Algebra I Lessons
Inequalities and Absolute Value
Using the Graphing Calculator
• Number Theory on the Graphing Calculator...The order of operations is a fundamental rule of mathematics, but it can be tricky to handle on a calculator. Calculators vary according to kind, model, and manufacturer; therefore, the process for entering a given expression can also vary. Be sure to check the instructions of the calculator you are using. Graphing calculators make the process easier, because the large screen allows an entire expression to be entered before evaluating, so you can see how the calculator will read the expression. The order includes parentheses (or brackets), exponents, multiplication and division, addition and subtraction. Don’t become confused by the order of the list itself. Division may precede multiplication, and subtraction may precede addition if they come first in the expression, working from left to right. Powers of negative numbers may also look confusing. An expression written -x^y indicates that x is raised to the power y, and the result is multiplied by -1. If a negative value is raised to a power, it must be written (-x)^y.
• Algebra on the Graphing Calculator...A graphing calculator presents numbers as pictures, in the form of points in space. This tool quickly shows how many different values relate to one another. Graphed equations have uses ranging from scientific research to video game animation. Additionally, many graphing calculators allow lines of equations and text to be displayed or stored in memory. Problems far too long for an ordinary calculator may be solved easily. Some graphing calculators are handheld. Other types exist as software on a computer. In addition, there are many Web sites with fully functioning graphing calculators; they are easy to access and simple to use.
• Exploration of Integers...The use of integers in algebra is reviewed, including number lines, absolute value, integer exponents, and consecutive integer problems.
• Stay Rational...Numbers as we usually think of them are rational. That is, they can be exactly defined and written down or placed precisely on a number line. However, not all math fits that description. Some numbers are irrational. The terms rational and irrational refer to ratios, but also to the psychological sense of these words. Is it rational to work with numbers that can never be completely measured? Thus, the study of rational numbers demonstrates the choices we make in defining how numbers work. The rules of math often seem as if they must have always been the way they are, yet they have come to us through a long process of arbitration between conflicting viewpoints. In ancient times, the idea of irrational numbers seemed revolutionary. In the modern world, new uses for mathematics still produce a need for new math. For instance, some computer programs still divide by zero. By learning to understand the building blocks of math, we are better prepared to use it.
• Be There or Be Square...Equations that can be graphed as a straight line are called linear equations. How can we use mathematics to deal with more complex situations? You can use expressions involving powers or roots to describe a wide variety of shapes and graphed data, beyond the scope of linear mathematics. The fundamental basics of exponentiation are best introduced by demonstrating the mathematics of a square. From the measure of a side, you can calculate the area. Starting with area, you can find the length of a side. Learning these concepts will allow you to master the more advanced techniques for calculating exponents of higher value than squares. The skills explained here lead to the concepts of nonlinear algebra and beyond.
• One-Step at a Time...Linear equations can be used to describe a variety of real-life situations. You probably solve them all of the time without even knowing it. When deciding how many pizzas to order for your friends, you are using a linear equation. When figuring out the mileage in your car, you are solving a linear equation. When doubling a recipe, you are solving a multiple linear equations. And so on. The trick is to recognize what makes all of these situations the same, and to develop general principles that allow us to solve any kind of linear equation.
• Equations with Multiple Steps...Linear equations can be used to describe all sorts of situations: how fast a car needs to travel to get to its destination on time, how many gallons of water a city needs to have stored in case of a drought, how much it will cost to give out a tax refund. Solving these equations gives us the information we need to make informed decisions. Sometimes solving is easy, and sometimes it is more complicated. Sometimes equations require more than one-step to solve them. While this will not be as easy as solving one-step equations, it will allow us to look at more interesting problems.
• Exploring Percentages, Ratios, and Proportions...A ratio is a relationship between two numbers. A ratio can be expressed in several ways, the most common of which is as a fraction. For example, the ratio of 1 to 2 can be written 1:2 or 1/2. It can also be written 2:4 or 2/4, as 1/2 = 2/4. In fact, any ratio in which the second number is double the first number is equivalent to this ratio, as the relationship between the numbers is the same. The ratio 1/2 can also be written as 0.5, 5/10, or 50/100, which means that 1/2 is equivalent to 50%. In other words, a ratio can also be expressed as a percentage. Taking a percentage of a number is the same as multiplying by a ratio.
• Exploring Odds and Variations...The study of probability involves not only gathering information and data, but also making predictions about what might happen in the future. Statistics, such as central tendencies, involve taking many pieces of data and reducing them to a simplified picture. Often only a sample of the possible range of data is considered. In both cases, the answers are subject to interpretation. In daily life, we frequently hear of statistical conclusions being disputed. When dealing with chance or statistics, it is important to understand the ideas behind the numbers, rather than simply the arithmetic involved. Variation is another aspect of proportion. It deals with how changing one value in an equation may affect other values as well.
Functions and Linear Equations
• Graphing Functions...If you have ever played any version of the game Battleship, you are already familiar with the idea of graphing. Aside from its use in games, graphing is actually very useful in math. It allows us to locate particular points and plot lines using those points. You will now learn how to set up, read, and use a graph.
• Slippery Slope...A line on a graph contains an infinite number of points. In order to describe the line in a simple, manageable way, we use concepts such as intercept and slope. We can compare lines with different coordinates by examining similar terms. For instance, the slope of some lines crosses (or intercepts) the y-axis of a graph. The x-value here is 0, so this number does not need to be included. Rather than writing (0, 3), we can simply write 3. This value may then be inserted into an equation. The concept of slope tells us many characteristics about a line. In fact, it describes one characteristic of a line without using coordinates at all. Imagine a hiker climbing a hill from the left to the right side of a graph. The steeper the slope, the higher the numerical value. When the hiker walks down a hill from left to right, a steeper slope has a greater negative value.
