IMPLEMENTATION STRATEGIES FOR “EQUATION GURU”

A User Friendly Intelligent Algebra Tutor

Senay Kafkas, Zeki Bayram

Computer Engineering Department, Eastern Mediterranean University

Famagusta, North Cyprus

Huseyin Yaratan

Educational Science, Faculty of Education, Eastern Mediterranean University

Famagusta, North Cyprus

Keywords:

Intelligent Tutoring Systems, Intelligent Learning Environments, Equation Solving.

Abstract:

We describe the implementation strategies of an intelligent algebra tutor, the “Equation Guru” (EG), which

is designed to help students learn the concepts of equation solving with one unknown. EG provides a highly

interactive and entertaining learning environment through the use of Microsoft Agents. It consists of two main

parts. The ﬁrst is the “Tutorial” part where students are guided through the steps of solving equations with

one unknown. The second, “Drill and Practice” part gives them a chance to practice their skills in equation

solving. In this part, equations are automatically generated by EG, and presented to the student. EG monitors

the student’s performance and adjusts the difﬁculty level of the equations accordingly.

1 INTRODUCTION

In the last decade, “computer-aided instruction”

(CAI) systems, which had the drawback of not be-

ing individualized to the learners’ needs and which

could not provide the same kind of attention that a

student receives from a human tutor, have been re-

placed with “intelligent tutoring systems” (ITSs). An

ITS is a complex software program that uses artiﬁcial

intelligence and pedagogical techniques to store and

use domain-speciﬁc knowledge, model the learner’s

behaviour, and tutor the student in its area of exper-

tise. The primary goal of an ITS is to achieve effective

teaching by emulating the behaviour of a human tutor.

It does so by “working” with the student on a one-to-

one basis, monitoring the student’s performance and

progress, making pedagogical decisions about how to

teach, and providing feedback and remedial material

when appropriate to the student.

ITSs have proven highly effective at increasing

students’ performance and motivation (Koedinger,

1998), (Martin et al., 2001), (Mayo, 2001), (Mitro-

vic et al., 2004). In this paper, we describe the im-

plementation strategies of an intelligent algebra tutor,

called Equation Guru (EG), that tutors in the domain

of equation solving with one unknown. EG is de-

signed for helping high school students at grade 8.

Microsoft Agents are used all throughout EG for

the interaction of the system with the student. These

animated characters can speak, make gestures, and

understand spoken words, although we have not used

speech understanding in EG due to the limited nature

of the speech understanding capability of the agents.

The agent both provides immediate feedback on the

current problem at hand, and also motivates the stu-

dent through encouraging words (e.g. “Well done,

Jane!”) and animations (e.g. clapping), leading to

a highly interactive learning environment.

EG consists of two main parts: The “Tutorial” part

and the “Drill and Practice” part. The goal of the

“Tutorial” part is to teach the student the concept of

equations, as well as the steps required to solve equa-

tions. It achieves this in an interactive manner with

Microsoft Agents and dialogs . The dialogs that are

used in this part are similar to the ones observed in a

real classroom setting. The “Tutorial” part is further

divided into sections. One of the sections includes a

game, based on the “beam balance” analogy. In the

other sections, step by step, the students are taught

how to solve equations and are prepared for the “Drill

and Practice” part. For the details of the “Tutorial”

part, the reader is referred to (Kafkas et al, 2005).

The “Drill and Practice” part runs in three modes.

The ﬁrst (main) mode makes use of the “Student Di-

agnostic” and the “Pedagogical” modules. It stores

the student’s actions and then makes pedagogical de-

cisions (like determining the level of next question)

according to the student model. In the second mode,

58

Kafkas S., Bayram Z. and Yaratan H. (2006).

IMPLEMENTATION STRATEGIES FOR “EQUATION GURU” - A User Friendly Intelligent Algebra Tutor.

In Proceedings of the Eighth International Conference on Enterprise Information Systems - AIDSS, pages 58-65

DOI: 10.5220/0002488500580065

Copyright

c

SciTePress

Figure 1: Overall architecture of the “Drill and Practice

part”.

the student can pose an equation to EG, view the step

by step solution and hear the explanation. In the third

mode, the student can choose a level, get an equation

from that level and try to solve it.

