Cayleys Sextic Essays - Algebraic Curves, Equations, Polynomials

Cayley's Sextic Essays - Algebraic Curves, Equations, Polynomials Cayley's Sextic The curve, Cayleys Sextic can be described by the Cartesian equation: 4(x^2 + y^2 ax)^3 = 27a^2(x^2 + y^2)^2. It is the involute of a nephroiod curve because of its slight kidney shape and because they are parallel curves. This curve was first discovered by a mathematician by the name of Colin Maclaurin. Maclaurin who was born in February of 1698, became a student at Glasgow University in Scotland during his early teen years. It was here that he discovered his abilities in mathematics and began working towards a future in geometry and mathematics. In 1717 Maclaurin was given the job as the professor of mathematics at Marischal College in the University of Aberdeen. Later during his mathematical career, Maclaurin wrote Geometrica Organica, a book which displayed early ideas of what later becomes known as the curve, Cayleys Sextic. The actual man credited with the distinct discovery of Cayleys Sextic is the man it is named after, Arthur Cayley. Cayley, who had a family of English ancestry, lived in St. Petersburg, Russia during his childhood where he attended his first years of schooling. In 1835 he began attending Kings College School in England because of his promise as a mathematician. After Cayley became a lawyer and studied math during his spare time, publishing papers in various mathematical journals. These journals were later looked at by Archibald and in a paper published in 1900 in Strasbourg he gave Cayley the honor of having the curve named after him. Cayleys Sextic The polar form of the equation for the curve, Cayleys Sextic, is shown as: r = 4a cos^3 (q/3). For the specific equation for the graph, the polar form is the equation of greatest ease of use. Use 1 in place of a and switch the calculator to polar form. The best viewing window for this graph is q min= -360; q max= 360; q step= 10; x-min= -5; x-max= 5; x scale= 1; y-min= -5; y-max= 5; y scale= 1. This window and equation will give an excellent picture of the curve, Cayleys Sextic. When a is increased in the equation for the curve, the entire curve increases in size, giving it a larger area. The value for x is greatly increased on the right side positive y-axis, while the value for x on the left side negative y-axis becomes gradually more negative at a much lower rate then that of the right side positive y-axis. The y values for the curve increase and decrease at the same rate on both sides of the x axis when the value of a changes. When the value of a becomes negative, the curve is flipped over the y-axis. When the value of a decreases to a lower negative number the area of the curve increases giving it a larger area. The value for x in greatly increased on the left side, negative y-axis, while the x on the right side positive y-axis becomes gradually more positive at a much lower rate then that of the left side negative y-axis. The y values once again increase and decrease at the same rate on both sides of the x-axis when the value of a changes.