• Review of Inequalities...An Inequality is a lot like an equation, but instead of showing an exact value, an inequality shows a range of possible values. This is a useful property to have because an inequality may show more information than an equation. Consider a formula where x = the speed of a car. If x equals 35 miles per hour, the equation is only true when the car is actually traveling at that speed. If x is defined as greater than 30 MPH and less than 60 MPH, however, this inequality is true at many different speeds. A computer graphics program may use inequalities to paint colors on the area defined by an inequality. Learning to use inequalities will open up new possibilities for using algebra and mathematics. The skills used to solve or simplify inequalities are familiar to anyone who can solve algebraic equations. Just watch out for the differences, such as those that occur when dividing or multiplying by a negative number.
• Advanced Inequalities...Compound inequalities show not only that a value is greater or less than another value, but they show where the value lies in relation to two other values. The range of values defined this way can be within a specific set, or everywhere on a number line except that specific set. Inequalities with two variables may be graphed on a Cartesian grid just as equations are. The variable x represents the x-value, and y is the y-value. If the graph produces a straight line, it is a linear inequality. The primary difference between the graph of an inequality and the graph of an equation is that the area on one side of the inequality’s line is shaded. This defines an area. For any point on the area, the inequality is true. For any point on the other side, the inequality is not true. Thus, an infinite number of points can be described by a single expression.
• Graphing Inequalities...Using inequalities is helpful when describing a possible range of values. We might know an average value and the amount by which the values could change. Writing the information as an inequality is a convenient way of expressing a number of situations. In music, scales are divided into octaves of notes, and further divided into sharp and flat notes. Each note is greater or lesser in pitch to adjacent notes. The need to precisely define this relationship led to the development of our understanding of inequalities. By expressing a musical note or a frequency as an inequality, you can construct an entire scale.
Polynomials and Factoring
• Monomials and Polynomials...At this point, you are probably familiar with the relationships between single terms. A variable, however, may represent an entire expression of values added or multiplied together. The prospect of working with these expressions may, at first glance, intimidate some. Fortunately, the familiar principles of mathematics apply. By taking apart a complex string of numbers, we can solve problems of enormous complexity.
• Taking in to Factor...Just as a single number may be factored into many smaller parts, an expression may be factored into component expressions. There are several techniques to accomplish the task, such as by finding common factors to each term. Factoring is the inverse of the way you might be accustomed to solving a math problem. Rather than performing operations to simplify an expression, factoring breaks the expression apart. Typically the result is a longer expression with a greater number of terms and operations. However each component expression is often shorter and simpler than the original polynomial. It could be useful to solve the whole expression by first solving a component expression. Another use is to find common factors to other expressions.
• Square Off...One method of factoring involves looking for expressions that are perfect squares. An expression in this form may be split apart by finding the root of one term, then by finding non-root factors that, together, make the expression true. You can use differences in squares to quickly compute area. For example, a window in a wall might consist of one large square, the outside edge, whereas the gap in the middle of the window is a smaller square. How can you determine the area of the wall alone? An elementary strategy would be to measure each portion of the of the wall as rectangle, find the area of each rectangle, then find the sum of the combined area. There are many steps to completing that operation, however. Calculating the problem as a difference of squares would require only two measurements and fewer operations.
• Introduction to the Quadratic...The basic forms of quadratic expressions are explained, as well as their applications with functions and graphs.
• Quadratic Equations...The study of algebra is really all about problem-solving; often, that means solving equations. We are about to begin the study of one type of equation: the quadratic equation. Quadratics involve a variable raised to the second power. You will find that they can be a little more complicated to solve than the linear equations you may be used to.
Rational and Radical Expressions
• Working with Rational Expressions...Most mathematical values may be written as rational expressions. The term is commonly used, however, to describe ratios between multinomial expressions. The techniques for solving problems like these are similar to the familiar ways we simplify fractions: factoring, finding common denominators, and so on. Rational expressions look abstract, but they are used for real-life applications such as in genetics. The probability of a certain gene sequence being found in a given section of DNA is a complex problem with many factors. Those factors may be rendered as a mathematical expression using ratios.
• Advanced Rational Concepts...When you were a child, the first set of numbers you became aware of was the whole numbers. You learned about 1, 2, and 3 as you counted your fingers and toes, even before you were aware that those quantities had names. Shortly thereafter, you learned about zero, as you realized you had none left after eating it. If you have siblings, your next discovery was probably the set of rational numbers or fractions. If there was one cookie left and two children, you learned about 1/2. If there were three of you, the fraction you became most familiar with was probably 1/3. Now that you are more mathematically sophisticated, you are ready for fractions that are more complicated. These new challenging fractions contain variables. You will be happy to learn that all of those former procedures about common denominators and canceling will still work.
• A Radical Standpoint...You may be familiar with techniques for working with exponents. The rules for handling radicals and roots are similar, but these rules are undertaken in reverse. The radical sign by itself indicates a square root, or x * x = √(y), whereas radicals with a higher index are indicated by a value outside the sign. This lesson will introduce you to the concept of equations having more than one solution. This feature is due primarily to the fact that the square root of a number has two solutions. (For example, the number 4 may have the square root 2 or -2.) Both possibilities must be accounted for; even though it is often possible to check and see which solution is correct. Expressions with radicals allow graphs to display curves rather than straight lines. This and other real-life applications, such as studies of population growth, make radical expressions a valuable tool of higher math.