Tuesday, March 24, 2020
4 Singing Exercises to Practice Dynamics
4 Singing Exercises to Practice Dynamics Suzy S. As a singer, understanding and utilizing dynamics can turn a mediocre performance into a great one. Read on to learn some singing exercises to practice, as shared by Augustine, FL voice teacher Heather L... Dynamics is a word that comes from the Greek word dynamo, meaning power. In the context of general music, we use it as a term for how loud or soft sounds are. Dynamics can make or break a song. Frankly, in the world of singing, sometimes the lack of or presence of dynamics can make a difference in whether a singer is regarded as great or not. Think of the best singers in the world and Ill bet that many have a tremendous sense of dynamics. The improvement of the use of dynamics in our singing voices comes right down to a word that I dont typically like to use as a voice teacher: control. The simple passing use of this word can often, maybe subconsciously, cause a singer to tense and strain. Control, however, is truly the thing that enables us to do much with our voices. Just make sure to maintain a free and open sound as you incorporate the concept of control in the following singing exercises. Learn and Practice Messa di Voce Perhaps the best singing exercise to help improve your vocal dynamics is messa di voce, an Italian phrase meaning place the voice. To sing a messa di voce, you sustain one single pitch, getting louder, then softer. The proper musical terms for getting louder and softer are crescendo and diminuendo, respectively. Decrescendo is a word created in modern times to mean become softer, but it is not an Italian word or a proper musical term. When using this exercise, be sure, as always, to use a warm, resonant sound and to stay connected to your breath. Remember, your sound begins at the bottom of the pelvic floor. Keep your shoulders floating in place; do not allow them to rise. Lip Trill Like a Revving Engine Another singing exercise to practice dynamics is singing lip trills, or lip bubbles, using messa di voce. This is a great one that I learned from Celine Dion. Begin with a low breath, filling up your torso all the way around. Start at a comfortable low/middle position in your voice, trill up to a comfortable high point, and begin to sail down to your lowest register. As you slowly sail down in pitch, make stops along the way. On those stops, crescendo and then diminuendo. Itll sound a lot like a old drag car revving its engine! Not only does this singing exercise improve your ability to sing dynamically, but it will also help to warm up and strengthen your singing muscles. Laugh Out Loud Speaking of strengthening exercises, its essential that the muscles of your torso are strong enough for the control thats needed for great dynamics. Sing short ha, ha, ha sequences on a descending scale, making each ha very short, but warm and round. Youll notice quite a jumpy motion from your tummy as you instinctively work to produce short bursts of air. Yow-wow-wow In this exercise, youll sing a single pitch on yow-wow-wow with a loose, open jaw, as if youve just touched a hot stove. This loosens the face, especially your joints, which is vitally important in preventing tension. Lastly, remember that singing dynamics comes not only from phonationthe pitched sounds that our vocal folds producebut also from articulation. Our articulators are our teeth, our tongue, and our lips. To be clear, crisp articulation should be an inherent part of your singing voice. You can use it, though, to sound even louder or softer than your phonation is. And as always, be yourself and sing with your own voice. Heather L. teaches singing, piano, acting, and more in Saint Augustine, FL, as well as through online lessons. She is a graduate of the prestigious Westminster Choir College in Princeton, New Jersey, and has performed with the New York and Royal Philharmonics, the New Jersey and Virginia Symphonies, the American Boy Choir, and the internationally renowned opera star Andrea Bocelli. Learn more about Heather here! Interested in Private Lessons? Search thousands of teachers for local and live, online lessons. Sign up for convenient, affordable private lessons today! Search for Your Teacher Photo by whartonds
Friday, March 6, 2020
Foil Method
Foil Method First multiply the first variable in each binomial. Multiply outside variable. Multiply the inside variables. Multiply last variable in each binomial. The parentheses are used for the multiplication sign. This method is used to multiply two. Example - Let us find the multiplication of the following binomials (a + x) (a-x) Solution - Applying FOIL method we get the following solution step by step. F stands for first which means that multiply the first variable in each binomial. a.a = a4 O stands for outside which means that we have to multiply outside variable. - x.x = -x I stands for inside which means multiply the inside variables. x.a L stands for last which means that multiply last variable in each binomial - x.a Combing the result of the FOIL a4 x.x + x.a - x.a = a4 x Hence the multiplication of two binomials (a + x) and (a-x) is a4 x Example - Let us find the multiplication of the following binomials (a + x) (a3 - ax +x) Solution - First terms of each binomial are multiplied a.a3 = a5 Multiply the outside terms. - a. ax + a x = - a4x + a x I stands for inside which means multiply the inside variables. xa3 - x.ax = xa3 - x3.a L stands for last which means that multiply last variable in each binomial X. x2 = x4 Combing the result of the FOIL a5 - a4x + a x + xa3 - x3.a + x5 Hence the multiplication of (a + x) (a3 - ax +x) is a5 - a4x + a x + xa3 - x3.a + x4
Where To Take Portuguese Lessons In Manchester
