I ran across this little flash widget which gives a great perspective on trigonometry:
You should always strive to learn efficiently. Ask the questions, How, what, where, why when learning a new topic. Always practice what you have learned. For kids the last is especially important. Without adequate practice, the knowledge fades away. Without
enough practice resources, kids get bored. It's just like at the gym, you have to alternate the weights you use so that your body doesn't get bored. How much more important is that with the brain when exercising and learning a new topic?
The unit circle is one of the most important concepts to understand in Trigonometry.
As a tutor who emphasizes understanding and comprehension over memorization, I try to make it as easy as possible for my students.
Here's the way I like to look at it:
1) First, realize that the unit circle is simply a few points drawn on an graph with an x-axis and a y-axis.
2) Recognize that there is an overall pattern.
Every 90 degrees (0, 90, 180, 270) is a combination of 0 and 1 (positive and negative).
Every 45 degrees (45, 135, 225, 315) is √2/2 (positive and negative).
Every 30 degrees (30, 60, 120, 150, 210, 240, 300, 330) are combinations of 1/2 and √3/2 (positive and negative).
This means that you only have to remember three numbers: 1/2, √2/2, and √3/2 (positive and negative).
The first quadrant (0-90 degrees), has all positive numbers, just like you'd expect in any other graph.
I'm going to list what I believe are the key concepts that you need to master across different math subjects. These are the tools that I have to use most often in order to solve problems, so you should get very familiar with the theory behind them and
very comfortable with applying them.
1. recognizing and factoring the three most common polynomial forms:
common factor expressions
difference of square expressions
2. synthetic division
3. Descartes's Rule of Signs
4. Rational Zero Theorem
5. Long division of polynomials
6. Factoring by grouping
7. Using the Quadratic Formula
understanding and using the unit circle
definitions of the trig functions "Soh-Cah-Toa"
factoring quadratic equations (using the quadratic formula, etc.)
My first year in college (a very long time ago...!), I came home for the Thanksgiving Holiday and learned that my younger brother, Chip, was struggling with Trigonometry. Chip was pretty smart, so I was a bit puzzled as to why he would be having trouble.
I sat down with him, and within about 15 minutes I discovered that he had missed one key concept early in the school year, and had been confused ever since. Once I explained that to him, the light went on in his head, and everything fell into place for
I was horrified that a bright, promising student like my brother would be left to flounder because his teacher did not have the time to sit down with him for just 15 minutes to figure out why he was struggling. But the truth was (and still is) that many
teachers are very overloaded, and really can't devote extra time to individual students. A typical high school math teacher may have four or more classes with 35+ students...
The majority of the students that I have often have the same problem -- they aren't grasping the information fast enough or they aren't really able to follow the lessons a teacher gives.
Sometimes, teachers aren't adaptive to every learning style for each student in their classroom. However, know that each student has the capability to learn math on their own. It is just necessary to have key characteristics to make it successful.
Every math student should have:
open communication between themselves and their teacher (inside and outside the classroom)
Always try to study outside of your home or dorm room. In our minds, those are places that we relax at and it can be difficult to turn your mind off from the distractions to study. Public libraries,
universities, coffee shops, and bookstores are the way to go...
I am happy to announce that all my students have passed the NY State Regents examinations, except one student. The subjects varied from Algebra 1, Algebra 11/Trigonometry, English, US and Global History and Living Environment. I am so proud of them.
Most of these students are students who struggled quite a bit. It was a long journey but one I would do again.
I am very proud of them as most of them will be graduating this year. The NY State Common Core examinations are next.
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...
I am not done with it yet. I still need to show how simple it is to do the "same" calculations in the second, third, and fourth quadrants.
In the spirit of giving, starting on 11/29/2013, I will be offering a few brainteasers/ trivia questions where the first 3 people to email me the correct answer will receive a free, one hour, tutoring session in any subject that I offer tutoring for (via
the online platform)! That's right free! Get your thinking hats on everyone!
