The Notes that accompany a problem frequently take a particular 'line' to emphasise one set of mathematical insights, skills, and opportunities which might emerge from time spent with the problem. Naturally there is no reason to assume that this is the only rich route through the intellectual space occupied by the problem. The 'line' taken here, in these Notes, is to emphasise the connection between visualisation and variables.
Ask the group to draw a circle of diameter 10cm and mark the diameter.
Direct students to pick a point on the diameter, other than the circle centre, and use that point as the centre of an inner circle which just touches the first circle, before creating a second inner circle to fill the remainder of the large circle's diameter.
Invite estimates for the area of all three circles, and allow discussion and justification of those estimates until a rough consensus is achieved.
Now pose the problem. Perhaps by simply showing the problem page and asking students to work in pairs, first to understand the challenge posed and then to think of a way to pursue the required result. Allow any method that has some validity, encouraging students to pursue their method through to a result and then to share their justification for their approach.
Try to ensure that by the end of this time all students have a sense for the larger inner circle expanding and contracting. Almost filling the outer circle before shrinking back to only half its diameter. Check that they can see that the space around the two inner circles can be almost nothing up to the two inner circles combined. And with this visualisation active ask them to consider the area of the different regions within the figure.
Secondly help students to see that this is a problem about proportion and that taking the initial diameter as the unit of length and the diameter of the larger circle as the controlling variable reduces the challenge to a one-variable problem (the smaller inner circle is defined by the size of its larger partner).
What is the challenge presented in this problem ?
How small and how big can the larger inner circle become ?
What portion of the outer circle's area will it contain at these extremes ?
Students who are not ready to engage with this problem directly should practice creating the diagram and calculating from their own measurements the areas of the three circles.