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...
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...
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...
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...
All my grade 8 & 9 students (10 students) passed the Algebra Core Regents exam. Only one student had to retake it in August and she passed with an 83%. In June she scored 53%. My two Trigonometry students passed the Regents, but only 2 out 4 students passed the Geometry Regents exams.
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,
Algebra 2/Trigonometry: http://www.nysedregents.org/a2trig/home.html
Math A, Math B, Integrated Algebra, Other Math: http://www.nysedregents.org/regents_math.html
Earth Science: http://www.nysedregents.org/EarthScience/
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...
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...
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...
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...
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!!!
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...
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...
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...
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).
Hello, if you are a student frantically searching for help with a math problem, take a second here and I will repost answers to any MATH related questions you may have.
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...
In high school geometry, we learned of the perfect right triangle. Both sides and the hypotenuse are integers (whole numbers). The perfect right triangle shown was 3, 4, 5. (3 sq + 4 sq = 9 + 16 = 25 = 5 sq). I wondered if there were others.
Years later, I seriously searched for other perfect right triangles. I began with a list of the squares of whole numbers and the difference between one square and the next (the "delta").
Discovery #1: The series of deltas = the series of odd numbers. I looked for deltas which are squares. Since the series of deltas is the series of odd numbers, the square of every odd number is the difference between two squares.
Discovery #2: Every odd number 3 and above is the side of a right triangle. Implication of Discovery # 2: Since there is an infinite number of odd numbers, there is an infinite number of perfect right triangles. Next, a formula for finding the other sides and hypotenuse of a right triangle was worked...
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...