Download Learning Objectives Unit Lesson Reading Assignments Learning

UNIT I STUDY GUIDE
Whole Numbers, Integers, and
Algebraic Expressions
Learning Objectives
Reading
Assignments
Upon completion of this unit, students should be able to:
1. Add, subtract, multiply, and divide whole numbers and integers.
2. Round, estimate, and order whole numbers.
3. Solve expressions and exponential notation following the rules for
Order of Operations.
4. Solve equations and applications with real-world problems with whole
numbers and integers.
5. Solve algebraic expressions.
6. Solve algebraic expressions involving like terms and perimeter.
Chapter 1:
Whole Numbers
Chapter 2:
Introduction to Integers
and Algebraic
Expressions
Learning Activities
(Non-Graded)
See information below.
Key Terms
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Absolute value
Additive identity
Additive inverse
Algebraic expression
Base
Constant
Equivalent
expressions
Exponential notation
Integers
Minuend
Multiplicative identity
Natural numbers
Place-value chart
Subtrahend
Value of algebraic
expressions
Variable
Unit Lesson
People learn in a variety of ways. For this reason, each section in each unit of
this course will include multiple resources, such as required reading assigned
in the textbook; optional media assignments that will include the hardest
concepts in the unit; homework with five help aids to the right of the problem,
which include ‘Help Me Solve This’; ‘View an Example’; ‘Textbook’, which
opens the e-textbook to the page where the concept is taught; ‘Ask My
Instructor’; and the option to print the question.
The best way to learn math is by practicing. Each unit will not only provide
instruction and guidance on how to solve problems, but also numerous
opportunities to practice solving them on your own through non-graded
assignments. At the end of each section will be an Exercise Set. Answers and
guidance for solutions are provided in indicated sections of the textbook, as
well as in the student solutions manual that accompanies the textbook.
When you click on Unit I in your course, you will see a ‘TO DO’ LIST to assist
you in starting your course.
Click Chapter 1 as listed in “Chapter Contents” in MyMathLab. A Chapter
Opener will be available for review. Next click on the sections within the
Chapter to open the following: video presentation, multimedia e-Text (e-book
with animations and videos to illustrate concepts), and your study plan. The
above holds true for every chapter and every section in the textbook. Each
Chapter also contains a mid-chapter review, chapter summary and review, and
a chapter test with test prep videos and after Chapter 2, there will be a
Cumulative Review that will cover all concepts up to and including the current
chapter.
Unit I covers Chapters 1 and 2 in the textbook. You will have a brief review of
arithmetic operations followed by intensive drill in basic algebraic concepts. In
arithmetic operations, you will give the meaning of digits in standard notation,
convert from standard notation to expanded notation, and convert between
standard notation and word names.
MA 1100, Basic Mathematical Fundamentals
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CHAPTER ONE: WHOLE NUMBERS
Section 1.1, Standard Notation
We start with Section 1.1, pages 1-8, reviewing standard notation, word
names, and expanded notations. Every time that we write a check, make a bid
on a job, or project populations for different countries, we are using these math
skills. When you click on 1.1, Standard Notation in your course, you will see
‘Chapter Opener,’ which opens the e-textbook. Up in the top right corner, you
will see left and right arrows on each side of a page number. If you click the
right arrow, it will take you to the next page, where you will find audio
definitions for key terms in this unit. You will also have animations and videos
that you can watch.
At the end of the section is a 1.1 Exercise Set for non-graded practice. All of
the odd-numbered problems are worked out in your student solutions manual
for your benefit. The synthesis problems at the end of each exercise set are
important, as they make sure that you understand the concepts and can apply
them in situations that are similar to what you have learned. Those types of
problems will be on the Assessments.
Section 1.2, Addition
In Section, 1.2, pages 9-13, we will review adding whole numbers. We will
explore writing an addition sentence that corresponds to a given situation and
using addition when finding the perimeter of a geometric figure. Do you know
the names of the individual parts of an addition problem?
3+
Addend
4
Addend
=7
Sum
Remember that in geometry, you write the formula for perimeter, substitute the
numbers into the equation, and solve for P.
