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 disconnecting numbers and letters that represent numbers. I will ask them to "al-jabr".
It was once known as the Cossic Art. "Coss" is Latin for "the thing". In modern Algebra we most frequently use "x" to represent "the thing". Start any word problem with labeling the unknown, "Let x = the number of ...." It is this great art that has so greatly advanced all the modern sciences. Think of it as the art that supports the sciences!
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 while on Kauai and have experience tutoring many students at various ability levels in the Tucson area to understand and solve problems in Algebra 2.
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 would be the height of the curtain at that instant. That is the fundamental theorem of Calculus: the rate of change of the amount of privacy at any instant is the height of the curtain providing the privacy.
Being able to work with a function, its derivative and its integral enriches the student's mathematical experience as if one can now operate on the surface of the earth (functions), under the water (derivatives) and in the air (integrals or anti-derivatives). The power of this new math brought society into the modern era. It will bring students to a new level of mathematical maturity.
I am currently tutoring 4 students in calculus in high school (3 in AP Calculus) and one in college. I have taught Calculus in a private high school and once taught 15 AP Calculus students in a public high school on Kauai because they couldn't understand their Calculus teacher and were all getting D’s and F’s. After several months of work, they all graduated at the top of their class and several became valedictorians. In graduate school I taught 1st year Calculus at Georgia Tech and at University of California at Irvine, as teacher’s assistant, I lead math labs for Calculus 1 that were so popular that there was standing room only.
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 true; it was a masterpiece of logical reasoning; it became the foundation of scientific discovery and it provided us with the hope that the universe is ultimately understandable. The Romans failed to appreciate that the Greek mathematics was more about "why" than is was about "what". So copying the facts without knowing why left them with no way to advance their knowledge.
High school Geometry remains much the same as it was developed over 2000 years ago by Euclid. But students of Geometry should focus on understanding why something is true and learn the inductive and deductive processes. This will allow them to continue to grow unlike the Romans.
Geometry has grown from Geometry as envisioned by Euclid, to Analytic Geometry by Descartes, to Curved Geometry by Riemann, to multidimensional topology. Basically Geometry is the mathematics of shapes.
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 can do this they must enter that strange garden and learn how to apply this knowledge to solve problems involving numbers, basic geometric shapes and probabilities. If they are good at it they may end up at the Mad Hater's tea party at the Large Hadron Collider.
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 needs a pallet. In this case the pallet is the coordinate plane and the student must learn to draw the faces (graphs) of all the functions and recognize them from their faces. Each member of a family looks like the parent of the family. Differences arise due to translations, stretches and reflections. Pre-Calculus is The Study of Functions. So, call it that!
I have taught Pre-Calculus from several different textbooks and successfully tutored many students through it. I am currently working with two students in the Tucson area, one in college and the other in high school.
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 of getting B, then there are n x m ways of A and then B. Counting progresses with the useful ideas of permutations (for ordered sets) and combinations (for sets without order). Using counting and simple probability facts students will learn of the Binomial Distribution and the Law of Large Numbers will lead to the "Bell Curve". The whole subject of Probability was started in a series of letters between Pascal and Fermat who were discussing the likelihood of winning a game that was prematurely stopped and the players wanted to distribute the pot based on the likelihood of ultimately winning had they been able to continue the game. Today the course is usually taught as Probability and Statistics or just as Statistics. Probability has found its way into all the sciences including the daily weather reports, biology and even theoretical physics, and that is why you need to know about it.
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 to take them within the allowed time limit. If a problem takes too much time, review it with your tutor (me please) to see if there is an easier way to get to the correct answer.
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 numbers can be used to approximate these unknown values within a range with a degree of confidence measured by the probability that the values lie within the range. There are various statistics that can be used to test the validity of a hypothesis. Some of the formulas are intimidating and your answers will sound something like "I don't know what the mean is but I don't reject what we have assumed the mean to be and I will probably be right in that assumption 95% of the time." Yea,...your going to need a tutor!
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 of any triangle.
A thorough knowledge of trigonometry is needed for the study of calculus which is the mathematics of motion. There are basically two types of motion: linear and rotational. Trigonometry is the lynchpin that connects these two and hence is essential for calculus.