This document introduces GeoGebra, a dynamic mathematics software that allows users to visualize and interact with geometric, algebraic, and calculus concepts. It can be used as a presentation, modeling, and authoring tool in education. The document provides examples of how GeoGebra allows students to visualize abstract concepts, represent mathematical ideas in multiple linked ways, and discover mathematics through experimentation. It encourages hands-on activities with GeoGebra to better understand its uses and potential for teaching.
Mathematical education software is designed for innovative, interactive and dynamic teaching in various areas of mathematics. Their application would be best in distance education, which implies the spatial distance between teachers and students. In this paper, the focus will be on GeoGebra, software that mathematically connects geometry, algebra and analysis. The method of software application and integration with the Moodle platform will be explained. Advantages of using it for students in the process of visual mastering of mathematical material. Increasing the interactivity between teachers and students, all with the aim of improving the quality of teaching. The work was created with the aim of popularizing the free GeoGebra software and the distance learning platform Moodle.
Slides for a 20 minute presentation about Julia, with a brief introduction to multiple dispatch/multimethods and how it is used for numerical linear algebra
ISTE2011 - Model Lesson. Be the student in three math lessons enhanced with different types of targeted-learning GeoGebra (freeware) applets: classroom, math practice, and one you will create. http://bit.ly/sdedBK
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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Mathematical education software is designed for innovative, interactive and dynamic teaching in various areas of mathematics. Their application would be best in distance education, which implies the spatial distance between teachers and students. In this paper, the focus will be on GeoGebra, software that mathematically connects geometry, algebra and analysis. The method of software application and integration with the Moodle platform will be explained. Advantages of using it for students in the process of visual mastering of mathematical material. Increasing the interactivity between teachers and students, all with the aim of improving the quality of teaching. The work was created with the aim of popularizing the free GeoGebra software and the distance learning platform Moodle.
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The headmaster of School No. 2 Marginea created a short presentation of the open resources on Geogebra platform in order to give teachers a way to use STEAM activities in a fun way.
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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Dear Dr. Kornbluth and Mr. Gorenberg,
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harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
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2. What is GeoGebra?
Dynamic Mathematics
Software
For Learning and
Teaching
in Schools
Geometry, Algebra and
Calculus
Freely available from
www.geogebra.org
4. GeoGebra as a Pedagogical Tool
Presentation Tool
dynamic visualizations
Modeling Tool
dynamic constructions
Authoring Tool
interactive online material
5. Examples of Using GeoGebra
Visualizing
students can “see” abstract concepts
Representations
students can make connections
Experiments
students can discover mathematics
8. Examples of Using GeoGebra
Visualizing
students can “see” abstract concepts
Representations
students can make connections
Experiments
students can discover mathematics
9. Multiple Representations
GeoGebra: Bidirectional Connection
Symbolic ⇔ Graphics
Examples
– Coordinates Point in Coordinate System
⇔
– Circle‘s Equation ⇔
Circle drawn in Coordinate System
– Function‘s Equation Graph of Function
⇔
12. Examples of Using GeoGebra
Visualizing
students can “see” abstract concepts
Representations
students can make connections
Experiments
students can discover mathematics
16. Hands-on activities
... of this module you should:
Know more about Geogebra.
Have constructed a worksheet and succeeded in
posting it online, adding it to your journal in
BLACKBOARD.
Didactics will follow in January.
Concretely, your journal, should have:
An uploaded geogebra worksheet.
A 300 word evaluation of the strengths and
weaknesses of Geogebra for teaching mathematics.
