Category Archives: Math

Irvine Math Tutoring: The Unit Circle

Irvine Math Tutoring Tips: The Unit Circle – Learning and Memorizing Made Easy!

The Unit Circle is a staple of trigonometry and precalculus classes. It is a circle with a radius of one that is centered at the origin of a two-dimensional coordinate system. Essentially the simplest circle that we can put on our grid – book your private Irvine math tutor today.

Nearly every class will require students to memorize specific angles and they’re coordinates on this circle. For example, the “top” of the circle is at 90° (the angle is measured from the right side of the x-axis, or the “East” stem if you think of it as a compass) which is the point (0 , 1) since it is straight up and the unit circle has a radius of one. Similarly, we get (0, -1) at 270° at the bottom of the circle. The harder memorization comes in when you look at some of the points that are don’t lie perfectly on our axes. See an image of a typical unit circle below.

the-unit-circle

image taken from Wikipedia, submitted by Jim.belk

Here, we see the points we mentioned, but also a lot of pi symbols, radicals, and many fractions. This image can look quite daunting since most teachers expect you to be able to draw it yourself on command. So, let’s dissect how to learn it more easily with much less memorization.

First, we need to know how to use radians (a way to measure angles without degrees). We won’t get into why radians are the way they are in this post, but you understand them on the unit circle. You’ll need to know two facts:

A circle is 360°

A circle is 2π radians

With these two facts, we can convert between the two with some dimensional analysis. It’s like how knowing that 12 inches is 1 foot allows you to figure out that 4 feet is 48 inches. For some examples, here is how to find 30° in radians:

Here we set up the fractions since we know that 2π is the same as 360°. You cross multiply and divide to find x, simplifying the fraction at the end. Here is the same concept except converting from radians to degrees. Let’s say we have π/4 and want to find it in degrees:

Here we had some more fractions to work with, but the pis cancel out to give us 45°.

Now back to the unit circle. The unit circle is better memorized as two circles instead of one. On one circle they count by 30° increments (which we just learned is equal to π/6 radians) and on the other, we count by 45° increments (which we also just learned is equal to π/4 radians). Here is circle number one:

unit-circle

Notice the bold terms.  They all have a denominator of 6.  This circle corresponds to the blue lines we see on Wikipedia circle.  But notice how much easier it is to memorize in increments of π/6.  One π/6, Two π/6, Three π/6, Four π/6, etc. up until all the way around the circle is Twelve π/6.  The unit circle is just simplifying the fractions!  12 π/6 is just 2π since 12/6 = 2.  Just count the π/6’s around the circle and simplify the fractions.  Much simpler than memorizing all of those fractions.

Now that we know the angles of the unit circle, we have to learn the coordinates at each angle.  The ones on the corners aren’t bad since those are just variations of -1, 0, and 1 and we can tell what the coordinate pair should be.  For the remaining 8 points, here are the only two numbers we need to memorize:

Again, we won’t go into why these are the numbers since we’re just focused on memorization. Notice here that they both have a denominator of 2. Then, notice that √3 is larger than 1. Every coordinate point will be a combination of these points, so just look for which side is bigger. If the x side looks bigger (like in π/6), then the x side gets the √3/2 and the y side gets the 1/2. For 10π/6, notice that the longer side is in the y-direction and is going down. This means the y coordinate get the √3/2 and it is negative: (1/2, -√3/2).

Notice now that the bold terms are all with a denominator of 4. Here we count by π/4’s instead of π/6’s. This circle corresponds to the red lines on the regular unit circle. Here we count increments of π/4 until we get to 8π/4 which is our full circle of two pi. Memorize that these are the two circles that are put on top of each other for the full unit circle. Both are just counting until you get to 2π.

Now we’ll learn the coordinate points for this circle. The “corners are still the same as the blue circle ((1,0), (0,1), (-1,0), and (0,-1)), and we only have one number to memorize for the diagonal angles in between:

All of the coordinates for these angles on the unit circle will be √2/2 for both x and y. Just don’t forget to include the negative signs when necessary. So, for example, 3π/4 will be (-√2/2, √2/2) and 5π/4 will be (-√2/2, -√2/2).

If you can keep these two circles separate in your head it will significantly help you when drawing your own: and without the brute force memorization of every reduced fraction that many teachers suggest. Notice too that the diagonals of the orange circle fit perfectly between the diagonals of the blue circle since 45° is halfway between 30° and 60°.

Though memorization is still necessary, hopefully this guide will save you from mindlessly cramming and consequently forgetting your unit circle as you delve deeper into trigonometry.

From trigonometry to statistics, our private Irvine math tutors are here to help. Call TutorNerds today to book you Irvine math tutor.

Michael C. is currently a private math, science, and standardized test tutor with TutorNerds in Irvine and Anaheim.

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Use Your Math Intuition | TutorNerds

Classroom parties are always a good time. Snacks, laughs, games, maybe even an unexpected educational moment or two. But how come everyone secretly can’t wait for it to end? Maybe it’s because there’s a whole lot of jellybeans at stake. The curiosity in the back of the classroom, a large mason jar filled with candy, has drawn more attention than Mr. Ludlow’s embarrassing dance moves.

