Proportional reasoning

 

Proportional reasoning takes fractions, decimals, ratios or percentages and places them in a problem solving context. It can present a different way of approaching some problems rather than immediately leaping to or looking for a formula that numbers can be plugged into without a real consideration for the relationships between quantities.

Many students find proportional reasoning problems difficult. This is often because they have not been introduced early enough to the multiplicative nature of proportion reasoning and struggle to recognise this. Instead, they use addition methods, or informal methods.

A nice science example that is relevant across both physics and chemistry is density. Being a compound variable, students can often get confused about what might happen to one quantity if another changed (whilst another remained the same). For example “if I had the same amount of gas in half the volume, what would happen to the density?”

Proportion problems can often be solved quite easily using these informal methods when the numbers involved are simple multiples of each other as with halving and doubling. However, students need to be able to deal with the general case using the operations of multiplication and division. So whenever possible encourage students to use a single step multiplicative calculation. For example a 20% increase means finding ‘120% of …’ which means multiplying by 1.20.

This will be particularly important when they come to finding reverse percentages, “the final cost after 20% increase was, what was the original cost?” This is reversed by dividing by 1.20 NOT by subtracting 20%. Students who are not fluent with using single step multiplies will struggle with this concept and will want try to use subtraction because it is the inverse of addition.

Remind students that a proportional increase/decrease can be represented by a decimal multiplier. For example to increase a quantity by 43%, multiply by 1.43, to decrease by 12%, multiply by 0.88.

Some students may think that a multiplier always has to be greater than 1.

 

The word ‘similar’ means something much more precise in this mathematical context than it may in their everyday experience and this can cause confusion.

Proportional Reasoning

An entire website dedicated to providing examples and support with teaching and learning proportional reasoning. There are many examples and a range of activities on the site including specific density and graphing ones.

Using a haemocytometer

This is an example of proportional reasoning used in the scientific context of a biologist who is interested in the number of cells in a sample of fluid. You may want to look at the Standards unit resource below in conjunction with this activity. This example also links to the list  principles of sampling applied to science.

 

The resource is interactive so make sure you check it on your system first. The links on the left hand side of the page link to additional guidance on the activity these include:

  • Teachers guide which includes a description of why to use this problems and suggested strategies for how to use it in the classroom. Key questions and suggestions for support and challenge. 

  • Guidance on how to get stared

  • Solutions many of which are sent in by students from schools using the resources.

  • There are also links to activities you can use to prepare or extend this one.

Developing Proportional Reasoning N6

Quality AssuredCategory:MathematicsPublisher:Department for Education

This Standards Unit resource is a very powerful resource for developing students understanding the multiplicative nature of proportional reasoning problems. It helps students move from their informal understanding into a more formal approach using single step multiplicative calculation.

Students are given four direct proportion problems to solve, Sheet 1 pages 7 and 8, taken from different areas of the mathematics curriculum. They then compare their methods for solving these with methods produced by other learners. This leads to a discussion that compares the use of more primitive informal methods that use adding, doubling and halving with the use of more sophisticated methods that use multiplication.

 

Students then use exemplar material that includes examples of the classic misconceptions that occur in proportional reasoning problems, sheet 2 sample work page 9. On pages 2 and 3 the material includes a detailed explanation of what each of the misconceptions is and their implication to the learner. Students are then taken through a simple way of developing their understanding of the multiplicative nature of proportional reasoning problems, page 4.

 

You could use this to check student’s ability to work with multipliers in proportional reasoning problems. If your mathematics department has already used this resource you could build on this by adapting the examples with examples from the science programme of study.

Carbon Footprints

This is an example of proportional reasoning using multipliers in the context of a carbon footprints.

The links on the left hand side of the page link to additional guidance on the activity these include:

  • teachers guide which includes a description of why to use this problems and suggested strategies for how to use it in the classroom. Key questions and suggestions for support and challenge.

  • guidance on how to get stared

  • solutions many of which are sent in by students from schools using the resources.

  • there are also links to activities you can use to prepare or extend this one.