Fundamentals of Math

Fundamentals of Math

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Achieve

Fundamentals of Mathematics

Study Guide

1 st Edition 1/26/2020

This study guide is subject to copyright.

Acknowledgements

We would like to thank the author, Matthew Stewart, for his patience, support, and expertise in contributing to this study guide; and our editor for her invaluable efforts in reading and editing the text. We would also like to thank those at Achieve whose hard work and dedication to fulfilling this project did not go unnoticed. Lastly, we would like to thank the Achieve students who have contributed to the growth of these materials over the years.

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Fundamentals of Mathematics

Table of Contents Chapter 1: Operations with Real Numbers...................................................................................................................6 1.1 Operations....................................................................................................................................................................6 1.2 Properties of Real Numbers ..................................................................................................................................7 1.3 Factors and Multiples ...........................................................................................................................................12 1.4 Operations with Integers.....................................................................................................................................15 1.5 Order of Operations...............................................................................................................................................18 Chapter 1 Summary .......................................................................................................................................................22 Chapter 1 Practice Quiz ................................................................................................................................................23 Chapter 2: Fractions, Decimals, and Percentages ...................................................................................................25 2.1 Introduction to Fractions .....................................................................................................................................25 2.2 Operations with Fractions ..................................................................................................................................29 2.3 Decimals.....................................................................................................................................................................33 2.4 Percentages...............................................................................................................................................................37 Chapter 2 Summary .......................................................................................................................................................40 Chapter 2 Practice Quiz ................................................................................................................................................41 Chapter 3: Beginning Algebra .........................................................................................................................................42 3.1 Exponents..................................................................................................................................................................42 3.2 Square and Cube Roots.........................................................................................................................................44 3.3 Expressions and Equations.................................................................................................................................47 3.4 Simplifying and Solving Linear Equations ....................................................................................................50 3.5 Ratios and Proportions ........................................................................................................................................52 3.6 Percent Change........................................................................................................................................................54 Chapter 3 Summary .......................................................................................................................................................55 Chapter 3 Practice Quiz ................................................................................................................................................56 Chapter 4: Intermediate Algebra...................................................................................................................................57 4.1 Introduction to Graphs.........................................................................................................................................57 4.2 Graphing Linear and Quadratic Functions ...................................................................................................60 4.3 Introduction to Functions ...................................................................................................................................67 4.4 Evaluating Functions.............................................................................................................................................67 4.5 Multiplying Binomials...........................................................................................................................................68 Chapter 4 Summary .......................................................................................................................................................69

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Chapter 4 Practice Quiz ................................................................................................................................................70 Chapter 5: Geometry ..........................................................................................................................................................72 5.1 Properties of Triangles.........................................................................................................................................73 5.2 Pythagorean Theorem ..........................................................................................................................................76 5.3 Perimeter and Area................................................................................................................................................77 5.4 Properties of Quadrilaterals...............................................................................................................................78 5.5 Similar Figures.........................................................................................................................................................80 5.6 Properties of Circles ..............................................................................................................................................82 Chapter 5 Summary .......................................................................................................................................................84 Chapter 5 Practice Quiz ................................................................................................................................................85 Chapter 6: Data Analysis...................................................................................................................................................87 6.1 Characteristics of a Good Graph .......................................................................................................................87 6.2 Line Graph .................................................................................................................................................................88 6.3 Pie Charts...................................................................................................................................................................91 6.4 Bar Graphs.................................................................................................................................................................91 6.5 Histograms ................................................................................................................................................................92 6.6 Scatter Plots..............................................................................................................................................................93 Chapter 6 Summary .......................................................................................................................................................97 Chapter 6 Practice Quiz ................................................................................................................................................98 Chapter 7: Statistics ......................................................................................................................................................... 102 7.1 Mean ......................................................................................................................................................................... 102 7.2 Median ..................................................................................................................................................................... 103 7.3 Mode ......................................................................................................................................................................... 104 Chapter 7 Summary .................................................................................................................................................... 106 Chapter 7 Practice Quiz ............................................................................................................................................. 107 Chapter 8: Logic ................................................................................................................................................................ 108 8.1 Truth Tables .......................................................................................................................................................... 109 8.2 Conditional Statements ..................................................................................................................................... 111 8.3 Counterexamples................................................................................................................................................. 112 Chapter 8 Summary .................................................................................................................................................... 113 Chapter 8 Practice Quiz ............................................................................................................................................. 114 Answer Key ......................................................................................................................................................................... 116

