Pre Algebra Lessons
• Exploring Integers...Basic algebraic operations are explored with integers for both problems and solutions. These include absolute value and the distributive property. Use of the distributive property for mental math is also covered.
• Solving Equations...Algebraic equations are generalized forms of numerical equations. In elementary school you spent most of your time learning how the numerical equations such as 4 + 2 = 6, 8 – 9 = 11, or 3 – 18 = 5(9 – 4) work. Now we deal with different types of equations that contain unknown variables such as x – 8 = 11, x = 11 = 9 – 3, or 3(x + 4) = 11 + x. In these equations, we will use the same principles and rules that we applied in arithmetic equations. Numbers replace the variables in the equations.
Algebra and Rational Numbers
• Number and Operations...Factors and powers are characteristics of numbers: 25 will always factor into 5 times 5; 25 squared is always 625. So when we see a problem that includes 25, knowing these characteristics is very useful. Learning how to take numbers apart and put them back together again is a good way to find the best method for solving a problem.
• Rational Numbers...Rational numbers are values that we can write as fractions. Sometimes, if we write the values as decimals, we are left with numbers that repeat forever as the decimal places get smaller and smaller. In this case, we may find it easier to work with the fractions.
• Monomials...The concept of polynomials is introduced with the monomial, as well as the rules for arithmetic and algebraic operations with these expressions.
Algebra and Data Analysis
• Get the Best Rate...As any shopper knows, percents play a large role in our finances. When we read the newspapers, we see ads about sales, news items on the unemployment rates, and information about what percent financing is being offered on the car of your dreams. We use ratios, proportions, and percents every day in our lives in just about every situation.
• Determing the Outcome...Values shown as percents are all around us in everyday life. 30% of voters prefer one candidate to another. 65% prefer chocolate flavor to vanilla. 95% of a bird population migrates with the seasons. These are quick, and easily understood ways of displaying information. By learning how to work with percents, we learn how to use the principles of math to communicate effectively. Percents are also a good way of learning how to use money. We may compare prices with a percent, or find a proportion of money for calculating a total price after taxes.
Linear Equations and Inequalities
• Multi-Step Equations...Solving linear equations is a systematic process and becomes easier as you practice. The distributive property is an important concept to employ when working on the equations. Grouping symbols direct how the components within an expression should be calculated. By combining like terms, a significant part of the job of solving for the variable is accomplished. We must take extra care when distributing a negative number to make sure the proper value(s) are affected.
• Inequalities...Linear Inequalities are a lot like the equations you are already familiar with. As you know, to solve an equation, you need to isolate the variable on one side of the problem. You can use that same skill when solving inequalities, adding only one or two additional steps. These new problems, however, will also require you to graph your answer on a number line.
• Introduction to Functions...So far, the types of linear equations we explored were all in one variable. We have dealt with equations such as 3x + 5 = 11 or (9x – 12) + 23 = -4(x + 2) which at most can have only one solution. Even in some equations we do not reach the same unique solution. Now we are going to deal with the same type of equations, that is, linear equation, but with two variables. In such cases, the answer of an equation is not limited to one; there can be an infinite number of answers for both variables. Linear equations in two variables are the most common mathematical models we use for applied purposes. Their structures are well-fit with many real data with linear relationships. In a linear relationship, two quantities are associated in which one variable is equal to the product of the other variable and a constant number plus a fixed value. The constant number is called the slope of the equation, and represents the rate of change in variables. For example, if y = -900x + 14000 represents the value of a car x years after purchase, it is the constant number or slope -900 that indicates how the car depreciated as time passed. Having such a model, such as the depreciation of a car, we can make reliable predictions about the value in the future. This can be done by drawing the graph of the function and finding its value in the upcoming years from the graph. The fixed value is the value of the car at the time of purchase; that is, when x = 0. This is called the y-intercept of the function. So, the y-intercept of this function is 14000.
• Down the Slope...A linear equation is a form of modeling that best presents a group of data. For example, a company might have a big ledger of employee salaries, as well as years of service. Although precise, this ledger does not give you any general idea of the company. If you were applying for a job at the company, you would definitely want to know that, for example, you will be better off after 10 years with that company. In such circumstances, linear equations can assist you. Formulating data through a linear equation, then, can help you with real-life decision-making. If you created this equation using the best fit line method, you can even figure out the rate of change in salaries over time.