Example: The perimeter of a rectangle has two formulas.
P=
P=
s+s+s+s+s
2l + 2w
Addition formula
Multiplication Formula
Let’s state that a rectangle has a length of 12 in and a width of 8 in. To solve
this, do the following:
P=s+s+s+s+s
P = 12 + 12 + 8 + 8 (Use the Rules for Order of Operations here.)
P = 24 + 8 + 8
P = 32 + 8
P = 40 inches
Second Way:
P = 2l + 2w
P = 2(12) + 2(8) (Use the Rules for Order of Operations here.)
P = 24 + 16
P = 40 inches
There are two laws in this section that are of particular importance. They are as
follows:
Associative Law of Addition: a + (b + c) = (a + b) + c
NOTE: Parentheses tell you what to do first (Order of Operation).
MA 1100, Basic Mathematical Fundamentals
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Suppose you want to add three numbers, such as: 2 + 3 + 6. According to the
parentheses, you can perform this operation in two ways:
2 + (3 + 6)
=2+9
= 11
OR
(2 + 3) + 6
=5+6
= 11
The second law is the Commutative Law of Addition, which states that we
can add whole numbers in any order:
a+b
2+3
=b+a
=3+2
or
or
a+b+c
2+3+4
=c+a+b
=4+2+3
What is the additive identity? Zero is the additive identity, which means that
any number added to zero is that number. Adding zero to a number does not
change the number, for example, 198 + 0 = 198.
Section 1.3, Subtraction
For Section, 1.3, pages 14-18, we practice the subtraction of whole numbers.
Along those lines, you will be given a subtraction sentence and be asked to
write a related addition sentence; and given an addition sentence, you will be
asked to write two related subtraction sentences. You will practice writing a
subtraction sentence that corresponds to a situation of “How much do I need?”
Do you know what minuends, subtrahends, and differences are?
6
Minuend
–2
Subtrahend
=4
Difference
Section 1.4, Multiplication
When we reach Section, 1.4, pages 19-25, we will review multiplying whole
numbers and use multiplication when finding the area of geometric figures.
Remember that in geometry, you write the formula for the area, substitute the
numbers into the equation, and solve for A. The formula for a rectangular
region is as follows: A = l * w
In algebra, we can also write this as A = lw, which is a shorthand way of writing
length (l) times width (w).
Let’s find the area of a standard table tennis table whose dimensions are 9 ft.
by 5 ft:
A = lw
A = 9*5
A = 45 sq. ft. or 45 ft.2
There are two laws in this section that are of particular importance. They are as
follows:
Associative Law of Multiplication: a * (b * c) = (a* b) *c
NOTE: Parentheses tell you what to do first (Order of Operation).
2 * (3 * 4) = 2 * 12
= 24
MA 1100, Basic Mathematical Fundamentals
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OR
(2 * 3) * 4 = 6 * 4
= 24
The second law is the Commutative Law of Multiplication:
a*b=b*a
3 * 6 = 18
6 * 3 = 18
NOTE: Any number multiplied times zero = 0, for example, 205 * 0 = 0
What is the multiplicative identity? When you multiply any number by 1, you will
not change that number. Therefore, 1 is the multiplicative identity.
Example: 25 * 1 = 25
Section 1.5, Division
In section 1.5, pages 26 – 34, we will concentrate on dividing whole numbers:
 If you divide a number by 1, you will get that same number: 15 ÷ 1 = 15.
 If you divide a number by itself, you get 1: 7 ÷ 7 = 1.
0
 Zero divided by any nonzero number is 0: 0 ÷ 14 = 0 and = 0.
3

Division by zero is undefined: 16 ÷ 0 is not defined, and neither is
16
0
.
At this point, you can practice what you have learned by completing the MidChapter Review on pages 35-36. Use the student solutions manual to check
your work.
Section 1.6, Rounding and Estimating; Order
In section 1.6, pages 37 – 47, we work with rounding numbers. We find this
handy for approximating numbers, as in the purchase of a new car. Let’s say
that your budget is $20,000. You have found the car that you want to buy at a
base price of $16,495. There are options that you want on your car. You can
round and estimate to see what options you can afford.
