Reference appendices, conversion charts, formula sheets, a complete glossary, answer key guidance, a full index, and MGU Library connections
The appendices that follow are reference and review tools. They are not additional chapters. Use them to prepare before beginning the book, to review a skill at any point during the course, and to look up formulas, symbols, or conversion values when solving problems. The Algebra I Readiness Check in Appendix J is a final self-assessment before moving into the next course.
This appendix is designed for students who want to check their preparation before beginning the textbook, or who need to revisit foundational skills at any point during the course. It is not a placement test. It is a self-check.
Work through each section honestly. If you find that a topic is unfamiliar or that you are making consistent errors, return to the relevant chapter before moving forward.
Can you do the following? Write and read whole numbers in standard form, expanded form, and word form. Compare whole numbers using the symbols <, >, and =. Add, subtract, multiply, and divide whole numbers without a calculator. Identify even and odd numbers. Round whole numbers to any given place value.
546 × 382,184 ÷ 6Answers may be placed in the answer key.
Can you do the following? Locate positive and negative numbers on a number line. Identify the opposite of a number. Find the absolute value of a number. Add, subtract, multiply, and divide signed numbers.
−8 + 3−4 × (−7)−36 ÷ 4Answers may be placed in the answer key.
Can you do the following? Identify the numerator and denominator of a fraction. Write equivalent fractions. Reduce a fraction to lowest terms. Add, subtract, multiply, and divide fractions. Convert between mixed numbers and improper fractions.
Answers may be placed in the answer key.
Can you do the following? Read and write decimals to the thousandths place. Compare decimals and order them on a number line. Add, subtract, multiply, and divide decimals. Round decimals to a given place value.
3.4 + 0.0755.28 × 0.3Answers may be placed in the answer key.
Can you do the following? Write a ratio in three ways. Convert between fractions, decimals, and percents. Solve a basic one-step equation. Evaluate a simple expression for a given value of a variable.
x + 7 = 153n − 2 when n = 4.Answers may be placed in the answer key.
These tables are provided as reference tools. Students are encouraged to know these facts from memory. Fluency with basic addition and multiplication supports speed and accuracy in all later work with fractions, decimals, percents, variables, and equations.
To find the sum of two numbers, locate one number along the top row and the other along the left column. The cell where that row and column meet gives their sum.
To find a product, locate one factor along the top row and the other along the left column. The cell where that row and column meet is the product.
Knowing multiplication facts through 12 × 12 by memory removes a significant mental burden from later work. If you are still learning these facts, start with the 2s, 5s, and 10s, then add the 3s and 4s, and finally fill in the remaining facts. Repeated, low-pressure practice over time is more effective than cramming.
A divisibility rule is a shortcut for deciding whether one number divides evenly into another, without performing full division. These rules are useful for factoring, reducing fractions, finding GCF and LCM, and simplifying calculations.
| Divisor | Rule | Example |
|---|---|---|
| 2 | The number ends in 0, 2, 4, 6, or 8 (it is even). | 348 is divisible by 2 because it ends in 8. |
| 3 | The sum of the digits is divisible by 3. | 4 + 2 + 3 = 9, and 9 ÷ 3 = 3, so 423 is divisible by 3. |
| 4 | The last two digits form a number divisible by 4. | 732 → 32 ÷ 4 = 8, so 732 is divisible by 4. |
| 5 | The number ends in 0 or 5. | 285 ends in 5, so it is divisible by 5. |
| 6 | The number is divisible by both 2 and 3. | 312: even, and 3 + 1 + 2 = 6 (divisible by 3), so divisible by 6. |
| 8 | The last three digits form a number divisible by 8. | 1,024 → 024 = 24, and 24 ÷ 8 = 3, so 1,024 is divisible by 8. |
| 9 | The sum of the digits is divisible by 9. | 8 + 1 + 9 = 18, and 18 ÷ 9 = 2, so 819 is divisible by 9. |
| 10 | The number ends in 0. | 4,730 ends in 0, so it is divisible by 10. |
There is no simple one-step rule for 7. The most reliable approach at this level is to divide and check. For those interested, one method is to double the last digit, subtract it from the rest of the number, and check whether the result is divisible by 7. This process can be repeated on smaller numbers. For most pre-algebra work, however, direct division or prime factorization is the more practical choice.
