Why do this problem?
This gives an interesting challenge in which students develop their
understanding of functions and skills at approximation and
This is a quick problem in which students should be encouraged to
try out function evaluation for various choices of numbers.
Remember, that the key point is whether the function is positive or
negative: we don't need to evaluate the exact values. Students
should be encouraged to focus on whether the function is positive
or negative, rather than computing the exact values.
- What do we know about the values of a function either side of a
- Can you think of functions for which this sort of approach
might not work?
- Does this method tell us anything about the number of roots of
Could students create an algorithm (i.e. recipe or clear
sequence of steps) to solve this problem for other functions? Can
they clearly express their algorithm so that someone else could
apply it for a function of their choice?
Note that at university this sort
of idea is extended in courses on Analysis, and this result is
called the Intermediate Value Theorem. It is actually a very useful
and powerful mathematical idea.
Suggest that students try key values of $0.5, 1, 1.5$ and so on.
Give them calculators.Suggest that they tabulate the results of