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Geometry Blogs

Thus far, all students that I have tutored for Regents and SAT exams (algebra, geometry, trigonometry) have passed their exams. Some students passed with distinction. Some students passed after failing their exams (sometimes twice) before my tutoring. I am also available for tutoring calculus, differential equations, numerical methods and other courses in mechanical engineering and applied math. I am certified in 44 subjects. I usually tutor adults at a public library near where they live. I shall tutor at a person's home if it is appropriate, and will not create wrong impressions. I shall tutor a minor (male or female) only at the student's home, with a parent or guardian present. If a parent or a responsible adult (family member) is not present, the tutoring session will be postponed or canceled. I shall tutor a minor at a public library, provided that the student is accompanied by a parent.

NOTE: For today's Math Journey, please refer to the image file “Image for Math Journey: Road Trip Around A Problem” under my WyzAnt files. Link: https://www.wyzant.com/resources/files/671706/image_for_math_journey_road_trip_around_a_problem Let's go on a road trip! When I teach geometry, especially geometry involving angle measures like this problem, I like to describe the process of solving a problem as taking a little road trip. I describe it this way because this is how I personally feel when solving a problem like this – my eyes rove around the figure from one intersection to the next, and I hop in my little math car and drive along lines and stop at intersections to figure out where I am. Geometry is a very visual discipline, and as a visual learner, I have the most fun when I can trace a physical journey around the problem, solving things as I go. So let's hop in our math car and chase this problem down! The first step in any problem like... read more

A student needed to draw a circle with a 2" diameter, then draw the following angles: 100°, 120º, and 140º. She had her compass but didn't have her protractor.   First she drew the circle, then she drew 2 perpendicular diameters. Since a circle encompasses 360º, each quadrant comprising 90º. We drew the 120º angle first using an entire 90º quadrant plus 1/3 of the adjacent quadrant, erasing the unneeded line, which leaves 60º in that second quadrant.   Then we found the circumference of the circle (C=πD, or 3.14x2"=6.28"). Next we found 1/4 of the circumference (6.28"/4=1.57"). We wanted to be able find the arc length in 10º increments, so we divided the arc of one quadrant by 9 (1.57"/9=0.174"). We converted this into 1/16ths of an inch by multiplying by 16 (0.174"x16=2.79 sixteenths of an inch).   Getting back to our angles, we measured the 100º angle next by taking our remaining 60º and adding 40º of... read more

Math Student's Civil Rights   I have the right to learn Math (Math is learnable like other subjects) I have a right to make mistakes, erase then, and try again (Failure points to what I have not learned yet) I have the right to ask for help (asking for help is a great decision) I have the right to ask questions when I don't understand (understanding is the primary goal) I have the right to ask questions until I understand (perseverance is priceless) I have the right to receive help and not feel stupid for receiving it (asking for help is natural) I have the right to not like some math concepts or disciplines (i.e. trigonometry, statistics, differential equations, etc.) I have the right to define success as learning no matter how I feel about Math or supporters I have the right to reduce negative self-talk & feelings I have the right to be treated as a person capable of learning I have the right to assess a helper's ability to... read more

As the school year ramps up again, I wanted to put out a modified version of a Memo of Understanding http://en.wikipedia.org/wiki/Memo_of_understanding for parents and students. It seems each year in the rush to get through the first weeks of school parents and students forget the basic first good steps and then the spiral downwards occurs and then the need for obtaining a tutor and then the ‘wish for promises’ from a tutor. Pay attention to your child’s folder or agenda book. A student is generally not able to self regulate until well into high school. Some people never quite figure it out. Be the best person you can be by helping your child check for due dates, completeness, work turned in on time. Not only will this help your child learn to create and regulate a schedule, it prevents the following types of conversations I always disliked as a teacher ("Can you just give my child one big assignment to make up for the D/F so they can pass"; "I am going to talk to... read more

I've been asking students the following question for years: "Why do you show so little work, and where are you completing the problem?" Most students I have worked with write less down than I do, and I have quite a bit of math under my belt. I still have not found the answer to this question. Some students say it’s because they don’t see the point, but they have been cheated if teachers have given them credit for answers without work. As math gets complicated there is more and more work that needs to be done, and if a student has bad habits of doing mental math, then this will be a hindrance to success. These are things that all students of higher mathematics should do: 1. Write the original problem down. When solving problems you want to make sure that you are staring at the actual problem. You don't want to look at your paper and then back to the book or sheet of paper that the problem is on. 2. Show your work just like your teacher does when they are introducing... read more