Astra 5-Door XE
Base price
4-speed automatic transmission
16-in. twin-spoke machined alloy wheels
Air conditioning
Dual-panel power sunroof
(Requires purchase of A/C)
Heated cloth front seats
StabiliTrak Stability Control
Estimated cost
PRICE
$16, 495
1,325
350
960
1,200
250
495
ROUNDED
PRICE
$ 16,500
1,300
400
1,000
1,200
300
500
$ 21,200
By rounding, you determine that the estimated cost is $21,200, which exceeds
your budget of $20,000. You will have to forgo at least one option.
In this unit, we will also work with inequality symbols “is less than” (<), and “is
greater than” (>) to order whole numbers. You will write true sentences for
problems such as these:
8 ___ 12
76 ___ 64
MA 1100, Basic Mathematical Fundamentals
8 < 12
76 > 64
4
Section 1.7, Solving Equations
In section 1.7, pages 48 – 53, we will solve one-variable equations. Remember
that we use a variable (a – z) to find the value of a missing number. It does not
matter which letter you use. Mathematicians usually use an ‘x’ or a ‘y’ to keep
things simple. However, you may use any variable that you choose to use. If
you have a statement, 9 = 3 + __, you know that 3 + 6 = 9. If you were writing
this algebraically, you would write 9 = 3 + x.
To find a solution for an equation is to find a replacement for the variable that
makes the equation true. When you are asked to solve an equation, you want
to find all of its solutions. The key to solving any equation is to get the variable
by itself on one side of the equation.
When we start solving equations, we must remember to show the addition,
subtraction, and/or division steps on both sides of the equation to receive full
credit. You must show your work algebraically. Remember to think of an
equation as a balance scale, where each side of the scale/equation equals the
other side at all times. For example, if you were to add 5 to one side of the
scale, that side would drop down because it is now unequal/heavier than the
other side. Therefore, if you add 5 to one side of the scale, you must also add 5
to the other side to keep the scale balanced and equal on both sides. When
you add, subtract, multiply, or divide in an equation, you must do the same
thing to the other side of the equation.
Let’s solve an equation. Here we want “x” by itself on the left side of the
equation. We will begin by subtracting 12 from both sides of the equation.
x + 12
x + 12 – 12
x+0
= 27
= 27 – 12
= 15
x
= 15
(You may omit this step in your work. You just need to
understand that 12 has been subtracted from both
sides of the equation.)
Section 1.8, Applications and Problem Solving
What is the purpose of math? Why do we bother learning all of these rules?
Let’s check out pages 54-70 and discover the answers to these questions.
Math is so integrated into our lives that many times we take it for granted
without realizing that without math, we would not have much of what we value
today. For example, we use math every day for checking account balances,
travel distances, total cost of merchandise that we sell or buy, area of rugs for
an open space in our homes, production of paper towels or anything
manufactured, or weight loss.
Check out page 62 for keywords, phrases, and concepts that will help you
solve application problems.
MA 1100, Basic Mathematical Fundamentals
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Section 1.9, Exponential Notation and Order of Operations
When a number is multiplied times itself several times, we can use a shorthand
notation to shorten the writing of the multiplication problem.
For example:
3 • 3 • 3 • 3 • 3 can be shortened to 35
5 factors
exponent
base
We would read the above example of exponential notation as “three to the fifth
power” or “the fifth power of three.”
If you work this problem out following the rules for Order of Operations
(page 72), the expression will look like this:
3•3•3•3•3
=9•3•3•3
= 27 • 3 • 3
= 81 • 3
= 243
The exponent (5) tells us to how many times to multiply the base times itself.
How do we simplify expressions? What is an expression? It is definitely not a
complete sentence. Since it is not a complete sentence, the first line will never
have an equal sign (=). However, all other lines will have an equal sign (=) at
the beginning of the line (from line 2 forward).
You will see the word ‘simplify’ at the beginning of most expressions.
Expression:
Simplify: 7 • 14 – (12 + 18)
= 7 • 14 – 30
= 98 – 30
= 68
First, carry out operations inside the ( ).
