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Ellen's Choice: Teach the Concept, Not the Algorithm

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

The Power Of Willpower: Five Tips To Strengthen Your Discipline

Willpower is unique to humanity. It is the keystone characteristic that is directly responsible for our technological advancement over the last several hundred thousand years. Willpower can be defined as the capacity to restrain our impulses and resist temptation in order to maximize our long-term success. It is the expulsion of energy to fight off innate survival based urges to exponentially increase future advantages and benefits. It is the driving force behind all civilizations, and it is what prods humankind forward to learn and grow. When we turn down a bite of cheesecake, step away from a mind numbing reality sitcom, or push off a nap to get some work done, the credit goes to willpower. It is this ghost like aura of control and discipline that we rely on to extend our existence and maximize our accomplishments. When we watch highly successful individuals exercise routinely, read voraciously, and work tirelessly, we are impressed with their ability to resist... read more

Different Centers in a Triangle

This week in geometry one of my students is learning about the different "centers" in a triangle (orthocenter, circumcenter, incenter, etc), as well as the midsegments theorem and triangle inequalities.  To help him visualize why all of these things are true, I had him cut out an acute triangle, an obtuse triangle, and a right triangle and use these to illustrate the concepts.  For triangle inequalities, we worked with different lengths of string to see why some combination of leg lengths and some do not. These are both quick, easy ways that help students understand beyond the words and definitions what we are talking about!

Parallel, Skew, and Perpendicular Lines

I've found that most students have little to no difficulty understanding the difference between parallel and perpendicular lines when only one plane is involved. Either they never touch, or they intersect at a 90 degree angle, or they just plain intersect. This concept is relatively easy to visualize because it is completely 2 dimensional.  Where the difficulty lies, is visualizing these same types of lines when different planes are involved, since it is 3d. To help, I utilize flash cards, or small pieces of paper. Have students draw a series of lines on each flash cards, making sure there is at least a set of parallel lines, perpendicular lines, and intersecting lines on each, and give each line a name. Then move the flashcards in different ways, either stacking them or making parallel planes, and quiz them about the new relationships between the lines. 

High School Geometry: For Some, Why Is This More Challenging Than Algebra 1?

Several of my current Geometry students have commented on this very contrast.  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

Favorite Math Resources, K-12 + GED

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... read more

Intersecting Circles

This blog concerns how to determine the intersection between two circles in the plane algebraically. It is a problem that can crop up in a variety of situations, from gaming to tools for computer aided design to astronomy.    This problem is interesting because is it a conceptually simple problem whose algebraic formulation is nonetheless apparently complex: a system of non-linear equations that are quadratic in both variables. However, by doing some geometric analysis of the problem, and applying tools from vector geometry, we are lead to a specific mathematical transformation of the problem that radically simplifies it. The key idea turns out to be a specific change of basis.   My exposition of this uses a few diagrams (which are not supported by the blog editor) and a lot of mathematical expressions (which are clumsy to create in the blog editor), so I put it in an Adobe Portable Document Format (PDF) document that you can access using the... read more

Proof that 3 non-collinear points determine a unique circle

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

How to find a distance between two parallel lines?-2

This is another way to find a distance between two parallel lines. This derivation was suggested to me by Andre and I highly recommend him and his answers to any student, who wants to learn math ans physics. This derivation requires the knowledge of trigonometry and some simple trigonometric identities, so this may be suitable for more advanced students.   Once again, we have two lines.   y=mx+b1  (1)--equation for the first line. y=mx+b2  (2)--equation for the second line.   Now recall that the slope of the line is the tangent of an angle this line forms with the x-axis. Indeed, m=(y2-y1)/(x2-x1), where x1, x2, y1, y2 are the x- and y-coordinates of any two distinct points on the line. If one draws the picture, it will be immediately obvious that m is the tangent of the angle between the line and the x-axis.   The difference b2-b1 gives the relative displacement along the y-axis of two lines... read more

How to find a distance between two parallel lines?

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

Christmas Present!!

