Multiple strategies for finding ratio of two variables in an equation: A learning study

This paper reports a learning study conducted in a school of high level of achievement. The school teachers suggested that ratio is a challenging topic for S1 students. A pilot test on this topic was set by the research team comprising three class teachers and one subject panel chairperson of the school, and two researchers of the partner university. The pilot test was taken by the S2 students as they had already learned this topic. The results of the pilot test indicated that students were good at many aspects of ratio but less satisfactory in finding the ratio of two variables in an equation (e.g. Find the ratio of x:y if 3x = 4y). After a series of meetings and thorough discussions, three mathematical methods (namely, the LCM, the substitution and the algebraic) of finding the required ratio were proposed. The LCM method tags into the fundamental concept of using common units for comparing two quantities. The substitution method counts on two aspects, namely a ratio is unaltered if both parts are multiplied by the same factor and an intelligent use of the given condition. The algebraic method is built on the understanding that x:y can be written as x/y and appropriate use of algebraic manipulation skills. Three S1 classes were taught in a row on this topic using about one hour and ten minutes. A pre-test was administered to the students before the experimental class and then a post-test soon after class. A significant improvement in student performance in this topic was found by comparing the pre- and post-test results. It shows that the S1 students have overcome this challenging part in learning ratio and are able to find the ratio of two variables in an equation by using the three methods.