Approaching functions with Casyopee (familiar)

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 apsect 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.

TDS

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.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Grade 11 students in a scientific stream

Age range show tooltip help

16-17 year 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).

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equiped with computers and video-projector. No internet connection necessary.

Institutional context show tooltip help

Senior high school. Grade 11. Scientific stream.

Socio-cultural context show tooltip help

Nothing special

Goals show tooltip helpexplode

Curricular goals show tooltip help

1. Associated functions 

2. Different expressions of the second grade polynomial

3.  Problem linking geometrical and algebariac aspects

4 .  Using of dynamical deometry software

See more on french curriculum here:

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

Cognitive goals show tooltip help

The scenario aim to help students to construct or enrich knowledge on two  aspects:
    1. algebraic function
    2. geometry

More specifically :
as for the notion of function, students should 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

  • 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 gemetric context.

Social-affective goals show tooltip help

Working by pairs

Instrumental goals show tooltip help

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

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The scenario uses Casyopée to allow:

- a balance between the consolidation of old knowledge and the introdction 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 

• Viability of the tasks in the Casyiopée environment

I.

Work plan show tooltip helpexplode

Setting show tooltip help

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.

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 functions

Identity show tooltip helpexplode

Authors show tooltip help

DIDIREM TEAM

Subject domains show tooltip help

algebre

Topics show tooltip help

functions

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

associated functions

  • differente expressions of a function

Description show tooltip help

three sessions about associated functions and different algebraic expressions of second grade polynomials

Rationale show tooltip helpexplode

In the french curiculum, 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.

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.

TDS

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.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Let f be a function, the associated functions are the following

h(x)= f(x)+a   ;   g(x)= bf(x)  ;   j(x)= f(x+c).

In the french curriculum, students are supposed to recognise the geometric transformation which permits to pass from the graphic of f to the graphics of the associated functions;

We use Casyopée to explore and/or to valid such tasks.

Population show tooltip helpexplode

School level show tooltip help

Grade 11 students in a scientific stream

Age range show tooltip help

16-17 year 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

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).

Context show tooltip helpexplode

Physical context show tooltip help

Classroom equiped with computers and video-projector. No internet connection necessary.

Institutional context show tooltip help

Senior high school. Grade 11. Scientific stream.

Socio-cultural context show tooltip help

Nothing special

Goals show tooltip helpexplode

Curricular goals show tooltip help

associated functions

different algebraic expressions  of second grade polynomials.

Cognitive goals show tooltip help

using affine transformation to interpret the role of the parameters in the functions associated 's expression.

Instrumental goals show tooltip help

to take in hand the alegebra part of Casyopée.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Three sessions

1. Introduction: Collective presentation of associated functions and Casyopée algebra part

2. Targetted function: individual trianing

3. introduction to diffents expressions of the second grade polynomial by using indivdually casyopée with the same process than in the prvious sessions.

Work plan show tooltip helpexplode

introduction

Identity show tooltip helpexplode

Authors show tooltip help

DIDIREM TEAM

Subject domains show tooltip help

Topics show tooltip help

    Language show tooltip help

    English

    Country show tooltip help

    France

    Keywords show tooltip help

    associated functions

    Description show tooltip help

    regular classroom, assocaited function, introduction to Casyopée (part algebra)

    Rationale show tooltip helpexplode

    Students have no knowledge on Casyopée

    Associated function are in the curriculum

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

    Population show tooltip helpexplode

    School level show tooltip help

    Grade 11 students in a scientific stream

    Age range show tooltip help

    16-17 year 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

    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).

    Context show tooltip helpexplode

    Physical context show tooltip help

    classroom eqipped with a computer for the teacher and a videoprojector

    Institutional context show tooltip help

    Senior high school. Grade 11. Scientific stream.

    Socio-cultural context show tooltip help

    Nothing special

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    associated functions

    Instrumental goals show tooltip help

    introduction to Casyopée (algebraic part)

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    vidéo projecteur, tableau blanc inter actif, casyopée

    Tool access show tooltip help

    NIL

    Work plan show tooltip helpexplode

    Setting show tooltip help

    Professeur manipule avec Casyopée

    Définit une fonction de référence  f(x) = x^2  + montrer son graphe

    Créer 3 paramètres

    Définit les fonctions associées a f(x); f(x) +  b; f(x+c)

    Travail papier crayon:

    les élèves anticipent les graphes pour certaines valeurs du paramètre.

    Un élève vient le corriger sur le tableau blanc

    Validation collective par le professeur à l’aide de Casyopée projeté sur le tableau blanc.

