Levels of difficulty of an exercise template following the constraints on parameters

The same wording with parameters can generate mathematical exercises with different level of difficulty.

For example, let us consider the generic wording :
Complete the equality: √a √b - √c = √ where a, b and c are positive integers.

Following the constraints associated to the parameters a, b, c of this wording, we obtain different exercise templates that we can order by increasing level of difficulty:

Constraints

Example and solution

a square, b prime, b=c

√16 √3 - √3 = 4√3 - √3 = 3 √3

a prime, b not square, c prime, a*b=k^{2}*c

√2 √6 - √3 = √2 √2 √3 - √3 = 2√3 - √3 = 1 √3

a not prime not square, b not square, c prime, a*b=k^{2}*c

It is thus very important that teachers can specify the constraints relative to the parameters in order to precise the level of difficulty of the exercises issued from the same template.

The constraints on the table above show clairly that it is not possible to define the parameters in a sequential way: a, b and c are in fact interdependant.
IMS QTI 2.1 doesn't allow the definition of interdependant parameters; we have thus proposed extensions to the specification.