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Algebra 1

I was always bothered that the word, "Algebra" had no intrinsic meaning to me. I know what algebra is, but then one might as well call it "Kaborsck" as far as I was concerned. After reading several math history books I pieced together a meaning that makes sense to me: The word "algebra" comes from the Arabic root word "al-jabr" which was borrowed from the medical profession to disconnect and reconnect bones. In Algebra students will learn to solve equations (1st and 2nd degree) by connecting and...
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Algebra 2

Algebra 2

In Algebra 2 students will extend the ideas of Algebra 1 which involved linear and quadratic equations to

1) systems of linear equations represented using matrices;

2) equations involving radicals (square and cube roots);

3) rational functions which are ratios of linear and quadratic expressions;

4) exponential and logarithmic equations; and

5) conic sections which involve parabolas, ellipses and hyperbolas.

I have taught Algebra 2 many times...
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Calculus

The French mathematician Lagrange once said that a teacher doesn't really understand his subject until he can explain it to a man on the street. I once took Lagrange's challenge to be able to explain the fundamental theorem of calculus to "a man on the street". I asked my barber to envision she was in a drugged stupor in a semi-private hospital room as the nurse pulls on the privacy curtain. She would experience in slow motion that the amount of additional "privacy" she was getting at any instant...
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Geometry

My Geometry teacher in the ninth grade told me I was the best student of geometry he had ever seen because of my ability to support each statement in a geometric proof with a reason, sometimes more reasons than he could think of. That complement set me on the road to becoming a math major in college.

Geometry means "Earth Measurement". Geometry as first envisioned by the ancient Greeks revolutionized the world. It was a mental discipline requiring us to ask and show why something was...
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Prealgebra

Prealgebra is all about fractions, "broken numbers". Percentages and decimals are fractions in a certain form. A ratio and a proportion involve fractions. The student must learn to deal with fractions: reduce and expand them in order to combine fractions by addition and subtraction.

Lewis Caroll, who was a math teacher, was thinking of adding and subtracting fractions when he wrote about Alice shrinking and expanding to get the key to get into the garden.

Once students...
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Precalculus

Pre-Calculus comes before Calculus. But what is it? To call it Pre-Calculus is like calling it "Post-Algebra-Geometry". To understand what it is all about we must understand what Calculus involves: derivatives and integrals which are functions derived from functions. So far math has been about numbers. Now the student must learn to see it from the perspective of functions: polynomial, rational, radical, exponential and trigonometric functions.

Calculus, as Algebra, is an art; the artist...
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Trigonometry

Trigonometry is the study of triangles. I starts with right triangles in which if one angle is the same as any other right triangle, then the triangles are similar and hence their sides are proportional. In particular the ratios of their sides is the same. The basic ratios are sin, cos and tan and their reciprocals are csc, sec and cot, respectively. Relationships are developed among these ratios as well as the law of sins and the law of cosines which are used to resolve all the sides and angles...
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Statistics

Probability is the mathematical study of the likelihood of events when the population is understood. Statistics is the mathematics of making inferences about a population that is not understood based on a random sample from the population. In probability we try to deduce from what we know; in statistics we try to infer things about what we don't know. Important concepts include measures of the sample data's center (mean and median) and the extent of its spread(standard deviation). The law of large...
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SAT Math

SAT tests mostly cover material from Algebra and Geometry. Statistics show the best way to improve your score by around 100 points is to take another year of math like Algebra 2. Working problems from an SAT prep manual can improve your score by around 50 points. Practice working the sample tests at first without any time pressure. Problems that you cannot work or that took too long should be reviewed with your tutor. After going through 3 or more sample tests with your tutor (me hopefully), begin...
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Probability

One of my teachers of graduate probability at UCI was Ed Thorpe, who wrote the book on how to beat Las Vegas at blackjack. Hopefully your teacher will be clearer than he was.

Probability is the mathematics of chance. Theoretical probabilities are based on a ratio of the number of ways a certain event can occur to the number of ways anything could happen. Therefore counting is very important. It starts with the fundamental theorem of counting: if there are n ways of getting A and m ways...
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Linear Algebra

Linear algebra is the study of simultaneous linear equations. They can be solved with one of two methods: substitution or elimination of a variable. With two equations in 2 unknowns they both have their advantages. However, with a 3x3 or higher system elimination becomes much more powerful and lends itself to computer solutions. A simple linear equation looks like: ax=b. A system of linear equations using arrays of numbers (or matrices) can be written as AX=B, and it begins to look like simple...
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