Research-based guidance and classroom activities for teachers of mathematics

Managing relationships between several variables

Managing relationships between several variables is a theme within ratio and proportional reasoning. Links to relevant activities and resources are on the right hand side of this page. 

Proportional relations depend on comparing ratios, thus handling four different quantities. In this question: ‘if 5 kilos of potatoes cost £3, how much will 8 kilos cost?’ there are three numbers and one unknown quantity. It is hard to juggle four quantities in the mind at the same time, and notations and diagrams help.

If learners depend on standard methods, it is common to make mistakes in choosing when to apply which method. They need to understand when and why to multiple or divide. Compressing two of the quantities into a ratio (5:8), or a unit (such as a unit price) makes it easier to deal with the remaining number and unknown. It seems, from neuroscience, that in early adolescence learners become more able to make these compressions which chunk separate elements together.