don steward
mathematics teaching 10 ~ 16

Wednesday, 22 September 2010

nth term rules from 'matchstick' growing sequences

this sequence grows, one step at a time, forever...

how could this sequence have been formed?
how could it (actually) have been drawn?

teaching (efforts) can helpfully focus student's attention on diagrammatic appreciations - the structure, rather than appealing, too quickly anyway, to a number pattern - obtained by counting sticks
in this way the nth term rule can be justified and a symbolic relation (club membership) is probably better appreciated

students need to appreciate what 'n' is:
the position of the shape/pattern/block in the growing sequence

students need to appreciate what the nth term rule enables you to find out (or tells you):
the number of matchsticks in that particular pattern for any ('n'y) position in the sequence

an important aspect of such work is to look at equivalent expressions that arise
for the example above, expressions that could be obtained are:

how can the sequence be viewed to see these rules?
why are they all the same rule? 

John Mason suggests providing just one pattern/block in a growing sequence (e.g. the 3rd)
following class agreement on how the shapes grow each time - the 'structure' of the growth, a generalisation can be sought

he suggests an important part of such work is this shared decision, possibly seeking alternatives, on what structure is being generalised:
what 'bunch' has been added each time?

Dave Hewitt proposes using a large particular value of 'n' (e.g. the 45th pattern) to see how the diagram can be deconstructed

he has emphasised that the question, "how might you have drawn this?" is a helpful one to direct attention onto the structures
this is explored in the NRICH 7 squares (8111) three videos:

deconstruction works

also in nth term work from number patterns, Dave suggests providing one input and output pair (e.g. the 4th term is 6) and asking students to consider possible generating rules that would include this pair  e.g.
  • 2n - 2
  • 10 - n
  • 3(1/2n)
  • 30 - 6n

I like the suggestion in Great Maths Teaching Ideas by William Emeny of using wipeboards with a row of students at the front of the classroom holding n = 1, n = 2 etc. on each and with the teacher providing a rule (e.g. 3n - 1), asking students to write their number in this sequence:

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