Virginia SOL Grade 8 Math

By Stephen Hearne

REA … Real review, Real practice, Real results. REA's Virginia Grade 8 SOL Math Study Guide! Fully aligned with Virginia’s Core Curriculum Standards Are you prepared to excel on this state high-stakes assessment exam? * Take the diagnostic Pretest and find out what you know and what you should know * Use REA's advice and tips to ready yourself for proper study and practice Sharpen your knowledge and skills * The book's full subject review refreshes knowledge and covers all topics on the official exam and includes numerous examples, diagrams, and charts to illustrate and reinforce key math lessons * Smart and friendly lessons reinforce necessary skills * Key tutorials enhance specific abilities needed on the test * Targeted drills increase comprehension and help organize study * Color icons and graphics highlight important concepts and tasks Practice for real * Create the closest experience to test-day conditions with a full-length practice Posttest * Chart your progress with detailed explanations of each answer * Boost confidence with test-taking strategies and focused drills Ideal for Classroom, Family, or Solo Test Preparation! REA has helped generations of students study smart and excel on the important tests. REA’s study guides for state-required exams are teacher-recommended and written by experts who have mastered the test.

• Language: English
• Publisher: Research & Education Association
• Release date: Jan 1, 2013
• ISBN: 9780738668734

Book preview

Index

PASSING THE VIRGINIA STANDARDS OF LEARNING (SOL) - GRADE 8 MATHEMATICS

ABOUT THIS BOOK

Our book provides excellent preparation for the Virginia Standards of Learning (SOL) Grade 8 Mathematics Test. Inside you will find reviews that are designed to provide you with the information and strategies needed to do well on this exam. We also provide a full-length practice test, so you can get a good idea of what you’ll be facing on test day. Detailed explanations follow the practice test. If you are having a problem with a certain question, we’ll tell you how to solve it.

Our Teacher’s Answer Guide contains full explanations to the Class and Homework Assignment questions in the diagnostic tests at the back of this book. Teachers may obtain the answer guide by contacting REA.

ABOUT THE TEST

Since 1998, the Virginia Department of Education has administered an eighth grade assessment to determine how well a student is advancing and whether the student is on course to perform well in high school. It is one of the key tools used to identify students who need additional instruction to master the knowledge and skills as defined by the Department of Education. Questions on the SOL are designed to test student mastery of these content strands:

Number and Number Sense

Computation and Estimation

Measurement and Geometry

Probability and Statistics

Patterns, Functions, and Algebra

HOW TO USE THIS BOOK

What do I study first?

Read through the review and our suggestions for test-taking. Studying the review thoroughly will reinforce the basic skills you will need to do well on the test. Our practice drills and diagnostic tests feature five answer choices, whereas the actual exam has only four choices. This results in a greater challenge and more rigorous preparation.

The SOL’s four-choice format is accurately reflected in our practice test, which you’ll find in the back of this book. Our practice test is designed to capture the spirit of the SOL, providing you with an experience that mimics the administration of the exam.

When should I start studying?

It is never too early to start studying for the exam. The earlier you begin, the more time you will have to sharpen your skills. Do not procrastinate! Cramming is not an effective way to study, since it does not allow you the time needed to learn the test material. The sooner you learn the format of the exam, the more time you will have to familiarize yourself with the exam content.

ABOUT THE REVIEW SECTIONS

The review sections in this book are designed to help you sharpen the basic skills needed to approach the exam, as well as to provide strategies for attacking each type of question. You will also find exercises to reinforce what you have learned. By using the reviews in conjunction with the drills and practice test, you will put yourself in a position to succeed on the exam.

TEST-TAKING TIPS

There are many ways to acquaint yourself with this type of examination and help alleviate your test-taking anxieties. Listed below are ways to help yourself.

Become comfortable with the format. When you are practicing, simulate the conditions under which you will be taking the actual test. Take the practice test in a quiet room, free of distractions. Stay calm and pace yourself. After simulating the test only a couple of times, you will boost your chances of doing well, and you will be able to sit down for the actual exam with much more confidence.

Read all of the possible answers. Just because you think you have found the correct response, do not automatically assume that it is the best answer. Read through each choice to be sure that you are not making a mistake by jumping to conclusions.

Use the process of elimination. Go through each answer to a question and eliminate those that are obviously incorrect. By eliminating two answer choices, you can vastly improve your chances of getting the item correct, since there will only be two choices left from which to make your guess. It is recommended that you attempt to answer each question, since your score is calculated based on how many questions you get right, and unanswered or incorrectly answered questions receive no credit.

Work quickly and steadily. Avoid focusing on any one problem for too long. Even so, you should never rush. Rushing leads to careless errors. Taking the practice test in this book will help you learn to budget your time.