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Chapter 1......................................................................................................................................................................... 116 Chapter 2......................................................................................................................................................................... 117 Chapter 3......................................................................................................................................................................... 118 Chapter 4......................................................................................................................................................................... 120 Chapter 5......................................................................................................................................................................... 122 Chapter 6......................................................................................................................................................................... 123 Chapter 7......................................................................................................................................................................... 125 Chapter 8......................................................................................................................................................................... 125 Index ...................................................................................................................................................................................... 127

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Chapter 1: Operations with Real Numbers Merriam Webster defines mathematics as, “the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations”. Pretty fancy talk for trying to say a scientific study of numbers and how numbers behave. Essentially all of mathematics from Algebra to Trigonometry to Calculus all gather their roots from the fundamental concepts defined by the real number system and the mathematical properties that follows. Sorry, academia talk again – basically, all of mathematics boils down to the real number system and its four primary operators: addition, subtraction, multiplication, and division. • Addition – combining two or more numbers (or items) together to make a new total. • Subtraction – taking one number away from another number. • Multiplication – a set value combined with itself a set number of times (repeated addition) • Division – splitting a number into an equal number of groups These four simple operators most of us initially learned in grade school are used in all domains of mathematics. Of course, the days of straightforward questions such as “what is 2+2?” are probably long gone. Today, most of what we encounter come in stories or worded questions like “If all red tags indicate the item is 20% off then how much will this red-tag shirt cost?” 1.1 Operations Let’s start by defining a few basic definitions so we are all on the same page.

So, let’s begin by reviewing a few phrases you are probably already familiar with and remind ourselves what they mean.

Figure 1.1.1: Common Verbal Operator Expressions Addition Subtraction Multiplication Division Combined Difference Multiplied Divided by Increased Decreased Product Into More than Less than Times Per Sum Minus Of Quotient Together Reduced Twice Out of Plus Fewer than Thrice Ratio of Added to Take away Split

Recognizing these key words can help us translate what we are hearing/asking in our everyday lives to turn our questions into actionable problems. For example, the manager knows that the restaurant has 24 tables in total and there will be four waitresses on staff today. How many tables per waitress will there be? By recognizing the number per number (or object per person) phrasing here helps us identify this is a division problem, since we know _____ per _____ means to divide.

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Practice 1.1 Directions: Circle the word (or words) in the questions below that indicate what operator is being mentioned. Then state what the operation would be if you were to actually answer the question.

1. Martin used to have four vehicles, but that has been reduced by one ever since he gave his oldest car to his grandson to drive. How many cars does Martin have now?

2. Phyllis needed to remember to stop by the bank on her way home from work to get her $10 bill split in half so she could give each of her girls their weekly allowance.

3. According to a recent survey by the U.S. Census Bureau there were almost 9.5 times more female nurses than male nurses. If there were 330,000 male nurses in the U.S. in 2016 how many nurses were female? 4. Brett and Christina decided to finally tackle their weekend project of getting the rotting boards on their back-deck replaced. They already had 10 pieces of lumbar in their garage and Brett brought home 15 more from the hardware store. How many pieces of lumbar do they have in total? 5. Emilio had gone to the gym 3 times this week, but his wife Alexandra went twice as many times as he did. She was determined to win their diet bet. How many times did Alexandra go to the gym?

Answer Key on Page 116

Visually we also need to recognize that each operator has different symbols to indicate what the problem is asking. Like the above phrases, most of them you probably already know, but take a moment to review figure 1.1.2 to refresh yourself.

Figure 1.1.2: Operator Symbols

Addition

Subtraction

Multiplication

Division

× or ( ) or ∗

+

−

÷ or /

1.2 Properties of Real Numbers A quick Google search asking for a definition of Real numbers will yield many results; however, they ultimately all draw the same conclusion.

