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Developing ratio tables to explore ratios

2019-08-01
SOZEN OZDOGAN, Sinem
AKYÜZ, DİDEM
STEPHAN, Michelle
The mathematics standards for many countries include teaching and learning ratios, typically with efficient algorithms and strategies (Australian Curriculum: Mathematics, Assessment and Reporting Authority, 2017; National Council of Teachers of Mathematics, 2014). In the 1990s, Cramer and Post (1993) claimed that school mathematics mostly involves teaching the cross-product algorithm to solve missing-value ratio problems. The ratio table--an organised table which has at least two rows and two columns to display ratios--is one of the tools which may become an efficient and meaningful tool when it is used with appropriate discourse and context (Shield and Dole, 2008). In this study, the authors present an example of an instructional sequence using a ratio table to support students to develop and strengthen their proportional reasoning. The sequence was designed by Stephan, McManus, Smith and Dickey (2015) and, according to their teacher's manual, includes the learning outcomes, instructional activities, mathematical tools, and possible topics of discourse to guide teachers. The Year 7 students in this work constructed proportional reasoning through interacting with their peers and by developing their own tools and strategies. The intention of the instructional sequence was to pose open-ended problems that might provoke students to organise their calculations in the form of a ratio table. For example, for the problem, "One food bar can feed three aliens; how many food bars will it take to feed 36 aliens?", students might write 1:3, 2:6, 3:9, 4:12, 5:15, 6:18, 7:21, 8:24, 9:27, 10:30, 11:33, 12:36 and conclude that 36 aliens will be fed by 12 food bars. The teacher would then capitalise on that personally meaningful strategy to introduce a more efficient organisational tool, called the ratio table. Then, as the instructional tasks proceed, students are prompted to reason with their tables in more efficient and multiplicative ways, leading towards more sophisticated proportional strategies.