Math from the Beginning Math can be confusing and intimidating, especially if we haven’t had to use math in a while. This course is going to start from the very beginning and give you a little bit of math at a time in a way that is hopefully easy to follow and understand. You may not realize it, but math will help you in a lot of different areas. You will probably be surprised at the number of places that these math skills can be useful in your everyday lives. If you measure things, build things, play games, need to double a recipe, and so on, these skills will help you. There are lots of branches of math, but we’re only going to cover a few specific areas. Most of our focus will be on what is called Algebra and we’re going to learn about the following mathematical operations in this math module: • Addition • Subtraction • Multiplication • Division • Parentheses • Exponents and what they mean We’ll learn that the people who invented these mathematical operations were basically “lazy” and were looking for a simple and easy way to do certain things. Math also has certain rules that we have to obey, and it’s easiest to think about learning math like you would think about learning a new game. Like any game, there are rules, and math has rules we have to follow. Like any game, it’s hard in the beginning to know which rule to use when, so math will come slowly in the beginning. But, the more we practice and play the game, the better and faster we will get at math. Mathematical Operations – Addition and Subtraction One easy way to think about addition and subtraction is to start with a number line, shown below. Place a dot at the starting point (0). Addition moves the dot to the right and subtraction moves the dot to the left. Rules and things to remember about addition • To add two numbers together, the units of measure must be the same. • You can add two numbers together if they are just numbers and there are no units of measure • You can add numbers together in any order • Addition problems can be given to us as numbers or written as words • The answer to an addition problem is called the sum. Subtraction is the addition in reverse. Using the number line, subtraction moves our counter to the left. Rules and things to remember about subtraction • To add two numbers together, the units of measure must be the same. • You can subtract if they are just numbers and there are no units of measure • You have to subtract numbers in the order given to us (you can’t change the order and get the same answer) • Subtraction problems can be given to us as numbers or written as words • The answer to a subtraction problem is called the difference. Mathematical Operations – Multiplication and Division Multiplication is a shortcut when we want to add a number to itself a bunch of times For example, the adding 5 to itself eight times would be written this way: 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40 But multiplication makes it much easier by letting us write it like this: 5 X 8 = 40 Rules and things to remember about multiplication • To multiply two numbers together, the units of measure don’t have to be the same, but the units of measure have to come along with the number and carry through to the answer. • You can multiply numbers if they are just numbers and there are no units of measure • You can multiply numbers in any order and the answer doesn’t change • There are many different symbols that can be used to show multiplication o X, *, ()(),[][], or two numbers separated by a dot • Multiplication problems can be given to us as numbers or written as words • The answer to a multiplication problem is called the product. Division is like multiplication in reverse. Rules and other things to remember about division • To divide one number by another, the units of measure don’t have to be the same, but the units of measure have to come along with the number and carry through to the answer. • You can divide numbers if they are just numbers without units of measure • You have to divide numbers in the specific order given or the answer will change • There are different symbols that can be used to show division: o / , ÷, −, or • Division problems can be given to us as numbers or written as words • The answer to a division problem is called the quotient. • If one number doesn’t evenly divide into another, there’s something left over and that is called the remainder • We can divide anything by 1 and it stays the same • If we divide anything by itself, the answer (quotient) is always 1 Some of these rules may not seem important right now, but they will become more important when we reach some of the other sections. For now, just keep these rules handy. Parentheses, Exponents, and Logarithms Parentheses are numbers that are put inside of (parentheses) or [brackets]. Examples are (5+6) or [12-4]. Follow the rules for each operation that appears inside of the parentheses or brackets. When there is just a single number left inside the parentheses, the parentheses can remain or it can be dropped. Exponents are another one of those “lazy” or timesaving operations that let us multiply a number by itself a certain number of times rather than writing out a long multiplication problem. The exponent indicates the number of times that a number is multiplied by itself. Logarithms are exponents taken in reverse. A logarithm is the number of times that a number is multiplied by itself. Rules of the Game of Math Here are some rules that apply to what we’ve just talked about • The order of operations is important when you have to use several mathematical operations in the same problem. Remember the phrase Please Excuse My Dear Aunt Sally to remember the order. The order is: o Parentheses o Exponents o Multiply o Divide o Add o Subtract • The Equalities rule applies to multiplication and division. If we have two things that are the same (such as when they are separated by an equal sign), we can multiply both sides by the same amount (or number) and they will still be equal. We can also divide both sides by the same amount (or number) and they will still be equal. • Multiplication by 1 rule says that you can multiply anything by 1 and it stays the same. (you’ll see how this rule will be used a little later on when we start solving problems) • Division by 1 rule says that you can divide anything by 1 and it stays the same (you’ll see how this rule will be used a little later on when we start solving problems) • Things