At the very beginning, the teacher should quickly show how one could accomplish the tasks of the *Familiarization Worksheet *(PP "familiarization") using a data projector. The objectives are:

- to show Casyopée features to those students who possibly did not succeed to complete the
*Familiarization Worksheet* in the previous section (or who were not present); - at the same time, to briefly remind each student the functioning of Casyopée; and
- to show how to display the graph of functions in Casyopée main window (there is not any specific task on that in the
*Familiarization Worksheet*).

The teacher distributes the worksheet containing the *"optimization problem 1".*

Before starting the problem solving session the text of the
problem is read aloud. If needed, the meaning of some terms
may be discussed and clarified, e.g. "given a triangle..." or "rectangle inscribed in a
triangle".

As remarked in the field "description" of the "rationale" (above): the problem is not posed in the frame of the
Cartesian Geometry, and the problem is formulated in general terms:
"given a triangle...".

That raises the need for the teacher to
guide a first "modelling" of the geometric situation described: from a
Euclidean coordinates-free frame to the Cartesian-like frame as
implemented in Casyopée.

With that respect the meaning of the expression "given a triangle..." has to be negotiated in the class, if not shared yet.

A
seemingly natural choice is to approach the problem in general terms,
that is to choose 3 free points in Casyopée Geometric Window as
vertices of the considered triangle. The consequence of such a choice
would be the construction of a model depending on at least 6
independent variables (the coordinates of the vertices), but Casyopée
supports only the construction of an algebraic model depending on one
variable.

The teacher could:

- (Recommended option)
anticipate this problem and initiate a class discussion for choosing 3
fixed points as vertices of the triangle (the same for anyone). This
choice on the one hand limits the richness of the problem, but on the
other hand makes easier the comparison among the students' solutions in
the following discussion. One could orient the discussion and take a
triangle with a side lying on one of the axes. This last option would
prepare the terrain for the "generalization" of te problem in the PP
"didactical cycle 2"
- anticipate the problem and ask(let) each
group of students to choose 3 fixed points as vertices of its own
triangle. This option could lead to a variety of geometric situation
and possible models.
- anticipate nothing and let students
possibly face the impossibility of constructing the desired algebraic
model. Once students face this impossibility, one could suggest them to
choose 3 fixed points as vertices of their triangle. This option would
be rewarding for the richness and the complexity which would emerge,
but it is also highly demanding for students and teachers and
time-consuming.

If option 3 is chosen, one can hypothesize that quite soon students will begin to meet one of the
following difficulties:

- the geometric construction of the inscribed rectangle is not
stable with respect to the dragging of the vertices of the
triangle;
- the system cannot handle the algebraic model because there are
too many algebraic variables.

The teacher's intervention is needed to face and
overcome these difficulties. The teachers could either interact with each group of students separately, or start a collective discussion (preferred option, but it would require that students experience the same difficulties nearly at the same time).

Be they "private" or "collective", the discussions with students should aim at analyzing the
source of the experienced difficulties and at constructing
different strategies for overcoming them, namely to reduce the number of
variables by taking a particular instanciation of
the geometric situation.

The question how to deal algebraically with the generality
expressed in the text of the problem will be reconsidered in
the second didactic cycle.

At the end, students are assigned as homework to produce individual written reports (Report2_request). Those students who
ended the problem solving session can (begin to) write down
their reports in the classroom, the others will be asked to produce
their reports at home.