Do all multiplications and divisions left to right.
Do all additions and subtractions left to right.
Sometimes we remove parentheses within parentheses which can have
different shapes: [ ] brackets, { } braces, or ( ) parentheses. When this
happens, computations for the innermost ones are to be done first.
Expression:
Simplify: [25 – (4 + 3 ) • 3] ÷ (11 – 7)
= [25 – 7 • 3] ÷ (11 – 7)
= [25 – 21] ÷ (11 – 7)
=4÷4
=1
Do the calculations on the innermost ( ) first.
Do the multiplication in the brackets.
Do the subtraction in the [ ], and the ( ).
Do the division.
Chapter 1 Summary and Review, pages 80-84
Chapter 1 Test, pages 85-86
At the end of each chapter, there will be a summary and review of key terms
and properties, important concepts, review exercises, and a chapter test. Each
summary and review will prove very valuable to you as you work your way
through the units.
MA 1100, Basic Mathematical Fundamentals
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Throughout the textbook, there will be an emphasis on real-life problem-solving
applications that will help build critical-thinking skills by requiring you to use
what you know to synthesize, or combine, learning objectives from the current
section with those from previous sections.
CHAPTER TWO: INTRODUCTION TO INTEGERS AND ALGEBRAIC
EXPRESSIONS
Section 2.1, Integers and the Number Line
Integers correspond to almost all real-world problems and situations. Some
examples are the rise and fall of stock shares and the location of a diver who is
1700 feet below sea level (-1700 ft.).
You will discover that absolute value is the distance a value is from zero; for
example, whether you have a negative 4 or a positive 4, the value is 4 units
from zero, which means that its absolute value is 4. The absolute value is
always positive.
Section 2.2, Addition of Integers
The basics for adding and subtracting integers are very simple:

If the signs are alike, keep the signs and add the numbers.
Examples:
3+5 =8
-6 + (- 9) = -15

Both signs are positive; add the numbers, and keep the positive
Both signs are negative; add the numbers, and keep the
negative.
If the signs are not alike, subtract the numbers, and take the sign of
the greatest absolute value.
Examples:
3 + (-5) = -2
11 + (-8) = 3
-7 + 4 = -3
-6 + 10 = 4

Signs are different; subtract the numbers;
-5 has the greatest absolute value; keep its negative sign.
Signs are different; subtract the numbers;
11 has the greatest absolute value; keep its positive sign.
Signs are different; subtract the numbers;
-7 has the greatest absolute value; keep its negative sign.
Signs are different; subtract the numbers;
10 has the greatest absolute value; keep its positive sign.
One number is zero: The sum is the other number.
Examples:
-8 + 0 = -8
25 + 0 = 25
Section 2.3, Subtraction of Integers
We cannot subtract integers. Therefore, we have to subtract by adding the
opposite, or the additive inverse, of the number being subtracted.
a – b = a + (-b)
MA 1100, Basic Mathematical Fundamentals
(Now we have an addition statement, and we follow the
rules for adding integers.)
7
Examples:
2–6
= 2 + (–6) (Follow the rules for adding integers.)
=–4
4 – (– 9)
=4+9
= 13
We use the addition and subtraction of integers to solve a variety of everyday
problems in our lives, from weight loss or gain to temperature changes to the
balance in our bank accounts.
Section 2.4, Multiplication of Integers and Section 2.5, Division of
Integers and Order of Operations
To multiply or divide two integers, follow these rules:
Multiply or divide the absolute values.
 If the signs are alike, the answer is positive.
𝑎
o [ a * b = ab] or [ -a * (-b) = ab] and [a ÷ b = ]
𝑏
o [ 6 * 4 = 24] or [ - 6 * (- 4)] = 24]
− 32
o [[
] = 8 or [-32 ÷ (-4)] = 8
−4

o
o
o
If the signs are not alike, the answer is negative.
𝑎
[ a * ( - b ) = - ab] or [ -a * b = -ab] and [ a ÷ (-b) = - ]
𝑏
[ 41 * (-3) = -123 ] or [ -8 * 3 = -24]
36
[36 ÷ (-6) = -6 ] or [ = −6 ]
−6
*Operations are to be performed in the order stated by the acronym.