The rest of the paper is organized as follows. In

section 2, we describe in detail the implementation

of EG’s “Drill and Practice” part. This includes the

user interface, domain expert, student diagnostic and

pedagogical modules. In section 3, we compare EG to

other equation solving ITSs. Finally, in section 4 we

have the conclusion and future research directions.

2 THE “DRILL AND PRACTICE”

PART

The aim of this part is to enable students to drill and

practice based on what they have learned in the “Tu-

torial” part. The architecture of this part is given in

Figure 1, which in fact is common to most intelligent

tutoring systems. There are four components in this

architecture: “Domain Expert,” “Pedagogical,” “In-

terface” and “Student Diagnostic” modules. Further-

more, this part runs in three different modes. We ex-

plain the modes, as well as the modules, in the fol-

lowing sections.

2.1 The Three Modes of the “Drill

and Practice” Part

The ﬁrst mode, named “Let my Guru ﬁnd my level,”

is fully automated. It includes all the modules de-

picted in Figure 1. The equation to be solved and the

action to be taken next (such as whether hints will be

generated, or whether some encouragement will be

given) is determined by consulting the student model

and making use of pedagogical knowledge about the

tutoring process. Figure 2 is the screenshot of this

mode.

In the second mode, “I will specify the equa-

tion,” the student poses an equation to be solved, and

Figure 2: Mode 1 of the “Drill and Practice” part - “Let my

Guru ﬁnd my level”.

watches the system solve the equation in a step-by-

step fashion and hears the explanation of each step.

In the third mode, “Let me specify my level,” the

student can choose the level of difﬁculty of the equa-

tion to be generated, and the equation is generated at

the speciﬁed level of difﬁculty for the student to solve.

The three modes of the “Drill and Practice” part

provide a very ﬂexible environment in which students

can improve their equation solving skills.

2.2 The “Domain Expert” Module

The “Domain Expert” module is the “technical ex-

pert” of EG. It acts as the lexical analyzer, problem

solver and evaluator. Its job is to solve equations with

one unknown. It uses the “Domain Knowledge-Base”

which stores knowledge about solving linear equa-

tions with one unknown in the form of rules in the

Prolog programming language.

2.2.1 Inner Representation of Equations

The “Interface” module of EG is capable of represent-

ing equations as strings. But these strings should be

passed to the Prolog side and understood by the sys-

tem somehow. The “Domain Expert” module con-

verts string equations to their corresponding inner

representations. Table 1 represents corresponding in-

ner representations of terms and expressions.

Some examples of equations and their correspond-

ing representations are provided by Table 2.

In addition to this, the inner representation of an

equation is converted back into string form by the

“Domain Expert” module, since the equations are pre-

sented to the user end in the string form.

IMPLEMENTATION STRATEGIES FOR “EQUATION GURU” - A User Friendly Intelligent Algebra Tutor

59

Table 1: Inner representation of expressions.

Term/Expression Inner Representation

N, N is positive integer n(N)

N, N is negative integer nn(N)

x v(x)

-x nv(x)

Cx, C is positive integer coeff(C,x)

Cx, C is -ve integer ncoeff(C,x)

+ plus(Term1,Term2)

- Minus(Term1,Term2)

* Times(Term1,Term2)

/ div(Term1,Term2)

Table 2: Some examples of equations and their inner repre-

sentation.

Equation Corresponding inner representation

5+x=10 eq(plus (n(5),v(x)), n(10))

5-2x=10 eq(minus (n(5),coeff(2,x)), n(10))

-5-2x=-x eq(minus (nn(5),coeff(2,x)), nv(x)

3x/7 =10 eq(div(coeff(3,x)),n(7)), n(10))

2.2.2 Lexical Analysis

The Equation Guru’s alphabet consists of the follow-

ing set of symbols: { 0,1,2,3,4,5,6,7,

8,9,x,=,+,-,/}. If an entry that contains

some symbol that is not in this set, the system will

warn the student. Figure 3 demonstrates how lexical

analysis is done. The predicate analyse takes the

student’s entry as input and returns the result of the

lexical analysis (Status). Simply it checks if the

provided string can be parsed (by predicate a ) and

has any unknown, x (by hasX) or not. The status is

either ok or lexical error.

