Some say that algebra 1 is all about identifying and practicing techniques for setting up and solving problems that are "harder" than what you have to work with in arithmetic. This is at least partly true. The problems you encounter in algebra 1 are more challenging than those you encounter in arithmetic. However, you often use the same techniques you used in arithmetic to solve algebra 1 problems! So really, algebra 1 is a lot like the kinds of things you have already worked with - it just "looks" different.
Algebra 1 topics include setting up and solving word problems, finding reciprocals, simplifying expressions involving a radical or square root, plus working with equations and inequalities - solving and graphing linear equations, quadratics and parabolas, polynomials, rational equations and functions, factoring, and more!
Algebra 2 is about identifying and practicing techniques for setting up and solving problems that are more challenging than what you had to work with in algebra 1. However, you often use the same techniques you're already familiar with to solve algebra 2 problems! So really, algebra 2 is a lot like the kinds of things you have already worked with - it just "looks" harder.
Algebra 2 topics include setting up and solving word problems, working with absolute value equations and inequalities, solving linear and quadratic equations and inequalities, working with polynomials and rational functions, understanding the concepts of scaling and translation of functions and understanding function inverses, working with exponential and logarithmic functions, and more!
At the simplest level there are two types of calculus - differential calculus (analysis of derivatives) and integral calculus (analysis of integrals). For functions of the form y = f(x), differential calculus is often associated with rates of change, while integral calculus is typically associated with being able to find the area under a curve. At their base, both differential and integral calculus depend on the concept of "the limit", and on the concept of continuity.
Things in calculus get more complicated when y cannot be cleanly separated out as a function of x, and they get even more complicated when you're dealing with problems that have more than two dimensions.
At the high school and beginning college levels, geometry can be divided into two basic categories: Coordinate geometry and plane geometry. If you're into more advanced mathematics you might be learning about something called non-Euclidean geometry - but that's a separate subject.
Generally speaking, coordinate geometry is about being able to compute values or find formulas associated with certain kinds of geometric objects - primarily lines and angles, various type of polygons (triangles and rectangles are examples of polygons), and various types of polyhedra (polyhedra is the plural form for the word polyhedron - spheres and cubes are two examples of polyhedra). Parameters you might need to find values for include length, area, volume, and etc.
Plane geometry, on the other hand, is more about being able to construct certain kinds of geometric objects using just a compass and a straightedge. Also in plane geometry, you typically learn how to classify certain attributes associated with geometric objects - you learn to use terms such as congruence, proportionality, similarity, regularity, and etc. with respect go various geometric objects.
Linear algebra (generally associated with analysis and solution of systems of linear equations) is used in a variety of engineering and business applications. Some examples of how I've used linear algebra in my personal experience include:
* Linear algebra is an essential component of least squares data reduction processing and covariance analysis, used in probability and statistics. In a variety of engineering and business applications, the ability to use probability and statistics is critical in defining usable and cost-effective solutions to real problems, and in creating simplified / usable "models" of environments that are too complicated to explain using non-statistical techniques.
* Aspects of linear algebra are fundamental in different types of motion analysis - the technical discussion of how collections of things move in space as a function of time. Experience in this area includes creation of graphical simulators of real mechanical systems (both rigid and non-rigid systems), and etc.
* Some of the more modern computer tools associated with solving linear algebra problems include software packages such as MATLAB. I've used and am familiar with many of these SW tools. In my early career I often had to create many of my own SW routines for performing these types of calculations, so I understand the "guts" of what's going on in these software packages.
Microsoft Excel is a powerful spreadsheet program that is commonly used in a variety of professional (e.g. business, engineering, medical, and etc.) settings. Many people also use Excel at home, for a host of different tasks ranging from balancing checkbooks to maintaining address lists.
Like many successful software programs, Excel gets updated a lot. I've used various versions of Excel including 2003, 2004 (on a MacIntosh), 2007, and 2010 for a variety of business, engineering, and home applications.