Divisibility rules work together. To test whether a number is divisible by 6, you test for 2 and for 3 separately. Both must be true.
Test for 2: 5,742 ends in 2, so yes, it is even.
Test for 3: 5 + 7 + 4 + 2 = 18, and 18 ÷ 3 = 6, so yes.
Since 5,742 is divisible by both 2 and 3, it is divisible by 6.
Answers may be placed in the answer key.
This chart provides reference conversions for the most commonly used fractions, decimals, and percents. Students who memorize this chart will work faster and more accurately on problems involving percents, proportions, and comparisons.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/100 | 0.01 | 1% |
| 1/20 | 0.05 | 5% |
| 1/10 | 0.1 | 10% |
| 1/8 | 0.125 | 12.5% |
| 1/5 | 0.2 | 20% |
| 1/4 | 0.25 | 25% |
| 3/10 | 0.3 | 30% |
| 1/3 | 0.333… | 33.3̄% |
| 3/8 | 0.375 | 37.5% |
| 2/5 | 0.4 | 40% |
| 1/2 | 0.5 | 50% |
| 3/5 | 0.6 | 60% |
| 5/8 | 0.625 | 62.5% |
| 2/3 | 0.666… | 66.6̄% |
| 7/10 | 0.7 | 70% |
| 3/4 | 0.75 | 75% |
| 4/5 | 0.8 | 80% |
| 7/8 | 0.875 | 87.5% |
| 9/10 | 0.9 | 90% |
| 1/1 | 1.0 | 100% |
This formula sheet collects the key formulas introduced throughout this textbook. It is a reference tool, not a substitute for understanding. Students who understand why each formula works will apply it correctly in new situations. Students who memorize formulas without understanding them will often apply them incorrectly when the problem changes slightly.
| Operation | Rule |
|---|---|
| Adding same signs | Add the absolute values. Keep the sign. |
| Adding opposite signs | Subtract the smaller absolute value from the larger. Keep the sign of the larger. |
| Subtracting | Change the sign of the number being subtracted. Then add. |
| Multiplying or dividing | Same signs → positive result. Opposite signs → negative result. |
| Rule | Formula |
|---|---|
| Product Rule | x^m × x^n = x^(m+n) |
| Quotient Rule | x^m ÷ x^n = x^(m−n) |
| Power Rule | (x^m)^n = x^(m×n) |
| Zero Exponent | x^0 = 1 (x ≠ 0) |
| Negative Exponent | x^(−n) = 1/x^n |
This appendix lists the mathematical symbols used throughout the textbook. When a symbol appears in a formula, expression, or problem, this reference can help you identify what it means.