2.2.3 Correctness Analysis

If the student’s entry is lexically correct, then it

will be evaluated to check for correctness. Figure

4 depicts how the “Domain Expert” checks correct-

ness. The predicate evaluate takes student’s in-

put (Studinput) and EG’s input (Tutinput)

then gives the evaluation results. The variable

Correctness shows if the submitted step is cor-

rect or not and Solved shows if the equation is

solved (i.e. x=“something”) by the entered step.

When the student provides his/her solution step

Figure 3: Lexical Analysis.

Figure 4: Correctness Analysis.

(Studinput) in string form, evaluate gets

this input, then the predicate FindWhich returns

the inner representation of the equation (Tuteq1)

and the required step (Stepname) to move ahead.

Stepname will be used by evaluate in order to

determine if the given equation is solved by the sub-

mitted step or not. The correctness analysis is done by

the predicates solution and dogrumudur. The

predicate solution is called two times. The ﬁrst

one gets the student’s solution step (Studinput)

and returns the ﬁnal result (Studout). The sec-

ond one gets the equation asked at the beginning to

the student (Tutinput) and returns the ﬁnal re-

sult (Tutout). The predicate dogrumudur sim-

ply checks if the student’s input is a right step on way

going to the solution or not. If it is a right step then

the argument (Correctness) will be equal to 1,

otherwise to 0. Similarly, if the equation is solved by

the submitted step, then Solved will be assigned to

1, otherwise to 0.

2.2.4 Equation Solving

Equation solving is an iterative process. Appropri-

ate mathematical operations (like dividing, adding

like terms, balancing the equation, expanding etc.)

are applied on each step until value of the unknown

has been found. Figure 5 is a piece of code from

EG which shows how the “Domain Expert” mod-

ule solves the equations in a recursive manner, sim-

ulating the iterative process. Notice that the predi-

cate solve calls itself. It gets the equation (Equ)

in its inner form, applies the appropriate operation,

and then returns resultant equation (NE) and solution

(S). The predicate GetEqMakeStr gets the equa-

tion (Equ) and converts it to the string form. Af-

ter the conversion process, the predicate FindWhich

gets the equation in the string form and returns the

appropriate step (Stepname) and inner representa-

tion of the equation (Equ2). Once the appropriate

step is obtained, the predicate operation gets the

step (Stepname) and equation (Equ2) and then

returns resultant equation (NE) and solution (S).

The resultant equation (NE) will be used by the pred-

icate solve until a solution has been found.

2.3 The “Pedagogical” Module

The “Pedagogical” module is the one that runs the

show, making use of the services of the “Domain Ex-

ICEIS 2006 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS

60

Figure 5: Equation solving in EG.

pert,” “Student Diagnostic” and “Interface” modules.

2.3.1 Deciding on the Next Action

The “Pedagogical” module applies pedagogical

strategies, stored in the “Pedagogical Knowledge”

component, to the current situation at hand to decide

on the next action. Whenever the student submits a

solution step in the form of a revised equation, it asks

the “Domain Expert” whether the submitted equation

is equivalent to the previous one (i.e. if it has the same

solution as the previous one). If indeed it preserves

the solution, this time the “Domain Expert” checks

whether the student’s response represents a move in

the right direction toward solving the equation or not.

If the student has moved away from ﬁnding a solu-

tion by making the equation more complex, s/he is

warned, but allowed to continue trying to solve the

equation. However, if the equation entered by the

student does not preserve the solution to the origi-

nal equation, then the “Student Diagnostic” module

is notiﬁed, which determines the error and updates the

“Student Model” accordingly. In such a case, the ser-

vices of the “Interface” module are used to notify the

student of the error and display hint messages.

As an example, the hint message generated for the

equation 3x+2=1 is “balance the equation.” An exam-

ple of an error message is “you should have balanced

the equation” where the “Pedagogical” module deter-

mines that the student couldn’t balance the equation.