Where To Take Portuguese Lessons In Manchester Business Portuguese Classes Manchester ChaptersWorking In PortugalBusiness Portuguese Lessons In ManchesterThere are many reasons why you would want to learn to speak a new language like Portuguese with language lessons: not only so that you can visit the coastal regions of Portugal and marvel at the gloriously sunny beaches, but also so that you can make use of business opportunities in this European country as an intermediate or near-native speaker.Most international business will have their main offices in or around the capital city, Lisbon (or Lisboa, in Portuguese) however, if you want to find work in other areas of Portugal or Brazil then you will no doubt find a wide range of businesses covering various industries, just like at home.The only difference in working for an international company and one whose workers communicate solely in Portuguese is that your language acquisition (particularly your comprehension/listening skills and your grammar and vocabulary) may need to be a notch or two higher in order to part icipate in a conversation and forge a relationship with natives and their fellow Portuguese contacts.So, if you are tired of Manchester and want to fulfil a dream of working abroad to gain a specific learning experience, or if you are seeking a job in Portugal to follow in the footsteps of family or a loved one, then here's all you need to know about working in a country like Portugal, how you might fare in terms of remuneration, and how you can embark on your journey as a learner seeking language programs adapted for Business purposes.Get set for jet-setting to Portugal with these tips for professionals. Photo credit: Zillay Ali on Visualhunt.com / CC BYClick here for more information.There you have it, so there is no excuse now not to brush up on your foreign language skills by getting reading and writing in Portuguese or attending language classes!Don't forget to learn about Portuguese language and culture, because understanding the culture is equally as important for immersion a s learning the native language effectively is during your placement in Portugal.As well as Manchester, you can learn Portuguese London, also there are lots of Portuguese lessons in Birmingham
Thursday, March 5, 2020
GMAT Math 5 Tips for Data Sufficiency Questions
GMAT Math 5 Tips for Data Sufficiency Questions GMAT MBA Admissions Blog GMAT Data Sufficiency problems present you with some initial information and a question, followed by two statements. You have to decide whether the information contained in each statement is sufficient, when combined with the initial information, to give a definite answer to the question. In this case, a definite answer means being able to answer either âdefinitely yesâ or âdefinitely noâ to the question. If you can only answer âdefinitely maybeâ then the statement is not sufficient by itself. If neither statement is sufficient by itself to give a definite answer, you then evaluate whether the two statements taken together are sufficient to find a definite answer. Some questions involve solving for a value; for these questions, a statement is sufficient if it allows you to solve for one, but no more than one, value. In case you havenât seen a Data Sufficiency question before, hereâs an example: If x is even and y is a positive integer, is the sum of x and y an odd integer? 1) x is a negative integer 2) the square of y is an even integer Notice that since we already know that x is even, the only way for the sum of x and y to be odd is for y itself to be odd. So we can rephrase the question as âIs y an odd integer?â Statement 1 is irrelevant. The fact that x is negative tells us nothing about whether y is odd or even. So statement 1 is insufficient to answer the question. Statement 2 is a different story. If y-squared is an even integer, that tells us that y itself is an even integer. Therefore we can answer âdefinitely notâ to the question âis y an odd integerâ, and statement 2 by itself is sufficient. Hereâs a list of tips to help you in solving GMAT Data Sufficiency questions: Itâs helpful if you can restate the question in simpler terms (as we did in the example above). Plugging in example numbers is also helpful. However, be careful not to make unwarranted assumptions about the kind of numbers you can plug in. For example, if the initial information in a question specifies that the quantity n is less than zero, be careful not to assume that n is necessarily an integer. Integers probably jump most readily to mind when making up examples, but data sufficiency questions often deal with non-integer values as well. Quickly look at both statements and try to evaluate which one will be easier to analyze. If statement 2 looks easier than statement 1, then test statement 2 first. If the first statement you test seems insufficient, remember to âforgetâ that statement when you move on to test the other statement. You have to test each statement alone for sufficiency before you test whether the statements are sufficient when combined together. On the other hand, if you feel really stuck on a Data Sufficiency question, go ahead and combine the statements to see if theyâre sufficient together to answer the question. HOWEVER, if they are sufficient together, you STILL need to look carefully and see if thereâs any way to take just one of the statements by itself and find a definite answer to the question. This is a risky technique if you donât have enough practice in separating the statements in your thoughts. But if youâre careful, it can pay off by making it easier to see how to restate the question or find the key connection between the statements and the initial information.