Real numbers have a value of a continuous quantity. Collectively these values can be represented as a distance along a line.

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Let's take a moment to unpack that statement, starting with "continuous quantity". To the average human most vales we interact with daily are rather fixed (e.g., $1.99, 20%, ¼ cup, ...), but numbers exists beyond these finite values. It is possible for a value to go on forever - like pi for instance.

≈ 3.141592653589793238462643383279502884197169399375105820974 …

Some continuous values may even form repeatable patterns.

1 3 ≈ 0.33333333333333333333333333333333333333333333333333333333 …

In essence, pretty much any number you can think of is a Real number, but because Real numbers contain so many types of numbers, we can further categorize them into parts to make them more discernible.

Figure 1.2.1 The Real Number System

Noting that the larger sets of numbers also include their small number counterparts. For example, by definition, we know -7 is an Integer it is also included in the set of Rational, Irrational, and Real Numbers.

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Figure 1.2.2 Number Classification

Name

Consists Of

Symbol

Examples

ℕ

Natural

The counting numbers

5, 27, 134, 2641

---

Whole

Natural numbers and zero

0, 1, 2, 3, 4

ℤ

Integer

Natural numbers, zero, and opposite natural numbers

−5, −1, 0, 2, 4

Natural numbers, zero, opposite natural numbers, and any number that results in a finite decimal or a repeating decimal.

1 2

7 9

ℚ

Rational

−3, 0,

,

, 7.5

4 23

---

Irrational

Non-repeating and non-terminating decimals

, √2, ℮,

ℝ

Real

All rational and irrational numbers

Any number

Now let's look at the second sentence in our definition: Collectively, these values can be represented as a distance along a line.

We organize Real numbers on a straight line with values placed in equal segments along its length; this line, known as a number line , can be extended infinitely in any direction. Values are placed numerically left to right with smaller quantities on the left and extending to larger quantities on the right.

Figure 1.2.3 A Generic Number Line

There is no set number line because a number line is just a visual tool to help users compare values and carry out operations like addition, subtraction, multiplication, and division.

Figure 1.2.4 Types of Number Line

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Practice 1.2.1 Directions: Identify the following numbers as natural, whole, integer, rational, irrational, or real - place a checkmark in each category that could be used to describe the number.

Natural

Whole

Integer

Rational Irrational

Real

1.

−8

2.

≈ 3.14 …

3.

5

4.

0

1 2

5.

Answer Key on Page 116

Let's take a look at some of the properties of Real numbers. These properties will prove useful when working with equations, functions, and Algebraic formulas.

*In the following definitions and examples, , , and are defined as Real numbers.

Commutative Property of ...

The commutative property states that when we add or multiply two real numbers, their order does not matter.

Addition

Multiplication

+ = +

× = ×

or

or

1 + 2 = 2 + 1

1 × 2 = 2 × 1

Associative Property of ...

The associative property states that while adding or multiplying more than two real numbers, then the effect of parentheses does not matter.

Addition

Multiplication

( + ) + = + ( + )

( × ) × = × ( × )

or

or

(1 + 2) + 3 = 1 + (2 + 3)

(1 × 2) × 3 = 1 × (2 × 3)

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Distributive Property

The distributive property applies to the cases that make use of both addition and multiplication. It states that if a term is multiplied by terms in parenthesis, we need to "distribute" the multiplication over all the terms inside.

× ( + ) = ( × ) + ( × ) or 2 × (3 + 4) = (2 × 3) + (2 × 4)

Identity Property of ...

Identity property describes the value that is added or multiplied to a number so that the original value of a number remains unchanged.

Addition

Multiplication

Zero (the additive identity) added to any number results in the same number.

One (the multiplicative identity) multiplied to any number results in the same number.

+ 0 =

× 1 =

or

or

1 + 0 = 1

2 × 1 = 2

Inverse Property of ...

The Inverse property describes the value you must add/multiply the original value by to bring you back to the identity value (0 for addition and 1 for multiplication).

