# Blogs

## Geometry Blogs

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

Here are 48 of my favorite math words in 12 groups of 4. Each group has words in it that can be thought of at the same time or are a tool for doing math.   between on over in   each multiply of many   ratio divisions distribution compartments   limit neighborhood proximity boundary   infinite infitesmal mark differentiation   graph width height depth   circle sphere point interval   hyper extra spacetime dimensional   geometry proportion sketch spatial   four table cross squared   target rearrange outcome result   area volume space place   What are your favorite math words? If you aren't sure, search for "mathematical words" and pick a few.

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

I am a mathematics tutor living in McKinney, TX just 25 miles north of Dallas. Some surrounding cities are too far away for me to drive to/from every tutoring session, however I am setup to tutor remotely if my students are interested. Here is how remote tutoring works: 1) For my first tutoring session we meet in person at the student's home. I do not charge for the long driving times for my first visit. 2) After our first face-to-face tutoring session, I setup their Internet connected computer for remote collaboration. 3) We then write down the ISBN numbers ant titles of each of the text books the student is using. I use these to search Amazon.com for a good used copy of the books. 4) For subsequent tutoring sessions we use the phone and Internet connected computers to learn. I can interact with their home computer, highlight text, draw graphs, etc. in real-time. I know this works since I use this method to tutor my 12th grade son in math when... 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

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

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

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

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

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

Here are some of my favorite Math resources. Check back again soon, this list is always growing! I also recommend school textbooks, your local library, and used bookstores. As a note, college-level math textbooks are often helpful for high school math students. Why is that? Isn't that a little counter-intuitive? Yes, it would appear that way! However, many college-level math textbooks are written with the idea that many college students may not have taken a math class in a year or more, so they are written with more detailed explanations. This can be particularly helpful for high school students taking Algebra, Geometry, and Trig. I have a collection of college-level math books that I purchased at a local used bookstore. The most expensive used math book I own cost \$26 used. Books that focus on standardized test prep (such as the SAT, AP, or GED prep) can be helpful for all core subjects, as they summarize key ideas more succinctly than 'normal' textbooks. These are GREAT... 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

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/

Here are some ways to help you more easily memorize volume equations: (1) If you can remember the area equations for shapes, for shapes like cylinders and shipping boxes the volume equation will be that area (of the shape's base) times the height. (2) If you are studying for the GED, the only pyramids on the test will be those with four-sided bases. The volume of such a pyramid is 1/3 times the area of the base times the height. An easy way to remember this complicated equation is that the volume of a four-sided pyramid is 1/3 the volume of a packaging box with the same base length and width and the same height. Although the GED provides these equations, you should still try to remember as many equations as you can to save you time during the test. (3) On the GED, if you are asked to calculate the volume of an irregular shape, first break the shape up into easy to manage parts, calculate each part individually, and then combine these parts with addition and or subtraction... 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

How Was This Possible? I hiked the Grand Canyon earlier this year, as I've done for many years. I started down the trail at exactly 7 am, and hiked at an irregular pace, slowing and stopping occasionally to enjoy the views. I’m not sure what time I got to the river, but I got there before dark and spent the night camped at the river. I started back up the same trail at exactly 7 am the next morning, hiking in the same leisurely and irregular way, and got to the top before dark. This year, I just happened to notice that there was a point on the trail that I reached at exactly the same time as the day before! How was this possible? Was that just a coincidence? Considering that I've hiked about 1200 miles in the Grand Canyon over 40 years, what are the chances that has happened before? (Try to observe your own thought processes as you work on this. How did you go about solving the problem? Did you approach this mathematically, trying to write... read more

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'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