2.3.2 Generating Explanations

The “Pedagogical” module of EG is capable of pre-

senting a step by step solution to the student by in-

teracting with the “Domain Expert” module. The ex-

planation of solution steps is speciﬁc to the equation

under consideration.

In Figure 6, we see the Guru explaining a step in

the solution of an equation.

2.3.3 Equation Generation

Another job of the “Pedagogical” module is to gen-

erate the equations at a suitable level for the stu-

dent. The equations are randomly generated at a given

level. The complexity of the equations increases from

level 1 to level 35. Figure 8, given in the appendix,

depicts these levels.

Figure 6: Explanations presented to the student.

When a new student logs on the system, his/her

level is started at level 1. If the student has logged into

the system before, the system remembers his/her last

level. In both cases, if the student provides two con-

secutive correct answers, then the level is increased

by one. If the student makes 5 errors of any type (see

Table 3 for the error types) then his/her level will be

decreased by one. Otherwise the level will remain

same.

The ﬁrst 4 levels are in the form of a+x=b. The

ﬁrst level is the simplest one since the unknown will

be obtained as a positive integer. In the second level,

again the unknown is a positive integer but this time

the numbers (a and b) are bigger than the previous

level’s. In level 3, the result (x), is a negative integer

and similarly in level 4, bigger numbers are used.

From level 5 to 8, the equations are in the form of

a-x=b. In the ﬁfth and sixth levels, b is greater than a,

but is the rest up to 8, a is greater than b.

Beginning with level 9, the equations are created

without any regard to how big the number is. If the

student has reached to this level then s/he should be

comfortable with big numbers.

From level 9 to 12, the equations are in the form of

b+ax=c. In level 9 and 10 the numbers are chosen in

such a way that the unknown x will be obtained as a

positive and negative integer respectively. In level 11,

the result will be a positive fraction and in level 12 the

result will be a negative fraction.

From level 13 to 16, the equations are in the form

of b-ax=c. In levels 13 and 14, the x will be obtained

as positive and negative integer respectively. In level

15 the unknown will be positive fraction and in level

16, it will be negative fraction. After level 16, the

equations are generated without taking into account

the result of the unknown. Similarly, it is assumed

that if the student has managed to reach that level than

s/he is comfortable with fractions and positive/ nega-

IMPLEMENTATION STRATEGIES FOR “EQUATION GURU” - A User Friendly Intelligent Algebra Tutor

61

tive results.

For the equations in levels 17 to 22, the operation

“add like terms” is required. The difference between

these levels is the places of unknowns. For the equa-

tions in levels 23 to 31, the operation “cross multipli-

cation” is required. The complexity is increased by

generating equations with negative numbers and un-

knowns in each level.

The equations that belong to levels 31-35 require

“fraction addition.” Similarly the complexity is in-

creased by changing + signs to - signs.

2.4 The “Student Diagnostic”

Module

The “Student Diagnostic” module maintains a model

of the student. This involves keeping track of the stu-

dent’s current level of knowledge, his/her progress as

well as the average solving time, the number of ques-

tions directed, the number of correctly solved ques-

tions, the number of could not be solved questions,

the kinds of errors he/she has been making at each

level.

2.4.1 Error Processing

Table 3 depicts the error types in EG. “Lexical” errors

are coded as E1. Error codes from E2 to E11 are as-

signed to operational errors such as division, adding

like terms, simpliﬁcation, etc. The error codes are

used during the modeling process of the student.

Table 3: Error Codes and Error Types in EG.

Error Code Error Type

E1 Lexical Error

E2 Add like terms

E3 Balance the equation

E4 Division

E5 Simpliﬁcation

E6 Multiply both side by -1

E7 Solution

E8 Expansion

E9 Cross multiplication

E10 Find common denominator

E11 Add fractions

2.4.2 Error Analysis

If the student provides a wrong solution step, then

the system will investigate the error type. In Figure

7, the predicate FindError gets the provided so-

lution step from the student (Studinput), as well

as his/her last correct action (Previnput) and re-

turns the error type (Errorname). In order to ﬁnd

Figure 7: Error Analysis in EG.

the error type, the predicate FindWhich is called

two times. It gets the equation in string form in its

ﬁrst argument, and returns (in its second argument)

the equation in its inner representation form, and the

expected step name (in its third argument) that should

be applied in order to solve the equation. Lastly, the

predicate fe gets these step names and determines the

error type (Errorname).