    Refaire l’exercice avec d’autres valeurs du paramètre; correction par un élève Sherpa

    2ème partie: problème inverse

    La fonction de référence est f(x) = x^2

    et le graphe de g(x) = 2x^2 +3 est caché.

    Résolution collective!

    Phase 1. Teacher using Casyopée: 

    1. Defines a reference function f(x) = x^2 and shows its graphic representation
    2. Creates 3 parameters
    3. Defines associated functions a f(x); f(x) + b; f(x+c).

    Phase 2 :The students with paper and pencil

    anticipate  the graphics reprsentations for some parameters values

    Phase 3 : A student come to the numeric board and draw his/her solution. Collective validation the teacherusing Casyopée.

    Phase 4 Same task, validation with a student (sherpa) using Casyopée.

    Phase 5 :  reciprocal problem : we know the graphic and we have to find the algebraic expression of the function (targetted function).

    reference function is f(x) = x^2

    tagretted function is :g(x) = 2x^2 +3

    Collective solving

    Time show tooltip help

    55mn

    Actors' roles show tooltip help

    Etape 1:
    Le prof manipule Casyopée. Les élèves anticipent papier/crayon

    Etape 2: manipulation par élève sherpa pour la validation.

    Etape 3 le prof fait un bilan en insistant sur la nécessité d'un travail organisé sur la variation des paramètres.

    targeted functions

    Identity show tooltip helpexplode

    Authors show tooltip help

    DIDIREM TEAM

    Subject domains show tooltip help

    Topics show tooltip help

    Keywords show tooltip help

    Description show tooltip help

    Rationale show tooltip helpexplode

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

    Population show tooltip helpexplode

    School level show tooltip help

    Grade 11 students in a scientific stream

    Age range show tooltip help

    16-17 year 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

    Students have already assist to the introduction session

    Teacher prerequisites show tooltip help

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

    Context show tooltip helpexplode

    Physical context show tooltip help

    laboratory class, students are working by pari on a computer equpped with Cassyopée.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    associated functions, reviewing on geometrical transformations.

    Cognitive goals show tooltip help

    Réinvestir les connaissances mathématiques présentées en séance 1

    •  toward a first distinction between variable and parameter  
    • using geometric transformations to interpret the movment of the graphs 
    • to manipulate a first expression of a second grade polynomial

    Instrumental goals show tooltip help

    Casyopée: Première appropriation individuelle de Casyopée.

    individual manipulation of algebraic part of Casyopée

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Work plan show tooltip helpexplode

    Setting show tooltip help

    séance en salle info
    un ordinateur par binome

    Ressources :

    Fichiers casyopée avec fonction cachée

    Fiche élève à ramasser à la fin.

    Tâches

    Activité cible 1

    Le fonction  h: x (x – 5)² – 2 (donnée dans le fichier Casyopée par son graphe) est cachée.

    Les élèves ont à créer les fonctions f et g et à grapher :

    Fonction de référence donnée f : x x²

    Fonction à créer g: x f(x+k) + a

    En manipulant les paramètres (entiers précisé dans la fiche) ils doivent superposer le graphe de g sur celui de h.

    Tache papier crayon:

    Donner la valeur des paramètres solution

    Interpréter quand c’est possible l’effet du changement du paramètre

    Pilotage de a, puis de k, puis de l

    Activité  cible 2

    Fonction cachée h : x -(x– 3)²

    Fonction de référence donnée f : x x²

    Fonction à créer g: x a f(x+k)

    Pilotage de a, puis de k

    Même tâche

    Activité Cible 3

    H(x)= -(x + 1)²  + 3

    Fonction de référence donnée f : x x²

    Fonction à créer g: x a f(x+k) + m

    Students are working by pair on Casyopée. They have 3 Casyopée files and a paper guideline to fill

    They have 3 tasks to do, all of them "targetted functions" as shown in the previous session. That is they ahve a graphic representation o a function and its algebraic representation with parameters suc as a(x+k)^2. They have to try and guess the adequate value of the parameters. They are notified that the parameters are integer 

    Task1

    The hiden function is   h(x)= (x – 5)² – 2 (donnée dans Students ceate reference function : f(x)= x² and targetted function g (x)= f(x+k) + a.

    they have to find the value of a and k

    Task 2

    Hidden Function h ( x)= -(x– 3)²

    Reference Function f(x)= x²

    Targetted Function g(x)= a f(x+k)

    Task 3

    Hidden function h(x)=-(x + 1)² + 3

    Reference Function f(x)= x²

    Targetted Function g(x)= a f(x+k)+m

    Task paper and pencil

    Note the right expression of each hidden function

    interpret , when it is possible, the signification of the parameter in term of geometric trabsformations.