English Conversation Classes Talking About Artist Constantin Brâncusi
English Conversation Classes Talking About Artist Constantin Brâncusi Those of you who have used Google this weekend will know that February marks what would have been the 135th birthday of the Franco-Romanian sculptor Constantin Brâncusi.Brâncusi was one of the leading lights in modernism in the field of sculpture at the start of the 20th century, and makes a great topic for an English conversation class.His is a form of abstract art.He was inspired by mythology, particularly Roman mythology, as well as folk tales and indigenous art from around the world.Talking About ArtWhen we talk about art and our views of it, there are a number of ways of expressing ourselves: We can sayI thinkâ¦I would sayâ¦In my opinionâ¦I believeâ¦I feelâ¦I would interpret this asâ¦.Iâm sureâ¦â¦Iâm certainâ¦If Iâm not mistakenâ¦.I assumeâ¦As far as I can tellâ¦As far as I can seeâ¦Discussing Brâncusi1. How would you interpret this sculpture by Brâncusi?Sleeping Muse, 1910.2. What do you think of this work?The Kiss, 1908.What is your opinion of this col umn?The Endless Column, 1938.Was this class interesting? Feel free to share it on Facebook and Twitter.
Language Marketplace Now open to Language Schools and Companies
Language Marketplace Now open to Language Schools and Companies Were excited that we have expanded the Language Marketplace to include language companies and schools. Now language learning companies can partner with italki and offer their services and products to italki users. We think this is a major step towards making italki a destination site for language learning. Our first partner company, Eleutian Technology, targets the largest language learning market in the world: Learning English. Eleutian uses American certified-school teachers and the highly-regarded Pearson English Language Learning Instruction System (ELLIS) to teach English online. click to go to marketplace Language Learning Ecosystem Our plan is to expand the marketplace to include every language, and to support a variety of services. By creating an ecosystem for language learning, we believe everyone benefits: students, independent teachers, language training companies, and application developers. Students benefit by having more choices. They can pay for premium services, or use the free community generated ones. Students are able to share information about different service providers and make informed choices. Students are also able to pick and choose a combination of services that may be useful to them. Most of all, a central location helps students find online providers of language education. Teachers and companies benefit by having an active market with demand for their service. Companies and teachers can reach out to students around the world, and transact without having to deal with individual payment issues. Quality teachers and companies also benefit by having their reputation socially confirmed by their customers. Again, were thrilled to be opening our expanded marketplace, and to have Eleutian join us as our first partner. If youre not learning English, just be patient were planning to add more companies and services for other languages soon. If you represent a company that might be interested in partnering with italki, please contact us at business at italki dot com. View the press release (PDF) The italki Team Language Marketplace Now open to Language Schools and Companies Were excited that we have expanded the Language Marketplace to include language companies and schools. Now language learning companies can partner with italki and offer their services and products to italki users. We think this is a major step towards making italki a destination site for language learning. Our first partner company, Eleutian Technology, targets the largest language learning market in the world: Learning English. Eleutian uses American certified-school teachers and the highly-regarded Pearson English Language Learning Instruction System (ELLIS) to teach English online. click to go to marketplace Language Learning Ecosystem Our plan is to expand the marketplace to include every language, and to support a variety of services. By creating an ecosystem for language learning, we believe everyone benefits: students, independent teachers, language training companies, and application developers. Students benefit by having more choices. They can pay for premium services, or use the free community generated ones. Students are able to share information about different service providers and make informed choices. Students are also able to pick and choose a combination of services that may be useful to them. Most of all, a central location helps students find online providers of language education. Teachers and companies benefit by having an active market with demand for their service. Companies and teachers can reach out to students around the world, and transact without having to deal with individual payment issues. Quality teachers and companies also benefit by having their reputation socially confirmed by their customers. Again, were thrilled to be opening our expanded marketplace, and to have Eleutian join us as our first partner. If youre not learning English, just be patient were planning to add more companies and services for other languages soon. If you represent a company that might be interested in partnering with italki, please contact us at business at italki dot com. View the press release (PDF) The italki Team
