Nix the Tricks: Fractional indices

This post on fractional indices is the second in a series featuring ideas from Tina Cardone’s book ‘Nix the Tricks’.

A recent class had difficulty in moving from integer to fractional indices. I used some of the ideas from ‘Nix the Tricks’ and began by clarifying language. 

A typical way of describing 23 is ‘Two multiplied by itself three times’ which could mean either

(2 x 2) x (2 x 2) x (2 x 2) 

or   ((2 x 2) x 2) x 2)

Given the lack of clarity students will sometimes use 2 x 3 as a fall back.

As an alternative indices can be described as indicating the number of factors of the base. So 23 is ‘three factors, each of which is two’.

A further advantage of this description of exponents is that it can be expanded to fractional indices and in doing so lessens the jump from integer indices.

Hence 81/3 is read as ‘one third of a factorisation of 8’. To find one third of something you must break the whole into three parts. A factorisation of 8 is

 8 = 2 x 2 x 2

One third of the factorisation gives the answer 2.

Representing  82/3  involves keeping two thirds of the factorisation giving the answer 2 x 2 = 4

This method struck students as being a natural progression between whole number and fractional indices. Linked to this post is a resource that develops this representation. Give it a try and reply to this post sharing how students get on.

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 5 of the book.

If you would like to consider further ideas for teaching through understanding join us on the following Enthuse funded courses.

  • 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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Subject(s)Mathematics
Age11-14, 14-16
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