Functional approach to equations and inequalities using ALNUSET (alien)

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebra and functions

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

  • function: x-value, function-value, variation
  • equation: solution, solving
  • inequation: solution, solving

Description show tooltip help

The main topic addressed in this scenario is “equations and inequations”. Since these notions are not new for the Grade 10 students, the aim is to propose remedial activities allowing to approach the notions related to this topic, namely the notions of equality between two expressions and solution of an (in)equation, since numerous research works report about students’ difficulties to grasp these notions. The hypothesis underpinning the scenario is that Alnuset, offering dynamic representation of the relationship between x and f(x), respectively independent and dependent variables, as well as the possibility to represent this relationship in different registers, will contribute to a better conceptual understanding of these notions.

The scenario is organised around three units. In the first unit, the students explore the new environment through initial activities about the notion of function and functional (in)equation. The second unit is devoted to exploring whether 2 expressions are equivalent or not. The last unit leads the students to “discover” different techniques for solving equations and inequations with Alnuset. The activities take advantage of the Alnuset potentialities to represent dynamically relationships between expressions and to articulate a representation of algebraic expressions on algebraic line and their graphical representation. Thus, ALNUSET will be also used to introduce solving equations and inequalities from both, algebraic and graphical points of views.

Rationale show tooltip helpexplode

In the Grade 10 class, the notion of function plays an important role and is linked with the notions of equation and inequation. In his PhD thesis, Erdogan (2006) [1] calls this "algebraic functional site of the Grade 10 class". The following excepts from the mathematics curricula [2] show the relationship to be developed between functions and equations:

"activities related to functions, equations and inequations will emphasize information provided by the form of an expression and will prompt searching for an appropriate form" (p. 34)

"the functional perspective can enrich the reflexion on the process of solving equations. These remarks concern even more solving inequations since the solution set in almost never reduced to a single value." (p.16).

Alnuset provides three representations of a function linked to each other: algebraic (algebraic expression), dynamic (point on the algebraic line in Algebraic line component) and graphical (curve in the Cartesian plane component). Thus, it allows to approach the notions of function and (in)equation from three different points of view whose articulation can foster conceptualisation of these notions.

Theoretical framework show tooltip help

First, anthropological theory of didactics (Chevallard, 1992) [1] is applied to an analysis of both the textbook used in the experimental Grade 10 class and of Alnuset, aiming at identifying praxeologies (types of tasks, techniques that can be used to solve these types of tasks, and technologies providing a justification of the techniques) related to the notions of function, equation and inequation existing in these two institutions.
Second, our educational goal is to help students conceptualise mathematical notions. We refer here to Vergnaud’s approach to conceptualisation, which situated knowledge within an action rather than within a theory (Vergnaud, 1996) [2]. He distinguishes between operational form of a knowledge, which is used in action of problem solving, and predicative form of a knowledge, oral or written, which shows only a part of the former form.

[1] Chevallard, Y. (1992). Concepts fondamentaux de la didactique : perspectives apportées par une approche anthropologique, Recherches en didactique des mathématiques 12(1), 73-112.

[2] Vergnaud, G. (1996). Au fond de l’action, la conceptualisation, in J.M. Barbier (Ed.): Savoirs théoriques et savoirs d’action (pp. 275-292), Paris: PUF.

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

For the reasons mentioned in the rationale, Alnuset appears as an appropriate tool to approach the notions of function, equation and inequation in the Grade 10 class.

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years old students

Student prerequisites show tooltip help

The students have to be familiar with the vocabulary related to the notions of function (x-value, function-value, domain, codomain, variations, curve...), equation and inequation (solution, solve...). However, since the pedagogical plan proposes remedial activities, all these notions will be revisited in the new environment of Alnuset.

Teacher prerequisites show tooltip help

Grade 10 mathematics.

Alnuset environment, particularly Algebraic line and Cartesian plane

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with 1 or 2 students per computer.

Institutional context show tooltip help

The contents at stake in the pedagogical plan are part of the French mathematics curriculum for the Grade 10 class. Link to the related document.

Goals show tooltip helpexplode

Curricular goals show tooltip help

Functions:

  • Identify a variable and its domain of definition for a function defined by a curve, a table of values or an expression;
  • Determine, in each case, the image of a given number.
  • Describe the behaviour of a function defined by a curve.

Equations, inequalities:

  • Solve an equation or an inequation of the degree 1;
  • Solve an inequation by using a sign table.
  • Solve graphically equations and inequations of the type: f(x)=k; f(x)<k; f(x)=g(x); f(x)<g(x)...

