Maths - South Africa Lessons
Introduction to Number Theory
Introduction to Decimals
• Introduction to Decimals...Numbers are an important part of life. You use numbers to calculate how much a new video game costs, how much you weigh, what your temperature is when your are sick, and how long to cut boards to build a tree house. When you think of your age, that is a whole number, but not every number is a whole number. Decimals are numbers that allow us to write numbers of all sizes smaller than one. We use a decimal point to show parts of a whole. Our decimal system of numbers is based on the number ten. In ancient times, people counted using their ten fingers or ten toes. This is where the decimal system began. A decimal is a special number; it contains a period or a point in the number. The portion to the right of the decimal point represents a fraction, or a smaller part of, of a whole number. For example, the numbers 2.1, 12.78, and 0.0124 are all decimals.
• Get to the Point; Decimals...Multiplying decimals is just like multiplying whole numbers. After you multiply, then you figure out where to put your decimal point. We multiply decimals all of the time. Has your class ever ordered pizza for lunch? Your teacher probably multiplied the number of students in your class by the number of slices they could eat. Then, multiply the number of pizzas you need to feed the whole class. Once you have that number, then you must multiply the cost of the pizza – which usually has a decimal point – by the number of pizzas you will order. In order to divide decimals, always make sure that the divisor is a whole number. There are easy steps to follow when solving division problems with decimals. Even long numbers are not difficult when learning the steps to dividing decimals. Exponents are a shortcut method that represents how many times a number is multiplied by itself.
Addition and Subtraction
• Properties and Problems ...It is important to understand the different relationships and properties of addition and subtraction.. Learning these properties helps us to solve problems that come up in daily life. This will be helpful in future math classes as well. Estimating an answer rather than solving an entire problem is helpful when hurrying through daily chores. It is also important to know basic addition and subtraction facts in order to complete tasks quickly and correctly.
Multiplication and Division
Welcome to Fractions
Exploring Basic Concepts of Algebra
Introduction to Geometry
Introduction to Measurement
Introduction to Data Analysis
• Introduction to Fractions...We use fractions in a variety of settings. Chefs measure ingredients in a recipe in fractions of a cup. Carpenters measure wood in fractions of an inch. Sale prices are reported in terms of fractions of the cost- as in buy one, get one half off. One of the most important skills to learn is how to compare fractions. When comparing fractions with the same denominator, the fraction with the bigger numerator is the larger fraction. When comparing fractions with the same numerator, the fraction with the smaller denominator is the larger fraction. When comparing fractions with both a different numerator and a different denominator, first you have to use multiplication to make the denominators the same.
Basic Algebra Concepts
Data Analysis and Probability
• Data Analysis...The world is full of information. Television, the Internet, magazines, books, and radio are sources of information about our world. Information is useless unless it is organized and accurate. Statisticians organize and interpret data which, to most of us, would be either too difficult to understand or too time-consuming to read. One measurement that helps us understand data is average. Think about your own grades. Averages are used to measure how well you are doing in each course. Based on that average, you are assigned a grade. Statistics and averages are used in almost any area of life you can think of – medicine, science, the news media, and elections are just some examples.
• Roll the Dice; Exploring Probability...What are the chances of rain tomorrow? What are the odds of winning the lottery? What is the likelihood of lightning striking? All of these questions refer to probability. A probability is a numerical measure of the likelihood of an event. For example, what are the chances of an event where you will meet someone who has the same initials as yours today? A probability can range from 0 to 1. If we assign a probability of 0 to an event, this indicates that the event will never happen, like the probability of rolling a 7 on a single dice. A probability of 1 means that the event will always occur; such as the probability of rolling a number less than 7 on a single dice. A probability of 0.5 or one-half means that it is just as likely for the event to occur as for the event not to occur. So the the probability of rolling an odd number on a single roll of a die is half, or 0.5. Probability is a discipline that is very useful in real life. In fact, just about everything you do involves a certain amount of risk. If you have ever had surgery, the doctor probably informed you about the chances of having complications. When you decide to get in the car, there is a chance that you will be involved in an accident. The chance of surviving that accident increases by wearing a seatbelt. Some occupations involve more risk than others do. For example, a dentist has a safer job than a police officer does.
• Customary Units...Have you ever heard of a fathom or a parasang? How about a rood? Believe it or not, these are older units of measurement. The metric system has replaced most of these older units, but some of them have survived today, including ounces, feet, inches and pounds. These are some of the U.S. customary units, which are the non-metric units of measurement used in the United States. Being able to take and record measurements using U.S. Customary units is an important skill. You will need to know the units used to describe each measurement and the relationships between them. U.S. Customary units of length and distance include inches, feet, yards, and miles; units of capacity include cups, pints, quarts and gallons, while units of weight include ounces, pounds, and tons. We use ounces to describe both capacity and weight.