“Closest Guess to Actual Amount Wins Candy,” reads the paper, crudely held on to the jar by two strips of scotch tape. Fair enough, I could guess that. Let’s see – from what I can count, there’s at least a hundred and fifty. A respectable strategy. After all, counting is one of the staples of mathematics. Mr. Ludlow would be proud. So how come, after hours of painstaking anticipation, it’s revealed that my guess was 335 jellybeans shy of a victory? And how come Jonas, who merely glanced at the jar, guessed five over? He doesn’t even like Jelly Beans.

Mathimage1

There’s three answers that come to mind: Jonas thought about the numbers intuitively, Jonas was lucky, Jonas cheated. Only the first one is relevant to learning, so let’s discuss. In Annie Murphy Paul’s article, How Guessing Games Help Kids Solve Math Problems this phenomenon of intuitive thinking is now being supported by science. In her words,

“a new study suggests that by playing games that involve quickly guessing how many items are in a group of objects, children can help themselves become better at traditional math problems.”

What the article calls “an intuitive sense of numbers” can be of great benefit to a student. In detail, intuition creates a personal connection to numbers  basic counting fails to do. Instead of approaching a difficult math problem as foreign, a student may now feel familiar with the units on the paper, and have a sense of familiarity with the problem. As a result, the student can answer the question faster and with more accuracy. Paul’s article draws on many studies to support these claims:

“Other research has shown that children who are better at intuitive number tasks also have higher math grades and perform better on math tests—but Hyde’s study first to provide a causal link. Their research shows that practice on intuitive number tasks actually causes better math performance in children.”

From an educator’s standpoint, it’s an important skill that’s easily practiced. Tutors, teachers, and parents can support this way of thinking by simply asking, “how many so-and-so’s do you think are in there?” Once the student begins using intuition to sense numbers, he or she, with practice, will begin to do so with accuracy.

grade-papers

Here at TutorNerds, we are committed to giving you an advantage. Whether you’re seeking help in Pre-Algebra,  Statistics, etc. our process of matching you with the right tutor is the first step to mathematical success. If you are concerned your intuitive sense of numbers isn’t what it should be, what better way to improve than a professional tutor, at your convenience, working directly with you? Read more about how we operate here.

Back to the mason jar. Turns out Jonas had a great tutor growing up.  Thanks to years of guessing, “how many so-and-so’s? ” His intuition for numbers is strong. More importantly, Jonas has higher math scores to compliment his pockets full of jellybeans.

Does Music Complement Math and Science Test Scores? | TutorNerds

Want Higher Math and Science Test Scores? Put on Your Headphones

From space travel to Silicon Valley, America has always been an innovative, progress forward country. Further, most of the world’s best Universities are found in our nation, as well as the biggest corporations, eager to hire their recent grads. So how come our children are posting flat test scores in science and math? Should we be concerned other nations, such as Singapore, are seeing test scores skyrocket, while ours remain flat? Of course we should.

With innovation comes the need for high-skilled workers. From engineers to statisticians, companies are beginning to rely on other countries to supply the brainpower where the U.S. lacks. To be fair, this isn’t a bad or unnatural thing, since globalization is the driving force of today’s economy, but it raises concerns in the realm of competitiveness. And where there is concern, there is debate.

slidebird1In an effort to pinpoint the ‘where and why’ American students aren’t learning as fast and efficiently as their contemporaries, schools have given (arguably) math and science most of their attention. Pressure is the keyword, as districts demand test scores are the main focus. Like a business consultant hired to eliminate overhead, schools are beginning to push subjects like music and art farther and farther down the priority list.

In Music Education for Creativity, Not A Tool For Test Scores, author Sarah McCammon reports that music educators are pleading music’s untestable benefits as the reason their classes should remain relevant. In her words,

 “some advocates say that rhetoric is missing the point and overlooking the virtues of music that can’t be tested.”

Few would argue music isn’t important to development, but it’s difficult to see, on paper that is, how it improves test scores. But what about that other buzzword, the one that usually describes the great minds who started the corporations hiring all the engineers? That’s right, creativity. Undoubtedly a key component in competitiveness, and a staple of America’s economic dominance, creativity is difficult to test for, but vital for success. That’s why many music teachers are arguing their classes should remain high on the priority list. Having a well-rounded education, including the arts, helps a student approach math, sciences, and life in general with more creativity and depth.

science-test-score All of this is good and well, but is it a bit of an overreaction? Are schools really putting music on the back burner? Some would argue no. For instance, the NPR article suggests schools, in general, aren’t ditching the pianos and violins so soon. To illustrate, Russ Whitehurst,

“points to a 2010 U.S. Department of Education report that found 94 percent of public elementary schools offer some kind of music classes, even if hours are being cut back in many places.”

As creativity becomes a focus for schools, especially higher education, there’s no surprise courses such as Stanford’s “Creativity: Music to My Ears” are becoming available to everyone, including non-musicians. Whether or not more or less music is the answer, the problem of America’s science and math drought is real and troubling.

Struggling With Math and Science?

As global competitiveness continues to grow, it’s crucial students don’t get too far behind in such topics. Luckily, we’re here to help. If you’re feeling overwhelmed by your math and science courses, don’t hesitate to contact us today. Whether you’re from Los Angeles, Orange County, or San Diego, we’ll match you with the perfect tutor. From Pre-Algebra to Physics, we’ve got your back.