Tasks to Develop Language for Ratio Relationships

Tasks to Develop Language for Ratio Relationships

Sue Pawula

Summary

Pre-service teachers build core concepts on ratio relationships by making connections through context, language, and drawings for better conceptual understanding in this article.  Two key ideas they will address with students in the classroom are: 1) “recognizing and expressing the inherent multiplicative comparisons in ratios”; and 2) “coordinating fraction language with ratio relationships.” In small groups the PSTs were asked to make comparisons in sentence form using precise language to compare two ribbons lengths.  They were asked to illustrate their sentences through diagrams, labeled units, and equations. They were prompted to produce statements that correctly compared the lengths in a reciprocal relationship. They discovered that through teaching learner to label the comparisons as additive or multiplicative, they could be referenced in two different ways of thought. Labeled diagrams particular helped language development of the multiplicative relationship when neither quantity was a whole-number multiple of the other. The tape or strip diagram model used in Asian curricula was introduced because of its simplicity, flexibility, and generalizability, and transferability to number lines.  Also, when labels were provided to distinguish the two way of thinking the additive comparison and multiplicative comparison became easier to discern. Learners need to use elements of the meaning of a fraction to understand ratios: 1) what is the whole, 2) identify the unit fraction, and 3) iterate the unit fraction to name the new amount in terms of the whole.  When learners experience language work this way it extends and solidifies fractions meaning and previews the multiplicative language needed for ratios.  PSTs then worked to identify part:part vs part:whole ratio language. Students need to learn that a critical idea of ratios is that there is a constant multiplicative relationship between two amounts. Appropriate tasks will provide student language development to be supported by the teacher and concept clarification.  When student comprehension is scaffolded through discussions like these, it establishes a firm base for language needed for scales, proportions, and probability.

Reflection

The diagrams produced by the PSD’s made the comparisons and language used much more relatable to ratios.  When you are addressing ratios with your students, different diagrams would greatly enhance student discussion and understanding.  Through drawing their diagrams and explaining their thoughts with their peers, students are allowed to use the math language necessary to build understanding of ratios.  The teacher should always be carefully listening to the discussions in order to direct student attention to incorrect conclusions or language usage while at the same time allowing the learners to teach each other.  This is an excellent introductory exercise for students and the Asian curricula model gives an easy added degree of support to students visualizing the language.

Rathouz, M., Cengiz, N., Krebs, A, & Rubenstein, R. N. (2014). Tasks to develop language for ratio relationships.  Mathematics Teaching in the Middle School, 20(1), 38-44.