France (English)

(Artigue, Le Feuvre, Lagrange, Meyrier Didirem)

Subject domain: Mathematics, Multi disciplinary approach,

Topics: 3D geographical position coordinates, 3D vector spherical coordinates, vector addition, Logo programming,

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Artigue, Le Feuvre, Lagrange, Meyrier Didirem

English

France

Cruislet Scenario

Global plan :

The global plan is to use Cruislet in the frame of a specific didactic design: that of TPE. TPE is a multidisciplinary project work that students prepare in small groups during the first half of the academic year in grade 11.

Three sessions are devoted to classroom work directed by the teacher. Students learn how to use Cruislet and to solve problems posed by the teacher.

Then they are are invited to design their own project, possibly involving the use of Cruislet.

3D deplacements and their representations are not a common subject in mathematics classes. This topic can provide students with a complementary understanding of mathematical notions like vectors, coordinates, trajectories...

It is also an opportunity to show the support that mathematics brings to other subject like geography and physics. It gives students another view of mathematics.

The themes are consistent with our concerns:

1. for the epistemological value of students' activity: we expect students to have a wider view on the notion of vector, including involvement of this notion into "real life" situations,

2. for students instrumental genesis: before entreing into a self designed project, students learn with their teacher the basic operation of the software and solve problems posed by the teacher.

TPE is a multidisciplinary project work that students prepare in small groups during the first half of the academic year in grade 11. Themes for TPE are freely chosen by the students but they have to be related to national themes. These are very general and thus do not constrain very much possible choices. Two hours per week are devoted to TPE, and TPE are supervised generally by two teachers from different disciplines.

11th grade, scientific stream

16, 17

Average, scientific stream

Vectors (adding, multiplying)

Basic knowledge about geographic coordinates, constraints of an air trip and of the movments of a camera.

The teacher should be ready for an interdisiciplinary project work, not necessarily linked to knowledge identified in the curriculum.

computer laboratory

videoprojector

TPE (see above)

avrage

Consistently with the choice of the TPE context, the goals are mainly methodologicial:

providing students with the opportunity of developing a multidisciplinary approach of questions which are not just school questions,

developing the abilities necessary for an effective search, selection and critical analysis of documentary resources,

helping students to mobilise their academic knowledge in such a context,

widening students ' intellectual curiosity,

Establishing more open relations between teachers and students.

developing students' autonomy,

helping them to acquire methods and the competencies required for working in groups,

The goal is to promote Cruislet as a tool for autonomous exploration of questions giving sense to notions like deplacments, vectore, coordinates

Outline of the session :

a) Collective presentation of the software using a video-projector. The collective presentation will explore the main characteristics of the DDA:

virtual exploration of Greece, location of important historical and touristic places by scrolling the 3D map and zooming in/out,

creation of an avatar, and presentation of the different existing modes for moving this avatar (entering a final position in (lat, long, height), a vector displacement in spherical coordinates or a city name),

presenting the camera system, and looking for reasonable parameters for it,

exploiting the interrelation between representations (for instance for getting the coordinates of a particular place),

exporting the displacement of an avatar into a Logo procedure.

b) Collective programming of a first trip with one escale, for instance a flight from Athens to Samos, with a stop in Mikonos or a circular flight (Athen, Iraklio, Rodes, Athen). Programming should first be done by absolute positions, then by deplacements. Fi could be 0 in a first step, then vary to produce a change in altitude.

c) Small group work for preparing variations of the initial trip.

d) Collective discussion: listing the questions raised this first activity and the solutions found to these if any by the groups.

Ouline of the session:

a) Collective discussion: coming back to the questions raised at the first session if necessary.

b) Small group work: each group completes at least one travel.

c) Collective discussion and synthesis : How to prepare a trip? What data are necessary? How to get these? How to program a trip? What has been learnt about the different commands?

d) Small group work on a new problem. Adding a turn around Olympe Mount

e) Collective discussion: Comparing the strategies used. How to make a circular trip at a given altitude? What to change to make an helicoidal trip? And to make a spiral trip at a given altitude?

Outline of the session:

a) Small group work on a new problem. Someone has prepared a flight from Athens to Heraklion. The altitude of cruise is 2000m. Program the flight and its visualisation. Unfortunately the programmer has not taken into account the wind. The wind comes from North-West and its force is 40km/h. The cruise speed of the plane is 200km/h. What is the real trajectory of the plane?

b) Collective discussion: identifying the difficulties met and the strategies developed. How to model the wind effect? How to multiply vectors by scalars with Cruislet?

c) Small group work on the inverse problem: a pilot has prepared a flight from Athens to Heraklion airport. He has planned that after 45mn of flight he would be above Milos Island. Visualise the flight. He arrives at Milos Island only after 40mn of flight. What is the direction and speed of the wind? How to correct the estimate hour for arrival at Heraklion airport?

e) A variant of this problem: a pilot has prepared a flight from Athens to Heraklion airport. He has planned that after 45mn of flight he would be just above the centre of Milos Island At the estimated time, he is above a small island, north east of Milos. How to correct the trajectory? Visualise the flight without and with correction.

Possible themes for project work:

- Programming a trip with given conditions in terms of places to be visited for a travel agency (with or without landing) and its visualisation. The program has to optimize fuel consumption (in order to fit better with the TPE national themes). The program can be more or less sophisticated according to the variables taken into account in the modelling process of the flights.

- Simulating plane acrobatics involving one or two avatars.