Addition

Multiplication

Multiplicative inverse of a would be −1 or 1 . The multiplication of a real number with its multiplicative inverse would be equal to 1,

Additive inverse is denoted by a negative sign. That is, the additive inverse of a would be (− ) . Addition of a real number with its additive inverse would be 0.

1

+ (− ) = 0

×

= 1

or

or

1 2

1 + (−1) = 0

2 ×

= 1

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Practice 1.2.2 Directions: Match the equations with the property being used

1 7

1. _________ Commutive Property of Addition

a. 7 ×

= 1

2. _________ Commutive Property of Multiplication

b. 4(1 + 2) = 4(1) + 4(2)

3. _________ Associative Property of Addition

c.

4(2) = 2(4)

4. _________ Associative Property of Multiplication

d, 5 × 1 = 5

5. _________ Distributive Property

e. 8 + (−8) = 0

6. _________ Identity Property of Addition

f.

3 × (5 × 9) = (3 × 5) × 9

7. _________ Identity Property of Multiplication

g. 7 + 9 = 9 + 7

8. _________ Inverse Property of Addition

h. 6 + 0 = 6

9. _________ Inverse Property of Multiplication

i.

5 + (6 + 1) = (5 + 6) + 1

Answer Key on Page 116

1.3 Factors and Multiples Mathematics is broken into multiple branches, with Arithmetic being one of them. Arithmetic focuses on properties and manipulations of real numbers. Let us look at two common manipulations known as factors and multiples. A factor is a number that divides evenly into a number with no remainder. Think of decomposing a number. Each number has a finite number of factors. While it's opposite, a multiple is a number that is the result of multiplying a given number by another. Unlike factors, multiples are infinite.

Figure 1.3.1 Factor and Multiple Comparison Factors

Multiples

Factor refers to an exact divisor of the given number. It’s a number multiplied to get another number.

Multiple refers to the product of a given number and another. It is the result of a product between a number and an integer.

Definition

What is it?

The Final Answer Can Be …

Finite

Infinite

Less than or equal to (≤) the given number.

Greater than or equal to (≥) the given number.

Result

Operator Used

Division

Multiplication

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Factors Factoring is like taking a number apart - the number becomes expressed as a product of its elements. The values that make up the product are either prime or composite.

Primes

Prime numbers are whole numbers that cannot be made by multiplying other whole numbers. Alternatively, a number whose only factors are 1 and itself.

Composites

Composite numbers are whole numbers that can be made by multiplying other whole numbers. Alternatively, a number which can be decomposed into factors other than 1 and itself. We ultimately strive to write a composite number as a product of all prime numbers. This product list is known as prime factorization .

Example 1.3.1

Find the prime factorization of 135.

1. 27 times 5 equals to 135, 5 is prime, but 27 is composite, so we can keep going.

2. 9 times 3 equals to 27, 3 is prime, but 9 is composite, so we can keep going.

3. 3 times 3 equals to 9, 3 is prime and there are no other composite numbers to factor, so we are finished.

The prime factorization of 135 is …

135 = 3 × 3 × 3 × 5

Returning to our original discussion of factors - factors can be prime or composite. Some questions may specifically ask you to decompose a number into a product of exclusively prime numbers (prime factorization). In this instance, the quickest method is to create a factor tree (see example 1.3.1); or they may state you should list all possible factors of a number. To list all the factors of a number, start with 1 and then place the number at opposite end.

Example 1.3.2

List all the factors of 42.

1 __________________________________________________42

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Next, count up to 2, see if it will divide evenly into the number. If it does then add 2 and the quotient to your list, and if not write nothing and progress again to the next step.

1, 2 __________________________________________________21, 42

Now, count up to 3, again see if it will divide evenly into the number. If it does then add 3 and the quotient to the list, and if not proceed to 4.

1, 2, 3 ___________________________________________14, 21, 42

You will eventually hit the halfway point where the number you are testing is already written in your list and that is how you know you have finished.

Multiples The product of any two factors is a multiple. To find the first multiples of any number start by multiplying the number by 1, then 2, 3, 4, ... . We list these numbers in a line separated by a comma - you will also often see an ellipse (...) written after the last number to indicate the list could continue, but you as the mathematician have chosen to stop listing the multiples. Remember, every real number has an infinite number of multiples.