• Systems of Equations...When solving one linear equation only, the task involved simple steps to reach the solutions. Here we will be dealing with a higher form of math: two equations in two variables. Pairs of equations might not be cooperative when we are struggling to find the unique solution. Sometimes they just do not work together! When solving two equations, if both of their left sides are the same, but their right sides are different, they are called inconsistent equations. If after simplification of one of the equations, you reach the other equations, then the system is called dependent. Think back to one linear equation containing two variables. Often, we can find an infinite number of solutions for x and y. For example, let the equation L + W = 45 represent the relationship between a length (L) and a width (W) of a rectangle. There are an infinite number of rectangles whose dimensions fit in this equation. But, if we are asked to find a certain rectangle whose dimensions not only fit in this equation but the ratio of its dimensions, for example 1:15. Solving this system yields only one solution for L and one solution for W. That is, the solution of this system will choose one out of thousands of the solutions of the first equation, which also satisfies the second equation. If in solving one linear equation, we are looking for an infinite number of solutions. In the case of systems of equations, we look for one solution from each equation that is the same. In other words, we look for a single solution that can be found among the infinite number of solutions of both equations. The difference between an inequality and an equation is only one aspect of algebra. One conveys one unique value; the other may convey an infinite number of values.
Algebra and Geometry
• Geometric Principles...The triangle is probably the most useful shape in all of geometry. Thanks to the shape’s many properties, it is often possible to find the measurement of a line or angle without physically measuring it. This is known as indirect measurement. This module will introduce you to the triangle, as well as to some rules and formulas that result from the properties of triangles.
• Geometric Figures...Concepts of area are explained, as well as theorems for congruent triangles. The basics of geometric transformations are also introduced.
• Going Around in Circles...We see shapes all around, from the windows in our classroom to the computer screen in front of us. Some of these shapes are familiar to us, and we can easily measure these. Others, such as a mouse to a computer, may have rounded edges, are a bit harder to measure. The area and circumference of a circle and a special value known as pi all help us calculate the area and perimeter of oddly shaped items.
• Scratching the Surface...Measurements of three-dimensional shapes are explained, including volume and surface area for various cylinders and related figures.
Algebra and Statistics
• Displaying Data...Various methods can represent data that has been gathered and arranged. The type of graph, chart, or table used makes an integral difference in the way data is analyzed and interpreted. Stem-and-leaf plots, box-and-whisker plots, and histograms are just three ways to present statistical information. Each focuses on an important aspect of the collected and classified numerical information. Statistics help us to understand many aspects of our society but it is always important to keep in mind the possibility for error or misrepresentation in statistical methods and depiction.
• Favorable Odds...We often hear people ask, “What are the chances?” When a couple with three sons is expecting another baby, for example, they may wonder what the chances are that they will have a girl this time. Someone who has entered a drawing for a prize may ask what the chances are that he will win. A high school student may wonder what her chance is of getting into a certain college. With a little knowledge of probability an answer is often not hard to find. In this module we will learn enough basic probability to calculate the chances of a number of common events.
• Introduction to Polynomials...We now turn to another important class of functions; polynomials. We already know that various algebraic operations may be applied to a group of numbers at the same time. For example, in -5(83) * (12 – 47) + 19, the operations addition, subtraction, multiplication, and exponentiation are applied to numbers -5, 8, 3, 12, 4, 7, and 19. If some or all of these numbers are replaced with parameters and variables, they form an algebraic expression. A similar algebraic expression to this numerical one is -4(x3) * (5 – y3) – z, where x, y, and z are variables. We will first study a special class of such expressions in which all the factors are in product form; in other words, all the components creating an expression are multiplied. The expression 4x3y is a routine example of such expressions. All the factors 4, x, and y are multiplied. We call such expressions monomials. Polynomials are simply the sum of a group monomials. When solving problems, more often you will come across combinations of many polynomials that need to be simplified. To do so, we will use addition and subtraction operations using the general properties of real numbers. Knowing the different operations of polynomials and various methods of manipulating polynomials are vital to solving many real life situations. From modeling the cost and revenue of a manufacturer or modeling the population of human being or a certain species in a community, a common mathematical concept is polynomials. In such real life phenomena, the rate of change is not a fixed value. The cost of producing a certain part may be reduced by increasing the number of parts or the population of a certain species in a habitat may be decreased as its size extended during a scientific investigation. All such phenomena can be described by polynomials. Using polynomials as a basic tool in events in which the rate of change is not a fixed value helps us to have relatively precise predication of variations and outcomes of such phenomena in the future.
• Linear and Nonlinear Functions and Graphs...When investigating linear functions, we observe a general pattern. Linear functions can be applied only to data sets with fixed rates of change. That is, both x and y increase at the same rate. In real life, however, the change in one quantity is not always proportional to the change in another quantity. For example, the rate of bacteria reproduction over time is not the same as the rate of change in time. For situations like this, linear functions obviously are not a good choice to use for modeling. We need another class of functions in order to describe those phenomena that are similar to the reproduction of bacteria. Exponential functions can have a very sharp inclination and are the best tool to describe such phenomena. In this Module, we will explore some of nonlinear functions that are suitable for certain real life phenomena such as the growth of bacteria, and will learn how to generalize such functions for real life data.