Hi All!   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!   Merry Christmas!!  Andrew L. Profile  

Math Equations, Formulas and Vocabulary

Area, Volume and Circumference equations: Area of a Square A=S2 Area of a Triangle A=1/2bh Area of a Rectangle A=LW Right Triangle/Pythagorean Theorem a2+b2=c2 Area of Parallelogram A=bh Area of a Trapezoid A=1/2h(a+b) Area of a Circle A=πr2 Circumference of a Circle c=πd or c=2πr Volume of a Sphere V=4/3πr3 Surface Area of a Sphere SA=4πr2 Volume of a Cube V=s3 Volume of a Rectangular Solid V=lwh Slope of a line Equations Slope-intercept form y=mx+b m is the slope b is the y-intercept y is a y coordinate on the graph (that coincides with the line) x is an x coordinate on the graph (that coincides with the line) Horizontal line y=b Vertical line x=a Finding... read more

Maybe I'm Just No Good At Math.

I've heard this sentiment over and over--sometimes from students, and sometimes, I'll admit, in my own head. Last night, I was working on my own math homework, and there was one problem I just couldn't get my head around. I read the book, looked back at my class notes, and even sat down with a tutor for a while, and still, when I tried a new problem of the same type on my own, it just didn't work! "Maybe I'm not as good at math as I thought," I told myself. "Am I REALLY smart enough for bioengineering?" It was hard, but I told myself "YES!" And I kept working. I laid the assigned problems aside and started doing other problems of the same type from the book. I checked my work every time. Each problem took at least ten minutes to solve, and the first three were ALL wrong! I kept going. I got one right, and it made sense! I did another, and it was half right, but there was still a problem. I did another, and it was right... read more

Need a Geometry/Chemistry Tutor?

I graduated with a BS degree in Chemistry from East Carolina University, Greenville NC.  Since then, I worked in pharmaceuticals as a chemist/lab analyst for 1.5 years performing drug analyzes with High Performance Liquid Chromatography, Ultra-Violet Spectrometry, FT-NMR IR, and moisture testing.  Also I tutored General Chemistry I and II /Math(pre-algebra, algebra I & II, calculus, geometry at a community college. Also I was a General Chemistry II Lab Instructor at the college and have taken Human Gross Anatomy.  If anyone of you need help with Anatomy, let me know!  I am a more a visual learner and use concept mapping frequently. I thoroughly enjoy helping students achieve their learning and for them to be able to reach their potential.  I look forward in working with you. XX

Teaching Angles - Use Your Arms

Most recently, I was working with a student on the angles in a triangle.  We started with a circle to show the angles all the way around, their sizes, and how they compare to each other.  Then, I set my arms hand to elbow in an L shape.  We worked on showing the sizes of various angles by moving my hand at 12-o'clock (90 degrees) to a position that would show the angle size when my other arm was at 9-o'clock (zero degrees).  It worked very well.

Things a good tutor should say and discuss with you:

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... read more

Proving Proportions

DEFINITIONS When given two ratios (in the form x:y) or two relations (in the form of fractions), if the ratios of each element are the same they're said to be proportionate. Example: 3/6 and 1/2 are proportionate because 3 out 6 is the same as 1 out of two (half). PROVING PROPORTIONALITY When given two fractions to prove as proportionate, such as 1 and 3 2 6 you solve through cross-multiplication. Cross multiplication involves multiplying the numerator (number on top) by the denominator (number on bottom) of the other fraction, and then comparing the results. If the values are the same, the fractions are proportionate. The set-up above will be set-up as such: 1 * 6 ? 2 * 3 (6) = (6). Because both values are the same, these fractions are proportionate. Example 2: 3/2 and 18/8 The cross-multiplication... read more

Summer: Relax your brain or retain your knowledge?

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... read more

Week 3 and I had a Breakthrough with the same Earth Science Student for Algebra I

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... read more

Frustration: a Part of the Learning Process

Although learning is an awesome thing, it can be a difficult and frustrating journey for many students. This difficulty, however, is often times quite normal although most feel it means that a child may not be able to learn or that he/she is so frustrated that learning is no longer taking place. This is where the experienced tutor steps in; for frustration in learning is a part of the learning itself. I have taught and tutored many students and have seen first hand how this frustration can leave some students, and their parents, feeling helpless and hopeless. But there is ALWAYS Hope!!! What they have failed to realize is that as the brain learns difficult concepts, it can only take in parts at a time, little parts at a time. So although it may seem no learning is taking place, it actually is, just in smaller segments. In fact, the most frustration comes right before a new concept is achieved. This is when most give up. Had they stayed focused for perhaps one or two more... read more

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