Example 1.3.3 Find the first five multiples of 13.

1 st

13 × 1 13 × 2 13 × 3 13 × 4 13 × 5

= 13 = 26 = 39 = 52 = 65

2 nd 3 rd 4 th 5 th

The first five multiples of 13 are …

13, 26, 39, 52, 65, …

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Practice 1.3 Directions: Use your knowledge of factors and multiples to answer the following questions.

1. Circle all the prime numbers in the list:

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

2. What is the prime factorization of 60?

3. List all the factors of 120.

4. List the first five multiples of 72.

Answer Key on Page 116

1.4 Operations with Integers As mentioned previously, one subset of Real numbers is Integers. Integers are like are just like whole numbers (no fractions or decimals), but they also include negative numbers:

. . . −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5. ..

For every positive integer, there is a negative counterpart. Negative integers have a " − " sign in front of the number to indicate that its value is less than 0. Positive numbers can have a " + " sign in front of the number to indicate its value is greater than 0. However, often if a number has no sign in front of it, then it is considered to be positive.

Zero is a neutral number. It is neither positive or negative.

Adding/Subtracting Integers We can add and subtract integers together using a number line. For example, −2 + 5

The first number (−2) tells you where to start on the number line. The sign of the second number tells you the direction you will travel in. For a positive number (+) you travel to the right, and for a negative number (−) you travel to the left. The second number (5) tells you howmany units to move.

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Therefore, we can conclude −2 + 5 is equal to 3.

Figure 1.4.1 Adding/Subtracting Integers on a Number Line

If you have two signs written next to one another, for example, −3 + (−2) you must first turn the double signs into one.

• Double negatives make a positive −(−) = + • A negative and a positive make a negative +(−) = − or −(+) = −

So −3 + (−2) = −3 − 2 = −5 .

Practice 1.4.1 Directions: Identify the sum or difference of each expression.

1. −2 − 5

2. −3 − (−2)

3. 6 + 3

4. 15 + (−21)

5. 15 + 0

6. 5 + (−10)

7. −12 + 16

8. 5 − (−4)

9. 8 − 4

10. −(−14) − 7

Answer Key on Page 116

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Multiplying and Dividing Integers We can multiply and divide integers together using a number line. For example, (3)(−2)

Starting at 0 , the first number (3) tells you how many jumps to make. The second number (−2) tells you how big your leap is. Then use the chart below to determine the final sign of your answer and which direction to leap on your number line (left for negative or right for positive).

Figure 1.4.2 The Product of Signs

Sign of the 2 nd Number Positive Negative

Positive (+)(+) + Negative (−)(+) −

Negative (+)(−) − Positive (−)(−) +

Positive

Sign of the 1 st Number

Negative

Tip:

Same Signs = POSITIVE Opposite Signs = NEGATIVE

Therefore, we can conclude (3)(−2) is equal to −6 .

Figure 1.4.3 Multiplying Integers on a Number Line

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To divide integers on the number line, like in −4 ÷ −2 begin by locating the first number (−4) on the number line. You will need to divide the distance between 0 and the first number (−4) into an equal number of parts indicated by the second number (−2) . Again, refer to figure 1.4.2 to determine the final sign of the quotient.

Figure 1.4.4 Dividing Integers on a Number Line

There, we can conclude −4 ÷ −2 is 2 .

Practice 1.4.2 Directions: Identify the product or quotient of each expression.