4 ways to prepare for your teach abroad job interview
4 ways to prepare for your teach abroad job interview After you make the decision to teach abroad, the next step in the process is to land a job at a school that meets your life and career goals. To that end, you might opt for applying for jobs online with a recruiter like Teach Away or decide to attend a teacher job fair or some in-person interview sessions. Regardless of which path you take, however, one thing is certain: You will be interviewed at some point in the hiring process - whether itâs over phone, Skype or in person. Congratulations, your teaching credentials and experience landed you an interview and now it's time to seal the deal. Depending on your level of experience, you may have attended many, many interviews in your time. Or maybe your last interview was your first. No matter which bucket you happen to fall into, I wanted to share my first-hand experience and advice for anyone interviewing for a job teaching abroad. Hopefully, this will give you some idea of what to prep for when interviewing for what (hopefully) will turn out to be your very own dream job teaching abroad. Iâm not going to give you a laundry list as there are so many resources out there already. For example, as I was preparing for the job fair, Teach Away hosted a webinar that gave advice and tips for job fair interviews (just an FYI that the job fair experience can be very different if you have never attended one). I also found these three posts helpful, so I recommend checking those out if youâve got time: 6 interview tips that will get you the job - straight from a teach abroad recruiter Preparing for an international teaching job interview 11 tips for a successful teacher phone interview Below, Iâve listed some of the things that I felt really helped me make a strong impression throughout the interview process: 1. Do due diligence on the school youâre interviewing with. Hopefully, you have already done a lot of research before the interview process to decide on where you want to teach and live. If you havenât, I recommend you do so. Each country and region has their own unique culture, and it is important to have an idea which fits you and/or your familyâs lifestyle. Next, you need to do the research to familiarize yourself with the prospective schoolâs vision, mission and curriculum philosophy. Research on the school can be beneficial in two areas. First, it gives you an opportunity to see if you would be a good fit for the school. Second, it will give you valuable information you can use and reference when answering questions in the interview. Teach Away will usually have a page on their site with key school information If not, you can also google and find the schoolâs official website. At the Teach Away job fair, I was also lucky enough to get the chance to sit in on info sessions held by the hiring schools in attendance. I made sure to take notes on key areas, such as the schoolâs philosophy, to ascertain how it matches my own strengths and values as an educator. Watch this video: This is what schools look for on your Teach Away profile 2. Donât be afraid to self-reflect and sell yourself. After doing your research, take a look back at your past teacher evaluations and your career as an educator. For some that might be a long career, others very short. Either way, it is important to self-reflect in order to truly understand your strengths and areas where you can improve as an educator. After all, we all have room to grow. Many of the schools that I interviewed with during my job hunt were interested in me teaching Advanced Placement (AP) classes or International Baccalaureate (IB) classes. If you are a secondary content teacher and have AP or IB experience that will increase your stock. I, however, do not have AP/IB experience, but many schools pay for professional development. I was open and honest in interviews about my lack of AP/IB experience. However, I made it clear that I was willing to attend PD sessions and continue to grow professionally in those areas. Being honest about the areas that you can improve when asked in an interview is really important, especially if those areas are bound to be brought up as the school is doing reference checks. Although you want to be honest about the areas that you can improve, it is also important to sell yourself. Two of the most common questions you will get in a teacher job interview are (although asked in different ways) are: Why do you want to work at our school? What would we see if we were to walk into your classroom? These are two unmissable opportunities to sell yourself as an educator. The first question is where your research on the school is crucial. As youâre answering, you have the chance to get across just good a fit for their school you are by conveying how your philosophies and strengths align with the schoolâs mission, vision, and curricular programs. The second question is where your self-reflection can help you sell yourself as a classroom instructor. Be prepared to give specific examples of teaching strategies you use in the classroom. You are good enough for this job, so donât be afraid to tell the school why. Sign up to Teach Away today for access to the latest teaching jobs around the world. 