 Link to the curriculum

Content-epistemological goals show tooltip help

Conceptualize a function as a dynamic relationship between two variables.
Construct relationships between algebraic and graphical representations of a function.
Conceptualize the notion of a solution of an equation or an inequation.

 

Social-affective goals show tooltip help

Develop the abiblity of

  • working collaboratively;
  • participating in class discussion.

Instrumental goals show tooltip help

Get familiarized with Alnuset, in particular with its Algebraic line and Cartesian plane components.

Learn instrumental techniques allowing to solve proposed activities.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The pedagogical plan consists of three units. The first unit is conceived to allow exploring two functions: square and inverse. The reasons for the choice of these functions are twofold:

  • in the French Grade 10 curricula, these are the first functions that are studied;
  • studying these functions will allow to get familiarized with the tools that will be needed in the subsequent units related to equations and inequations solving.

Solving simple functional equations and inequations, involving square and inverse functions, will complete the study of these functions.

The second unit aims at visualizing with Alnuset the equivalence of two expressions.The aim is twofold:

  • help the students grasp the notion of identity as an equality, which is true for any value of the variable;
  • approach the notion of domain of a function.

The last unit aims at solving equations and inequations with Alnuset. Didactical variables are set in a way to  allow, in Example 1, the technique based on horizontal movement of the variable on the algebraic line, and prompt, in Example 2, the use of techniques based on commands E=0, respectively a<b.

ALNUSET

ALNUSET (ALgebra on the NUmerical SETs) is a new dynamic digital artefact devoted to the learning and teaching pre-algebra and algebra. It includes three components: Algebraic Line , Algebraic Manipulator and Cartesian Plane. Algebraic line component allows dynamic represention of algebraic expressions on a number line highlighting a functional relationship between variables. In Cartesian plane component, graphical representations of expressions can be shown in articulation with Algebraic line. Finally, Algebraic manipulator allows transforming expressions by applying rewriting rules, either pre-defined or defined by a user. The three components are inter-related.

In this pedagogical plan, the Algebraic manipulator is not considered.

 

Tool access show tooltip help

NIL

Algebraic line [Component]

The description below is extracted from the pedagogical plan "Equality and equivalent notions in treating algebraic expressions and equations (familiar)" by G. Chiappini et al.

Algebraic line component provides the opportunity to use letters as names of mobile points on the line. It provides geometrical models for graphically constructing sums and differences, products and ratios, powers and roots of the numbers, letters and expressions already represented on the line.

Every new expression constructed in this way is associated to a point on the Algebraic line that indicates the result of operations performed in sequence. The algebraic line is based on a representation built by mathematicians over previous centuries and is distinct from the number line, which is of a profoundly different nature. Indeed, the very nature of the Algebraic Line is algebraic, while the very nature of a number line is arithmetic. This algebraic nature is given by operational and representational features, i.e. by the fact that in ALNUSET this artefact is able to:

  • manage mobile points corresponding to algebraic variables;
  • manage the construction of literal expressions involving variables and their representation as points on the line;
  • manage the dragging of mobile points corresponding to algebraic variables and the movement of the points of the expressions involving those variables.
Cartesian plane [Component]

Cartesian plane component allows visualizing a graphical representation of algebraic expressions previously defined in Algebraic line component. The point (x, f(x)) is represented on the curve with the current value of x, so that when x is dragged on the algebraic line, the point on the curve moves accordingly.

Work plan show tooltip helpexplode

Setting show tooltip help

The setting described below takes into account material constraints of the experimental class.

The first session will take place twice with a half class. In both half-classes, the students will work in pairs on a computer during one hour. One worksheet per pair of students will be gathered at the end of the session. Interactions of a few pairs of students will be audio recorded.

The second session will take place with the whole class and will last for two hours. The students will work on computer in pairs following the instructions on a worksheet. The same pairs of students will be audio recorded.

Time show tooltip help

3 hours

Actors' roles show tooltip help

Students:

  • solve the assigned tasks
  • provide written answers to questions asked
  • participate to classroom discussions

Teacher:

  • supervise the students' work
  • manage class discussions
  • manage synthesis and institutionalization phases

Observer:

  • observe the students' work
  • take field notes

 

What to do and how show tooltip help

The pedagogical plan proposes activities organised into three units:

  1. Studying functions
  2. Comparing expressions
  3. Solving equations and inequations

The units and activities are further described in parts of the plan related to these units. 

Exploring functions

Identity show tooltip helpexplode

Subject domains show tooltip help

Topics show tooltip help

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

Based on the considerations described in the following section (cf. theoretical framework), we hypothesize that grasping the notion of function requires be able to:

- perceive a function as a relation of dependence between variable quantities;

- dissociate the notion of function and its graphical representation.

