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Spaces for Exploration
Stage: 3 and 4
Article by Alf Coles
Alf wrote this article in 2006 in response to a request to choose a favourite problem.
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, as well as how to write about it, hard. 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:
$5 \to 12 $
$3 \to$ ... 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
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 \to \cdots$ and may have to explain I want someone to write the rule they were using.
Typically there will be many different rules:
$ x \to x \times 2 + 2 $
$x \to x + 1\times 2$
$x \to xx + 2$
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 $x^2$). 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.:
$x \to xx + 2$
$5 \to $
$3 \to $
$0 \to $
$-1 \to $
$-4 \to $
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.
In 2010, Alf Coles and Tracy Helliwell wrote an article
Kingsfield School - Building on Rich Starting Points
which elaborates on these ideas.
Transformation of functions
Functions and their inverses
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