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.
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