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The GED Mathematics TestIntroduction to AlgebraMargaret A. Rogers, M.A.ABE/GED TeacherAdult School AdministratorEducation ConsultantCalifornia Distance Learning Projectwww.cdlponline.org1

GEDVideo Partner#38 Passing the GED Math TestAlgebra and money are essentially levelers: the firstintellectually, the second effectively.Simone Weil (1909 - 1943)Video 38 Focus: how you use algebra to simplify equations and solve for variables.You Will Learn From Video 38:!!!!!How to use algebra to solve equations.How to simplify algebraic expressions.How to isolate variables as a rule for solving equations.The rules for operations with signed numbers.That algebra has a language of its own.Points to Remember: Words You Need to Know:While viewing the video, put the letter of the meaning by thecorrect vocabulary word.Answers are on page 21.1. signed numbers2. algebra3. variable4. equation5. inverse operationa. statement that two expressions areequalb. letters used to substitute fornumbersc. the set of positive and negativenumbersd. the opposite operation as additionis to subtraction and multiplication isto divisione. branch of mathematics2 Algebra is a branch ofmathematics that usesrules to strategicallysolve for variables.You need to knowsome basic rules ofalgebra for the GEDMath Test.Less than one-fourthof the GED Math Testmust be solved withalgebra.Some of the simpleproblems in algebracan be solved usingbasic arithmetic andlogical thinking.Algebra can be fun!

Introduction to AlgebraAlgebra is the branch of mathematics where the object is to use rules strategically to solve forvariables. Algebra has a symbolic language that is used to express relationships. Many of thesame rules and algorithms that we use in arithmetic we also use in algebra. However, in algebra,these rules are often used to solve equations. Equations are statements that two expressions areequal. An example of such an equation is:3 x 8 x 6In algebra, we are trying to find out which solution will make both sides of the equation equal.We are trying to balance the equation. There are many solutions to this equation. A solution willbe any number or expression that can fill the blank to make the right side of the equation equal tothe left. The simplest solution is the number 4. However, we could also fill the blank withexpressions such as (2 2) or (9 - 5). After we fill the blank we want to test to make sure thatboth sides of the equation are equal.3 x 8 x 63x8 4x624 24In algebra, letters are used to stand for unknown numbers. These letters are called variables. Avariable can stand for a single number or a complete expression. In the example above, we canreplace the blank line with a letter to stand for the variable answers.3x8 Ax6On the GED Math Test you will have to simplify equations, solve for variables, and useoperations with signed numbers. There are many other skills that are associated with thebranch of mathematics. However, if you are comfortable with these skills, you will be well onyour way to answering most of the algebra questions correctly.It is also important to understand the properties that allow you to manipulate an equation as yousimplify or solve for variables. In this Video Partners workbook, you will learn about andpractice with the following properties:"""Commutative properties of addition and multiplicationAssociative properties of addition and multiplicationDistributive property of multiplicationAlthough algebra is more abstract than arithmetic, it is important not to be afraid of it. Algebra isfull of step-by-step procedures. If you learn the steps one by one and then systematically applythem when you are simplifying or solving equations, you will be successful on the algebraquestions on the GED Math Test. And remember, there are only about 10 algebra questions outof the 50 questions on the GED Math Test.3

Balancing EquationsEquations are statements that two expressions are equal. In algebra, often some part of theequation is missing. The object of solving the equation is to discover what part(s) will balancethe equation and make the two sides equal to one another.Even though there are certain steps that are recommended to balance equations using the rules ofalgebra, it is often possible to balance equations just by using arithmetic skills. In this equation, itis easy to see that 4 is the only number that will make a true statement.3x8 Ax6Using your arithmetic skills, find one number which will balance each of the followingequations. Answers are on page 21.3 x 8 68 x 3 48 - 12 6 x 61,000 B104A 2425 a22 3x5 X-3(2 3)5 50/y 421/2 x 1/3 k/36.2 .2 %1 dozen b x 3VariablesA variable is a letter or symbol used to represent an unknown quantity in an equation orformula. The value of a variable can change. Sometimes the value is dependent on otherquantities and which quantities are known or unknown in the equation or formula. For example,the formula for finding the area of a rectangle is A LW. If the area is known to be 24, there areseveral solutions for L and W. If L 6, W 4. If L 8, W 3. The solution for one variable isdependent on the value of the other.Write at least three solutions for the variables in the following formulas:Answers are on page 21.A LWA 12A 1/2 BHA 24P 2L 2WP 36V LWHV 300Often there is a single solution for a variable. Find the solution for the missing value in theformulas below:A LWL 10, W 5A C !D4D 3C