Learn the directions and format for the test. Familiarizing yourself with the directions and format of the test will not only save time, but will also help you avoid anxiety (and the mistakes caused by getting anxious).

Work on the easier questions first. If you find yourself working too long on one question, make a mark next to it on your test booklet and continue. After you have answered all of the questions that you can, go back to the ones you have skipped.

Avoid errors when indicating your answers on the answer sheet. Marking one answer out of sequence can throw off your answer key and thus your score. Be extremely careful.

Eliminate obvious wrong answers. This ties in with using the process of elimination. Sometimes a question will have one or two answer choices that are a little odd. These answers will be obviously wrong for one of several reasons: they may be impossible given the conditions of the problem, they may violate mathematical rules or principles, or they may be illogical. Being able to spot obvious wrong answers before you finish a problem gives you an advantage because you will be able to make a better educated guess from the remaining choices even if you are unable to fully solve the problem.

Work from answer choices. One of the ways you can use a multiple-choice format to your advantage is to work backwards from the answer choices to solve a problem. This is not a strategy you can use all the time, but it can be helpful if you can just plug the choices into a given statement or equation. The answer choices can often narrow the scope of responses. You may be able to make an educated guess based on eliminating choices that you know do not fit into the problem.

THE DAY OF THE TEST

Before the Test

On the day of the test, you should wake up early (hopefully, after a decent night’s rest) and have a good breakfast. Make sure to dress comfortably, so that you are not distracted by being too hot or too cold while taking the test. Also plan on arriving at school early. This will allow you to collect your thoughts and relax before the test, and will also spare you the anguish that comes with being late.

During the Test

Follow all of the rules and instructions given by your teacher or test supervisor.

When all of the test materials have been passed out, you will receive directions for filling out your answer sheet. You must fill out this sheet carefully since this information will be printed on your score report. Fill out your name exactly as it appears on your identification documents, unless otherwise instructed.

You can write in your test booklet or on scratch paper, which will be provided. However, you must be sure to mark your answers in the appropriate spaces in the answer folder. Each numbered row will contain four ovals corresponding to each answer choice for that question. Fill in the oval that corresponds to your answer darkly, completely, and neatly. You can change your answer, but be sure to completely erase your old answer. Only one answer should be marked. This is very important, as your answer sheet will be machine-scored and stray lines or unnecessary marks may cause the machine to score your answers incorrectly.

ARITHMETIC REVIEW

1. INTEGERS AND REAL NUMBERS

Most of the numbers used in algebra belong to a set called the real numbers or reals. This set can be represented graphically by the real number line.

Given the number line below, we arbitrarily fix a point and label it with the number 0. In a similar manner, we can label any point on the line with one of the real numbers, depending on its position relative to 0. Numbers to the right of zero are positive, while those to the left are negative. Value increases from left to right, so that if a is to the right of b, it is said to be greater than b.

If we now divide the number line into equal segments, we can label the points on this line with real numbers. For example, the point 2 lengths to the left of zero is - 2, while the point 3 lengths to the right of zero is + 3 (the + sign is usually assumed, so + 3 is written simply as 3). The number line now looks like this:

These boundary points represent the subset of the reals known as the integers. The set of integers is made up of both the positive and negative whole numbers: { ... - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, ... }. Some subsets of integers are:

Natural Numbers or Positive Numbers

Whole Numbers

Negative Numbers

Prime Numbers—the set of positive integers greater than 1 that are divisible only by 1 and themselves: {2, 3, 5, 7, 11, ... }.

Even Integers—the set of integers divisible by 2: { ..., - 4, - 2, 0, 2, 4, 6, ...}.

Odd Integers—the set of integers not divisible by 2: { ..., - 3, - 1, 1, 3, 5, 7, ... }.

RATIONAL AND IRRATIONAL NUMBERS

where a is any integer and b is any integer except zero. An irrational number is a number that cannot be written as a simple fraction. It is an infinite and non-repeating decimal.

The tree diagram below shows you the relationships between the different types of numbers.

EXAMPLE

Here are some examples of some rational numbers.

EXAMPLE

Here are some examples of irrational numbers.

PROBLEM

List the numbers shown below from least to greatest.

3, 0.3

SOLUTION

3

PROBLEM

List the numbers shown below from greatest to least

e, π

SOLUTION

e

PROBLEM

List the numbers shown below from least to greatest.

SOLUTION

PROBLEM

Classify each of the following numbers into as many different sets as possible. Example: real, integer ...

SOLUTION

Zero is a real number and an integer.

9 is a real, natural number, and an integer.

is a real number.

½ is a real number.

⅔ is a real number.

