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WHAT I SUPPOSED TO KNOW

  • What I supposed to know

    I have been working with Spanish and Dutch students since 2012. I always  appreciated their will to learn, the flexibility of their competences  and the desire to be part of our community.

    Once I could read the Maths tests that my first Dutch student had to sit.

    For all these reasons I accepted to prepare a lesson for a multinational class to be delivered at Quercus school at the end of September 2015.

    What lesson

    I told Cristina Vallini, the Italian coordinator, that I could prepare a lesson on geometry where the main tool to be used was Geogebra, a freely-downloadable software for dynamic geometry. She replied that the idea could be good since one of the themes of the Erasmus plus Project was the use of technologies in teaching practice.

    I wanted to try geometry & Geogebra even if I knew that the students I would meet, and the Italian students too, didn’t know anything about the usage of Geogebra and what kind of problems could be solved through that tool. I hoped that the Italian students could get some expertise in a couple of hours’ drill. I chose particularly simple problems too.

    I supposed that the following problems could be solved in ninety minutes’ time:

    1. The length of the perimeter of an isosceles triangle is 12 and the perperdicular distance of the vertex to the base is 3. Determine the length of the sides of the triangle.

     

    1. The length of the base AB of an isosceles triangle ABC is 10. Let CD the altitude to the base. If the length of CD+BC is 9, determine the length of CD.

     

    1. Determine the area of a right-angled triangle where the length of one of the two (mutually perpendicular) sides is 10 and the length of the altitude to the hypotenuse is 6.

     

    1. Determine the area of a right-angled triangle ABC  where AB is the hypotenuse, CD the altitude to the hypotenuse, AD the projection of AC on AB, DB is the projection of CB on AB,  AC-AD=9.6 and DB=25.6

     

    A slight surprise

    On the bus to Salamanca, where the groups from Italy and Holland met, I realized that the students from the Low Countries were much younger than those I met in Italy. The Spanish students too were quite young.  So I tried  to understand if what I prepared was too difficult. I concluded that it definetly was.

    I made the previously chosen problems be preceded by a set of simpler ones and a visual glossary.

    The following warm-up constructions were then given:

    Determine the hypotenuse of a right-angled triangle where the lengths the two (mutually perpendicular) sides are 8 and 4 respectively

     

    The base of an isosceles triangle is 4 and the altitude to the base is 9. Determine the lengths of  the congruent sides.

    Determine the area of a right-angled triangle where the length of one of the two (mutually perpendicular) sides is 7 and the length of the the hypotenuse is 10.

    The lengths of the sides of a triangle are 4, 5 and 7 respectively. Determine the area of the triangle.

     

    The lesson

    Then the lesson began. The students formed multinational groups,  I showed the first construction and let things flow, observing their work and the communication within each group, giving some advice.

    Some groups managed to solve all the warm-up constructions; the second part laid untouched.

    Ninety minutes’ time wasn’t enough for absolute beginners.

    MASSIMO PAMINI