1. −2 × −5

2. −6 ÷ −2

3. 6(3)

4. 15 ÷ −3

5. 15 × 0

6. 10 ÷ 5

7. −2(7)

8. −12 ÷ 4

9. 1 × −4

10. 14 ÷ −7

Answer Key on Page 116

1.5 Order of Operations Let's say you are presented with a problem like "what is 5 + 3 × 4 "? At first glance, you are left with a conundrum because you could, in theory, solve this two ways:

1. Add 3 to 5 and then multiply by 4 which results in 32 -- OR 2. Multiply 3 times 4 and then add 5 which results in 19

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They cannot both be correct, so which way do you solve the problem? Well, mathematicians (as early back as the 1800s acknowledged this problem years ago and collectively began testing and organizing which operators took precedence over another. There have been some modifications over the years, but the general principles remain the same. The order of operations state: 1. Do everything inside the parenthesis first. 2. Then take care of any exponents or radicals left to right (whichever comes first). 3. Next, you should multiply and divide left to right (whichever comes first). 4. And finally, add and subtract - again left to right (whichever comes first). So how do you answer "what is 5 + 3 × 4 "? Multiply 3 and 4 before adding 5; making the correct answer 19. It's also important to note that operators can be displayed very differently from problem to problem.

Figure 1.5.1 Order of Operations

P

E

M

D

A

S

P lease

E xcuse

M y

D ear

A unt

S ally

Mnemonic

Parenthesis Exponents Multiplication Division

Addition

Subtraction

Parenthesis

(anything in here) or [here] or {here}

Exponents/ Radicals

or under here √

anything up here

Multiplication /Division

9 3

× ∙ ∗ 2(3) 8 ÷ 2

7 |21̅̅̅̅

Addition/ Subtraction

+ − ±

You will often notice too that when working lengthier problems, your expressions start to resemble an inverted triangle as you find your solution.

Example 1.5.1

1. There are no parenthesis or exponents (or radicals), so begin with multiplication and division. Division occurs first (left to right), so start by diving 30 by 5 (which is 6). 2. With the simplified expression you still need to take care of the multiplication before you add, so 6 times 2 is 12.

30 ÷ 5 × 2 + 1

6 × 2 + 1

3. Now add the remaining to values.

12 + 1

The solution is 13. 13 Now let's consider the case when you have multiple sets of parentheses (or nested brackets). In this instance, you should begin with the innermost set of parentheses and work your way out. Once all the operations have been completed inside the parenthesis, you can drop the parenthesis (or brackets) altogether.

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Example 1.5.2

[3(15 + 3) + 27]/3

1. Start with the inner most parenthesis 15 plus 3 which equals 18. In this case, we do not drop the parenthesis yet because we still need to distribute (a.k.a. multiply) the 3 – the parenthesis themselves turn into an operator.

[3(18) + 27]/3

2. Next me move to the multiplication 3 times 18 which equals 54.

[54 + 27]/3

3. Since the sum of 54 and 27 is inside a bracket you do that before dividing by 3 – the sum is 81.

4. Lastly, you can divide by 3. The quotient of 81 divide by 3 is 27.

81/3

The solution is 27.

27

Fractional form, "everything on top" over "everything on the bottom", means all the stuff "on top" is divided by all the stuff "on the bottom". And "all the stuff" or "everything" implies those expressions go together. Think of it as the top part of the fraction has a master set of parentheses around it and the bottom part has its own master set of parentheses around it. It's just that, when fractions are typeset vertically (as opposed to typed out horizontally like in an email or a Math forum) we don't include the grouping symbols (a.k.a. parenthesis). The vertical layout actually makes things easier to understand.

If you have a problem written in a fractional form, then it is implied that you should do everything up top and everything on the bottom before dividing.

Example 1.5.3

(6 + 2 2 ) × 7 2 × 15 − 25

1. Remember, we do everything up top and everything on the bottom before dividing.

a. Top: Exponents first 2 2 = 2 × 2 = 4 b. Bottom: Multiply first, 2 × 15 = 30

(6 + 4) × 7 30 − 25

2. Again, simplify the top and bottom completely before dividing. a. Top : 6 + 4 = 10 b. Bottom: 30 − 25 = 5 3. We still must finish simplifying the top before we divide. a. Top: 10 × 7 = 70 b. Bottom: ---

10 × 7 5

70 5 14

4. Now you can divide, 70 ÷ 5 = 14 .

The solution is 14.

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Practice 1.5 Directions: Simplify the following expressions. Each question will match to one of the answers in the answer bank below.

