amt 67 (1) 20116

Proportional reasoning has been recognised as a crucial focus of mathe-matics in the middle years and also as a frequent source of difficulty for students (Lamon, 2007). Proportional reasoning concerns the equivalence of pairs of quantities that are related multiplicatively; that is, equivalent ratios including those expressed as fractions and percents. Students who do not learn to reason proportionally are unequipped to learn mathematics topics such as similarity, scaling, and trigonometry. Proportional reasoning is also essential to understanding rates and hence many science concepts such as speed, density and molarity. Ratios express the proportions of components in a combination. They can relate the sizes of parts of a single whole (e.g., the number of boys and girls in a class), two wholes (e.g., the numbers of grade 5 students and grade 6 students), or one part to a whole (e.g., the number of girls in a class to the total number of students in the class). The last of these is the type of ratio most commonly represented as fractions but in fact each type can be represented in this way. A key part of teaching about ratio is helpin