It Stands to Reason
Hone your pupils' reasoning skills about ratio and proportion
Understanding ratio is one of the nine basic maths skills that can cause major headaches in the classroom.
In the New GCSE Curriculum, “Ratio and Proportion” has now been defined as a skill area all of its own within the GCSE subject content and assessment objectives, such is its importance. As in all skill areas, pupils must:
- develop fluent knowledge, skills and understanding of mathematical methods and concepts
- acquire, select and apply mathematical techniques to solve problems
- reason mathematically, make deductions and inferences and draw conclusions
- comprehend, interpret and communicate mathematical information in a variety of forms appropriate to the information and content.
The 16 objectives within the new curriculum clearly define the links between ratio and many other areas of mathematics, highlighting the interconnected nature of the subject matter: for example, links are made at the higher level with trigonometric ratios and compound interest.
William Emeny (@Maths_Master) has written a superb blog at Great Maths Teaching Ideas on the use of bar modelling and its application to ratio and proportion problems. His presentation is beautifully clear and legible: handwriting to die for! (go to the blog for a larger version of the examples below).
The blog begins with an embedded video that is well worth watching. The content is for primary pupils, but the excellent inquiry-based pedagogy that is modelled by the presenter is applicable right through to KS5, and beyond.
Having used several of the approaches I’ve drawn together in this article with pupils and with trainee teachers, I can see the huge benefit there is to having these tools to use. In the case of the bar model, it is a tool that gives consistency of approach to many maths problems, from easy to complex levels.
Bar Modelling, Algebra Tiles and Arrow Diagrams are not in themselves a panacea to solving all problems, including world peace, they are certainly very helpful concrete visualisations that give pupils routes into and out of problems that are usually found to be difficult.
Here you can see bar model solutions to a higher level compound interest question as well as a lower level equivalency and simplifying ratios. Both are beautifully demonstrated and drawn by @Maths_Master himself:
Examples of arrow diagrams can be clearly seen here in exam questions answered by Mr Barton (not quite the legibility of @Maths_Master but still good - again, go to the link for larger versions of the examples). The lower level skill of finding the missing amount given the ratio is aided with the use of the arrow diagram and then recognising the multiplying factor.
In this next example the arrow diagram is used to identify the multiplying and dividing factors so that the pupil can then apply them to solve an unstructured problem:
Here - from Inquiry Maths - you can see the Arrow Diagram in action as students react to the Inquiry Prompt Question “What do the arrows represent?”
Durham Maths Mysteries
The Ratio and Proportion Mystery is an old favourite but an excellent one, with 11 clues covering ratio, fractions, percentages and average. The mysteries cover a lot of maths content, and they deepen pupils’ understanding as they use the clues to work backwards to find the answers.
Nuffield Free Standing Maths Ratio Bingo & Matching Cards
Practise the skills required within a good game of bingo: all the threes, two little ducks, clickety click...
NRICH and ratio
101 things to do with ratio & proportion
A challenging functional-style question.
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