Suppose, one have two parallel lines given by the equations:   y=mx+b1 and y=mx+b2. Remember, if the lines are parallel, their slopes must be the same, so m is the same for two lines, hence no subscript for m. How would one approach the problem of finding the distance between those lines?   First, if one draws a picture, he or she shall immediately realize that if a point  is A chosen on one of the lines, with coordinates (x1, y1), and a perpendicular line is drawn from that point to the second line, the length of the segment of this new line between two parallel lines give us the sought distance. Let us denote the point of intersection of our perpendicular line with the second line as B(x2,y2).   What do we know of point A and B?   First, since A lies on the first parallel line, its coordinates must satisfy the equation for the first line, that is,   y1=mx1+b1       (1)   Same... read more

Alegbra: http://www.nysedregents.org/algebraone/   Algebra 2/Trigonometry: http://www.nysedregents.org/a2trig/home.html   Geometry: http://www.nysedregents.org/geometrycc/   Math A, Math B, Integrated Algebra, Other Math: http://www.nysedregents.org/regents_math.html   Chemistry: http://www.nysedregents.org/Chemistry/   Earth Science: http://www.nysedregents.org/EarthScience/   Physics: http://www.nysedregents.org/Physics/

Proof of the Assertion that Any Three Non-Collinear Points Determine Exactly One Circle This is an interesting problem in geometry, for a couple of reasons. First, you can apply some earlier, basic geometry principles; and secondly, you can choose two different strategies for solving the problem. The basic geometry underlying: any three non-collinear points determine a plane, somewhere in 3D space. Once that has been done, imagine that the plane has been rotated into the x-y plane, which will make the problem much easier to solve! The two strategies for solution are: (Proof A) actually solve to find the circle. This is equivalent to finding the center of the circle (finding the equation of the circle is simple from there). But, you actually have to do some math to get this! If, while doing this, there is no possibility to obtain other values for the coordinates of the center of the circle, you have proved the assertion as well as obtained a method (and perhaps... read more

1. No one was born to lose. The best of my students understand this principle like the backs of their hands. No, there is no inherent genetic formula or organic compound you can use to get an A in a class. We are all products of our hardwork and investments. Whoever decides to put in excellent work will definitely reap excellent results. 2. Always aim for gold. Have you heard that there is a pot of gold lying somewhere at the end of the rainbow? It's true! Okay, I'm just joking, but my best students always aim for the gold. The very best. As, not Bs, or Cs, or Ds. Just the very best. The one thing people don't think they are capable of achieving is the best. The top of the class. Or the valedictorian. 3. Never settle for less. My best students are innovative, inquisitive thinkers. They tend to think outside the box, never settling for "just what they got from class." They love to use real life examples and explore how theory comes alive in their personal experiences... read more

Reading Formulas can make or break how a student comprehends the formula when alone - outside the presence of the teacher, instructor, tutor, or parent. Formula For Perimeter of Rectangle:  P = 2l + 2w How To Read:  The Perimeter of a Rectangle is equal to two (2) times the Length of the longer side of the rectangle (L) plus two (2) times the Width of the shorter side of the rectangle (W). When is reading formulas like this necessary?  At three particular moments, reading this formula in this manner can be effective.  When students are initially learning what the formula means When student are learning what it means when they should already know (remediation).  When students want to remind themselves (basics learning study skill habit) Remember, Formulas at their introduction are complete statements or thoughts.  Students cannot and will not recall complete thoughts or statements... read more

As you may know, I am a big fan of the well-known author and brain specialist, Dr. Daniel Amen. He mentions in several of his books that Physical Exercise is good for the brain. I have read of research studies that showed a clear correlation between IMPROVEMENT in students' test scores in math and science, and their level of physical activity (for example, when math class followed PE class, the students had significantly higher scores). Maybe we should schedule PE before all math classes in our schools. What do you think about that idea? This morning I read an online article on the myhealthnewsdaily site, entitled "6 Foods That Are Good for Your Brain," and another article about how Physical Exercise helps maintain healthy brain in older adults too. The second article, "For a Healthy Brain, Physical Exercise Trumps Mental Workout" was found under Yahoo News. The remainder of this note is quoted from that article: Regular physical exercise appears to... read more

