# Efficiency of Dynamic Computer Environment in Learning Absolute Value Equation

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Multiple Representations

#### 2.2. Multiple Representations in Computer Based Environment

#### 2.3. GeoGebra Software Package

#### 2.4. Multiple Representations of Functions and Equations

#### 2.5. Research Question

## 3. Materials and Methods

#### 3.1. Background and Participants

#### 3.2. Instruments and Procedures

#### 3.2.1. Instruments and Procedures of the Tests

#### 3.2.2. Instruments and Procedures of Students’ Practice Work

#### 3.2.3. Instruments and Procedures of Students’ Practice Work in a GeoGebra Environment

#### 3.3. Analysis of Students’ Practice Work on the Tasks

## 4. Results

#### 4.1. Data Analysis

#### 4.2. Statistical Analysis of the Pre-Test Results

#### 4.3. Statistical Analysis from Exercise Results

#### 4.4. Statistical Analysis of the Post-Test Results for Each Task

#### 4.5. Statistical Analysis of the Post-Test by Groups of Tasks

#### 4.6. Statistical Analysis of Post-Test Results of the Group B of Tasks by Representations

- (1)
- (B1) The frequency of using algebraic representation in solving the task (AR);
- (2)
- (B2) The frequency of using graphic representations in solving the task (GR); and
- (3)
- (B3) The frequency of using algebraic and graphic representations in the solving task (AGR).

#### 4.7. Statistical Global Analysis of the Post-Test Results

#### 4.8. Analysis of the Post-Test Results by Representations

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Tasks for the Pre-Test

- Match the graphs with the corresponding equations. ${f}_{1}\left(x\right)={\left(x-1\right)}^{2}-1$, ${f}_{2}\left(x\right)=-\left|x\right|+1$, ${f}_{3}\left(x\right)={x}^{2}-1$, ${f}_{4}\left(x\right)=\left|x-1\right|$, ${f}_{5}\left(x\right)=-{\left(x-1\right)}^{2}+1$, ${f}_{6}\left(x\right)=-\left|x\right|+1$
- Sketch the graphs of the functions $\left|{f}_{3}\left(x\right)\right|$, ${f}_{5}(\left|x\right|)$ on the corresponding graphs of the functions in the previous task.
- A function is defined as $f\left(x\right)=-\left|x\right|+1$. Write down the x-intercept.
- Find the value of parameter p so that the equation $-\left|x\right|+1=p$ has two solutions? Explain.
- How many solutions are there to the equation $\left|{f}_{3}(x)\right|=1$? Explain.

## Appendix B. Tasks for the Practice

- Sketch the graphs of the functions ${f}_{1},\text{}{f}_{2},\text{}{f}_{3},\text{}{f}_{4},\text{}{f}_{5}$. Solve equations ${f}_{i}=d,\text{}i=1,2,3,4,5$ by using algebraic and graphical representations $a,\text{}b,\text{}c,d\in R$.
- (1)
- ${f}_{1}\left(x\right)=a\text{}\left|x\right|$
- (2)
- ${f}_{2}\left(x\right)=\left|x-b\right|$
- (3)
- ${f}_{3}\left(x\right)=\left|x\right|+c$
- (4)
- ${f}_{4}\left(x\right)=a\left|x-b\right|+c$
- (5)
- ${f}_{5}\left(x\right)=\left|a\left|x-b\right|+c\right|$

- Sketch the graphs of the functions ${f}_{6}$, ${f}_{7}$. Solve equations ${f}_{6,7}=d$ by using algebraic and graphical representations, $a,\text{}b,\text{}c,d\in R$
- (6)
- ${f}_{6}\left(x\right)=\left|a{x}^{2}+bx+c\right|$
- (7)
- ${f}_{7}\left(x\right)=a{x}^{2}+b\left|x\right|+c$

## Appendix C. Tasks for the Post-Test

- Find the equations of the functions ${f}_{2},\text{}{f}_{3},\text{}{f}_{4},\text{}{f}_{5}$ and sketch the graphs of ${f}_{1}$, ${f}_{6}$
- Sketch the graphs of $\left|{f}_{3}(x)\right|$, ${f}_{2}\left(\left|x\right|\right)$ in the previous task. Explain.
- A function is defined as $f(x)=\left|x-2\right|-2$. Write down the x-intercept?
- Find the value of parameter p so that the equation $\left|x-2\right|-2=p$ has two solutions? Explain.
- Find the value of parameter p so that the equation $\left|{\left(x+1\right)}^{2}-1\right|=p$ has two solutions? Explain.