Introduction to Algebraic Functions
Integers and Rational Numbers
Basic Geometry Concepts
• Introduction to Geometric Properties...The study of the shapes in the world around you is just beginning. This module has introduced you to some of the basic geometric shapes and their properties. You know a little about some pairs of angles, and about how to find the area and circumference of a circle. Look at the shapes that surround your world. As you do so, you will probably see some more complex shapes, formed by a combination of the basic shapes you have begun to study. You may begin to see that these shapes share some of the properties we have discussed.
• Exploring Polygons...Perimeter and area are commonly used measurements. You can find the distance around any shape by measuring the length of each side and finding the sum of the lengths. You can find the area of figures by using a formula. Shapes that are not regular polygons can be divided into pieces that are regular polygons. Then you can find the area of each piece and add all the areas. Landscapers, architects, artists, and engineers are some of the professions that use perimeter and area measurements. Look around and you will see geometric figures everywhere.
Ratios, Proportions, and Percents
• 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
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 flavour 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 maths 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.
• 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.
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.
Polynomials and Factoring
• All Alone, Monomials...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.
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.
• 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.
• 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.
• 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.
• 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.
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.
Coordinate Geometry and Transformations
Locus and Constructions
• Inside the Circle of Trust...Studying the intersection of lines and circles is a useful way of finding the characteristics of both. After learning the fundamentals of this interaction, we can find ways to apply that knowledge to everyday life. The number of revolutions required by gears in a machine, the composition of subjects in a photograph, and the design of a building are all examples of arc measurement. Additionally, the basic concepts of circles will prepare you to learn the more advanced operations of trigonometry.
• Comparing Polygons...Ratio and proportion are familiar concepts from arithmetic and algebra. Applied to geometry, they allow us compare objects of different sizes but similar shapes. The component segments of a given figure may be compared to other segments of the figure, allowing us to define its characteristics. An architect can draw a blueprint that is only a few feet across but that represents a large building. The drawing could even be revised to show buildings of different sizes but similar proportions. In the past, engineers often used scales in multiples of 12 and 8, based upon the divisions of an inch ruler. An inch on a 1:12 plan could be instantly converted to 1 foot when working with the actual object; likewise, a 1/8-inch hatch-mark on a 1:96 plan could be instantly known to represent a foot.
• Exploring Inqualities in Geometry...The uses of inequalities in geometry are discussed, as well as several methods of negative proofs.
• Properties of Quadrilaterals...The properties of parallel lines may be applied to quadrilaterals, or four-sided figures. Polygons made from two sets of parallel lines are called parallelograms. Parallelograms are convenient to use because they allow us to apply the characteristics of parallel lines to representations of real-life objects. Parallel lines share mathematical data, such as slope or angle. After finding the data relevant to one line, we can find similar data for lines parallel to the first line. When this principle is applied to a shape, such as a polygon, the information becomes a useful way of making calculations. For example, the parts of a building, such as bricks or wooden beams, fit together one on top of the other to form parallel lines. This means that that architects and builders must often find parallel lines and that the study of parallel lines has many applications.
• Comparing Attributes of Triangles...In your studies of mathematics you have encountered many laws that govern operations. Oftentimes, the law is quoted as a rule without an explanation of why the rule is true. The study of congruent triangles opens the door on the laws of math, illustrating how it all works. Obviously, this is useful knowledge to have for solving geometry problems; however, geometric proofs can be applied to algebra and arithmetic. They historically have been the basis for much of our mathematical knowledge. The concept of a proof, or how we know what we know, has applications beyond pure mathematics. It is how all manner of logic may be rendered into a systematic process and how philosophical arguments are analyzed. The methods for proving congruence in basic triangles are an easy-to-comprehend way of leaning how a logical proof is constructed.
Definitions, Postulates, Theorems
• Basic Geometric Figures...Figures created by geometry can be intricately complex. However, they may be broken down into simpler parts. The most fundamental components are points, and the lines between points. In fact, these concepts are so simple and basic that points and lines are thought of as undefined. A point has no size or shape. A line has no thickness. Therefore they occupy no real space, and so we cannot define them in the way that we define a real object. Lines, rays, planes, and angles extend infinitely, another abstraction, which is unlike real objects. By studying and exploring these concepts, we may define the characteristics of real-life objects. That is why geometry is one of the most ancient and culturally widespread disciplines of mathematics. Whether you want to plant a field with mielies or build a house, it all breaks down into points and lines.
Geometry and Algebra