3. Practice makes perfect. Like anything else, interviewing is a skill that can be mastered with practice. Of course, just like most other things, interviewing comes easier to some than to others. Whether you feel like youâre a natural or youâve come to dread the interview process, practicing before a real interview will not only help prepare you for the possible questions that could be asked, it will also go a long way in helping battle those pesky nerves that tend to sneak up as the interview date looms. The questions I shared above are a great place to start. Have a friend, spouse or colleague ask you questions specific to the teaching job youâre in the running for and practice responding to them. You donât have to memorize your answers verbatim but itâs a good idea to jot down your responses so you can work on refining them as you do your interview preparation. Itâs also worth practicing for an interview with a formal administrative colleague that can share some valuable insight into the common questions they like to ask during the interview process. You can either review these questions or conduct a mock interview with them. Iâm very fortunate to have a father who is a former administrator who is always happy to help me run through my interview responses. During past interviews, I have foundIdo have either a tendency to get caught in those annoying ummm...uhhhhh...pauses when questions catch me off guard, or I. Oy to avoid any awkward silence so I s at allstart speaking before Iâve really thought my answer through. With enough practice, however, I was I can start tonate my awkanyrd pauses and keep my cool if I happened to momentarily thrown by a tricky question. 4. Practice getting comfortable in front of the camera. Video interviews are common practice when recruiting teachers from overseas, largely due to convenience, flexibility and the money-saving benefits for schools. As a result, practicing interviewing in an online setting is important. Interviews online or over Skype have a completely different feel from in-person interviews. For example, depending on how reliable your internet connection is, the video quality might be spotty and you might find it difficult not to talk over each other in the interview. Due to financial and family constraints, my wife and I were unable to attend the teacher job fair together. Instead, I flew out to the fair while Grace stayed home with our two boys. However, Grace was able to interview with the same schools I did - just over Skype. We even interviewed as a team in some instances - I was physically in the room and while Grace was sat next to me on a laptop (which was a unique experience for everyone involved!) Grace and I also had multiple Skype interviews following the fair. Sheâs an elementary teacher and Iâm a secondary teacher, so it stood to reason that both principals wanted to meet with us. I think itâs safe to say weâre old hands at Skype interviews by now! So here are some of our basic tips when interviewing over Skype: Dress professionally, just as you would for an in-person interview. Make sure your technology is up to scratch. Test your interview location and camera angle by calling a friend. (The last thing you want is something embarrassing being visible in the background.) Be on time. Make sure you know whoâs calling who. Take care to articulate your words clearly and speak a touch louder than you normally would in an in-person interview. Be sure to look right into the camera as much as you can. (This is the Skype version of making eye contact, however unnatural it might feel!) Take the quiz: Where should I teach abroad in 2018? If youâre currently gearing up for an interview to teach abroad, remember that preparation is key for a successful interview, whether itâs face-to-face or via Skype. Hopefully, the tips and tricks provided above will help you land your dream job teaching abroad this year. And if youâve already had a call saying youâve gotten the job offer, thatâs awesome! I know first hand that accepting a new job teaching abroad is a life-changing decision, so be sure to read this blog about what should be included in your teach abroad contract before you sign on the dotted line! In the meantime, if youâre not preparing for an interview to teach abroad, I recommend you continue to self-reflect and build your instructional knowledge by incorporating best practices into your current classroom. That way, when you do start to prepare for your very first teach abroad interview, youâll be able to draw on those instructional strengths in your interviews as well as your future classroom.