    Time show tooltip help

    55mn

    Actors' roles show tooltip help

    Les élèves effectuent les tâches avec Casyopée et remplissent la fiche élève

    Le prof aide les élèves individuellement et fait un bilan.

    students have 3 casyopée task and 1 paper and pencil task (see setting for more details.

    If enough time an interactive game is organised (guess my function)

    teachers help students individually for Casyopée manipulation as for mathematical reflexion.

    What to do and how show tooltip help

    The teacher orchestrates the classroom activity. The students in pairs on exercises indicated on the guideline  The teacher interrupts the students’ work when necessary in order to organise a short whole class discussion.

    After each activity, a whole class discussion and a synthesis phase are foreseen

    different expressions of a function

    Identity show tooltip helpexplode

    Authors show tooltip help

    DIDIREM TEAM

    Subject domains show tooltip help

    Topics show tooltip help

    Keywords show tooltip help

    Description show tooltip help

    Rationale show tooltip helpexplode

    it is in the curriculum

    it is the following of the two previous sessions

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

    Population show tooltip helpexplode

    School level show tooltip help

    Grade 11 students in a scientific stream

    Age range show tooltip help

    16-17 year 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

    students have assist to the 2 previous sessions, in particular they know :

    • associated functions
    • the process of the targetted function,
    • how to manipulate the algebraic part of Casyopée

    Teacher prerequisites show tooltip help

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

    Context show tooltip helpexplode

    Physical context show tooltip help

    class laboratory, students are working by pair on Casyopée

    Video projectof for task 3.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    In the curriculum students are asked to be familiar with three expressions of the second grade polynomial:

    • factorised form a(x-u)(x-v)
    • canonical form (x-a)^2 +k
    • developped form ax^2+bx+c

    Cognitive goals show tooltip help

    Séance introductive à l'écriture du polynome sous différentes formes.

    Le principe est le même qu'en séance 2 : une fonction cachée est donnée les élèves doivent trouver son expression sous forme factorisée et sous forme canonique. Ils doivent en outre interpréter, quand c'est possible, le rôle des paramètres.

    Introductive session to the 3 forms of the second grade polynomial.

    toward an interpretation when it is possible of the different parameters.

    Instrumental goals show tooltip help

    consolidation des connaissances de manipulation de Casyopée côté algèbre.

    reviewing manipulation of the algebraic part of Casyopée

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Work plan show tooltip helpexplode

    Setting show tooltip help

    Tâche 1

    Fonction cachée h(x) = 2x^2-4x-6 donneé dans un fichier Casyopée par son graphe

    Les élèves doivent créer trois fonctions de la forme

    f(x)=a(x-d)^2+e     g(x)=a[(x-k)^2+m] et p(x)=a(x-u)(x-v)

    Il y a plusieurs méthodes possibles : soit les élèves créent les trois fonctions avec des paramètres et font varier les paramètres pour retrouver le graphe de h, soit ils le font pour une seule et retrouvent les deux autres par des calculs algébriques.

    Tâche 2

    Effectuer la même tâche quand c'est possible avec les fonctions

    h(x)=-5x^2+4x-1 et h(x)=3x^2+2x+6

    Tâche 3 :

    discussion collective : que représentent géométriquement les paramètres dans chacune des formes, quand c'est possible.
    Pourquoi ça ne marche pas toujours  (introduction au discriminant).

    Task 1
    same process than previously but with 3 targetted functions

    • hidden function : h(x) = 2x^2-4x-6
    • Targetted functions: 

    f(x)=a(x-d)^2+e     g(x)=a[(x-k)^2+m] et p(x)=a(x-u)(x-v).

    There are 2 possible methods :

       - to do the same process (manipulate the parameters as far as the graphics of the hidden function and the targetted function are the same) for each function

       - to do the same process for one function and find the others by algebraic manipulations (with or with out Casyopée).

    Task2

    find , when it is possible, the 2 others forms of the following functions
    h(x)=-5x^2+4x-1 et h(x)=3x^2+2x+6
    (NB Here no hidden functions to induce algebraic methods)

    Determine the abcisse of the extremum of the function

    Task 3

    Collective dicsussion and validation , institutionnalisation of the 3 forms, interpretation when it is possible of the parameters.

