Academic journal article Australian Primary Mathematics Classroom

REFractions: The Representing Equivalent Fractions Game

Academic journal article Australian Primary Mathematics Classroom

REFractions: The Representing Equivalent Fractions Game

Article excerpt

Stephen Tucker presents a fractions game that addresses a range of fraction concepts including equivalence and computation. The REFractions game also improves students' fluency with representing, comparing and adding fractions.

Students frequently encounter concepts that are difficult to grasp without significant interactions with concrete manipulatives. Although students often explore fraction concepts using manipulatives and activities, games usually lack effective recording components to transfer concepts and outcomes from concrete to symbolic forms. REFractions: The Representing Equivalent Fractions Game sheds light on concrete manipulation of fractional representation, comparison, and addition, connects them to pictorial representations, and extends them to symbolic representations. This supports the Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority, 2012) goal that that students must "make connections between related concepts and progressively apply the familiar to develop new ideas. They develop an understanding of the relationship between the 'why' and the 'how' of mathematics" (p. 5). The game also connects to the Standards for Mathematical Practice of The Common Core State Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) in the United States, which indicates that fraction concepts are part of making "mathematically proficient students who can apply what they know [and] are comfortable making assumptions and approximations to simplify a complicated situation, realising that these may need revision later" (p. 7).

REFractions covers concepts including representing, comparing, adding, and finding equivalencies between fractions, all of which require reasoning skills and flexibility with various models for greater understanding. Throughout the game, students form what Clements and McMillen (1996) call "integrated-concrete knowledge" where the objects, their interactions with the objects, and the generalisations they create form useful mental structures (p. 271). The discourse and multiple representations involved in REFractions also parallel parts of Van de Walle, Karp and Bay-Williams' (2009) assertion that one should use pictures, written symbols, oral language, manipulative models, and real-world situations to model mathematics, and that the interactions between these representations helps develop new concepts (p. 27).

This article describes the necessary materials for gameplay, how to play the game, and a group of students' first experience with the game. It then discusses evidence of fraction strands and standards from gameplay, as well as possible variations or modifications to the game. Students need background knowledge of representing and illustrating fractions, an awareness that one can compare fractions and find equivalencies between fractions, and an understanding that they can add fractions. During gameplay over multiple sessions, students will develop each of these skills, as well as their mathematical discourse abilities.


The purpose of this game is to increase students' fluency with representing, comparing and adding fractions in concrete and symbolic forms. Students use dice, representation mats, tiles and their recording sheets to develop and transfer their conceptual understanding of the concrete forms of fractions to the symbolic representations. To play REFractions each pair of students will need a recording sheet (Figure 1) and representation mat (Figure 2), a pencil, an eraser, tiles, and one double dice, which are translucent, coloured dice, each with a second, smaller opaque white die contained within. If double dice are not available, substitute two standard dice. Different colours of tiles are important so students can use a different colour for each fraction they represent, compare and add, as shown in Figure 3. This intentional use of manipulatives aligns with Sowell's (1989) and Clements' (1999) claims that manipulatives strengthen the sequence of concrete-toabstract understanding but are most effective when specifically purposed. …

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