E-Learning Futuristic Education Resources for Today and Ideas for Tomorrow
E-Learning Futuristic Education Resources for Today and Ideas for Tomorrow This article discusses ideas for integrating new technology with education, what the future may hold for education, and real high-tech resources for students and educators. Smart Phones and Tutoring? The smart phoneâ¦screen size may present problems for students. Most students require a lot of room to perform mathematical calculations. Perhaps they could write on a tablet at home/school, then save a reduced copy of it on their phone for later study purposes. The phone should have a vertical and horizontal scrolling type feature to enable them to go back and review or edit previous work in a connected fashion. Concept mapping is a very useful teaching tool in many subjects and also requires this type of navigation. Voice recognition would be a great way to handle the notation for mathematics. Iâm not aware if there is such software available yet, but if not, it may be soon. Speaker phone is nice for conversation, but a closer microphone (less interference) might be needed for voice recognition of symbols. It will save so much time, and verbalizing concepts is a productive element for the teaching and learning process as well. If the instructional device could interface with the studentsâ PD, then they would be totally free to think without worrying about taking notes, etc. They could manipulate these notes on their own PD to arrange them according to their personal learning style. Students think about things in so many different ways. Enabling them to interact with the concepts personally is very important. In the schools (and maybe even in the home) there may be a problem with the applicability of the smart phone if it was not in some way a âsecuredâ learning environment. There should be a way to ensure that students are engaged in learning â" which could be argued is always the case, no matter what they are doing. Iâm personally not a advocate of censorship, but as a teacher, I have noticed that distractions to the learning process have become a very real issue since cell phones. Itâs not that Iâm bothered so much by distractions (they can be valuable at times) itâs just that many students do not, for whatever reason, decide to use their time effectively. Itâs frustrating to see very intelligent young people âwasting their timeâ on ânet junkâ⦠things that may not benefit them in the future. Iâm aware of screen monitoring systems, but they require the attention of the instructor whose time is much better spent teaching students. What is needed is a semi-close d system that affords the child room to explore, but also blocks irrelevant interference. In a perfect world, students will learn that they control the technology and their ability to learn from it. It should not control them, or draw them away from opportunities to learn and advance intellectually. The structure of the environment is extremely important. Negative Aspects of E-learning Regarding security and technology, let me give you a personal example of possible negative aspects âe-learningâ. One of my wealthy friends sends her children to an exclusive private school where laptops are issued on enrollment. She thought that by doing this she was giving her child the very best. The school had preloaded certain educational software for the students, so she thought he had to work on that particular computer, and since they didnât have administrative rights, they could not install anything. Without her knowledge, her preteen son became involved with some very scary, unsavory folks during times that she thought he was working on his schoolwork. It will take a very long time for their family to get over this incident â" the childâs personal information is also out there. These things should not happen. The school proceeded with their laptop initiative without anticipating problems. Future Hopes for Education Technology and education will soon become mutually interdependent. Each person will need to learn how to use technology to improve their education and expand opportunities for the future. I would like to see the development of community-based technology learning centers (for lack of a better term right now) to connect community resources, families and schools to achieve this level of learning. [Honestly, Im afraid that the schools, with a slow-paced institutionalized approach, cannot respond appropriately to our needs] These centers could provide quality technical support to the schools, information and training for parents, students, and teachers. They could bridge the gap between the home and different levels of schooling. [This gap seems to widen as children get older and access to technology increases. This is the exact opposite of what should be occurring. Parents should be empowered to guide their own children.] Staffed with educational technology personnel, assisting the devel opment of more effective tools for the teaching and learning process. This would be a community-based forum for collaboration, innovation, and technical support. A bank of âbest lessonsâ could be available for teachers or homeschoolers to practice with and/or use in teaching. These centers could also serve as e-learning satellites for distance learning projects as we discussed earlier. Like the public library system began years ago, these centers could inject an educational technology renaissance now. I hope to see many changes happening in education in the very near future. Our children are depending on us. Resources Smartboard, Smart Technologies (https://smarttech.com) Complements As high school teacher who uses a SMART board, from SMART technologies, in the classroom I think it offers a great platform for interaction between students and their teacher. My district has three high schools, so the SMART board technology has been extremely useful for me to communicate with other mathematics teachers at the other buildings. Itâs nice to just be able to write the symbols, and not have to try to type everything up to email it. I think that there is a lot of potential for teachers and students in e-learning environments to communicate more effectively with a SMART board. Criticism I found myself flipping back and forth between screens when I first started using the SMART board. Students had it all right their on their paper and tried to be patient with me as I fumbled around. What is GeoGebra? (https://www.geogebra.org/cms/) GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On the one hand, GeoGebra is a dynamic geometry system. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards. On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum. These two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice-versa.
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