that will help us play (and WIN) at the game of math • Write down and carry through the units of measure • Knowing which formula to use to solve a specific problem • Understanding Variables, Substitution, Fractions, and Scientific Notation Units of Measure are important to keep track of when playing the game of math. • Examples of numbers with units of measure are: o 375 gallons o 12 miles o 30 minutes o 2,200 cubic feet • To add and subtract numbers, the units must be the same • To multiply and divide, the units don’t have to be the same. They will either o Carry through o Cancel each other out • Always write down the units of measure and keep track of them until the very end because if the units of measure match the units of measure of the answer, you have correctly solved the problem! Formulas and knowing which formulas to use. A formula and an equation show that two things are equal. These two equal things are separated by an equal sign (=). Examples of some equations we will need to know are: • Area equations • Volume equations Area calculations • Area of a rectangle = length X width, or L X W. • Area of a circle = 𝜋R2 or 0.785D2 Volume Calculations • Volume of a rectangular or square box = Length X Width X Height, or L X W X H. • Volume of a cylinder = 𝜋R2H or 0.785D2H • Volume of a cone = 1/3 X 𝜋R2H or 1/3 X 0.785D2H Don’t worry, you aren’t going to have to memorize these equations and formulas. Variables and Substitution A variable is a letter that is used instead of a number to show us something that may change or “vary” from one situation (or problem) to the next. As an example, to carpet a room, we would need to know the length and width of the room. Written as an equation, here is what it would look like: Area of room = Length X Width Area of room = L X W where the letters L and W are the variables because the numbers will change from one room to the next. Substitution is where we take the number or numbers given to us in a specific problem and we “substitute” or replace the variables (the letters) with the numbers given to us in order to solve the problem. Fractions are extremely important to understand and master. They will be very helpful when it comes to solving problems and helping us keep track of units of measure. Fractions are commonly used to describe a part of a whole. The numerator is the number on the top of the fraction. The denominator is the number on the bottom of the fraction. In order to add or subtract fractions, the denominators have to be the same To multiply or divide fractions, the units don’t have to be the same, but they will either carry through or cancel each other out. We can divide any number by 1 to convert it to a fraction (so we can use them to solve a problem) Using Fractions – When we see a word problem, look for the word “per” as a hint because the number and units that appear before the word “per” go in the numerator and the number and units that appear after the word “per” go in the denominator. Here’s an example: Neil drove down the road at 45 miles per hour. Writing this as a fraction goes like this: Let’s sidetrack for just a minute to talk about reciprocals, which is when we swap the numerator and denominator. Reciprocals are important if we have to divide a fraction by a fraction to solve a problem. If we ever run into one of these problems, we take the fraction in the denominator and instead of dividing the numerator by the denominator, we multiply the numerator by the reciprocal of the denominator as shown below. Scientific Notation is another one of those timesaving shortcuts that we can use to more easily write either really large or really small numbers. Scientific notation uses a combination of both numbers and exponents to write out a number. • It takes a long time to write out 23,000,000,000,000,000 or 0.00000000000015 Let’s take a simple example to illustrate how scientific notation works by writing 255 using scientific notation • We would normally write it as 255 • For scientific notation, we need to show where the decimal point is (255.0) • We could also write it as 25.5 X 10 • We could also write it as 2.55 X 10 X 10 • Every time we moved the decimal point to the left, we multiplied by 10 • Since 255 = 2.55 X 10 X 10, and since 10 X 10 = 102 • We would 255 using scientific notation as 2.55 X 102 Here are the rules for Scientific Notation • Start with the number and add in the decimal point. • Count the number of spots you have to move the decimal point until the number you reach a number between 1 and 10 • Raise 10 to the number of places you moved the decimal point and that’s it. Let’s write this number in scientific notation. 340,000,000 • We need to move the decimal point 8 spots to get a number between 1 and 10 • 340,000,000 would be written as 3.4 X 108 Really small numbers are a little different because we have to move the decimal point to the right instead of to the left, which means we have to divide by 10 instead of multiplying by 10. Let’s look at a simple example using a small number (0.016) that we wouldn’t usually write using scientific notation. • This time, to get a number between 1 and 10, we need to move the decimal point to the right • Every time we move the decimal point to the right, we have to divide by 10 instead of multiplying by 10 • So we could write 0.016 as 0.16/10 • But we could also write it as 1.6/(10 X10) • Because we divided by 10, the exponent is negative, so • Scientific notation for 0.016 would be 1.6 X 10-2 Let’s write this number using scientific notation: 0.0000001 • The decimal point needs to move 7 spots to the right to get a number between 1 and 10 • Raise 10 to that negative power and • Scientific notation for 0.0000001 would be 1.0 X 10-7 • Remember this number because it will become important when we start talking about pH Playing the Game of Math – How to use the rules we just learned to solve problems. While there are lots of rules, we usually only need one or two of the rules to solve any given problem. In the beginning, it will take time, practice, and repetition as you learn which rules to use and how to use the rules – just like when you learn to play a new game. With a new game, you may refer to the rules often until you understand them better. The most important thing to understand are the specific types of questions that you will be asked on the Grade 1 or 2 licensing exam. Specifically, you will need to understand and be able to do the following math problems