FAMILIAR : VERSION 4: Approaching functions with Casyopee

Identity show tooltip helpexplode

Authors show tooltip help

DIDIREM TEAM

Subject domains show tooltip help

  • Mathematics

Topics show tooltip help

  • Functions, Geometry

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • associated functions, geometric functions, optimization, parameters, semiotic registers

Description show tooltip help

The plan proposes a succession of tasks exploiting the potential a priori offered by Casyopee for approaching and studying the notion of function, and especially:

- the role played by functions for solving problems arising from geometrical situations,

- the role played by parameters for studying family of functions and accessing generalization.

Specific importance is given to the construction of tasks where students can choose different variables for exploring functional dependencies  and to the connection between algebra and geometry. This connection is supported in Casyopee by geometric expressions which allow to express magnitudes in a symbolic language mixing geometry and algebra. Moreover, according to the choices made for the independent variable, the resulting algebraic expression of functional dependence automatically produced by Casyopee can be of very different complexity. The scenario aims at exploiting these didactical functionalities of Casyopee and the different associated feedbacks, coherently with the theory of didactic situations.

The instrumentalisation process is initiated in each phase of the scenario through a collective phase orchestrated by the teacher, which also serves as a devolution phase for the type of task which is considered.

The scenario is built around three main types of tasks :

- finding targetted second grade functions by acting on parameters (five different tasks according to the semiotic forms used for these functions),

- functional modelling of a geometrical situation for solving a problem of relationships between areas,

-functional modelling of a geometrical situation for solving an optimization problem.

 

Rationale show tooltip helpexplode

Consolidating the approach of functions already developed in grade 10 is a main aim of the first term in the grade 11 curriculum before the teaching of calculus begins. 

1) Specific emphasis has to be put on the systematic study of "associated functions", on the diversity of semiotic forms possibly attached to the same function, and on the geometrical interpretation of algebraic transformations.

2) More autonomy on the functional modelling of different situations is also asked from students (choice of variables, interpretation of the obtained results in the situational context...).

The use of Casyopee , which provides feedbacks and a link between algebraic an geometric functions permits to work these aspect in an innovative way.

 

Theoretical framework show tooltip help

 

Anthropological theory of didactics (Y. Chevallard)

the dialectics between ostensive and non-ostensive will be helpful to deal with semiotic aspects of functional dependency. The experimentation has also to support the general goals of the mathematics teaching at this academic year. Problems of cultural or institutional compatibility, as well as ergonomic difficulties, had to be avoided in the building of these situations.

Theory of didactical situations 

In the building of these situations, in coherence with the TDS, we will be especially sensitive to the potential of actions and retroactions offered by the ‘milieu’. This is dependent on representational characteristics of Casyopée but not only on these. The task itself and the way it is framed is essential.

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School level show tooltip help

Grade 11, students in a scientific stream

Age range show tooltip help

16-17 years old

Population description show tooltip help

 

Students aimed at are scientific students in grade 11. They have already some basic knowledge of functions and algebra (factoring and developing expressions, solving first grade equations). They have met affine functions, some exemplars of simple second grade, third grade and homographic functions. Approach has been mainly graphical and numerical. Functions have already been used for modelling geometrical situations but the modelling process was strongly guided. For instance, the independent variable was always given. They have only worked on particular examples non dependent upon parameters.

Student prerequisites show tooltip help

 

Some basic knowledge of functions and algebra as explained above, and of elementary geometry (calculation of areas, Pythagoras and Thalès theorem).

No specific prerequisite on Casyopee.

Teacher prerequisites show tooltip help

Familiarity with Casyopee, and management of classrooms in computer environment (both individual and collective orchestration).

 

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Physical context show tooltip help

Schools are equiped with computers classrooms and in a classical classroom, it is possible to use a videoprojector connected to a computer.

 

Institutional context show tooltip help

There is a pression from the institution to integrate ICT in the learning and teaching process. 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

1. Associated functions (phase 1)

"On travaillera à l'aide de grapheurs sur des familles de courbes représentatives de fonctions associées à deux fonctions données u et v : x --> u(x)+a ; x --> u(x+a) ; x ---> u(ax) ..."

2. Different expressions of the second grade polynomial (phase 1)

"On proposera plusieurs écritures d'une même fonction trinôme"

3.  Problem linking geometrical and algebariac aspects (phase 2 and 3)

"La problématique des lieux géométriques sera présente dans tous les paragraphes de géométrie (...) . On s'appuiera (...) sur une démarche d'analyse-synthèse"

4 .  Using of dynamical geometry software (phase 2 and 3)

"Les logiciels de géométrie dynamique serontutilisés pour visualiser certains lieux"

See more on french curriculum here:

http://www.education.gouv.fr/bo/2000/hs7/vol5mathsc.htm

 

Content-epistemological goals show tooltip help

 

Concerning the notion of function, students should construct and consolidate:

  • the meaning of variable
  • the distinction between variable and parameter
  • the meaning of function of one variable
  • the fact that a same function may have several algebraic expressions

As for geometry, students should construct and consolidate:

  • the ability to experiment and anticipate in front of a dynamic geometric situation
  • the ability to modelling a geometric situation by  a geometric then algebraic calculus
  • the ability to interpret an algebraic result in the geometric context.

Cognitive goals show tooltip help

The scenario aim to help students to construct and enrich knowledge on two  aspects:

    1. algebraic function

    2. geometry

 

Social-affective goals show tooltip help

Working by pair

 

Instrumental goals show tooltip help

 

Casyopée technique for entering functions, computing and optimizing areas, studying respective positions of curves.

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Rationale show tooltip helpexplode

 

The scenario uses Casyopée to allow:

- a balance between the consolidation of old knowledge and the introduction of new knowledge

- a dialectic between algebraic and dynamic geometric situation

Theoretical framework show tooltip help

 

TSD framework : a specific attention is turned on the bulding of situations and on the use of the feedbacks provided by Casyopée.

Instrumental approach : a specific attention is turned on the management of Casyopée by the students and the teachers

Anthropological theory of didactics : a specific attention is turned to the proximity of the designed tasks with the curicula and the viability these tasks in the Casyopée environment

Resources show tooltip helpexplode

ressource 2: targetted functions [Resource for students]
ressource 3: second grade polynomial functions [Resource for students]
ressource 4: areas of triangles [Resource for students]
ressource 5: cutting areas [Resource for students]
ressource 6: problem of optimization [Resource for students]
ressource 1 : associated functions [Resource for students]

Work plan show tooltip helpexplode

Setting show tooltip help

The scenario is built around three phase organised by three main types of tasks:

- phase 1 : finding targetted second grade functions by acting on parameters (five different tasks according to the semiotic forms used for these functions),

- phase 2 : functional modelling of a geometrical situation for solving a problem of relationships between areas,

- phase 3 : functional modelling of a geometrical situation for solving an optimization problem.

 

Time show tooltip help

Phase 1: 5h; Phase 2: 3h; Phase 3 : 2h

Process documentation show tooltip help

 

1. For the whole cohort of students (2 classes) : analysis of local and global assesment

2. For the two teachers involved in the experimentation :reflexive interview after each session of the scenario

3. Observation of each session by a researcher

4. For 4 to 5 students analysis (by software Noldus) of the recording of all each of their actions on Casyopée . The film obtained permits to "replay" the sequence.

associated function

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introduction

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targeted functions

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different expressions of a function

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functions and geometry: variables and equations

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Introduction (to divide a triangle in piece of fixed area)

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Application (to divide a rectangle in piece of fixed area)

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function and geometry: optimization

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