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'Golden Thoughts' printed from http://nrich.maths.org/
Why do this problem?
offers opportunities to experience several key
problem-solving processes including making a useful transformation
into a simpler task and making use of algebra to represent the
situation before interpreting results as meaningful solutions
within the problem context.
Any teaching approach always involves a choice about the level of
problem-solving process students are to experience. For example
this problem could be used as a guided example of algebraic
modelling, but the problem might equally successfully be presented
as a simple diagram and students left to make their own progress
with the question over time. Both approaches are valid, but the
outcomes for the student will be very different with each.
Taking a route between those two approaches :
Ask a student to explain what the problem requires, then ask
the group for any thoughts about where to start or what might work.
Students may choose to work on specified (and simple) rectangle
dimensions, hoping to generalise later once they have some results.
This is a good and valid strategy.
Before long, try to catch someone questioning whether the rectangle
dimensions actually matter. This is a key moment and it is worth
stopping the group to allow discussion. We can work with the unit
square (which simplifies the task) because areas are only required
to match, and any stretch (to create a rectangle) scales all areas
by the same factor. This transformation of the problem into a
simpler problem is a key mathematical moment. Although students who
can see how to proceed without this reduction should be allowed to
It is possible to use a spreadsheet to explore the change in
Through questions and discussion guide students into nominating an
independent variable (for example the distance from R to X), and
use this to go on to expressing the areas of interest in terms of
that variable. If this is pursued algebraically a quadratic
equation emerges with two distinct solutions only one of which fits
the context being modelled.
Key questions :
- What does this problem require ?
- Where might you start - what might work ?
- Would it help to fix on one specific rectangle, or is there
another way to start ?
If students have settled on using the Unit Square :
Possible extension :
For more on algebraic modelling try
For more reasoning about triangle areas try Triangle in
More extension work on many other problems relating to the golden
ratio, and articles on the subject, are collected in the package
Possible support :
is still a challenging task but might make a good 'warm up'