The order of operations is best remembered by the acronym BEDMAS:
MA 1100, Basic Mathematical Fundamentals
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NOTE: Some students have difficulty with negative and positive powers of
integers. If you are one of those, please study carefully page 108, examples
14-17, as well as the box below those examples.
Mid-Chapter Review, pages 118-119
Section 2.6, Introduction to Algebra and Expressions
Think of an algebraic expression as a phrase such as a prepositional phrase,
like “in the park” or “to the mountain.” Since it is not a complete sentence, it will
not have a left side or a right side like an equation does. Therefore, follow this
rule for all expressions:
 No equal sign (=) on the first line
 Place an equal sign (=) at the beginning of each line from line 2 forward.
To evaluate an expression, substitute the value for the variable:
10n, for n = 2
= 10 * n
= 10 * 2
= 20
Use the distributive property to write an equivalent expression:
7 (a – b)
=7*a–7*b
= 7a – 7b
We use algebraic expressions to convert Fahrenheit temperatures to Celsius
temperatures, to find the amount of an investment after 4 years, and to
calculate the distance that an object falls in 5 seconds. We will discover more
applications of expressions as we work through the textbook.
Section 2.7, Like Terms and Perimeter
What are like terms, and how do we combine them? A term is a number, a
variable, a product of numbers and/or variables, or a quotient of numbers
and/or variables. Terms are separated by addition signs. If there are
subtraction signs, we have to find an equivalent expression that uses addition
signs (by adding opposites or additive inverse).
Identify the terms: 3xy – 4y +
𝟐
𝐳
Answer: The terms are 3xy, 4y,
𝟐
𝐳
Identify the like terms in the following expression: 7x + 𝟓𝒙𝟐 + 2x + 8 + 𝟓𝒙𝟑 + 1
In the following, like terms from the expression above have been highlighted
with colors to help you visualize like terms.
Like terms: 7x + 2x, 8 + 1 , 𝟓𝒙𝟐 , and 𝟓𝒙𝟑
If we were to work this problem, we would work this as an expression,
arranging the terms in descending order and combining like terms.
7x + 𝟓𝒙𝟐 + 2x + 8 + 𝟓𝒙𝟑 + 1
= 𝟓𝒙𝟑 + 𝟓𝒙𝟐 + 7x + 2x + 8 + 1
= 𝟓𝒙𝟑 + 𝟓𝒙𝟐 + 9x + 9
MA 1100, Basic Mathematical Fundamentals
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You will not have to use different colors when showing your work, but you must
show your work for expressions as above.
Perimeter: A polygon is a closed geometric figure with three or more sides.
The perimeter of a polygon is the distance around it, or the sum of the lengths
of its sides. Please review pages 129-130 for examples of solving for
perimeter. One formula for finding the perimeter of a rectangle that is very
useful is:
P = 2l + 2w
Section 2.8, Solving Equations
A solution of an equation is a replacement for the variable that makes the
equation true. The purpose for solving an equation is to find the value of the
variable, which means that the variable needs to be by itself on one side of the
equation.
Equivalent Equations: Equations with the same solutions are equivalent
equations. Are the two equations below equivalent?
a. 5x + 1
b. 2x – 4 + 3x + 5 (Combine like terms) = 5x + 1
Yes, they have the same solution; therefore, they are equivalent equations.
The Addition Principle: a=b is equivalent to a + c = b + c
Example:
Solve: x – 7 = - 2 Remember to solve this as an equation. To “undo” the
subtraction of 7 on the left side, we must add 7 to both sides of the equation.
x–7+7=-2+7
x+0=5
x=5
Add 7 to both sides of the equation, then combine like terms.
You may omit this step when showing your work. It is only
included in order to make sure that you understand what
happened to -7 + 7.
The value of x is 5.
