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# Secondary Magazine - Issue 14: Focus on Sequences

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Created on 07 July 2008 by ncetm_administrator
Updated on 14 July 2008 by ncetm_administrator
 Welcome to issue 14 of the NCETM Secondary Magazine. Fortnightly features include: The Interview, Around the regions, An idea for the classroom, 5 things to do, The Diary and Focus on. Issue 14 focuses on Sequences.

Focus on… Sequences

 • In Dave Hewitt's article in MT163 (the journal od the Assocation of Teachers of Mathematics) entitled Approaching Arithmetic Algebraically, he makes what I consider a very thought provoking comment concerning the algebraic statement 3x + 1:"But let us not confuse the two words involved - algebraic and statement. The algebra is the work you do in order to get yourself in a position where you could make a statement (such as the example given above where there is a way of seeing the matches so that the formula is just a way of expressing what is 'seen'). '3x + 1' is the statement part, and only hints at the algebra which may have taken place to make such a statement."From the Maths Café community discussion What is algebra? • If you came for an interview at my school, you might well be asked to teach a lesson based on this question:If this is shape 3, what does shape 4 look like?  We are looking for teachers who give the kids a chance to think for themselves and be creative.We want them to draw out what is the ‘three-ness’ of the shape and how this translates into ‘four-ness’ and ‘two-ness’ and so on. We would be impressed if the candidate then got the pupils to talk about, say shape 10, shape 100 and so. This can then be generalised into shape n.We would not be so impressed if the candidate reduced the problem to a difference table method, as this loses the ‘generalisation’ feel.So this is a plea: by all means teach number sequences, but don't be fooled into thinking this will lead to an understanding of what generalisation really means. Our experience is that most children simply “don't get it” from number sequences.From the Secondary Forum discussion Sequences – finding the nth term. • What I found helpful from John Mason's remarks is to remind me of the distinction between spotting pattern and spotting structure. It is structure that one is searching for when attempting to predict the nth term.While it is useful to utilise the link between a sequence and a graph containing ordered pairs representing that sequence (as Keith suggests), both of these are representations of pattern, not structure.A further comment from the discussion Sequences – finding the nth term. • Starting with 5, every alternate Fibonacci number is the length of the hypotenuse of a right-angled triangle with integer sides; or in other words, the largest number in a Pythagorean triple. • It may be that the first mathematical use of the word ‘sequence’ is considerably more recent than you’d expect. The OED2 shows a use by Sylvester in 1882 in the American Journal of Mathematics giving the definition of "a succession of natural numbers in order".The word ‘sequence’ is also found in an 1891 translation of Axel Harnack’s An introduction to the study of the elements of the differential and integral calculus. • The formula for finding the nth term of the Fibonacci Sequence $n^{th} \ term \ = \ \frac{1} {\sqrt{5}}\left( \left( \frac{1+\sqrt{5}} {2} \right)^n - \ \left( \frac{1-\sqrt{5}} {2} \right)^n \right)$ was discovered in 1843 by Jacques Philippe Marie Binet.

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