If you need a good solid introduction to the basic capabilities of Excel, or you need to know how to do more advanced things in Excel, give me a call!
Prealgebra is about word problems, knowing the correct order of operations for evaluating an expression, distance and time problems, relative scale problems, solving equations in one variable, understanding the basics of probability and basic geometry, and more!
Precalculus explores functions and equations that go beyond those you encounter in algebra 1. It expands on what you already know (or are at least aware of) about linear functions, quadratic / parabolic equations, factoring, circles, rational functions, inequalities, and etc.
Beyond the basics, ideas often encountered in precalculus include working with and graphing function inverses, exponential functions (including compound interest problems), and logarithms or logarithmic functions. In some precalculus courses you may encounter identities that help you solve problems associated with triangles - things like the law of sines and the law of cosines. In even other precalculus courses, you'll have to work with what are called "conic sections" - circles, ellipses, parabolas, and hyperbolas. Many of these problems are expressed as word problems, that you need to set up and solve.
Even though these are considered to be "pre-calculus" topics, they are important in their own right. They are often encountered in business, medicine, engineering - even in the legal arena! So even if you aren't planning to go on to calculus, being aware of them can help you in your future.
Probability is closely related to statistics. In fact, at the introductory levels, probability and statistics are often taught as a single class. One of the key differences is that statistics relies on having data; you use the data to compute values associated with mean, standard deviation, and etc. With probability, you can create a theoretical distribution (which may or may not be based on attributes of collected data) that you believe is a good model for the types of outcomes you might expect to see in a particular experiment.
These models can be either discrete (like the binomial distribution) or continuous (like the normal distribution). You can then use these models to calculate values for parameters such as mean, standard deviation, and etc. You can also use these models for more sophisticated types of analysis - such as computing the likelihood that the outcome of an experiment will fall in some particular range of possible outcomes.
One of the powerful attributes of probability theory, combined with statistics, is that it provides you with a mechanism you can use to determine if a given set of data supports a given hypothesis or not. This is called hypothesis testing.
A typical definition is "Statistics is the science of planning studies and experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions based on the data."
Basic descriptive attributes of data include the number of data values, plus the mean, median, mode, standard deviation, variance, range, quartiles, and etc. of the data. Given a set of data, these values are often easy to determine using a calculator or a software package such as Excel.
Another descriptive aspect of statistics might be used to describe the relationship between a given sample and the population that the sample was drawn from. You might compare information about the population mean and standard deviation with a given sample value, and use the combination to calculate a z-score - and then you might use that z-score to determine if a specific sample element is "typical" for the population, or is unusual.
Or, you might use some simple kinds of regression analysis to determine, based on a set of data, whether the data is correlated or not across a set of independent and dependent parameters.
Statistical summaries of data are often presented using graphs. Again, many of these graphs are easy to generate using software packages such as Excel.
Once you get into the area of hypothesis testing and inferential statistics, statistics and probability become very closely coupled. An understanding of both discrete and continuous probability distributions becomes important when doing hypothesis testing.
Some say trigonometry is all about understanding what are called "circular" functions ... that is, being able to work with the sine, cosine, tangent, cosecant, secant, and cotangent functions and their inverses. From my experience, it's a lot easier to understand and work with these functions if you pay attention to some basic concepts about circles, triangles, and the Cartesian coordinate plane. Some simple rules, the same kinds of things you learned in algebra 1, will help you through trigonometry.
Ideas encountered in trigonometry include angles (in degrees, radians, or revolutions), the length of an arc for a piece of a circle, the definitions of the circular functions identified above, and a series of what are called "identities" for the circular functions. In trig courses, these identities typically include the half angle and double angle addition rules, plus the angle sum and difference rules.
From a professional perspective, these ideas are heavily used in physics and in a large variety of engineering applications. On the "more fun" side, they are also at the heart of describing spatial motion in a number of different computer games and in many animated films, especially when combined with linear algebra.