−4

25 −6

−55

4

32 35

2

3

10

1. (8 − 3) × 2

2. 12 ÷ 2 − 4

4. (9 + 3) − (16 ÷ 4) 2

3. 4 + 7 × 3

5. 5 × 2 2 + 12

6. [(11 − 6) 2 + 6] + 4

7. 14 − 5 − (24 ÷ 3) 2

8. (4 + 9 + 16 ÷ 4) − 8 − 3 × 5

9. (48÷2)−4 2 +4 2+5−4

10. 20−3×2 2 +1 8+14÷7×3−11

Answer Key on Page 117

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Chapter 1 Summary Vocabulary

Addition

Integers

Inverse Property

Subtraction

Rational Numbers Irrational Numbers

Factor

Multiplication

Multiple

Division

Number Line

Prime Number

Real Numbers

Commutative Property Associative Property Distributive Property

Composite Number Prime Factorization Order of Operations

Natural Numbers Whole Numbers

Formulas

Sign of the 2 nd Number Positive Negative

Positive (+)(+) + Negative (−)(+) −

Negative (+)(−) − Positive (−)(−) +

Positive

Sign of the 1 st Number

Negative

Tip:

Same Signs = POSITIVE Opposite Signs = NEGATIVE

Don’t Forget

• All of the rules, as mentioned earlier, apply not just to integers—they also apply when you are performing operations on any signed numbers, including signed fractions and signed decimals. • PEMDAS o Parenthesis (or brackets)

o Exponents/Radicals – left to right, whichever comes first o Multiplication/Division – left to right, whichever comes first o Addition/Subtraction – left to right, whichever comes first

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Chapter 1 Practice Quiz

4. Which of the following statements is FALSE? a. All Natural Numbers are Whole Numbers b. Irrational Numbers contain all decimals

1. What mathematical operation would you need to use to solve the problem?

Every Halloween Smallville sees about 394 trick or treaters, and on average, they collect about 163 pieces each. On average, how much candy is given out on Halloween in Smallville?

c. All Integers are Whole Numbers d. Fractions are Rational Numbers

a. Addition b. Subtraction c. Multiplication d. Division

5. What is the prime factorization of 64? a. 2 × 2 × 2 × 2 × 2 × 2 b. 1, 2, 4, 8, 16, 32, 64 c. 64, 128, 192, 256, 320, ... d. 2, 4, 2, 4

2. Which of these expressions do NOT indicate the operation is division? a. Quotient b. Take away c. Out of d. Into

6. Which list is made up of multiples of 4? a. 4, 6, 8, 10, ... b. 1, 4 c. 2 × 2 d. ... 12, 16, 20, 24, ...

3. Which equation below showcases the Identity Property of Multiplication?

a. 8 × 0 = 0 b. 3 × 1 = 3 c. 4 × 6 = 6 × 4 d. 1 × (2 × 3) = (1 × 2) × 3

7. The temperature at 3 am on January 1st was −2°F . By 11 am the temperature had risen 10 degrees. What was the temperature at 11 am? a. −12°F b. −8°F c. 8°F d. 12°F

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8. What is the next number in the sequence -6, -12, -18, -24, ...?

9. How could you arrange the numbers 6, 9, and 4 in the sequence, so the answer is 30?

a. −28 b. −30 c. −32 d. −34

a. 9 − 6 × 3 b. 3 × 9 + 4 c. 4 × 6 + 9 d. 9 × 4 − 6

10. Locate the error in the problem below.

2 3 − 12 ÷ 4 × 3 + 1

Line 1

8 − 12 ÷ 4 × 3 + 1

Line 2

8 − 12 ÷ 12 + 1

Line 3

8 − 1 + 1

Line 4

7 + 1

Line 5

8

a. Line 1 – 2 3 ≠ 8 . b. Line 2 – Divide 12 by 4 before you multiply times 3. c. Line 3 – 8 − 1 ≠ 7 . d. Line 4 – Multiply 7 and 1 instead of adding.