I have been working with a few students who are ready to learn math much, MUCH faster than allowed by the traditional classroom model in which math is taught over 6 to 8 years. Based on this experience I believe that many students as young as 4th grade and as old as 8th grade (when starting in the program) can master math in 2 years from simple addition through the first semester of Calculus, with Arithmetic, Algebra 1, Geometry, Algebra 2, Precalculus, Probability, Statistics, and Trigonometry in between. This is significantly faster than the traditional approach and is enabled by a combination of one-on-one teaching and coaching and a variety of media that I assign to students to complete in between our sessions. This is a "leveraged blended learning" approach that makes use of online software, selected games, and selected videos with guided notes that I have created that ensure that students pick up the key points of the videos, and which we discuss later. The result... read more

Hello Wzyant Academic Community and welcome to my blog section! This is where I am available for online chit-chat, educational assistance free of charge, business discussions & arrangements, and more! I am always eager to help and love to talk turkey with all realms of academia, so don't be shy and feel free to ask many questions!!!       P.S.  ∫∑∞√−±÷⁄∇¾φΩ

Several of my current Geometry students have commented on this very distinction.  This has prompted me to offer a few possible reasons.   First, Geometry requires a heavy reliance on explanations and justifications (particularly of the formal two-column proof variety) that involve stepwise, deductive reasoning.  For many, this is their first exposure to this type of thought process, basically absent in Algebra 1.   Second, a large part of Geometry involves 2-d and 3-d visualization abilities and the differences in appearance between shapes even when they are not positioned upright.  Still further, for a number of students, distinguishing the characteristic properties amongst the different shapes becomes a new challenge.   Third, in many cases Geometry entails the ability to form conjectures about observed properties of shapes, lines, line segments and angles even before the facts have been clearly established and stated... read more

I hear a lot about math teachers from my students, and while every teacher is unique, some comments are repeated over and over. By far the most common one I hear is that their teacher didn't really explain something, or was incapable of elaborating when questioned and simply repeated the same lecture again. As a tutor, my first priority is to make sure the student understands the material, and if they're still confused, to find another way to explain it so that it makes sense. In order to do that, I need to have a thorough understanding of the concepts myself, so that I am not simply reading from a textbook but actually explaining a concept. In my years of tutoring math, I've developed a point of view and approach to math that I refer to as “teaching the concept, not the algorithm.” An algorithm is a step-by-step procedure for calculation. The term is used in math and computer science, but the concept of an algorithm is universal. I could tell you that I have an algorithm... read more

I wanted to take a moment to share a recent "success story". Recently, a Student contacted me because he needed to pass a formal standardized exam, known as the "Praxis I". The Praxis tests are used by State Governments and Colleges of Education to ensure they bring only quality students into their programs to be trained as educators. My Student had unfortunately previously failed all 3 components of the Praxis test, and was now "under the gun", since a second failing score would have resulted in his expulsion from his School. In my home State, students must achieve a combined Praxis I Score of at least 522 to be eligible for School. The passing score for the Reading test is 176, the Writing test 173, and the Math test 173. The minimum score on each test is 150, and the maximum score is 190. It should be noted that this is a fairly difficult exam series; the median scores (175-179) are barely above the minimum passing scores (173-176). My Student,... read more

Should I get a tutor? Will it help my child? These are some of the most common questions posed to tutors by parents of students struggling in school. Tutoring can be expensive and difficult to schedule so parents must decide whether the time and money will be well spent. Instead of relying on a crystal ball, use these factors to help make the decision. 1. Does the student spend an appropriate amount of time on homework and studies? While it can help with study skills, organization, and motivation, tutoring cannot be expected to keep the student on track unless you plan on having a session every night. If you can make sure the student puts in effort outside of tutoring, she will be more likely benefit from it. 2. Does the student have difficulty learning from the textbook? If this is the case, the student will probably respond to one-on-one instruction that is more personalized. A tutor will help bring the subject to life and engage the student. A good tutor will explain... read more