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Proportional reasoning is a skill that develops naturally, but can be accelerated with targeted teaching. Engaging students in proportional reasoning situations gives them an opportunity to use fraction and ratio skills in authentic situations.

Proportional reasoning is an important aspect of numeracy. It requires students to understand the relationship between any two variables in a mathematical problem. The two most common operations used in proportional reasoning are multiplication and addition, however multiplication is the desired operation as it demonstrates higher order thinking and allows students to solve problems quickly.

Students who have developed proportional reasoning skills will benefit when solving many of the NAPLAN test questions. Below are examples of the common types of proportional reasoning.

Direct proportional reasoning occurs when any two given variables maintain a constant ratio. A common example can be seen when buying petrol: if one litre costs $1.50, then ten litres cost $15.

An inverse proportional relationship occurs when one variable increases if the other decreases. For example, if car A travels a distance at twice the speed of car B and arrives at its destination in half the time.

It is important to expose students to questions that seem proportional, but are not. Students need practice recognising when a multiplicative relationship exists.

For example, Bill is 6 years old and Mary is 12 years old. When Mary is twice as old as she is now how old will Bill be? (18 years old.) A proportional thinker will recognise Bill’s age is not doubled.

The transition from additive to multiplicative thinking can be difficult for some students. Questions like the one below test whether students see the multiplicative response required (15) or the incorrect additive response (12).

My cake recipe uses 4 cups of sugar and 10 cups of flour. If I increase my recipe to use 6 cups of sugar, how many cups of flour do I need?

Proportional reasoning on a linear scale can be used to help visualise relationships. For example, in the scale below what number does X represent? (53).

More complex questions involving enlargement or reduction of 2 and 3 dimensional figures are important and challenging.

___ 47 ___ ___ *X* ___ ___ ___ ___ ___ ___ ___ 67

Absolute thinking requires students to take an additive approach to a problem while relative thinking requires a multiplicative approach. The question below requires students to differentiate between the two types of thinking.

A class of 20 boys and 10 girls was surveyed for their preferred sport. Twelve boys and eight girls preferred swimming. Was swimming more popular with boys or girls? An absolute response is that more boys than girls in the class preferred swimming. A relative response is that a higher proportion of girls preferred swimming (80%) to boys who preferred swimming (60%).

Part-part-whole involves fractional thinking and is a valuable means of determining students’ understanding of ratios. For example, which has the strongest chocolate taste: A full glass of milk with 3 teaspoons of chocolate, half a glass of milk with 2 teaspoons of chocolate, or a third of a glass with 1 teaspoon of chocolate? The half glass would have the strongest chocolate taste because proportionally 4 teaspoons of chocolate are needed to make a full glass.

As teachers it is important to ‘spot’ proportional reasoning opportunities across the curriculum and draw them to the attention of our students. Proportional reasoning appears across many curriculum areas including:

**History**– in the form of timelines,**Geography**– when using population data,**Art**– when creating work,**Science**– the opportunities are exhaustive.

Engaging students in opportunities to think proportionally helps to develop this important concept for students’ greater access to subjects such as science, technology, engineering and mathematics as well as for critical numeracy required for active citizenship. You, the teacher, can open this door.

Geoff Hilton, Annette Hilton, Shelley Dole, Merrilyn Goos, Mia O’Brien. (The University of Queensland.)