundreds—and their knowledge of measurement concepts.
This blog entry by The Elementary Math Maniac describes a game that helps students understand that fractional pieces do not have to be congruent in order to be equivalent. I love the use of student discourse and the requirement for them to talk through disagreements. A set of Common Core tasks from North Carolina also offers some good ideas for developing students’ early understanding.
Use the Ready® Mathematics lessons on grade 3 fraction standards to start with unit fraction and add visual representations to help make sense of fraction relationships. For example, in Grade 3 Lesson 16, have students explore the concept of equivalent fractions with fraction strips (rectangular area model) and number lines (linear model).
When working with fraction strips, I recommend having students make their own set to make fractions tangible—you can use this template (the one without labels). It is a good idea to probe their understanding with queries such as, “How many 1/5s make one whole?” and “Prove in two ways which is larger, 2/5 or 2/3.”
The goal of all of this early work is to have students develop a more coherent understanding of the meaning of fractions, so that rather than seeing 3/5 as “3 over 5” they think of it as “three one-fifths” or “1/5 + 1/5 + 1/5.” Instead of teaching tricks or procedures to compare fractions, students should be encouraged to use sense-making strategies such as those outlined in “Laying the Foundation for Success in Algebra,” which includes “fraction tents” that can be used for a fraction clothesline (number line) ordering activity. Having used this activity many times, with children and adults, I can attest that it generates a lot of good mathematical reasoning. The extension into algebra is quite powerful and can be used with students in grades 7 or 8.
Once students recognize fractions as numbers with meaning, introduce them to operations with fractions. While standard algorithms for fraction operations are often efficient, they should not be taught to students in the absence of sense-making and understanding why they work. Ideally, students’ own thinking will form the basis for what they come to know as standard algorithms. In addition to Common Core aligned materials such as Ready Mathematics, the Rational Number Project offers research-based activities that support learning fraction operations with understanding.
Based on my own experiences as a middle school teacher, I believe that students must be given opportunities to develop their own ways of reasoning about fraction operations, typically starting with visual models and then moving to abstract numerical representations. These activities are supported by honing your skills with eliciting and extending students’ thinking and having students engage in error-analysis tasks aimed at addressing common mistakes or misconceptions. These activities get students communicating with and about fractions as they refine their understanding and also help to develop their academic language.
One of my favorite topics to teach is division by a fraction. It is a joy to see the light come on when learners understand why division by a fraction between 0 and 1 “makes things bigger.” To start this investigation, ask students what situation is represented by the expression “5 divided by 1/3.” First dividing a whole number by a unit fraction lets students focus on the meaning of the expression and worry less about the quantities involved. This teacher blog shares a similar approach, and a lesson from the Charles A. Dana Center nicely illustrates how to build on this beginning.
- The Illustrative Mathematics site offers a set of short videos aimed at teachers and other adults that illustrate the fraction progressions in the Common Core.
- The Ontario Ministry of Education has produced materials documenting teachers’ development and implementation of lessons on fraction division including video clips.
- Professor Hung-Hsi Wu of UC Berkeley has prepared an in-depth guide for Teaching Fractions According to the Common Core Standards.
This blog is the third in a series from Curriculum Associates about the most challenging math standards. Stay tuned, later this month Curriculum Associates will release a new white paper on the toughest math standards.
For more see:
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Measurement
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Modeling
Mark Ellis is a National Board Certified Teacher and professor of education at California State University at Fullerton. He is an author of Curriculum Associates’ Ready® Mathematics program.