The GED Mathematics Test

[Pages:24]The GED Mathematics Test

Introduction to Algebra

Margaret A. Rogers, M.A. ABE/GED Teacher

Adult School Administrator Education Consultant

California Distance Learning Project

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GED Video Partner

#38 Passing the GED Math Test

Algebra and money are essentially levelers: the first intellectually, 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.

Words You Need to Know:

While viewing the video, put the letter of the meaning by the

correct vocabulary word.

Answers are on page 21.

_____1. signed numbers _____2. algebra _____3. variable _____4. equation

_____5. inverse operation

a. statement that two expressions are equal b. letters used to substitute for numbers c. the set of positive and negative numbers d. the opposite operation as addition is to subtraction and multiplication is to division e. branch of mathematics

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Points to Remember:

? Algebra is a branch of mathematics that uses rules to strategically solve for variables.

? You need to know some basic rules of algebra for the GED Math Test.

? Less than one-fourth of the GED Math Test must be solved with algebra.

? Some of the simple

problems in algebra can be solved using basic arithmetic and logical thinking.

? Algebra can be fun!

Introduction to Algebra

Algebra is the branch of mathematics where the object is to use rules strategically to solve for variables. Algebra has a symbolic language that is used to express relationships. Many of the same 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 are equal. An example of such an equation is:

3 x 8 = _____ x 6

In 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 will be any number or expression that can fill the blank to make the right side of the equation equal to the left. The simplest solution is the number 4. However, we could also fill the blank with expressions such as (2 + 2) or (9 - 5). After we fill the blank we want to test to make sure that both sides of the equation are equal.

3 x 8 = _____ x 6 3 x 8 = 4 x 6

24 = 24

In algebra, letters are used to stand for unknown numbers. These letters are called variables. A variable can stand for a single number or a complete expression. In the example above, we can replace the blank line with a letter to stand for the variable answers.

3 x 8 = A x 6

On the GED Math Test you will have to simplify equations, solve for variables, and use operations with signed numbers. There are many other skills that are associated with the branch of mathematics. However, if you are comfortable with these skills, you will be well on your 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 you simplify or solve for variables. In this Video Partners workbook, you will learn about and practice with the following properties:

" Commutative properties of addition and multiplication " Associative properties of addition and multiplication " Distributive property of multiplication

Although algebra is more abstract than arithmetic, it is important not to be afraid of it. Algebra is full of step-by-step procedures. If you learn the steps one by one and then systematically apply them when you are simplifying or solving equations, you will be successful on the algebra questions on the GED Math Test. And remember, there are only about 10 algebra questions out of the 50 questions on the GED Math Test.

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Balancing Equations

Equations are statements that two expressions are equal. In algebra, often some part of the equation is missing. The object of solving the equation is to discover what part(s) will balance the 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 of algebra, it is often possible to balance equations just by using arithmetic skills. In this equation, it is easy to see that 4 is the only number that will make a true statement.

3 x 8 = A x 6

Using your arithmetic skills, find one number which will balance each of the following equations. Answers are on page 21.

3 x 8 = _____ + 6

8 x 3 = 48 ? _____

_____ - 12 = 6 x 6

1,000 = B 10

4A = 24

25 = a2

2 + 3 x 5 = X - 3

(2 + 3)5 = 50/y

_____ = 42

1/2 x 1/3 = k/36

.2 + .2 = ____ %

1 dozen = b x 3

Variables

A variable is a letter or symbol used to represent an unknown quantity in an equation or formula. The value of a variable can change. Sometimes the value is dependent on other quantities 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 are several solutions for L and W. If L = 6, W = 4. If L = 8, W = 3. The solution for one variable is dependent on the value of the other.

Write at least three solutions for the variables in the following formulas:

Answers are on page 21.

A = LW A = 12

A = 1/2 BH A = 24

P = 2L + 2W P = 36

V = LWH V = 300

____________ ____________ ____________

____________ ____________ ____________

_____________ _____________ _____________

_____________ _____________ _____________

Often there is a single solution for a variable. Find the solution for the missing value in the formulas below:

A = LW

L = 10, W = 5

A = _____

C = !D D = 3 C = _____

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Operations with Signed Numbers

Before using the basic rules of algebra to solve for variables, it is essential to know how to perform the four operations, addition, subtraction, multiplication, and division, with positive and negative numbers. Positive numbers are those to the right of zero on the number line. Negative numbers have values less than zero and are found to the left of zero on the number line.

negative

positive

... -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 enough space to write all of the whole numbers, fractions, decimals, etc. They are all theoretically sitting in their proper place on the number line. Also, the number line is infinite. It extends in both directions with no end.