1.5 is a real number.

PROBLEM

Write each integer below as a product of its primes.

2 12 5 22 18 36

SOLUTION

A prime number is a number that has no factors other than itself and 1. For example, the numbers 1, 3, 5, and 7 are prime numbers. Write each integer as a product of its primes.

PROBLEM

Find the greatest common divisor (GCD) for the following numbers: 12 and 24.

SOLUTION

Step 1 is to write out each number as a product of its primes.

3 × 2 × 2 is the common factor in both sets of prime factors.

Step 2 is to multiply the common factors together. 3 × 2 × 2 = 12

Therefore, 12 is the GCD.

ABSOLUTE VALUE

The absolute value of a number is represented by two vertical lines around the number, and is equal to the given number, regardless of sign.

The absolute value of a real number A is defined as follows:

EXAMPLE

| 5 | = 5, | - 8 | = -(- 8) = 8.

Absolute values follow the given rules:

(A) | - A | = | A |

(B) | A | ≥ 0, equality holding only if A = 0

(D) | AB | = | A | x | B |

(E) | A |² = A²

Absolute value can also be expressed on the real number line as the distance of the point represented by the real number from the point labeled 0.

So | - 3 | = 3 because - 3 is 3 units to the left of 0.

PROBLEM

Classify each of the following statements as true or false. If it is false, explain why.

| - 120 | > 1

| 4 - 12 | = | 4 | - | 12 |

| 4 - 9 | = 9 - 4

| 12 - 3 | = 12 - 3

| - 12a | = 12 | a |

SOLUTION

True

False, | 4- 12 | = | 4 |- | 12 | |- 8 | = 4 - 12 8 ≠ - 8 In general, | a + b | ≠ | a | + | b |

True

True

True

PROBLEM

Calculate the value of each of the following expressions:

|| 2 - 5 | + 6 - 14 |

SOLUTION

Before solving this problem, one must remember the order of operations: parenthesis, multiplication and division, addition and subtraction.

|| - 3 + 6 - 14 | = 3 + 6 - 14 | = | 9 - 14 | = | - 5 | = 5

PROBLEM

Find the absolute value for each of the following:

zero

4

- π

a, where a is a real number

SOLUTION

(1)

(2)

(3)

(4)

PROBLEM

Which of the number lines below correctly graphs the following points: -8, 5, and 20?

Number Line A

Number Line B

Number Line C

SOLUTION

The correct answer is Number Line C. The numbers in Number Line A are incremented correctly, but -5 is graphed instead of 5. Number Line B is incorrect because only 5 is graphed. Number Line C is correct because the numbers are incremented correctly and the points are correctly graphed.

PROBLEM

Using the number line below, graph the solution to -5 - (-3).

SOLUTION

Step 1 is to graph point -5 on the number line.

Step 2 is to move 3 units to the right of -5. In this problem we move to the right of -5 because a negative number is being subtracted. Since -2 is 3 units to the right of -5, graph -2 on the number line.

POSITIUE AND NEGATIUE NUMBERS

To add two numbers with like signs, add their absolute values and write the sum with the common sign. So,

To add two numbers with unlike signs, find the difference between their absolute values, and write the result with the sign of the number with the greater absolute value. So,

To subtract a numberbfrom another numbera, change the sign of b and add to a. Examples:

(1)

(2)

(3)

To multiply (or divide) two numbers having like signs, multiply (or divide) their absolute values and write the result with a positive sign. Examples:

(1)

(2)

To multiply (or divide) two numbers having unlike signs, multiply (or divide) their absolute values and write the result with a negative sign. Examples:

(1)

(2)

PROBLEM

Identify the sign resulting from each operation.

(+)(+)

(-)(-)

(+)(-)

(-)(+)

SOLUTION

The correct answer for problem a is a positive number. When multiplying two positive numbers, the product is always positive.

The correct answer for problem b is a positive number. When multiplying two negative numbers, the product is always positive.

The correct answer for problem c is a negative number. When multiplying a positive and a negative number, the product is always negative.

The correct answer for problem d is a negative number. When multiplying a negative and a positive number, the product is always negative.

According to the law of signs for real numbers, the square of a positive or negative number is always positive. This means that it is impossible to take the square root of a negative number in the real number system.

ORDER OF OPERATIONS

When a series of operations involving addition, subtraction, multiplication, or division is indicated, first resolve any operations in parentheses, then resolve exponents, then resolve multiplication and/or division, and finally perform addition and/or subtraction. One way to remember this is to recite Please excuse my dear Aunt Sally. The P stands for parentheses, the E for exponents, the M for multiplication, the D for division, the A for addition and the S for subtraction. Now let’s try using the order of operations.