2.5 The “Interface” Module

The “Interface” module consists of two parts. The

ﬁrst part is designed for students’ interactions and the

second part is designed for the instructor.

2.5.1 Student Interaction with EG

In a learning environment, life-like interaction is a

very important aspect. Early ITSs were implemented

without this feature. To remedy this situation ped-

agogical agents were developed. In computer-based

learning environments, pedagogical agents appear to

the students as animated characters and facilitate their

learning process by interacting with them (Shawn et

al., 1999). They achieve this by engaging the student

in a continuous dialogue, emulating the aspects of di-

alogue between a instructor and student, giving the

impression of being life-like and believable, appear-

ing to the user as natural, knowledgeable, attentive,

helpful, concerned, etc (Lester et al., 1997). The ef-

fect of pedagogical agents on students’ learning has

been investigated in (Lester et al., 1997), where it is

shown that life-like characters have a strong positive

effect on the learning process in an interactive learn-

ing environment .

EG uses Microsoft Agents as its main communica-

tion medium. These animated characters are able to

motivate the student in many ways. They congratu-

late the student for a correct move. They warn the

student in a friendly way in case of mistakes, without

discouraging him/her and guide him/her towards a so-

lution through helpful hints. Furthermore, the agents

display interesting characteristics and use body lan-

guage, both to entertain, and engage the student to the

equation solving process.

ICEIS 2006 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS

62

2.5.2 Instructor Interaction with EG

The second part of the “Interface” module is designed

as a tool for the instructors. They can initiate the

student model of a student and view their students’

progress. Initiating the student model of a student cor-

responds to recording his/her personal information.

3 COMPARISON OF EG WITH

OTHER EQUATION SOLVING

TUTORS

This section compares Equation Guru with other

equation solving tutors. For this comparison only

equation solving tutors in the domain of linear equa-

tions are considered.

Some equation solving tutors’ domains are re-

stricted to equations in some speciﬁc form. For in-

stance, Equation Solving Tutor (EST) (Ritter et al.,

1995) helps students in solving linear equations only

in the form of ax+b=c, while in E-Sit (Prince,2004),

the equations are only in the form of ax+bx=c. An-

other tutor, E-Tutor (Razzaq et al., 2004) supports

cross multiplication and expansion in the domain

of linear equations while AlgeBrain (Alpert et al.,

1999) has these features in the domain of linear and

quadratic equations. Different from these tutors, EG

supports all forms of linear equations, including equa-

tions with fractions. Furthermore, the format of equa-

tions is not restrictive - anything is accepted, as long

as it represents a valid equation.

Motivation and attracting the attention of the

learner are important aspects of the learning process.

In order to motivate the student, E-Sit uses a game.

The game window appears automatically whenever

the student gets a speciﬁc mark from the posed ques-

tions and the duration of the game depends on the stu-

dent’s success. AlgeBrain uses a character in its tutor-

ing process. Animation features of this character are

limited and it has no speech capability. EG uses Mi-

crosoft Agents for motivating and attracting the atten-

tion of the student which provides a highly effective

learning environment. These characters have a wide

range of animations and they can speak and easily en-

gage the student to the learning process.

E-tutor tutors in a dialog-based manner. Similarly,

the “Tutorial” part of EG is designed in this manner.

In all of the above mentioned equation solving tu-

tors, next problem selection is based on the informa-

tion available in the student model. EG also works

this way. In E-Sit, however, next problem selection

is based on a utility function which does not support

exactly student dependent tutoring.

AlgeBrain is a web-based and collaborative equa-

tion solving tutor while other tutors, including EG are

standalone applications.