Andrew L. Profile
I was working with a student today, and as we worked through the section in his book dealing with Trigonometric Identities and Pythagorean Identities, we stumbled across a problem that gave us a bit of trouble. The solution is not so complicated, but it
sure had us stumped earlier.
The problem was presented as such:
Factor and simplify the following using Trigonometric and Pythagorean Identities:
sec3(x) - sec2(x) - sec(x) + 1
We tried a couple of different approaches, such as factoring sec(x) from each term:
sec(x) * [ sec2(x) - sec(x) - 1 + 1/sec(x) ]
and factoring sec2(x) from each term:
sec2(x) * [ sec(x) - 1 - 1/sec(x) + 1/sec2(x) ]
We followed these approaches through a few steps, but nothing we were attempting led to the solution. After doing some reading online, I found that the solution required a...
Buckle up readers, it's Trig time!
Trigonometry can be scary to many students, and in my opinion, a lot
of that is because one of the most confusing concepts in trigonometry
occurs right at the very beginning, in the form of the Unit Circle
Let's start at the beginning. Give yourself a circle with a radius of 1. Now center that circle on the origin of a coordinate plane, so that the line of the circle itself passes through the points (1,0) (0,1) (-1,0) and (0, -1). Got that?
Now, this circle is referred to as the Unit Circle, because the radius is one unit and it is therefore easier for us to do various manipulations and calculations with it.
Now choose any point on the circle (we'll call the coordinates of
that point (x,y)), draw the radius to it (which will still be a
length of 1), and drop a line back perpendicular to one of the axes.
Do that and you'll have a right triangle with the...
I was very fortunate to have been taking electronics at the same time I was learning high school trigonometry. Like most folks, I was never good at abstract math, but being able to see physical demonstrations of math principles was a huge help.
One of the real "bears" for most math students is the concept of the imaginary number...and yet it's crucial to working with all alternating current electrical and electronics problems.
When I teach electronics, I have students ranging from postgraduate math majors to middle school students who want to take a few shop classes before high school. (Nice thing about working at a technical arts extension of the local University!)
I introduce practical trigonometry without even using the term, using nothing more than a straightedge, some graph paper, and a protractor. I avoid trig tables entirely UNTIL they understand how to manipulate the Pythagorean theorem inside out and upside
I then demonstrate...
I was a fairly typical young person and, like my peers, counted down the days until summer. My mother was a math professor, so I never stopped doing math during the summer, but felt like other parts of my brain became a little mushy in the summer. Come September,
it was difficult to get back into the swing of writing papers and studying history and memorizing diagrams. I was out of practice and lost my routine. As an adult, I have almost continually taken classes, because I enjoy learning and find that from class to
class, I need to maintain a routine, i.e. a study area and a time of day that I complete my assignments. I have also found that reviewing material a week or two before the course begins helps me to start the class with more confidence and competence. I am
a big believer in confidence fueling success and I wonder if younger students practiced assignments in the week or two prior to return to school, if that confidence would help the transition to the school year...
I was excited on Tuesday, July 16th, 2013. This was my third meeting with this student and I finally had a breakthrough with him. On the first meeting it was clear that he saw Algebra I almost as a foreign language. I began with one of the test packet, and
had him do 10 questions and reviewed the questions he had done wrong. So this continued for a while, and of course sometimes he would say that he understood, but it was clear that he did not. Anyway, after reviewing the entire packet I began a teach and learn
session, in which I picked a variety of topics and had him practice various equations. After which I gave him a quiz.
He failed the quiz miserably, so of course he still did not understand. Anyway, I gave him another packet for homework. When I saw the student again, I reviewed with him, but still not much improvement, but at least he tried. I did the teach and learn session
again, of which some of the questions were from the previous session, and I gave him...
The Summer session has just begun. The stress has already begun to set in, but this week I had a break through with a few of the students. So this is my second week with a student who I am tutoring for both Algebra I and Earth Science. So far he seems stronger
in Earth Science but still needs much practice, before I can be very confident about his ability to pass the Regents exam in August. After the first session of Algebra, I walked away thinking about how am I going to get him ready by August 13th. I recommended
an additional session to the parents, but so far they have said no. I did several practice examples, and made the second session mainly a teaching and learning session. Then I ended the session with a quiz, but he failed :(.
