An experiment on the use of Apprenti Géomètre in junior highschool André VANDENBRUAENE

1. An experiment on the use of Apprenti Géomètre in junior highschool André VANDENBRUAENE CREM, Nivelles Institut Sainte-Marie, La Louvière, Belgium

2. Apprenti Géomètre (version 2) • Easycreation of geometricshapes • Movements : to slide, to turn, to return • Operations : to divide, to cut, to duplicate, • to merge • New features for the secondaryschool : • lines, parallel and perpendicularlines, strips, • arcs, sectors of circles, etc.

3. Standard shapes and free shapes • Standard shapes • cannotbedeformed • are gathered in families • welladapted to the learning of measurement of • geometrical magnitudes and fractions

4. Standard shapes and free shapes • Free shapes • canbedeformed • are gatheredaccording to theirgeometrical • characteristics • keeptheircharacteristicswhendeformed

5. The research (CREM, 2005-2007) « Measuring areas of usualgeometricshapes, from 10 to 14-year old » • Learning sequencesusing AG2, to come across • the concepts of area and measure of an area • Impact of AG2 on the waypupilsconceptualize • First part (10 to 12 year-old) : installing the area • concept and the formula for the parallelogram • Second part (12 to 14-year old) : building some • area formulas

6. The research (CREM, 2005-2007) • An epistemological frame • Qualitative perception • Quantification • Computation • A didactical frame • AG2 is an environment (Brousseau, 1998) requiring new • instrumental knowledge • Attention paid to the system of instruments (Rabardel, • 1999) and to the mixing of « paper and pencil » tasks and • manipulations withAG2 (AssudeangGelis) • A cognitive frame • Four ways of seeing (Duval, 2005) : « botanist », • « surveyor », « constructor » and « inventor » • Most of the time, in the classroom, only « botanist » and • « surveyor » • AG2 allows to develop the « constructor » and the • « inventor » ways of seeing

7. Main objectives • The area formulas for the commonpolygons • Updating and reconstruction work, to go beyond sole memorization • Giving new geometricalrepresentations (strips) • The area concept as a visualization and proof tool • « (…) le thème des aires est un outil performant et irremplaçable tant dans l’apprentissage de la démonstration géométrique, que pour une compréhension de ce qui dans un problème relève du géométrique, et de ce qui relève du numérique »

8. Someactivities • 1. Introduction to the software and geometryrevisions • A littlesurvey of the functions of AG2 • First tasks : reproducingdrawings, tesselations, cutting and merging • shapes, etc. • Twosecantstrips (searching for quadrilaterals)

9. Someactivities • 1. Introduction to the software and geometryrevisions • Parallelogramswith the same base, in the samestrip

10. Someactivities • 1. Introduction to the software and geometryrevisions • Write down all the shapesyousee… • Parallelogramswith the same base and the sameheight have the • same area

11. Someactivities • 2. Assemblingisometric triangles • Create a common triangle, duplicate it and • realize a quadrilateral • Parallelograms and deltoids

12. Someactivities • 2. Assemblingisometric triangles • Formulas

13. Someactivities • Assemblingisometrictrapeziums

14. Someactivities • The area of the rhombus • Create a rhombus and cutit up in order to realize a rectangle with the same area • Somemisunderstandings and troubles

15. Someactivities • The area of the rhombus • Otherideas • Formulas

16. Someactivities • 5. The area of a regularpolygon • Cut a regularpolygon in order to realize a quadrilateralwith the same area • A well-known solution • Solutions found by the children for the hexagon

17. Someactivities • The area of a regularpolygon • Solutions found by the children for the pentagon • Solutions found by the children for the octogon

18. Someactivities • The area of a regularpolygon • Some troubles

19. Someactivities The area of a regularpolygon : towards a formula

20. Someactivities The area of a regularpolygon : towards a formula

21. Someactivities • The area of a disk • Regularpolygons in a circle • Severalways to realizethis figure

22. Someactivities The area of a disk : the formula

23. Someactivities • Enlarging and reducingshapes • Working on a squaredgrid • Comparing the areas of similarshapes

24. Someactivities • Enlarging and reducingshapes • Completing the table (dimensions, scale, ratio of the areas) • Finding the relation between the scale and the ratio of the areas

25. Impact of AG2 • Attitude of the students • changed in a positive way ; computer wasfinallyseen as a toolallowing to work, not only to play • studentsincreasedtheirautonomy • Positive effects on the student’slearning • « It is question of the ability to see a geometricfigure, to decomposeit in constituent elements, to complete itbydrawingadditionallines, linesonecannot draw without havingimaginedthemmentally, in an abstract way. (...) The studentswhoworkedwithApprentiGéomètreseem to have acquired a bettergeometricsight, what is notastonishingaccording to the dynamiccharacter of the activitieswith the computer. » • « Duringthis experiment carried out by 12-year oldchildren, the impact of ApprentiGéomètremainlyshoweditself in twodirections : a bettergeometricintuitiondue to the familiaritywith the movements of the shapes, and the resultingperception of isometries ; a more active attitude in the face of a newsituation, of a problem to solve. The previous attitude is speciallyinteresting, as far as onetries to promoteanactiveway of learning, basedon the treatment of problematicsituations. »

26. Contact To download “ApprentiGéomètre 2” www.crem.be To download the final report of the research ( Impact du logicielApprentiGéomètre sur certainsapprentissages. Rapport de recherche 2005-2007.) www.enseignement.be/index.php?page=25995 andre.vandenbruaene@hotmail.com