Hint messages in Cognitive Tutor (Koedinger et al.,

2000) are generated sequentially to the student. It al-

ways provides a strong hint, by telling exactly what

to do at the end of the sequence. But this approach

contradicts with the principles of effective teaching

identiﬁed in (VanLehn et al., 1998). That is, the tu-

tor should not provide strong hints for the solution

of equations when students need them. If so, then

they may miss the opportunity to learn how to solve

equations when they are provided an answer and not

allowed to reason for themselves. EG generates an

appropriate hint message to the student when needed

but this message will never be strong.

In E-Sit, the next expected action (next step) from

the student is speciﬁc. For example, for the equation

3x-5=15, the next expected action is 3x=15+5. Also,

the name of the solution step (like transformation, ad-

dition, etc) must be submitted to the system by the stu-

dent. The correctness analysis of the student’s action

is based on this assumption. In such a system, if the

student submits a correct solution step ahead of the

expected one (x=20/3, for the above example), then

the ITS will consider this unexpected step as wrong

solution step. Therefore, the wrong evaluation will

yield a wrong student model. Furthermore, wrong

modeling will yield wrong pedagogical decisions and

strategies. In AlgeBrain, the next expected action is

a set of possible actions that can be applied to sim-

plify the considered equation. This discussion brings

us to a major point of strength in EG, when compared

to other equation solving tutors. In EG, the expected

solution step from the student is not speciﬁc. As long

as the student’s action results in an equation with the

same solution as the equation provided by the system,

it is assumed to be a correct move, and the student is

allowed to carry on. However, the student is warned if

his/her solution step takes him away from obtaining a

correct answer. Also, there is no need for the student

to provide the step name to the system.

Furthermore EG supports linear equations in a va-

riety of forms, including those with fraction additions

(this feature is missing in many other tutoring sys-

tems), and a blackboard at the right top corner of the

screen where solutions are displayed, creating a fa-

miliar medium, similar to a classroom learning envi-

ronment.

4 CONCLUSION AND FUTURE

RESEARCH DIRECTIONS

In this paper, we described the implementation strate-

gies of an intelligent tutoring system, called Equation

Guru (EG), which is designed to help high school

students at grade 8 with algebra. EG is an ITS that

IMPLEMENTATION STRATEGIES FOR “EQUATION GURU” - A User Friendly Intelligent Algebra Tutor

63

teaches cognitive skills needed for the solution of lin-

ear equations with one unknown. Equation Guru tu-

tors in a truly interactive manner. This is achieved

through the use of life-like animated characters, Mi-

crosoft Agents, which provide a unique, entertaining

experience to the users of the system.

There are two main parts in EG. In the ﬁrst (“Tu-

torial”) part, the students are taught how to solve lin-

ear equations with one unknown. Students are then

directed to the “Drill and Practice” part which pro-

vides three modes for students to practice their skills

in equation solving. This “Drill and Practice” part has

been the main focus of this paper.

Future work on EG includes adapting it to be Web

enabled in order to support distance education, and

adding robust natural language processing and under-

standing capability to it in order to make the experi-

ence of using it even more realistic and enjoyable.

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APPENDIX

A Difﬁculty Levels in Equation

Guru

Please refer to the next page for the table containing

the levels of difﬁculty of equation problems in Equa-

tion Guru.