Consider

Notice that 25 ÷ 5 could be evaluated at the same time that 4 × 10 is evaluated, since they are both part of the multiplication/division step.

DRILL: INTEGERS AND REAL NUMBERS

Addition

1. Simplify 4 + (- 7) + 2 + (- 5).

2. Simplify 144 + (- 317) + 213.

3. Simplify | 4 + (- 3) | + | - 2 | .

4. What integer makes the equation - 13 + 12 + 7 + ? = 10 a true statement?

5. Simplify 4 + 17 + (- 29) + 13 + (- 22) + (- 3).

Subtraction

6. Simplify 319 - 428.

7. Simplify 91,203 - 37,904 + 1,073.

8. Simplify | 43 - 62 | - | - 17 - 3 |.

9. Simplify - (- 4 - 7) + (- 2).

10. In Great Smoky Mountains National Park, Mt. LeConte rises from 1,292 feet above sea level to 6,593 feet above sea level. How tall is Mt. LeConte?

Multiplication

11. Simplify - 3 (- 18) (- 1).

12. Simplify |- 42 | × | 7 |.

13. Simplify - 6 × 5 (- 10) (- 4) 0 × 2.

14. Simplify - | - 6 × 8 |.

15. A city in Georgia had a record low temperature of -3°F one winter. During the same year, a city in Michigan experienced a record low that was nine times the record low set in Georgia. What was the record low in Michigan that year?

Division

16. Simplify - 24 ÷ 8.

17. Simplify (- 180) ÷ (- 12).

18. Simplify | -76 | ÷ | -4 |.

19. Simplify | 216 ÷ (- 6) |

20. At the end of the year, a small firm has $2,996 in its account for bonuses. If the entire amount is equally divided among the 14 employees, how much does each one receive?

Order of Operations

5-1

22. 96 ÷ 3 ÷ 4 ÷ 2 =

23. 3 + 4 × 2 - 6 ÷ 3 =

24. [(4 + 8) × 3] ÷ 9 =

25. 18 + 3 × 4 - 3 =

26. (29 - 17 + 4) ÷ 4 + | - 2| =

27. (- 3) × 5 - 20 ÷ 4 =

29. | - 8 - 4 | ÷ 3 × 6 + (- 4) =

30. 32 ÷ 2 + 4 - 15 ÷ 3 =

2. FRACTIONS

The fraction, alb, where the numerator is a and the denominator is b, implies that a is being divided by b. The denominator of a fraction can never be zero since a number divided by zero is not defined. If the numerator is greater than the denominator, the fraction is called an improper fraction. A mixed number is the sum of a whole number and a fraction, i.e., 4⅜ = 4 + ⅜.

OPERATIONS WITH FRACTIONS

A) To change a mixed number to an improper fraction, simply multiply the whole number by the denominator of the fraction and add the numerator. This product becomes the numerator of the result and the denominator remains the same. E.g.,

PROBLEM

SOLUTION

into an improper fraction.

The whole number 2 gets multiplied by the denominator 8.

2 × 8 = 16

Next, add the numerator to the previous result.

16 + 12 = 28

Find the greatest common divisor (GCD) of 28 and 8. This is done by writing the numbers as products of their primes.

The GCD of 28 and 8 is 4.

Next, divide the numerator and denominator by the GCD.

To change an improper fraction to a mixed number, simply divide the numerator by the denominator. The remainder becomes the numerator of the fractional part of the mixed number, and the denominator remains the same. E.g.,

To check your work, change your result back to an improper fraction to see if it matches the original fraction.

B) To find the sum of two fractions having a common denominator, simply add together the numerators of the given fractions and put this sum over the common denominator.

Similarly for subtraction,

PROBLEM

Find the solution to the following problem. Simplify the answer, if possible.

SOLUTION

Step 1 is to subtract the numerators.

-8 - (-4) = -4

are equal, keep the common denominator.

To simplify the answer, write out the numerator

and denominator as products of their primes to find the greatest common divisor (GCD).

The GCD of 4 and 16 is 4. The next step is to divide the numerator and denominator by the GCD.

4 ÷ 4 = 1 16 ÷ 4 = 4

C) To find the sum of the two fractions having different denominators, it is necessary to find the lowest common denominator, (LCD) of the different denominators using a process called factoring.

To factor a number means to find two numbers that when multiplied together have a product equal to the original number. These two numbers are then said to be factors of the original number. E.g., the factors of 6 are

1 and 6 since 1 × 6 = 6.

2 and 3 since 2 × 3 = 6.

Every number is the product of itself and 1. A prime factor is a number that does not have any factors besides itself and 1. This is important when finding the LCD of two fractions having different denominators.

we must first find the prime factors of