    Time show tooltip help

    55mn

    functions and geometry: variables and equations

    Identity show tooltip helpexplode

    Authors show tooltip help

    DIDIREM TEAM

    Subject domains show tooltip help

    • Mathematics

    Topics show tooltip help

    functions et geometrie

    Language show tooltip help

    English

    Country show tooltip help

    France

    Keywords show tooltip help

    area, variable, function, distance

    Description show tooltip help

    Two one hours'sessions on the choice of a good variable in Casyopée among acceptable variables, the work in Casyopée and the links beetween its results and paper and pencil work, the different work on parameters and variables.

    One introduction session on the geometrical part of Casyopée.

    One session about the notion of variable and the importance of the choice of a good variable.

    Rationale show tooltip helpexplode

    Autonomy on the functional modelling of different situations is  asked from students (choice of variables, interpretation of the obtained results in the situational context...). This session is devoted to the choice of variables.

    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.

    TDS

    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.

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

    Students are asked to produce a construction in the geometric module "bonnet d'ane". A mobile point  M have to be fixed by the students so that an area is equal to an other area.

    The situation change according the choice of parameters and variables.

    The students are supposed to construct abilities in the choice of good variables.

    Population show tooltip helpexplode

    School level show tooltip help

    Grade 11 students in a scientific stream

    Age range show tooltip help

    16-17 year 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).

    Context show tooltip helpexplode

    Physical context show tooltip help

    Classroom equiped with computers and video-projector. No internet connection necessary.

    Institutional context show tooltip help

    Senior high school. Grade 11. Scientific stream.

    Socio-cultural context show tooltip help

    Nothing special

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    to use a  dynamical geometric software

    to model a geometriacl situation

    Content-epistemological goals show tooltip help

    Cognitive goals show tooltip help

    enrich the concept of variable

    distinguish variable from parameter

    model a geometrical situation

    reinterpret the algebraic result in the geometrical context

    reviewing simple area calculus

    Instrumental goals show tooltip help

    manipulation of geometry part of casyopée

    reviewin manipulation of the algebraic part  of casyopée.

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Introduction :
    Consolidating knowledge on simple geometrical situations.
    Introduction to Casyopée’ geometrical module and to the transition to the symbolic module.

    Application:
    dividing a triangle into figures of fixed area

    Work plan show tooltip helpexplode

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

    Identity show tooltip helpexplode

    Authors show tooltip help

    DIDIREM TEAM

    Subject domains show tooltip help

    • Mathematics

    Topics show tooltip help

      Language show tooltip help

      English

      Country show tooltip help

      France

      Keywords show tooltip help

        Description show tooltip help

        Collective session involving simple problems solving in modelling geometry . it permits to shows How Casyopée works and why y may be useful

        Rationale show tooltip helpexplode

        Target show tooltip helpexplode

        Rationale show tooltip helpexplode

        Population show tooltip helpexplode

        School level show tooltip help

        Grade 11 students in a scientific stream

        Age range show tooltip help

        16-17 year 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

        Thales theorem and simple formula of arera calculus ( triangle and parallelogram)

        Teacher prerequisites show tooltip help

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

        Context show tooltip helpexplode

        Physical context show tooltip help

        a computer and a video projector for the teacher

        Institutional context show tooltip help

        Senior high school. Grade 11. Scientific stream.

        Socio-cultural context show tooltip help

        Nothing special

        Goals show tooltip helpexplode

        Curricular goals show tooltip help

        simple situation of modelling geometry and dynamic geometry

        Instrumental goals show tooltip help

        introduction to the geometry part of Casyopée and the link between algebraic and geometric parts.

        Specifications show tooltip helpexplode

        Rationale show tooltip helpexplode

        to show what may be the advantage of using a dynamic geometry software linked with a CAS

        Work plan show tooltip helpexplode

        Setting show tooltip help

        Définir deux paramètres a \in [0,10] et b \in [0,10]

        Placer A(a,0) et B(a,b) et C(0,b) et un point M dans le plan.

        Et alors, on en fait quoi collectivement ?

        Time show tooltip help

        2h

        Actors' roles show tooltip help

        Phase 1

        The students do task 1 (For one cut, one takes the line between a vertex and the middle of the opposite side. One can also cut the line from a vertes perpendicular to a side at a ration of √2. With two cuts, I being the middle point of [AB] one cuts in OI and IA.

        Collective discussion with Casyopée. The teacher shows how to enter coordinated points, the middle point, the other lines and the geometrical calculations.

        Phase2

        The students do tasks 2. Task 2.1 is easy because M has to be the middle point. Task 2.2 is less easy. Students will become aware of Casyopée’s help for solving equations.

        The teacher shows how to create a free point on a segment, then a student will operate Casyopée’s geometrical module then create a function to show a solution for task 2.1.

        for task 2.2, a student operates Caysopée for an exploration and, if possible, gives a solution. For task 2.3, the teacher explains the rest of the procedure, particularly about the variable (choice, feedbacks)

        for task 2.3, the solution is similar. The teacher shows how to create points with parametric coordinates.

