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Math Help Online for Algebra & Geometry - Practice Math Online |
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The whole numbers are numbers we use everyday. They form amazing patterns that continue to intrigue everyone, including mathematicians. As you study this module remember that to understand math you have to understand and be fluent in whole number arithmetic. Take the time to play with the Concepts on the computer, they will help you to understand the underlying ideas. Be sure to concentrate when you do your homework problems. The practice will help you gain the fluency that will make all of the rest of math much easier to learn.
In Module 1 you learned about whole numbers. Here we deal with part numbers, numbers that do not represent wholes, but instead represent parts or fractions of wholes. You will find the arithmetic of fractions very different from the arithmetic of whole numbers. For example, multiplying fractions generally makes the result smaller, and dividing makes it larger. Multiplying and dividing fractions are generally easier to do and learn than adding and subtracting, so we study them first. These are some of the things you will come to understand by visualizing fractions as fraction bars.
Decimals or, more formally, decimal fractions are fractions whose denominators are powers of 10. They are valuable because they can be written and manipulated in our base 10 place value system just like ordinary whole numbers. Instead of dealing with fractions we can now deal with whole numbers and decimals points. Decimals are used instead of ordinary fractions in everything from money to the price of stocks, from the number of gallons of gas we put in our tanks to measurement in the metric system.
Ratios and proportions may well be the most important ideas in all of arithmetic; they are the ones we use most often in the real world. If ratio and proportion have confused you try out our visualizations. If you experiment with them, they can help you to understand these powerful ideas and use them in your real world activities.
Picture ratio as a triangle and picture proportion as two similar triangles.
Everyday we are bombarded with charts and graphs depicting information gathered from this survey or that telephone marketing poll. To make sense out of all the information swirling around us we use a branch of mathematics called statistics. Scientific research relies on statistical measures to determine whether or not an experiment is successful or whether one procedure is better than another one. This module will introduce you to data into tables, graphing data, and descriptive statistics (mean, median, mode).
In this module we do the arithmetic of the Real Numbers. The real numbers are built up from: the counting numbers, the whole numbers, the integers, the rationals, and the irrationals. As you go through these lessons think about how the operations on each of these sets of numbers are the same and are different from each other.
Most of the visualizations that will be key to your understanding of algebra are introduced in this module. Play with them until you understand their patterns.
Linear equations are the simplest equations; yet they are the most useful equations in all of mathematics because you can always solve a linear equation. In linear equations the variable is to the first power. They are called linear because when we graph them, they produce a straight line. We use them everywhere to calculate a tip, interest on a loan, or speed for a car. If you really understand how to solve linear equations, then you will understand algebra.
The graph of a linear equation in two variables is a straight line, which is the reason that this form of equation is called linear.
This module covers elementary graphing, the slope of graphs of linear equations, and the graphing of inequalities as well as equalities.
Graphs enable us to visualize these equations. By the time you are finished you should be able to look at a linear equation or inequality and picture its graph in your mind.
In the real world, most situations require more than two variables to describe them. For example, we can describe the simple motion of a ball in two variables: distance and time. But to describe the weather, we need many more variables. We seek to describe such situations with a system of linear equations, one for each variable we have. This enables us to find a solution for each variable. So, if we have two equations in two variables, we can find a solution, or the exact values of the variables that solve both.
Geometry, from the Greek Geo for earth and metry for measure, is the study of shapes. Here we introduce you to some of its basic concepts. Every closed shape has a perimeter, an edge, and an area. Every shape made of straight lines has angles associated with it. The simplest of closed shapes that can be made of straight lines are triangles. At the end of this modules you will learn the Pythagorean Theorem and use it to calculate lengths and distances.
Thus far, we have been dealing with linear expressions and equations. These monomials have variables to the first power. Nomial is another word for term, and poly means many. A monomial is a single term and a polynomial can have many terms. The simplest polynomials are in one variable and each term has that variable raised to a different power or exponent. Remember, only like variables raised to the same power can be added.
Polynomials can be combined through multiplication. Factoring is a method for reducing polynomials into a product of simpler polynomials. The factoring process undoes multiplication, enabling us to put a polynomial into its simplest form. Once a polynomial is in its reduced form we will solve for the roots of some of them. In this module you will investigate the critical steps involved with factoring polynomials, determining their roots, and solving equations. We approach the factoring of polynomials systematically. The key is to building up a repertoire of polynomials, so that you can quickly see both the form and the expected factors.
The word rational stems from ratio and is not about making sense. Rationals are ratios or fractions. Rational equations are expressions or equations that involve fractions. Rational equations can be simplified. To add or subtract rational numbers we must find their least common denominators. To divide them we invert the divisor and multiply. You might want to revisit 6.3 Operations on Rationals before you start this section. Rational equations may look complicated and scary but you can solve them by taking one step at a time.
Radical expressions are more than just finding square roots, cube roots, and so on. They can be manipulated in equations and used to solve application problems. Radical expressions can be written in several forms, as well as added, subtracted, multiplied and divided. When radicals are expressed in exponential form, all of the rules of exponents apply. We will begin studying exponents and radicals by learning to translate between forms and to express each in reduced form. You will draw on your skills of working with fractions and exponents from previous modules.
Quad means square in Latin. A quadratic equation is polynomial where the highest order term is a square; it is thus said to be a second order polynomial. Quadratic equations are very important for two reasons: 1) They can represent a wide variety of things like motion in the real world, and 2) We can find solutions for them. While there is a general formula for solving quadratic equations, it requires a good deal of calculation and taking square roots. Therefore, we look for tricks, special cases that are easier to solve.
The concept of a function is considered one of the most important concepts in mathematics. A function is a unique relation that transforms an input to an output. The input can be a numerical value, an algebraic expression, a quantity, or some other mathematical object. A functional transformation is a unique, one-to-one correspondence, meaning that each function input maps to one, and only one output. In math we visualize functions as graphs, tables, charts, or symbols. In this module you will develop your understanding of functions and how to manipulate their representations.
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