Answer Key on Page 117

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Fundamentals of Mathematics

Chapter 2: Fractions, Decimals, and Percentages Now that have you have a thorough understanding of integers and order of operations, you may have noticed some difficulty when dividing certain numbers. For instance, 18 divided by 6 or 10 divided by 2 may be relatively simple, but something like 24 divided by 5 causes some trouble: 5 doesn't go into 24 evenly--the quotient is somewhere between 4 and 5! (We would, therefore, say that 24 is not divisible by 5.) This uncertainty with the divisibility is definitely a conundrum, but this type of division is not far-fetched from what people experience in the real world. Such as a recipe calling for one and a half pound of beef or a pizzaiolo needing to cut the pizza into eighths. 2.1 Introduction to Fractions From our previous chapter, we know Integers are all the counting positive and negative counting numbers (. . . , −3, −2, −1, 0, 1, 2, 3, . . . ) . However, between any two integers (such as 0 and 1 or −5 and −4 ) is an entirely new set of numbers (rational) known as fractions. When we discuss fractions, we will refer to a number such as one-fourth ( 1 4 ) as a fraction rather than the decimal notation 0.25. The term fraction, however, simply tells us howmany parts of a whole we have - whether it is written as a decimal, percentage, or with traditional fraction notation. ≠ 0 and and are both real numbers. The top number ( ) is referred to as the numerator , and the bottom number ( ) is the denominator (remember "D" for the number down below). 3 ← Numerator 4 ← Denominator The line seen between the numerator and denominator is treated as the division operator (like ÷ ). Let's take a look at this relationship. Imagine a circle divided into six equal parts (the denominator), and we shaded in one part (the numerator), then we are left with one-sixth of the circle (the shaded portion). Fractions are written in the form , where

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Equivalent Fractions Some fractions may look different, but are really the same ( equivalent fractions ), for example:

“One-Half”

“Two-Quarters”

“Four-Eights”

4 8

1 2

2 4

You should always simplify your fractions into their smallest forms; however, unless the problem explicitly asks you not to.

Increase Fractions

Simplify Fractions

Divide both the number and denominator by the same values (number must divide into both sets of numbers evenly).

Multiply both the numerator and denominator by the same values.

1 2

1 × 3 2 × 3

3 6

16 28

16 ÷ 4 28 ÷ 4

4 7

=

=

=

=

Practice 2.1.1

Directions: State which fractional number ( ) you would multiply each fraction by to create the equivalent fraction. 1. 1 6 × _____ = 6 36 2. 13 24 × _____ = 78 144

3. 2 5

× _____ = 8 20

4. 9 2

× _____ = 81 18

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Directions: Simplify the following fractions into their lowest forms.

5. 12 15

6. 20 30

=

=

4 18

8. 15 45

7.

=

=

Answer Key on Page 117

Types of Fractions Like people, fractions come in many forms - proper, improper, and mixed. A proper fraction is one whose numerator is less than its denominator (the top number is smaller than the bottom). Improper fractions are the opposite - their numerators are larger than their denominators (the top number is bigger than the bottom). Then there is the mixed case - with mixed numbers you have a fraction (traditionally proper) with its whole number counterpart.

Proper Fraction

Improper Fraction

Mixed Numbers

2 3

1 9

27 4

8

There is a rumor going around the mathematical world that improper fractions should always be converted to mixed numbers, and that is simply not true. Mixed numbers help us better understand the size of a number (for example you need 4 1 3 cup of flower vs. 13 3 ), but from an algebraic point of view, there is no real benefit to mixed numbers.

To Convert from an Improper Fraction to a Mixed Number

1. Divide the denominator into the numerator.

2. The quotient is the whole number counterpart, the remainder is the new numerator, and the denominator (the divisor) remains the same. Then simplify your remaining fractional part if possible.

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To Convert from a Mixed Number to an Improper Fraction

1. Multiply the denominator and whole number together

3 × 5 = 15

2. Add the numerator to the product.

15 + 2 = 17

3. The sum becomes the new numerator, and the denominator remains the same. Then simplify the fraction if possible.

2 5

17 5

3

=

Practice 2.1.2

Directions: Shade in the boxes appropriately to indicate if the fraction is a mixed number, proper fraction, or improper fraction.

1 3

8 11

16 10

1 2

13 8

9 7

15 20

1

9

Mixed Number Proper Fraction Improper Fraction

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