Operations with Signed NumbersBefore using the basic rules of algebra to solve for variables, it is essential to know how toperform the four operations, addition, subtraction, multiplication, and division, with positive andnegative numbers. Positive numbers are those to the right of zero on the number line. Negativenumbers have values less than zero and are found to the left of zero on the number line.negativepositive -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 Remember, the number line is a representation of all numbers even though there is not enoughspace to write all of the whole numbers, fractions, decimals, etc. They are all theoretically sittingin their proper place on the number line. Also, the number line is infinite. It extends in bothdirections with no end.Practice this exercise to review your understanding of the number line. Answers are on page 21.negativepositive -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 On the number line above:a.b.c.d.e.circle zerodraw a box around -8add 4.5 in the correct placeput a star above -2add -1/2 in the correct placef. draw a triangle around 10g. add 7 3/4 in the correct placeh. add the next whole number to the left and righti. add - 2 1/2 in the correct placej. shade 9There are special rules to add, subtract, multiply, and divide signed numbers. These rules are notdifficult, but you must be able to perform these operations with confidence in order to succeed inalgebra.When solving algebraic equations, you must be able to move terms from one side of the equalssign to the other in order to isolate variables. In order to move terms, you will make use of therules for operations with signed numbers. You will also be using inverse operations as well.Inverse operations are the opposite operations. Addition and subtraction are opposites, andmultiplication and division are opposites. Later you will learn to eliminate terms using inverseoperations.However, before we follow the basic rules of algebra to solve equations, we must practice usingthe rules for operations with positive and negative numbers. Different math books explain therules in slightly different ways, but the result is always the same. Read and practice the rules foreach of the four operations, addition, subtraction, multiplication and division. When you find youare comfortable with these methods, then you will be ready for the basic rules of algebra.5

Operations with Signed ivisionThe RulesChecks and BillsChange the sign of the number being subtracted.Checks and BillsMultiply ignoring the signs.Give a sign to the answer: Like signs Unlike signs Divide ignoring the signs.Give a sign to the answer: Like signs Unlike signs -Addition - Checks and BillsOne way to think of the rules for adding signed (positive and negative) numbers is to just thinkof checks and bills. An illustration for checks and bills is found in postman stories.Although my father, who was born in 1908, clearly remembers when the mail was deliveredtwice a day, that has not been the case since the 1950s. There is now only one delivery each day;and yet we still use the phrase, “I’ll put it in the morning mail.” Every day is a new day whenanalyzing postman stories.Answers are on page 21.MondayOn Monday the postman brought two checks, one for 56.00 and one for 10.00. Assuming youhave no other money in the world, what is your financial situation after this delivery?TuesdayOn Tuesday, the postman brought a check for 25.00 and a bill for 13.00. Assuming you haveno other money in the world, what is your financial situation after this delivery?WednesdayOn Wednesday, the postman brought two bills. One was for 68.50, and the other was for 16.00. Assuming you have no other money in the world, what is your financial situation afterthis delivery?ThursdayOn Thursday, the postman brought two checks and one bill. The checks were for 24.75 and 34.00. The bill was for 100.00. Assuming you have no other money in the world, what is yourfinancial situation after this delivery?6

FridayOn Friday, the postman brought three checks and one bill. The checks were for 14.00, 39.50,and 45.00. The bill was for 64.00. Assuming you have no other money in the world, what isyour financial situation after this delivery?SaturdayOn Saturday, the postman brought only bills. Alas, there were three of them. There was anelectric bill for 44.50, a magazine subscription for 24.95, and a parking ticket for 6.00.Assuming you have no other money in the world, what is your financial situation after thisdelivery?There are no mail deliveries on Sunday.Challenge problem: If every day were not a new day, how much money would you have or oweat the end of the week?As adults experienced with money, we may have used different ways to arrive at the total eachday. No matter what we did, including subtraction, we were still adding signed numbers as wecompleted this exercise. Continue to practice with some numerical problems:Add (Checks and Bills):Note: Any number without a sign is positive (a check). 36- 5- 8-17 100- 25-100 25 32-176040-40-60 81-36 3-7 8-2( 17) ( 5) (-6) (-2) 15 12 (-17) ( 8) (-75) (-8) During the holiday mail season, bills were mounting up for Carrie. She had placed many catalogorders that were arriving C.O.D. She kept her checkbook right by the front door so she would beready to settle the bills and accept the items. On December 10th, she received C.O.D. orders for 12.64, 39.57, and 19.11. She wrote checks for all of them. The same day, she received a giftcheck for 50.00 from her Aunt Tilda. Did she spend more or come out ahead that day? Howmuch?7

SubtractionThe rules for subtracting signed numbers are completed in two steps:1. Change the sign of the number being subtracted.2. Add (Checks and Bills)When you are subtracting (-4) from ( 16), you would first change the sign of the number beingsubtracted (-4). Then add (checks and bills). In this case, you now have two checks.Subtract: 16- 4Subtract: 16 4 20Answers are on page 22.-12 3 16-5 38 2 19-6-37 5-85-50 8-3 3-8 50-275-60-7560-75-60( 6) - (-9) - ( 3) - ( 7) (-6) - ( 9) - (30) - (-7) MultiplicationThe rules for multiplying signed numbers are completed in two steps:1. Multiply ignoring the signs.2. Give a sign to the answer: Like signs Multiply:Unlike signs -Answers are on page 22.(8) (-3)(-3) (-5)( 5) (-11)( 5) (-3)(-2) (-12)( 3) (-5)( 6) ( 8)(-10) (-10)(-7) ( 9)(-20) ( 5)(-5) (20)(-9) ( 6)8