Practice this exercise to review your understanding of the number line. Answers are on page 21.

negative

positive

... -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. circle zero b. draw a box around -8 c. add 4.5 in the correct place d. put a star above -2 e. add -1/2 in the correct place

f. draw a triangle around 10 g. add 7 3/4 in the correct place h. add the next whole number to the left and right i. add - 2 1/2 in the correct place j. shade +9

There are special rules to add, subtract, multiply, and divide signed numbers. These rules are not difficult, but you must be able to perform these operations with confidence in order to succeed in algebra.

When solving algebraic equations, you must be able to move terms from one side of the equals sign to the other in order to isolate variables. In order to move terms, you will make use of the rules 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, and multiplication and division are opposites. Later you will learn to eliminate terms using inverse operations.

However, before we follow the basic rules of algebra to solve equations, we must practice using the rules for operations with positive and negative numbers. Different math books explain the rules in slightly different ways, but the result is always the same. Read and practice the rules for each of the four operations, addition, subtraction, multiplication and division. When you find you are comfortable with these methods, then you will be ready for the basic rules of algebra.

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Operations with Signed Numbers

Operation

Addition Subtraction

Multiplication

Division

The Rules

Checks and Bills Change the sign of the number being subtracted. Checks and Bills Multiply 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 Bills

One way to think of the rules for adding signed (positive and negative) numbers is to just think of 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 delivered twice 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 when analyzing postman stories.

Answers are on page 21.

Monday On Monday the postman brought two checks, one for $56.00 and one for $10.00. Assuming you have no other money in the world, what is your financial situation after this delivery? _______________

Tuesday On Tuesday, the postman brought a check for $25.00 and a bill for $13.00. Assuming you have no other money in the world, what is your financial situation after this delivery? _______________

Wednesday On 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 after this delivery? _______________

Thursday On 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 your financial situation after this delivery? _______________

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Friday On 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 is your financial situation after this delivery? _______________

Saturday On Saturday, the postman brought only bills. Alas, there were three of them. There was an electric 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 this delivery? _______________

There are no mail deliveries on Sunday.

Challenge problem: If every day were not a new day, how much money would you have or owe at the end of the week? _______________

As adults experienced with money, we may have used different ways to arrive at the total each day. No matter what we did, including subtraction, we were still adding signed numbers as we completed this exercise. Continue to practice with some numerical problems:

Add (Checks and Bills):

Note: Any number without a sign is positive (a check).

+36

- 8

- 5

-17

+100

-100

+32

60

- 25

+ 25

-17

40

-40

+81

+3

-60

-36

- 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 catalog orders that were arriving C.O.D. She kept her checkbook right by the front door so she would be ready 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 gift check for $50.00 from her Aunt Tilda. Did she spend more or come out ahead that day? How much? _________________________________

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Subtraction

The 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 being subtracted (-4). Then add (checks and bills). In this case, you now have two checks.

Subtract: +16

+16

- 4

+ 4

+20

Subtract:

Answers are on page 22.

-12

+16

+38

+19

-37

-85

+3

-5

+2

-6

+5

-50

+8

+3

+50

75

-75

-75

-3

-8

-2

-60

60

-60

(+6) - (-9) - (+3) - (+7) = _____

(-6) - (+9) - (30) - (-7) = _____

Multiplication

The rules for multiplying signed numbers are completed in two steps:

1. Multiply ignoring the signs. 2. Give a sign to the answer: Like signs + Unlike signs -

Multiply:

(8) (-3) _______

Answers are on page 22.

(-3) (-5) _______

(+5) (-11) _______

(+5) (-3) _______

(-2) (-12) _______

(+3) (-5) _______

(+6) (+8) _______

(-10) (-10) _______

(-7) (+9) _______

(-20) (+5) _______

(-5) (20) _______

(-9) (+6) _______

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