to pass the exam: • Converting from one unit of measure to another • Know to use decimals • Calculate Percentages • Calculate Averages • Paint a building or a retention tank • Calculate the volume of a cylindrical or conical bottom tank Fortunately, you will not need to memorize any equations or conversion factors. All of the equations, conversion factors, and abbreviations will be provided to you when you walk in to take the licensing exam. And we have an exact copy of the sheets you will be given when you walk into the exam, given to you with their permission of course, and appearing on the next several pages. Keep these sheets handy for use on math problems. The formulas and equations are listed in alphabetical order, so make sure you know where to find the ones you need. You will need to know which formula to use and how to use them You will be able to bring a calculator with you, but it has to be a simple one, one without the ability to connect to the internet. I strongly suggest getting a calculator that you feel comfortable with, using it with these math problems, and bringing it with you to the exam. It is important to understand that the same set of equations and conversion factors are given to everyone taking a licensing exam, regardless of whether they are taking the municipal or industrial exam and regardless of whether they are taking the Grade 1, Grade 3, or Grade 6 exam. This means that there are a lot of equations that you won’t need for your exam, so use these sheets now so you know where to find the equations you need. It will save you a ton of time on exam day. Converting Units of Measure from one set of units to another Conversion Factors are equalities that help us convert from one unit of measure to another. We start with an equality, and here are a few examples of equalities • 1 foot = 12 inches • 1 day = 24 hours • 1 minute = 60 seconds We have to know how to convert from one unit of measure to another to pass the licensing exam. The ABC formulas and conversion factors give us all the conversion factors. One foolproof way to convert unit is to use fractions. Let’s see how everything works by looking at a simple example: converting 48 hours into days. We start with the conversion factor: Using our dividing by itself rule, we could write two conversion factors, with both equal to 1: And we want to convert from hours to days, so we want to use the conversion factor with days in the numerator as we write 48 hours as a fraction: Because we have hours in the numerator and in the denominator, they cancel each other out and disappear, leaving only days in the numerator. Doing the multiplication gets us to the answer: Decimals and Percentages A decimal point helps us distinguish between whole numbers and portions (or fractions) of a whole number. For the number 28.5, the decimal is everything to the right of the decimal point (the dot), which would be .5. When we divide one number into another and the don’t divide evenly or exactly, we are left with a decimal to the right of the decimal point. If you remember, this is called the remainder. Percentages represent another way to express a number that is part of a whole. A percentage is always expressed as how many parts per hundred there are. Multiplying a decimal by 100 converts the number to a percentage. Percentages, fractions, and decimals can be used to express the same value in different ways. For example, 0.5 (decimal) could be written as 5/10 (fraction) or as 50% (percentage). Averages are obtained when we add several different numbers together and divide by how many numbers were added together. Averages require us to use parentheses, addition, and division. Calculating Area to Paint a Building or a Retention tank requires us to calculate area and it usually requires us to convert from one set of units to another (from square feet (ft2) to gallons) Solving these problems takes several steps, so it may be helpful to break things down into smaller sections to make it easier to keep track of things. See the example in the video. There are a lot of steps to solving these types of problems and it can be confusing in the beginning, so we have some help in the form of a worksheet to prompt you for everything you’ll need to do. If you haven’t downloaded the worksheet yet, please do so now and print off a few copies that you can use to help solve the problems. You won’t be able to bring this worksheet with you when you go to take the test, but this sheet will help you remember all the steps, which will be helpful in the beginning. Once you get some practice under your belt, you’ll start to remember what to do and eventually, you won’t need the worksheet any more. Calculating Volumes involves several steps and the worksheet will help us remember everything we need to do to solve the problem. Remember to use the worksheet until you get more comfortable with solving these types of problems on your own, which will take practice. The most challenging problem that will be asked on the wastewater exam is to calculate the volume of a conical bottom cylindrical tank, so it makes sense to practice solving these problems using the practice questions provided. Solving Word Problems Most of the math problems on the licensing exam will be word problems, which will require us to use the information that we are given in the wording of the problem. Here are the steps: • Read each problem carefully. You may need to read a problem a few times to fully understand what they are looking for. • Look for the information that is given in the problem • Determine the formulas needed to solve the problem • Look for the units of measure they want in the answer and determine if the units need to be converted. • Find any conversion factors needed • Use the worksheet as a guide Use the worksheet, formulas, equations, and conversion factors to solve the following problems 1) Convert 30 cubic feet to gallons 2) A building measures 60 feet long, 50 feet deep, and 15 feet tall. A gallon of paint covers 400 square feet. How much paint is required to paint the outside walls of the building? Ignore any doors and windows. 3) A water conservation program reduces the flow rate of a process from 700 GPM to 400 GPM. What is the percent reduction? 4) Calculate the volume in cubic feet of a cylindrical tank that is 40 feet in diameter and 25 feet tall. 5) Calculate the volume in cubic feet of a cone that is 25 feet in diameter with a cone depth of 15 feet.