Thus, in the pedagogical plan, consistently with this hypothesis, functions under study are first explored in the Algebraic line component, where the dynamic relationship between a variable x and an expression dependent on x can be observed. The notions of image, pre-image, domain and co-domain of a function are approached, as well as variations of functions are studied. Graphical representation of the functions in Cartesian plane component is proposed later, in connexion with Algebraic line component.

 

 

Theoretical framework show tooltip help

Numerous research works report about difficulties encountered by students with the notion of function. Il appears that a large part of them are due to obstacles of both epistemological and didactical nature (Chauvat 1999) [1]  as will be shown in what follows.

Epistemological considerations

Historically, the notion of function took a long time to fully develop. First, the notion of curve appeared in Greek mathematics. Their notion of curve, mainly as a geometric locus, served as a tool for solving geometric problems until Descartes (17th  century). Descartes starts considering a curve as a set of points characterized by a distinctive property, called symptom, which can be expressed by means of an algebraic relation. A symptom emphasizes an algebraic relationship between the involved quantities, but conceals the functional relationship of dependency between them. With Leibniz, Newton and Bernoulli, the notion of function becomes a mathematical tool, and later, with Euler in the 18th century, it becomes an object of study. New types of problems, such as searching for minimums or maximums or study of tangents to curves, gave rise to the development of differential and integral calculus and led to the notion of numerical function where two conceptions coexist: function=analytical expression and function=curve. As Sierpinska (1992) [2]  points out, they are both obstacles to the general notion of function. Indeed, the conception function=curve (i.e., its graphical representation) does not allow to perceive a dynamic process the function models: one quantity varies in relation with variations of an other one. The perception of this relational dynamics is eclipsed by the statics of a finished drawing.

Didactical considerations

The teaching of functions in the French schools seems to reinforce this obstacle (Chauvat 1999) [1]. In fact, students first encounter curves and functions through the example of linear function linked to the study of proportional relationship between quantities (Grades 7 and 8). The general notion of function is taught later (Grade 10) and is based on the notion and use of graphical representations, which leads to assimilation by the students of a function with its curve.

Theoretical frameworks

First, anthropological theory of didactics (Chevallard, 1992) [3] is applied to an analysis of both the textbook used in the experimental Grade 10 class and of Alnuset, aiming at identifying praxeologies (types of tasks, techniques that can be used to solve these types of tasks, and technologies providing a justification of the techniques) related to the notions of function, equation and inequation existing in these two institutions.

Second, our educational goal is to help students conceptualise mathematical notions. We refer here to Vergnaud’s approach to conceptualisation, which situated knowledge within an action rather than within a theory (Vergnaud, 1996) [4]. He distinguishes between operational form of a knowledge, which is used in action of problem solving, and predicative form of a knowledge, oral or written, which shows only a part of the former form.

[1] Chauvat, G. (1999), Courbes et fonctions au collège, “Petit x” 51, 23-44.

[2] Sierpinska, A. (1992), On understanding the notion of function. In The concept of function: Aspects of epistemology and pedagogy, MAA Notes n°25, 25-58.

[3] Chevallard, Y. (1992). Concepts fondamentaux de la didactique : perspectives apportées par une approche anthropologique, Recherches en didactique des mathématiques 12(1), 73-112.

[4] Vergnaud, G. (1996). Au fond de l’action, la conceptualisation, in J.M. Barbier (Ed.): Savoirs théoriques et savoirs d’action (pp. 275-292), Paris: PUF.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Grade 10

Age range show tooltip help

15-16 years old students

Population description show tooltip help

Student prerequisites show tooltip help

The students have to be familiar with the vocabulary related to the notions of function (x-value, function-value, domain, codomain, variations, curve...), equation and inequation (solution, solve...). However, since the pedagogical plan proposes remedial activities, all these notions will be revisited in the new environment of Alnuset.

Teacher prerequisites show tooltip help

Grade 10 mathematics.

Alnuset environment, particularly Algebraic line and Cartesian plane

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with 1 or 2 students per computer.

Institutional context show tooltip help

The contents at stake in the pedagogical plan are part of the French mathematics curriculum for the Grade 10 class. Link to the related document.

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Curricular goals show tooltip help

  • Identify a variable and its domain of definition for a function defined by a curve, a table of values or an expression;
  • Determine, in each case, the image of a given number.
  • Describe the behaviour of a function defined by a curve.

 

Content-epistemological goals show tooltip help

Conceptualize a function as a dynamic relationship between two variables.
Construct relationships between algebraic and graphical representations of a function.
Conceptualize the notion of a solution of an equation or an inequation.