So when I had to meet him again for Earth Science, my mind was swirling as to how I can help him, and will I at least be successful with this subject. When I checked the homework, there was a slight improvement but not enough to celebrate....
Hello Miss Gil, I received a 96% in Global History. I was so excited to hear these words from my student! At first she did not want to be tutored. Her father dropped her off at the Library. So I told her that if she did the practice test, and did well, she
would never have to see me again. Well, she scored a 58%, and there were so many events and topics that she did not know.
We scheduled 3 additional three hour sessions. By the last session, her essays had improved and her overall score was an 83%. I told her that I believe that she can score as much as a 95% on the Regents Exam. She laughed and said "Yeah right". Well she scored
a 96% and I am very proud of her.
Humans have a tremendous capacity to learn and adapt. However, we consistently build barriers that hinder our natural ability to change and grow. Many people, regardless of age, perceive themselves as not being talented enough to excel at math and science.
They view math and science as the realms in which only scientists, engineers, mathematicians, and geniuses truly soar.
Nothing could be further than the truth. Sure, possessing a natural affinity towards these subjects helps. Yet, a supposed lack of talent does not prevent you from learning. The path may be more arduous. The journey may be longer. Nevertheless, you possess
within you the fire to endure. Willpower, dedication, self belief, and an open mind can compensate for any lack of ability.
Bruce Lee was a legendary martial artist, actor, and philosopher who continues to inspire millions with the sheer intensity which he pursued his endeavors. Frail, sickly, and small as a child, Bruce Lee overcame many physical...
Now that students, teachers, parents and tutors have had a chance to catch their breath from final exams, it's time to make use of the weeks we have before school starts back. Consider all that could be accomplished in the next few weeks:
Areas of math that students NEVER REALLY GRASPED could be fully explained. This could be
elementary skills like adding fractions, middle school topics like systems of equations, or
high school areas like sequences and series.
Students could have a TREMENDOUS HEAD STARTon topics that will be covered in the first few weeks of school. Imagine your son or daughter being able to raise their hand to answer a question in the first week of school because they had worked
several problems just like the ones that the teacher is demonstrating.
ENORMOUS PROGRESS could be made in the area of preparation for the standardized tests (PSAT, SAT, ACT and more) that are so important to getting into a great college.
A wise man once told me: "You can continue to beat your head against that rock, but you will not chip the rock, your head (on the other hand) will be deformed." I guess I should have seen it coming, my being summarily fired from a tutoring job - The parent
(in this case the mother) demanding extra "busy-work" for her son between sessions, the lack of discipline, on the student's part (especially his inability to do homework or speak to his subject teacher) and his continual lack of attention during sessions.
The call came, "You are not coming here anymore, Billy Ben (not his real name) ONLY got an 81 on his Geometry test. We want top performance, 95 or better, YOU failed." Did I tell you that this student, previous to my seeing him, was working on a solid average
of 40? So, it was over. Had I failed? I'm not so sure. First, I didn't take HIS test, and second, knowing the student as I did, I actually thought that an 81 was pretty good and we might...
Hi math students :)
When preparing for a mathematics tutoring session, try to have the following things at hand...
Textbook (online or e-text)
Syllabus, assignment, tips/hints/suggestions, answer sheet/key
Pencils, pens, erasers, paper (graph paper, ruler, protractor)
All necessary formulas, laws, tables, constants, etc.
Calculator that you will use on tests
Do I really need my calculator? I can do most of my work in my head.
Having your calculator is just as important as paper and a pencil in most cases. You'll be using it on your test and if you don't know how to input what you want, you won't do very well. Have your tutor teach you about your calculator's functions beforehand.
Learn how to check your simple math and how to input exponents, logarithms, or trigonometric functions before your test.
Why do I need my book, notes, or answer key? Isn't the tutor supposed to know everything?
Yes :), but even the most experienced tutor...