| Symbol | Meaning | Example |
|---|---|---|
+ | Addition | 4 + 3 = 7 |
− | Subtraction or negative sign | 9 − 2 = 7, or −5 |
× | Multiplication | 4 × 3 = 12 |
· | Multiplication (dot notation) | 4 · 3 = 12 |
÷ | Division | 12 ÷ 4 = 3 |
/ | Division (fraction bar or slash) | 12/4 = 3 |
= | Equals | 2 + 3 = 5 |
| Symbol | Meaning | Example |
|---|---|---|
< | Less than | 3 < 7 |
> | Greater than | 9 > 4 |
≤ | Less than or equal to | x ≤ 5 |
≥ | Greater than or equal to | x ≥ 0 |
≠ | Not equal to | 3 ≠ 4 |
≈ | Approximately equal to | π ≈ 3.14 |
| Symbol | Meaning | Example |
|---|---|---|
( ) | Parentheses, grouping | (3 + 4) × 2 = 14 |
[ ] | Brackets, outer grouping | [3 × (4 + 1)] = 15 |
{ } | Braces, set notation | {1, 2, 3} |
| | | Absolute value | |−5| = 5 |
√ | Square root | √16 = 4 |
ⁿ√ | nth root | ³√8 = 2 |
π | Pi (approximately 3.14159…) | C = 2πr |
| Symbol | Meaning | Example |
|---|---|---|
a, b, x, y, n | Variables — unknown or changing quantities | x + 3 = 7 |
x^n | x raised to the power n | x² means x × x |
∞ | Infinity | Used in describing number lines and sets |
| Symbol | Meaning | Example |
|---|---|---|
10^n | Ten raised to the power n | 10³ = 1000 |
× | Multiplication in scientific notation | 3.2 × 10⁵ |
E (calculator) | Exponent key, replaces × 10 | 3.2E5 = 3.2 × 10⁵ |
These formulas were introduced in Chapter 14. This appendix collects them for quick reference. Students are encouraged to understand the meaning of each formula, not only how to compute with it.
Perimeter is the total distance around the outside of a figure. It is measured in linear units such as centimeters, inches, feet, or meters.
| Figure | Formula |
|---|---|
| Square | P = 4s |
| Rectangle | P = 2l + 2w |
| Triangle | P = a + b + c |
| Regular polygon (n sides) | P = n × s |
| Circle (circumference) | C = 2πr or C = πd |
Area measures the space inside a two-dimensional figure. It is expressed in square units such as cm², in², or ft².
| Figure | Formula |
|---|---|
| Square | A = s² |
| Rectangle | A = l × w |
| Parallelogram | A = b × h |
| Triangle | A = (1/2) × b × h |
| Trapezoid | A = (1/2)(b₁ + b₂) × h |
| Circle | A = πr² |
Volume measures the space inside a three-dimensional figure. It is expressed in cubic units such as cm³ or ft³.
| Figure | Formula |
|---|---|
| Rectangular prism (box) | V = l × w × h |
| Cube | V = s³ |
| Cylinder | V = πr²h |
| Triangular prism | V = (1/2) × b × h × l |
The variable r always refers to the radius, the distance from the center of a circle to its edge. The variable d refers to the diameter: d = 2r. The variable h always refers to the height, measured perpendicular to the base. Area is always in square units and volume is always in cubic units.
A calculator is a tool, not a substitute for understanding. This appendix introduces the most important calculator skills for students using a standard four-function or scientific calculator. Before relying on a calculator, always estimate the answer mentally so that you can catch input errors.
Entering negative numbers: On most calculators, a button labeled (+/−) or (−) changes the sign of a number. This is different from the subtraction key. To enter −7, type 7, then press (+/−).
Entering fractions as decimals: Most basic calculators do not handle fractions directly. Convert the fraction to a decimal first by dividing the numerator by the denominator.
Order of operations: Standard four-function calculators do not follow order of operations automatically. They compute from left to right. A scientific calculator with algebraic logic will follow order of operations correctly. Always know which type of calculator you are using.
Press: ( 3 + 4 ) × 5 = → Result: 35
Without parentheses on a basic left-to-right calculator, compute the grouped part first, write down the result, and then continue.
On most scientific calculators, the exponent key is labeled x^y, ^, or y^x. To compute 3⁴: press 3, then the exponent key, then 4, then =. Result: 81.
The square root key is labeled √. To compute √144: press √, then 144, then = (on some calculators, press the number first). Result: 12.
The key for entering scientific notation is usually labeled EE, EXP, or E. It means "times ten to the power of." To enter 3.2 × 10⁶: press 3, period, 2, then EE or EXP, then 6. The display may show 3.2E6 or 3.2 × 10⁶, depending on the calculator.
Do not press the multiplication key or the 1 and 0 separately when entering scientific notation. Using the EE key is the correct method.
Before accepting a calculator result, always ask: Is the answer in the right ballpark? Did I enter the numbers in the correct order? Did I handle negative signs correctly? Did I use parentheses where the problem required grouping? A calculator will give you an exact answer to whatever expression you enter. It will not tell you whether you entered the right expression.
Word problems are the place where arithmetic connects to real life. Many students find them difficult, not because they lack mathematical knowledge, but because the translation from words to mathematics is a separate skill that must be developed.
Throughout this textbook, the following five-step method is used. This is not a rigid formula. It is a mental habit. Over time, students who practice this method internalize it and move through these steps naturally.
Read — Understand the problem fully before writing anything.
Represent — Draw a diagram, define a variable, or set up a table.
Solve — Carry out the mathematics, showing each step.
Check — Verify the answer using estimation or substitution.
Explain — Write a complete sentence that answers the question in context.
| Signal Word or Phrase | Likely Operation |
|---|---|
| sum, total, in all, combined, more than | Addition |
| difference, less than, decreased by, how many more | Subtraction |
| product, times, of (with fractions), multiplied by | Multiplication |
| quotient, divided by, per, ratio, split equally | Division |
| is, equals, the result is | Equals |
| at least, no more than, minimum, maximum | Inequality |
The problem context always takes priority over signal words alone. The word "more" sometimes signals addition and sometimes signals a comparison requiring subtraction. Read carefully.
Answering the wrong question. Re-read the question after solving. Make sure your answer addresses exactly what was asked.
Ignoring units. Track units throughout the problem. Miles per hour times hours gives miles, not miles per hour.
Forgetting to define the variable. Writing an equation with x before saying what x represents leads to confusion in multi-step problems.
Solving only one step of a multi-step problem. Re-read the problem and ask whether the computation you did is sufficient to answer the question, or whether more steps are needed.
Using a signal word blindly. Context matters. Do not substitute an operation for careful reading.
This appendix gives students a final chance to assess their readiness before moving forward into Algebra I. It draws on skills from every part of this textbook. Work through all problems without a calculator unless otherwise noted. Review any section you find difficult.
−14 + (−6) + 3−5 × 4 ÷ (−2)x³ × x⁵y⁸ ÷ y³(2³)²4x² − 3x + 1 when x = 2.5a + 3b − 2a + 7b3x − 7 = 142x + 1 ≤ 93 + 4² × 2 − (6 ÷ 3)(8 − 2)² ÷ 4 + 52 × [3 + (7 − 4)²]Answers may be placed in the answer key.
If you answered most problems confidently, you are prepared to begin Algebra I. If you found several sections difficult, return to the corresponding chapters in this textbook before moving forward. A solid arithmetic foundation will make every topic in Algebra I faster and clearer to learn.
This glossary defines key mathematical terms used throughout the textbook. Terms are arranged alphabetically. Each definition is followed by the chapter where the term is first introduced.
The answer key for this textbook is maintained separately as a supplementary document and as a digital resource in the MGU Library system. Answers to chapter checkpoints, appendix practice sets, and the Algebra I Readiness Check in Appendix J are included in that document.
Students are encouraged to attempt all problems independently before consulting the answer key. Checking answers after genuine effort reinforces learning. Checking answers before attempting problems does not.
The index lists major topics, terms, and concepts with the chapters and sections where they appear. Section references use the format chapter.section.
This textbook is part of the Malone Global University Library Mathematics Series. The following resources are available through the MGU Library system and are related to the content in this volume.
The MGU Library textbook editions are the expanded source versions of the corresponding ebook-courses. They are designed for students who benefit from deeper explanation, additional examples, broader context, and more extensive practice. Both formats cover the same mathematical content. The ebook-course provides a structured, paced learning path. The Library textbook becomes the complete, reference-ready volume. Students and instructors may use both formats together, or either one alone.