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**Figure 4.**(

**a**) solution of student B from group C on task 2a; (

**b**) solution of student B from group C. on task 2b.

**Figure 6.**(

**a**) The paper of student C from the experimental group on task 2. (

**b**) The paper of student D from the experimental group on task 2.

**Figure 7.**The percentage of correct answers by tasks; (* significant difference at p < 0.05, ** significant difference at p < 0.01).

Group | Number of Students | Arithmetic Means | Standard Deviation | Test of Difference between Arithmetic Means | |
---|---|---|---|---|---|

n | M | SD | t | p (2-tailed) | |

Experimental | 113 | 15.310 | 5.978 | 0.0453 | 0.9639 |

Control | 113 | 15.345 | 5.764 |

Group/Task | 1. Task | 2. Task |
---|---|---|

Experimental | 3.8 | 3.0 |

Control | 1.5 | 1.2 |

p | p < 0.00001 | p < 0.00001 |

Group/Task | 1. Task | 2. Task | 3. Task | 4. Task | 5. Task |
---|---|---|---|---|---|

Experimental | 74.9% Ab | 62.8% Ac | 86.7% Aa | 61.1% Ac | 61.1% Ac |

Control | 71.8% Aa | 46.0% Bb | 76.1% Ba | 45.1% Bb | 38.9% Bb ^{1} |

^{1}Capital letters show significant differences between groups (p < 0.05). Lower case letters show significant differences between tasks within a single group.

Group/Group of Tasks | A | B |
---|---|---|

Experimental | 4.13 Aa | 4.18 Aa |

Control | 3.53 Ba | 3.20 Ba |

Group/Task | 3. | 4. | 5. | ||||||
---|---|---|---|---|---|---|---|---|---|

Representations | AR | GR | AGR | AR | GR | AGR | AR | GR | AGR |

Experimental | 37.8% | 48.0% | 14.3% | 43.1% | 49.2% | 7.7% | 10.1% | 68.1% | 21.7% |

Control | 65.1% | 32.6% | 2.3% | 58.5% | 40.0% | 1.5% | 61.4% | 34.1% | 4.5% |

abs(E-C) | 27.4% | 15.4% | 12.0% | 15.4% | 9.2% | 6.2% | 51.2% | 34.0% | 17.2% |

p | 0.0002 | 0.034 | 0.0041 | 0.0801 | 0.2891 | 0.0949 | 0.0000 | 0.0004 | 0.0000 |

Group/Representations | AR | GR | AGR |
---|---|---|---|

Experimental | 30.3% | 55.1% | 14.6% |

Control | 61.6% | 35.5% | 2.8% |

abs(E-C) | 31.3% | 19.6% | 11.8% |

p | 0.0001 | 0.00001 | 0.00001 |

Group | Number of Students | Arithmetic Means | Standard Deviation | Test of Difference between Arithmetic Means | Effect Size | |
---|---|---|---|---|---|---|

n | M | SD | t | p (2-tailed) | Cohan’s d | |

Experimental | 113 | 20.796 | 6.486 | 4.7797 | 0.00003 | 0.6359 |

Control | 113 | 16.681 | 6.456 |

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Jokić, M.; Takači, Đ.
Efficiency of Dynamic Computer Environment in Learning Absolute Value Equation. *Symmetry* **2020**, *12*, 473.
https://doi.org/10.3390/sym12030473

**AMA Style**

Jokić M, Takači Đ.
Efficiency of Dynamic Computer Environment in Learning Absolute Value Equation. *Symmetry*. 2020; 12(3):473.
https://doi.org/10.3390/sym12030473

**Chicago/Turabian Style**

Jokić, Marina, and Đurdjica Takači.
2020. "Efficiency of Dynamic Computer Environment in Learning Absolute Value Equation" *Symmetry* 12, no. 3: 473.
https://doi.org/10.3390/sym12030473