Ratios, Proportions, and Proportional Reasoning in the Quantitative Reasoning Pathway

Proportional reasoning “involves maintaining a sense of multiplicative scale in a relationship between quantities” (Gaze, 2019, p. 90). It includes understanding proportion, ratio, percent, and linearity; constant multiple/ratio and scaling; constant multiple/ratio of changes and scaling changes; and multiplicative reasoning. Gaze (2019) suggests “the concept of ratio can provide a common theme to convey the interrelated meanings of fractions, percentages, proportions, decimals, and rates” (p. 91). Proportional reasoning helps students make comparisons in realworld contexts, supports covariational reasoning, and assists students in recognizing reasonable answers to numerical problems.

Faculty at the MIP Quantitative Reasoning Initiation Workshop in May, 2021 recommended that instructional resources developed by CoRDs and ARCs addressing Ratios, Proportions, and Proportional Reasoning for success in the Oklahoma Quantitative Reasoning Pathway should:

1. Base problems about ratio, proportion, and proportional reasoning in contextual problems, affording students an opportunity to address the social context of the data (e.g. including multiple values/perspectives/needs, attending to implicit biases)

2. Support students in understanding a quantity as a measurable attribute of an object, and a ratio as a relationship between two quantities showing the relative amount of one that is associated with the other

3. Define what it means for a student to understand ratio, percentage, and proportion, and develop a wide variety of tasks to build the concepts of ratio, proportion, percentage, and proportional reasoning from real world problems.

4. Support students in recognizing proportional relationships as distinct from nonproportional relationships and justifying why a relationship is/is not proportional

5. Leverage proportional reasoning to understand covariation as reasoning about two quantities changing simultaneously

Addressing Components of Inquiry

Participants of the MIP Workshop on Quantitative Reasoning suggested the following ways resources about Ratios, Proportions, and Proportional Reasoning could address the three MIP components of mathematical inquiry:

Active Learning: Proportional reasoning tasks afford opportunities for students to explore problematic situations and choose the appropriate tools to describe multiplicative change. Because ratio is a unifying idea, it provides a common structure to connect mathematical ideas.

Meaningful Applications: Proportional reasoning tasks afford opportunities to explore and identify mathematical relationships between quantities. Ratio in its various forms is a common mathematical structure, and CoRDs can create tasks with ratio as a foundation to help students see various forms of ratio as representing the common mathematical structure of multiplicative change.

Academic Success Skills: Ratios and proportions can be difficult for students to understand. As such, it affords an opportunity to build students’ perseverance. Additionally, because students have possibly struggled with this topic in the past, CoRDs should attend to task design principles that decrease students’ mathematics anxiety. Collaborative work can support students in mathematical success in this sometimes difficult topic, and foster classroom community.

References

Common Core ratio and proportion standards and examples: http://www.corestandards.org/Math/Content/6/RP/

Gaze, E. (2019). Thinking quantitatively: Creating and Teaching a Quantitative Reasoning Course. In Tunstall, L., Karaali, G., and Piercey, V. (Eds.), Shifting Contexts, Stable Core: Advancing Quantitative Literacy in Higher Education. (p. 89-106). MAA Press.

Thompson, P.W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: NCTM.