 

Cognitive goals show tooltip help

Social-affective goals show tooltip help

Develop the abiblity of

  • working collaboratively;
  • participating in class discussion.

Instrumental goals show tooltip help

Get familiarized with Alnuset, in particular with its Algebraic line and Cartesian plane components.

Learn instrumental techniques allowing to solve proposed activities.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Square function

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Inverse function

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

Based on the considerations described in the following section (cf. theoretical framework), we hypothesize that grasping the notion of function requires be able to:

- perceive a function as a relation of dependence between variable quantities;

- dissociate the notion of function and its graphical representation.

Thus, in the pedagogical plan, consistently with this hypothesis, functions under study are first explored in the Algebraic line component, where the dynamic relationship between a variable x and an expression dependent on x can be observed. The notions of image, pre-image, domain and co-domain of a function are approached, as well as variations of functions are studied. Graphical representation of the functions in Cartesian plane component is proposed later, in connexion with Algebraic line component.

 

 

Theoretical framework show tooltip help

Numerous research works report about difficulties encountered by students with the notion of function. Il appears that a large part of them are due to obstacles of both epistemological and didactical nature (Chauvat 1999) [1]  as will be shown in what follows.

Epistemological considerations

Historically, the notion of function took a long time to fully develop. First, the notion of curve appeared in Greek mathematics. Their notion of curve, mainly as a geometric locus, served as a tool for solving geometric problems until Descartes (17th  century). Descartes starts considering a curve as a set of points characterized by a distinctive property, called symptom, which can be expressed by means of an algebraic relation. A symptom emphasizes an algebraic relationship between the involved quantities, but conceals the functional relationship of dependency between them. With Leibniz, Newton and Bernoulli, the notion of function becomes a mathematical tool, and later, with Euler in the 18th century, it becomes an object of study. New types of problems, such as searching for minimums or maximums or study of tangents to curves, gave rise to the development of differential and integral calculus and led to the notion of numerical function where two conceptions coexist: function=analytical expression and function=curve. As Sierpinska (1992) [2]  points out, they are both obstacles to the general notion of function. Indeed, the conception function=curve (i.e., its graphical representation) does not allow to perceive a dynamic process the function models: one quantity varies in relation with variations of an other one. The perception of this relational dynamics is eclipsed by the statics of a finished drawing.

Didactical considerations

The teaching of functions in the French schools seems to reinforce this obstacle (Chauvat 1999) [1]. In fact, students first encounter curves and functions through the example of linear function linked to the study of proportional relationship between quantities (Grades 7 and 8). The general notion of function is taught later (Grade 10) and is based on the notion and use of graphical representations, which leads to assimilation by the students of a function with its curve.

Theoretical frameworks

First, anthropological theory of didactics (Chevallard, 1992) [3] is applied to an analysis of both the textbook used in the experimental Grade 10 class and of Alnuset, aiming at identifying praxeologies (types of tasks, techniques that can be used to solve these types of tasks, and technologies providing a justification of the techniques) related to the notions of function, equation and inequation existing in these two institutions.

Second, our educational goal is to help students conceptualise mathematical notions. We refer here to Vergnaud’s approach to conceptualisation, which situated knowledge within an action rather than within a theory (Vergnaud, 1996) [4]. He distinguishes between operational form of a knowledge, which is used in action of problem solving, and predicative form of a knowledge, oral or written, which shows only a part of the former form.

[1] Chauvat, G. (1999), Courbes et fonctions au collège, “Petit x” 51, 23-44.

[2] Sierpinska, A. (1992), On understanding the notion of function. In The concept of function: Aspects of epistemology and pedagogy, MAA Notes n°25, 25-58.

[3] Chevallard, Y. (1992). Concepts fondamentaux de la didactique : perspectives apportées par une approche anthropologique, Recherches en didactique des mathématiques 12(1), 73-112.

[4] Vergnaud, G. (1996). Au fond de l’action, la conceptualisation, in J.M. Barbier (Ed.): Savoirs théoriques et savoirs d’action (pp. 275-292), Paris: PUF.

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

Solving functional equations and inequations

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

f(x)=k

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

f(x)<k

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

f(x)=g(x)

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

f(x)<g(x)

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Comparing expressions

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Solving equations and inequations

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Solving equations

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Example 1

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Example 2

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Solving inequations

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Example 1

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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

Example 2

Identity show tooltip helpexplode

Authors show tooltip help

Jana Trgalova, Hamid Chaachoua, MeTAH-LIG, Grenoble, France

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

France

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