ACT Reading Tutor

What Makes for An Effective SAT/ACT Reading Tutor SAT / ACT Prep Online Guides and Tips You'd think that with a name like â€Å"Critical Reading† or just plain â€Å"Reading† that the best tutor for these sections of the SAT and ACT would just be someone who is good at reading critically. Alas, it is not that simple. For instance: I love to read and aced the SAT Critical Reading section when I took the test lo so many years ago, but because it comes naturally to me, I haven’t ever had to think of alternate ways to approach questions. I actually have to work much harder at tutoring Reading than I do at tutoring subjects that I had to put more effort into learning (for me, SAT Math). Read on for more specifics on what makes the most effective SAT/ACT Reading tutor. And if you want specific advice on SAT/ACT Math tutoring as well, don't worry – you can read more about that here. feature image credit: Reading by Pedro Ribeiro Simà µes, used under CC BY 2.0/Cropped from original. Effective SAT/ACT Reading Tutors Scored Well On The Test The best SAT/ACT Reading tutors are the 1%: at a bare minimum, they've scored in the 99th percentile on the test. At PrepScholar, we take it one step further. Not only do we exclusively hire 99th percentile scorers, but many of these tutors were perfect scorers on the sections that they tutor. Why is this important? Because in order to be able to customize teaching, a tutor must have mastered the material. This doesn’t just mean knowing the content (like vocabulary). Tutors need to understand HOW the SAT/ACT will question you on reading. SAT/ACT Reading in particular is a difficult subject to improve, because it requires skills the student has practiced over a decade. It's not easy to teach critical reasoning and logic in a short amount of time. How do effective SAT/ACT Reading tutors do it? Read on! Effective SAT/ACT Reading Tutors Can Pinpoint Your Weaknesses The best SAT/ACT Reading tutor will be able to understand exactly WHERE students are making mistakes. When all is said and done, a particularly tricky reading question can involve over half a dozen components, any of which could be your Achilles' heel. Let’s use the following passage and question as an example. There are many complicated steps you have to take to answer this question Understand the question: what’s this weirdly-worded question actually asking? Where in the passage is the information you need? How much time are you spending reading the passage? What information does the passage give you to answer the question? How do you use this information to answer the question? What answer choices can you eliminate? Why? How can you confirm the right answer is correct at the very end? A student who misses the question could be tripped up by any one of these steps. How should a tutor address this? A mediocre tutor will just make sure the student understands how to get the answer to this particular question and move on. Why is this bad? It’s inefficient – it’s highly unlikely this particular question will show up on the SAT/ACT, and so the assistance given is not generalizable. Furthermore, it doesn't identify the student's actual weaknesses - it just patches over serious problems with a quick fix. The best tutor will pinpoint exactly where the student is having issues, then work on weaknesses until there are no longer any issues. A student who doesn't understand how to identify the information in the passage that answers the question needs very different help from a student who is spending too much time re-reading the passage when answering this question. If the tutor doesn't identify these weaknesses and address them, the student will find it very hard to improve her score. Darts by Richard Matthews, used under CC BY 2.0. All your tutor's efforts should be directed towards the bullseye of your weaknesses. Effective SAT/ACT Reading Tutors Will Customize Strategies To You The best tutors will understand optimal strategies and be able to teach them to students, customizing the strategies to a student's particular level. Read on for an example of two different strategy suggestions: one suggestion is for someone aiming for a 600 on the SAT or 26 on the ACT and the other is for someone aiming for a perfect score on the SAT/ACT). Students who are aiming for a 600 or a 26 or a can skip hardest 20% of questions and instead focus on easier ones. For more strategies like these, read our articles on how to improve your SAT Reading Score or how to improve your ACT Reading score. If you're aiming for the perfect score, on the other hand, you might need to take note of time management. Take a timed practice test, and if you run out of time, mark the questions you answer with extra time. Compare your scaled score with extra time with your realistic score (the score that only includes questions answered during the normal time period). If your score differs by more than 50 points (on the SAT) or 1 point (on the ACT), you have serious time management issues. Are you generally slow at reading passages, or were there particular questions that slowed you down? For more time management tips and other strategies, read our articles on scoring an 800 in SAT Critical Reading or 36 in ACT Reading. Actions To Take The best way to get a sense of any tutor’s teaching style is to get her to demonstrate how she’d explain the process of answering a difficult question. PROTIP: Pick out a question ahead of time that you previously missed and have had trouble with from a real SAT and use it every time, so you can compare the explanations of multiple tutors. I’ve also compiled a list of sample questions parents and students might want to ask potential tutors. Click to download the PDF, or click on the image below. Use the answers to these questions to assess how a tutor teaches and whether or not her particular teaching style and method will be effective for you. What’s Next? If you haven’t already, read our article on the key qualities of an effective SAT/ACT tutor. How do you find the best tutor for you? Learn about what tutors actually do during SAT/ACT tutoring and how to find the best tutoring match for your situation. Want to get serious about improving your SAT/ACT score? Our tutoring service is designed to be the most effective in the nation. We hire the best tutors from across the country. This is then paired with our exclusiveonline drillingprogramthat gives your tutor complete insight into your progress to design the most effective sessions. Think about what improving 160 points on your SAT, or 4 points on your ACT, can do for you. Learn more about tutoring now: