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**Unformatted text preview: **Understanding and Mastering Elementary Algebra c 2019 by Aaron Wong All rights reserved.
Copyright Contents
Preface 5 Unit 0. Essential Skills
1. Addition and Subtraction on the Number Line
2. Addition and Subtraction on the Number Line with Negatives
3. Multiplication as Area and as “A Groups of B”
4. Division as Counting Groupings (Measurement Division)
5. More Multiplication and Division Concepts
6. Fraction Concepts: Parts of a Whole and the Number Line
7. Addition and Subtraction of Fractions
8. Multiplication and Division of Fractions
9. Decimal Concepts: The Number Line and Fractions
10. Decimal Arithmetic
11. Order of Operations
12. The Algebraic Properties of Arithmetic 7
8
19
30
44
55
66
79
87
99
109
115
126 Unit 1. Variables and Algebraic Expressions
1. Variables as Placeholders
2. Combining Like Terms
3. Multiplying Monomials
4. Simplifying Expressions with Variables
5. The Distributive Property (Part 1)
6. The Distributive Property (Part 2)
7. Simplifying More Expressions with Variables
8. Factoring out a Common Factor
9. Factoring By Grouping
10. Factoring Monic Quadratics
11. Factoring General Quadratics
12. Special Factorizations 135
136
146
152
160
164
175
181
185
190
195
201
206 Unit 2. Algebraic Equations and Inequalities
1. Variables as Unknown Quantities in Equations
2. The Logic of Solving Equations
3. The Algebraic Properties of Equations
4. Solving Basic Linear Equations
5. Solving General Linear Equations
6. Inequalities
7. Clearing the Denominator
8. The Zero Product Property
9. Factoring Review
10. Solve Equations by Factoring
11. Properties of Exponents
12. More Properties of Exponents 211
212
223
235
247
255
264
276
289
295
301
308
316 Unit 3. Applications
1. Translating Between English Expressions and Mathematical Relationships
2. Ratios
3. Unit Conversions
4. Proportions
5. Percents as Ratios
6. Percents as Numbers
7. Scientific Notation 325
328
340
348
355
364
374
386 3 4 CONTENTS 8. The Cartesian Plane
9. Plotting Lines
10. Solutions to Linear Equations in Two Variables
11. Equations of Lines 396
408
422
431 Preface
This textbook is designed to cover the content typical of Prealgebra and Elementary Algebra level remedial
courses. However, the emphasis is not so much about pure computation as it is about conceptual development.
For example, unit zero is primarily about the calculations one would find in a Prealgebra course, but we have
side-stepped the usual arithmetic in columns and long division in order to create space to develop intuition
using a number line and grouping concepts, as well as exposing students to algebraic manipulations that they
will see later (such as the distributive property).
The mindset of this textbook is that for any moderately long “real life” calculations, students will have
access to a calculator or other computational program. Similarly, we would expect students to have a calculator
for any calculations they might see in a science or statistics course. Therefore, we will not force students to
perform moderately long calculations and practice manipulations that they will not use. This does not deny
the role of practice and repetitions of calculations in developing mathematical thought. All the calculations will
still need to be done correctly. It is merely a de-emphasis on manual computations.
Aside from an increased emphasis on concepts, there is also more emphasis on presentation. What and
how students write impacts how they think. All the examples and all the solutions are presented in the exact
manner that students are expected to present them. Many textbooks simply present answers for students,
which implicitly suggests that the answer is the most important part of a math problem. However, we want to
emphasize both the process and the communication of the process, so full solutions to the problem are presented.
In the end, the most valuable skill that students should gain from remedial mathematics is not computational
proficiency, but mental organization. We must recognize that most students taking these classes are mathphobic and are unlikely to require computational proficiency to be successful in their chosen careers. Forcing
them into too many purely symbolic manipulations only drives them further away from math. However, mental
organization is a skill that we can develop and that they can actually take with them. By enforcing presentation
expectations, we can help students transition from the scattered mental processes represented by the sloppiness
of their work, and move them closer to being precise and orderly (i.e., mathematical) thinkers. Even though we
may not be able to get them all the way there during our time, any progress we make in this area is extremely
valuable.
This is the fifth textbook I’ve written for Nevada State College. It started with a desire to create a Prealgebra
textbook in the $30-$40 range that placed a clear emphasis on conceptual development over raw computation.
Over time, it expanded into a set of three textbooks to span the range from Prealgebra through Intermediate
Algebra (and a corresponding modularized approach to the content delivery). Due to issues regarding funding
and other anticipated policies, we were required to reorganize the courses again, which has led to the creation
of this textbook.
I would like to thank Carissa Berge-Sisneros and Kristina Mehaffey for their contributions to the textbook.
Together, they wrote the majority of the problems and solutions, and assisted with the proof-reading process.
Without their help, it would have been impossible for this book to come together as quickly as it has (approximately 8 months from beginning to end), especially with much of my time being devoted to my responsibilities
as the Department Chair of the Physical and Life Sciences. Although the text, problems, and solutions have
been through a collective process of review, in the end, I own all the mistakes that this book contains. As errors
are found, I intend to distribute them to the students through our online course content management system.
To all of our students, I wish you all the best of luck with this course and all of your future courses. I hope
that this book prepares you for future classes the way I intended.
Aaron Wong, Associate Professor of Mathematics, Nevada State College 5 UNIT 0 Essential Skills 7 8 0. ESSENTIAL SKILLS 1. Addition and Subtraction on the Number Line Learning Outcomes:
• Students will be able to locate integers on a number line.
• Students will be able to perform the addition and subtraction of positive integers on a number
line.
• Students will be able to perform the addition and subtraction of positive integers without drawing
a number line.
• Students will present their work for addition and subtraction calculations according to the expectations laid out in the section. The
√ number line is a visual representation of the numbers. Although it includes some more exotic numbers
like 2 and π, we will start by focusing on the integers. The integers are the numbers 1, 2, 3, and so on, their
negatives, and the number 0. They are arranged along a line in order so that the numbers get larger moving to
the right and smaller moving to the left. The number line is infinitely long, but for practical reasons we’ll just
be drawing parts of it instead of the whole thing.
−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 We will focus on the positive integers in this section before discussing the negative integers in the next one.
For this reason, we will be drawing the number line with 0 located all the way to the left side of the line.
0 1 2 3 4 5 6 7 8 9 10 11 12 We are not restricted to using step sizes of 1 when we draw the number line. Instead of counting by 1s, we can
count by 2s, 5s, 10s, or any other number we want.
0 5 10 15 20 25 30 35 40 45 50 55 60 The number line is one of the fundamental concepts in mathematics, and it appears over and over again at
all levels. For example, the number line can be thought of as a way of measuring the distance between numbers.
But we will begin by using the number line as a tool to help us to visualize addition and subtraction. This is
best understood through examples.
Example #1:
Compute 3 + 5 on the number line.
At the beginning of each calculation, we will start at 0 facing to the right. We will indicate the direction
you’re facing with an arrow below the number line at your current location.
0 1 2 3 4 5 6 7 8 9 10 11 12 To do the calculation 3 + 5, we need to translate the symbols into a series of instructions for how to move.
We will read the math from left to right, just as we would an English sentence. The first symbol we see is the
number 3, which tells us to move 3 steps forward. Since we’re facing to the right, this means 3 steps to the
right.
3
0 1 2 3 4 5 6
3 +5 7 8 9 10 11 12 1. ADDITION AND SUBTRACTION ON THE NUMBER LINE 9 The next symbol indicates which direction we will face for the next step. If it is a + sign then you will face
to the right. If it is a − sign then you will face to the left. In this case, it’s a + sign, so we will add an arrow
pointing to the right at our current location (which is the number 3).
3
0 1 2 3 4 5 6 7 8 9 10 11 12 3 + 5
Lastly, we will look at the last symbol. This is the number 5, and that tells us to move 5 more steps forward.
This means 5 more steps to the right. Notice that at the end of this movement, we are standing at the number
8.
3
0 1 5
2 3 4 5 6 7 8 9 10 11 12 3+ 5
An important element of this course is to learn to present information in an orderly and organized manner.
The following is what this example looks like when presented fully and completely. Notice that you can see all
the steps of the number line and the final calculation is included in the box below the picture.
3
0 1 5
2 3 4 5 6 7 8 9 10 11 12 3+5=8 Example #2:
Compute 7 − 4 on the number line.
The process for subtraction is almost the same, except that after the first motion, we need to face to the
left. Beyond that, it’s just a matter of taking the correct number of steps.
4 7
0 1 2 3 4 5 6 7 8 9 10 11 12 7−4=3
The direction of the arrow is very important to the calculation. This will become even more apparent in
the next section when we work with negative quantities. We always start by facing to the right, and from that
point forward, the direction of the arrow depends on the calculation. For addition, the arrow should point to
the right and for subtraction the arrow should point to the left.
We expect students at this level to be able to do these basic one-digit calculations without the aid of the
number line. Calculations like 3 + 5 should be at the level of automatic recall. Automatic recall is the ability to
recall information without thought. For example, if someone asks you for your name, you probably don’t have
to stop and think about it. The answer is right there. Your one-digit addition and subtraction calculations
should be at that level.
If it’s not at that level, the good news is that this is something that can be learned through practice. The
internet is full of online flashcards and other ways to practice your arithmetic. A key technique is to practice
in quick bursts. It’s better to spend 15 minutes a day for a week than it is to spend 2 hours in one day trying
to cram.
The benefit of having these facts accessible through automatic recall is that it frees your conscious brain to
learn other things rather than having to focus its attention on performing small calculations.
However, when numbers get bigger, we can use the number line to help us think about the calculations to
help us do them efficiently and correctly. We will discuss this in the next example. 10 0. ESSENTIAL SKILLS Example #3:
Compute 45 + 29 using a two-step presentation.
We can draw this picture schematically by leaving out the individual markings for the individual numbers.
This lets us use the number line as a thinking tool and not a counting tool.
45 29 0 45 ?? But in order to get the right answer, we need to do a calculation. Nobody really wants to count 29 steps
past 45, so we need to think of a better way to do this calculation. There are several approaches, and we will
introduce one of them here.
Instead of trying to take all 29 steps at once, we can break it up into two motions. We can move 20 steps,
followed by 9 more steps. What this does is it breaks the number of steps down into pieces that are simpler
to add. In fact, with a little practice, you should be able to do these calculations mentally with very little
difficulty.
45 20 0 45 9
65 74 This gives us the result without having to do a calculation that’s too difficult. Furthermore, it’s in a form
that’s much easier to do mentally than trying to add in columns mentally, and it’s even faster than writing out
the addition in columns as well. For these reasons, we will not be working with addition in columns in this
book.
When writing up your final presentation, it’s important to practice communicating the thought process in
addition to communicating the final result. In fact, math is ultimately about the reasoning process and not so
much about the number at the end of the calculation.
In order to do make the presentation match the thinking, we will use a “two-step” presentation.
45 + 29 = 45 + 20 + 9
= 65 + 9
= 74
Notice that the presentation communicates exactly what happened. First, we broke the 29 steps into 20
steps followed by 9 more steps, and then we performed the calculation. Also, notice that the equal signs are
lined up vertically and that the original problem appears on the left of the first line. This is an important for
organization, as in later sections this will prevent you from losing terms and making other errors of that type.
Example #4:
Compute 72 − 26 using a two-step presentation.
We will start with a picture of this calculation.
26 72
0 72 ?? Once again, the second movement can be broken down into steps that are easier to work with.
6 72
0 46 20
52 72 And from here, we can write up the calculation.
72 − 26 = 72 − 20 − 6
= 52 − 6
= 46
For problems where you’re simply asked to compute the result, you do not need to write out the two
steps. However, it’s worth practicing thinking through the steps because it will enhance your ability to do these
calculations more and more quickly. 1. ADDITION AND SUBTRACTION ON THE NUMBER LINE Example #5: 11 Compute 28 + 51.
28 + 51 = 79 Notice that the entire problem and presentation is condensed into a short equation. However, this does not
mean that smaller steps were not done to get the answer. It is important to pay attention to the presentation
expectations so that you write the correct amount of work.
When adding and subtracting multiple terms, we work from left to right.
Example #6:
Compute 25 − 11 + 38.
The individual steps are explained below. This is then followed by the completed picture.
• The starting arrow is placed at 0 and always points to the right.
• We move 25 steps forward, moving from 0 to 25.
• The next arrow points to the left because the next calculation is subtraction.
• We move 11 steps forward (which is now to the left) and end up at 14.
• The last arrow points back to the right because the final calculation is addition.
• We move 38 steps forward (which is to the right again) and end up at 52.
38
11 25
0 14 25 52 When writing up the presentation, each calculation should be its own step. You will not be asked to write
up two-step calculations when you’re given a longer expression like this. Instead, you can just do one step per
line of work.
25 − 11 + 38 = 14 + 38
= 52
It is useful to be able to think about these pictures at an intuitive level. Consider the following number line
addition calculations:
0 0
(BIG) + (small) (small) + (BIG) The relative sizes of the movements give you a sense of the relative importance of the two numbers in the
calculation. If we try to do the same thing with subtraction, we run into an interesting situation. 0 0
(small) − (BIG) (BIG) − (small) Notice that in the “(small) − (BIG)” calculation, we ended up on the left side of zero. We will explore this
situation as well as several others in the next section. Problem Sets
Problem Set #1:
Perform the calculations using a number line. Include the arrows and the final equation.
#1)
#2)
#3) 4+5
7+2
3+8 #4)
#5)
#6) 8−4
9−7
7−2 12 0. ESSENTIAL SKILLS Perform the calculations using a number line where the spacing is 4. Include the arrows and the
final equation.
#7)
#8) 12 + 28
8 + 24 #9) 36 − 12
#10) 28 − 16 Perform the calculations using a two-step presentation. You do not need to draw the number
line.
#11)
#12)
#13)
#14)
#15) 44 + 24
51 + 51
71 + 43
44 + 44
85 + 51 #16)
#17)
#18)
#19)
#20) 77 − 29
68 − 56
69 − 66
42 − 24
53 − 41 Perform the calculations. You may write up your presentation as one line.
#21)
#22)
#23)
#24)
#25) 97 + 61
83 + 54
30 + 48
76 + 43
81 + 87 #26)
#27)
#28)
#29)
#30) 61 − 37
54 − 39
55 − 22
78 − 55
70 − 39 Perform the calculations. Write one step per line of work.
#31)
#32)
#33) 32 + 19 − 11
28 − 21 + 54
44 − 13 − 15 #34)
#35)
#36) 21 + 49 + 38
84 − 65 + 31
37 + 29 − 42 Problem Set #2:
Perform the calculations using a number line. Include the arrows and the final equation.
#37)
#38)
#39) 9+3
4+3
6+4 #40)
#41)
#42) 9−5
5−3
6−4 Perform the calculations using a number line where the spacing is 7. Include the arrows and the
final equation.
#43)
#44) 14 + 21
28 + 42 #45)
#46) 77 − 56
84 − 49 Perform the calculations using a two-step presentation. You do not need to draw the number
line.
#47)
#48)
#49) 39 + 38
65 + 49
89 + 56 #50)
#51)
#52) 78 − 29
57 − 23
92 − 34 1. ADDITION AND SUBTRACTION ON THE NUMBER LINE 13 Perform the calculations. You may write up your presentation as one line.
#53)
#54)
#55) 89 + 48
92 + 63
76 + 37 #56)
#57)
#58) 51 − 22
43 − 19
94 − 75 Perform the calculations. Write one step per line of work.
#59)
#60) 52 + 38 − 20
98 − 29 + 46 #61)
#62) 68 − 23 − 15
45 + 79 + 13 Problem Set #3:
Perform the calculations using a number line. Include the arrows and the final equation.
#63)
#64)
#65) 6+5
4+8
5+2 #66)
#67)
#68) 8−5
6−2
9−3 Perform the calculations using a number line where the spacing is 12. Include the arrows and
the final equation.
#69)
#70) 36 + 12
60 + 36 #71)
#72) 120 − 72
144 − 48 Perform the calculations using a two-step presentation. You do not need to draw the number
line.
#73)
#74) 56 + 37
82 + 34 #75)
#76) 74 − 55
59 − 33 Perform the calculations. You may write up your presentation as one line.
#77)
#78)
#79) 84 + 27
35 + 44
19 + 82 #80)
#81)
#82) 72 − 43
88 − 45
99 − 28 Perform the calculations. Write one step per line of work.
#83)
#84) 52 + 18 − 37
76 − 25 − 32 #85)
#86) 15 + 32 − 43
23 − 16 + 48 14 0. ESSENTIAL SKILLS Problem Set #4:
In this problem set, we will learn a different way of performing mental calculations.
As people become more experienced with mental arithmetic, they can learn various useful tricks to perform
calculations more quickly. To demonstrate this, consider the following calculations: 49 + 38 and 50 + 37. The
second calculation is much easier to do. The trick is to use a two-step calculation, but instead of breaking the
second number into tens and ones, it’s broken up so that after the first step we end up at a multiple of 10.
Here’s how it looks.
49
1
37
0 49 50 87 49 + 38 = 49 + 1 + 37
= 50 + 37
= 87
This trick works particularly well when the first number ends in a 7, 8, or 9 because you only need to change
the second number by 3, 2, or 1. But it would technically work in any situation that involves a “carrying” step.
Perform the calculations using the technique described above. Write up your calculation using a
two-step presentation.
#87)
#88) 49 + 37
38 + 44 #89)
#90) 29 + 75
57 + 57 Perform the calculations using the technique described above. You may write up the calculation
as one line.
#91)
#92)
#93) 45 + 29
39 + 37
58 + 79 #94)
#95)
#96) 47 + 38
42 + 19
78 + 55 Problem Set #5:
In this problem set, we will learn a different way of performing mental calcu...

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- Fall '19
- Addition, Subtraction, Negative and non-negative numbers