One of my favourite problems
I tend to think more about
'spaces for exploration' than 'problems to solve' when I am
planning teaching, so I have found deciding on a favourite
problem hard, as well as how to write about it. But the problem I
imagine I have used with more groups, for longer, than any other,
is one I call 'Functions and Graphs'.
The challenge is simple to state:
given any equation, predict what its graph would look like.
It is harder to set up and is never-ending to work on. I start by
playing a 'function game'.
I'll write, in silence, as dramatically as possible, something
like:
2® 6
5 ® 12
3 ®
... and offer the pen (in silence still) to a student to come and
fill in the gap.
I have a rule in mind and will respond to what is written with
a
or a
trying to maintain
silence for as long as possible.
If a student gets a then they choose the
next starting number.
I'll use a linear rule to start off year 7; quadratic if it is a
year 8 I have worked with before.
At the point of getting descriptions of the rules I will get as
many as I can from the class:
I'll write x ®... and may have to explain I want
someone to write the rule they were using.
Typically there will be
many different rules:
x ® x ×2 + 2
x ® x + 1×2
x ® xx + 2
I will accept these rules exactly as the students write them and
explain how a mathematician would interpret them if there is any
ambiguity (e.g. xx means x squared). I offer creating a picture for
each rule as a mechanism for deciding which ones are the same or
different and which fit our game.
Students write each rule across the top of a page and choose a
random selection of 5 numbers (including some negatives) to
list under each one, e.g.:
x ® xx + 2
5 ®
3 ®
0 ®
-1 ®
-4 ®
Having filled in these inputs (many conversations usually
necessary for the negatives - I am deliberately evasive at this
point; one of the powers of the activity for me is that it
offers students a chance to try and make sense of operations
with negatives) we re-write input and output as co-ordinates
and plot them all on the same grid. A discussion of what is the
same or different usually throws up many rich questions for
exploration; e.g. "Which rules give curved lines and which give
straight?"; "How do I get parallel lines?".
The power of the activity, for me, lies
in the scope it offers for students' choice and exploration; the
naturalness of the questions it throws up; the interest it tends to
generate and the range of curriculum areas students will be working
on (negatives, co-ordinates, substituting into formulae, indices,
writing and interpreting algebra, not to mention all the using and
applying issues). This wide range of topic areas allows me to
work happily on this problem with a class for a month. This is a
problem I will work on with classes from year 7 to year 13 Further
Mathematicians (who by then are on to graphing rational functions).
Year 8 students can work on gradients of curved lines (having
sorted out y=mx+c) and deriving gradient functions in the same room
as other students grappling with how to label axes and add a
negative number. There is also, of course, a natural link in to
using ICT and graphical calculators. In the end, what I like most
about this problem is its familiarity; I have worked on it with
every class I have ever taught. Rather than this becoming
monotonous, it seems to mean that each new class goes further than
the one before, as I become more and more attuned to what students
say and to what possibilities there are for exploration. And I
imagine this is true for everyone's 'favourite problems' - in the
end it does not matter so much what they are as what use you have
made of them.
Alf Coles is the Head of
Mathematics at Kingsfield School in Gloucestershire,
UK.