The Division Principle:
For any numbers a, b, and c {c ≠ 0}, a = b is equivalent to
𝑎
𝑐
=
𝑏
𝑐
Examples:
Solve: 9x = 63
9x
9x
9
= 63
= 63
9
x
= 7
(Using the division principle, divide both sides by 9)
Solve: 48 = –8n
48
48
8
–6
= -8n
= -8n
8
=n
(Using the division principle, divide both sides by –8 (watch your signs)
Page 137 shows you how to check your solutions.
MA 1100, Basic Mathematical Fundamentals
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The general rule of thumb when using these principles together is to solve, or
“undo,” addition/subtraction first and save multiplication/division for last. Please
study pages 139-140, examples 9 and 10.
Summary and Review, pages 144-148
Chapter 2 Test, pages 149 - 150
As stated earlier, at the end of each chapter, there will be a summary and
review of key terms and properties, important concepts, review exercises, and
a chapter test [At the very end of every Lesson Guide for every chapter in this
book, there will be a multimedia summary and review plus a multimedia
chapter test (Practice) with a video showing how to work every problem
correctly.] Each summary and review will prove very valuable to you as you
work your way through the units.
Throughout the textbook, there will be an emphasis on real-life problem-solving
applications that will help build critical-thinking skills by requiring you to use
what you know to synthesize, or combine, learning objectives from the current
section with those from previous sections.
Reference
Bittinger, M. L., Ellenbogen, D. J., & Johnson, B. L. (2012). Prealgebra
(6th ed.). Boston, MA: Addison-Wesley.
Learning Activities (Non-Graded)
Practice What You Have Learned
After reading Chapter 1 and Chapter 2, improve your mastery of the content by
working the odd-numbered problems on the following pages:










Section 1.1 Exercise Set, pages 6-8
Section 1.2 Exercise Set, pages 12-13
Section 1.3 Exercise Set, pages 17-18
Section 1.4 Exercise Set, pages 24-25
Section 1.5 Exercise Set, 32-34
Section 1.6 Exercise Set, pages 43-47
Section 1.7 Exercise Set, pages 52-53
Translating for Success, page 63
Section 1.8 Exercise Set, pages 64-70
Section 1.9 Exercise Set, pages 77-79
Once you have completed the problems, you can check your answers in the
back of the textbook to see how well you did.
These are non-graded learning activities, which means that you do not have to
submit them. If you experience difficulty in mastering any of the concepts,
contact your instructor for additional information and guidance.
Review What You Have Learned
Before attempting the Homework and the Unit Assessments, study the chapter
summaries, review the concepts taught in the chapters, and work the oddnumbered problems in the review exercises:
MA 1100, Basic Mathematical Fundamentals
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


Mid-Chapter Review, pages 35-36
Chapter 1 Summary and Review, pages 80-84
Chapter 1 Test (Practice) pages 85-86; Note: See the top of page 85 for
how to access videos showing the step-by-step solutions for each
problem in the test.
These are non-graded learning activities, which mean you do not have to
submit them. If you experience difficulty in mastering any of the concepts,
contact your instructor for additional information and guidance.
Study Plan for Unit I
Once you have accessed the MA 1100 course in MyMathLab, click on your
chapter study plan to see your progress and work the practice exercises.
Your study plan is updated each time you take an Assessment. The study plan
is optional and is generated specifically for each student. Its contents are
based on each student’s results in order to provide practice where it is needed
for each student to obtain mastery of the unit concepts.
Other Resources and Activities
If you need additional guidance or information, you may use all the resources
located within the MyMathLab.
For example, clicking on ‘Tools for Success’ on the toolbar on the left will bring
up the Multimedia Library, which will provide access to video lectures given by
the authors of the textbook, PowerPoints, animation, and interactive figures on
a variety of topics.
When you click on ‘Tools for Success’ at the very top of the page, it will also
provide links to a variety of helpful aids ranging from Translating for Success
and Visualizing for Success Interactive Animations to Basic Math Review Card
and the Introductory Algebra Review Card, which give a brief summary of
many key math concepts.
In the ‘Chapter Contents’ you will also be able to view the Answer section from
your textbook, view the Glossary from your textbook, and view the Index from
your textbook.
MA 1100, Basic Mathematical Fundamentals
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