ICEIS 2006 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS

64

Level Sample Equation General shape of Properties of

the Equation the Equation

1 5+x=6 a+x=b a <b,0 <a,b <10

2 12+x=16 a+x=b a <b,10 <=a,b <=20

3 5+x=3 a+x=b a >b,0 <a,b <10

4 17+x=14 a+x=b a >b,10 <=a,b <=20

5 7-x=8 a-x=b a <b,0 <a,b <10

6 12-x=16 a-x=b a <b,10 <=a,b <=20

7 4-x=1 a-x=b a >b,0 <a,b <10

8 13-x=11 a-x=b a >b,10 <=a,b <=20

9 10+2x=14 b+ax=c c >b, (c-b)/a is +ve integer,

1 <a <=20,0 <b,c <=20

10 19+4x=3 b+ax=c c <b, (c-b)/a is -ve integer,

1 <a <=20,0 <b,c <=20

11 7+6x=8 b+ax=c c >b, (c-b)/a is +ve fraction,

1 <a <=20,0 <b,c <=20

12 11+17x=4 b+ax=c c <b, (c-b)/a is -ve fraction,

1 <a <=20,0 <b,c <=20

13 12-4x=16 b-ax=c c >b, (c-b)/a is +ve integer,

-20 <=a <-1,-20 <=b,c <=20

14 17-10x=7 b-ax=c c <b, (c-b)/a is -ve integer,

-20 <=a <-1,-20 <=b,c <=20

15 3-5x=7 b-ax=c c >b, (c-b)/a is +ve fraction,

-20 <=a <-1,-20 <=b,c <=20

16 19-14x=7 b-ax=c c <b, (c-b)/a is -ve fraction,

-20 <=a <-1,-20 <=b,c <=20

17 12x-15x=19 ax+bx=c -20 <=a,b,c <=20

18 -6x+2x-8=7 ax+bx+c=d -20 <=a,b,c,d <=20

19 -2x-5-5x=-7 ax+bx+c=d -20 <=a,b,c,d <=20

20 11x=-19-13x ax=c+bx -20 <=a,b,c,d <=20

21 15-17x=-11x b+ax=cx -20 <=a,b,c,d <=20

22 -9+12x= 16- 10x b+ax=d+cx -20 <=a,b,c,d <=20

23 (2x+3)/5=1 (ax+b)/c=d 0 <c,d <10, 1 <=a,b <=20

24 (4x-9)/4= 1 (ax-b)/c=d 0 <c,d <10, -20 <=b <-1, 1 <=a <=20

25 (-8+16x)/6=-4 ((a+bx)/c=d -10 <c,d <10, -20 <=a <=20, 1 <=b <=20

26 (-1-14x)/-5= 3 (a-bx)/c=d -10 <c,d <10,

-20 <=b <-1, -20 <=a <=20

27 (-10x+4)/-9=x/2 (ax+b)/c=x/d -10 <c<10, -20 <=d,

1 <=b <=20 , a <=20

28 (-18x-6)/-7=x/11 ((ax-b)/c=x/d -10 <c<10,

-20 <=b <=-1,-20 <=a,d <=20

29 (13x+10)/5=(9x+ 6)/20 (ax+b)/c=(a

2

x+b

2

)/d -10 <=c<=10,

-20 <=a,a

2

,d <=20, 1 <=b,b

2

<=20

30 ((-8x+12)/10=(10x-15)/-2 (ax+b)/c=(a

2

x-b

2

)/d -10 <=c<=10, <=a,a

2

,d <=20,

-20 <= b

2

<=-1,1 <=b,b

2

<=20

31 (-8x-17)/3=(17x-11)/-10 (ax-b)/c=(a

2

x-b

2

)/d -10 <=c<=10,

-20 <=a,a

2

,d <=20, -20 <=b,b

2

<=-1

32 (-13x+19)/-4+(4x+10)/2=-9 (ax+b)/c + (a

2

x+b

2

)/d=f -10 <=c<=10,

-20 <=a,a

2

,d,f <=20, 1 <=b,b

2

<=20

33 (-4x+11)/-3-(15x+8)/4=-1 (ax+b)/c - (a

2

x+b

2

)/d = f -10 <=c<=10,1 <=b,

-20 <=a,d,f <=20,b

2

<=20, -20 <= a

2

<-1

34 (2x-8)/-10+(15x- 8)/10= 5 (ax-b)/c + (a

2

x-b

2

)/d = f -10 <=c<=10, -20 <=d,f <=20,

-20 <=b,b

2

<=-1, 0 <= a,a

2

<20

35 (-4x-16)/4-(5x-9)/8= 3 (ax-b)/c - (a

2

x-b

2

)/d=f -10 <=c<=10, -20 <=d,f <=20,

b

2

<=-1 ,-20 <=b, -20 <= a,a

2

<-1

Figure 8: Difﬁculty Levels in Equation Guru.

IMPLEMENTATION STRATEGIES FOR “EQUATION GURU” - A User Friendly Intelligent Algebra Tutor

65