Nix the Tricks: Areas of quadrilaterals, triangles

Areas of quadrilaterals

A great deal has been mentioned in recent years about ‘cognitive load’ and reducing this for students studying mathematics. Tina Cardone’s book ‘Nix the Tricks’ has some interesting ideas around this in relation to area formulae.

When teaching area the order in which I have previously approached topics has been; rectangles, right-angled triangles, and then non-right-angled triangles.

Tina suggests switching this order and teaching parallelograms after rectangles. Since any triangle is half of a parallelogram (and a rectangle is a special case of a parallelogram), this eliminates the need to treat right-angled and non-right-angled triangles separately.

Making parallelograms more of a focus has an additional pay-off when considering trapeziums. By making a copy of a trapezium, rotating it 180 degrees, and placing the result adjacent to the original, a parallelogram is formed. Finding the area of the trapezium is reduced to halving the area of the resulting parallelogram.

Taking this thinking to its logical conclusion there may even be an argument for teaching parallelograms first, and deriving the area of the rectangle as being a special case. If you do give this a go then please let the community know how your students get on by leaving a comment to this post.

A final consideration is around language. The usual description of calculating the area of a rectangle is unique amongst shapes in using the words ‘length’ and ‘width’. Switching to ‘base multiplied by height’ the area for each of the shapes is then a variation using the same words.

We have recently added the ‘Nix the Tricks’ book both in its entirety and divided into chapters. The discussion in this post arises from Chapter 4 of the book.

  • Mastering mathematics at key stage 3: It is important that mathematics at key stage 3 builds upon the mathematical experiences students experience at primary school. Designed for teachers of mathematics at key stage 3, explore what is meant by mastery, consider the transition between primary and secondary school and the mapping of progression through key stage 3. 
  • Teaching mathematics GCSE content with understandingExplore the content of the mathematics GCSE and gain an understanding of the importance of mathematical reasoning and problem solving. Develop problem solving skills and resilience in the context of hard to teach GCSE topics. Topics covered include trigonometry, linear graphs, proportional reasoning, standard form and powers, frequency trees, Venn diagrams, equations of circles, turning points, iterative processes, quadratic sequences, vectors and proof.


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