        He announces the next session on this topic.

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

        Identity show tooltip helpexplode

        Authors show tooltip help

        DIDIREM TEAM

        Subject domains show tooltip help

        • Mathematics

        Topics show tooltip help

        Keywords show tooltip help

        Description show tooltip help

        Rationale show tooltip helpexplode

        Target show tooltip helpexplode

        Rationale show tooltip helpexplode

        Population show tooltip helpexplode

        Context show tooltip helpexplode

        Goals show tooltip helpexplode

        Specifications show tooltip helpexplode

        Rationale show tooltip helpexplode

        Work plan show tooltip helpexplode

        Setting show tooltip help

        Problem In orthogonal normed axis, create A(0, a); B (b ; 0) et C (a; b). (a and b being parameters in [0,10]). How to choose a point M in order that triangle BMC’ area is the third  of rectangle ABCD’s ?

        Task 1 Draw the figure

        Task 2 Create a geometrical function representing the area of MNPQ

        Free exploration, moving M

        Task 3 (to be explicited)

        Students have to fill in a table with choices for the variable and the two feedbacks before and after the definition of a function. The goal is to make students aware that xM and yM are variables, and that yM is a suitable variable whereas xM is not.

        Task 4 Solve the problem in the symbolic module

        Task 5 Back to the geometric module for the interpretation.

        Task 6 Double check by moving M.

        Further work

        1 Draw from the previous work a method to cut the rectangle into three aqual area figures:a triangle and two trapezes.

        2. Same problem without help : the area has to be the fifth or by replaing the triangle by a trapeze (to be checked with Casyopée in case of horizontal parallel sides)

        Time show tooltip help

        2h

        Actors' roles show tooltip help

        Phase 1

        The students  do the task;

        The teacher helps them individually

        Phase2  about 40mn before the end of the session. Discussion. The goal is to highlight and compare the different expressions of the function depending on the choice of the variable.

        function and geometry:optimisation

        Identity show tooltip helpexplode

        Authors show tooltip help

        DIDIREM TEAM

        Subject domains show tooltip help

        • Mathematics

        Topics show tooltip help

        function, geometry, optimization

        Language show tooltip help

        English

        Country show tooltip help

        France

        Keywords show tooltip help

        function, geometry, area, variable, optimization,

        Description show tooltip help

        A Two hours'sessions to reviewing Casyopée uses in a solving geometric problem.

        Rationale show tooltip helpexplode

        Autonomy on the functional modelling of different situations is  asked from students (choice of variables, interpretation of the obtained results in the situational context...). This session is devoted to all this work.

        More over, this session use all the work done before because the factorised form (view in sessin 3) is necessary to find the maximum of a function.

        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.

        TDS

        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.

        Target show tooltip helpexplode

        Rationale show tooltip helpexplode

        Population show tooltip helpexplode

        School level show tooltip help

        Grade 11 students in a scientific stream

        Age range show tooltip help

        16-17 year 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

        a convenient familairity with the two part (geometric and algebraic) of Casyopée)

        Teacher prerequisites show tooltip help

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

        Context show tooltip helpexplode

        Physical context show tooltip help

        Classroom equiped with computers and video-projector. No internet connection necessary.

        Institutional context show tooltip help

        Senior high school. Grade 11. Scientific stream.

        Socio-cultural context show tooltip help

        Nothing special

        Goals show tooltip helpexplode

        Curricular goals show tooltip help

        problem solving of geometrical optimisation.

        Content-epistemological goals show tooltip help

        Cognitive goals show tooltip help

        The scenario aim to help students to construct or enrich knowledge on two  aspects:
            1. algebraic function
            2. geometry

        More specifically :
        as for the notion of function, students should 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

        • 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 gemetric context.

        Social-affective goals show tooltip help

        Working by pairs

        Instrumental goals show tooltip help

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

        Specifications show tooltip helpexplode

        Rationale show tooltip helpexplode

        Work plan show tooltip helpexplode

        Setting show tooltip help

        Problème

        • a, b et c, positive parameters. A (-a, 0) et B (0, b) et C(c, 0). Find a rectangle (MNPQ) with M on [OA], Q on [OC], N on [AB] and P on [BC] with a maximal area ?

        Free problem solving, drawing on the work in session 5 (definition of geometrical function) and of session 3 (completing the square)

        Further work

        How to choose M in order that MNPQ will be a square?

        Time show tooltip help

        2h