Prealgebra 2e - Comprehensive Study Notes
Preface
OpenStax provides free, peer‑reviewed, openly licensed textbooks for introductory college and AP courses. Textbook content is licensed under CC BY 4.0.
OpenStax mission: increase student access to high-quality materials; maintain academic rigor at little to no cost.
Prealgebra 2e is designed to meet one‑semester scope and sequence for prealgebra; emphasizes a student-support approach with small, progressive steps.
Be prepared discussions and errata process: Be Prepared, How To, Try It features; errata corrections are tracked on openstax.org.
Pedagogical foundation includes Learning Objectives, Narration, Examples, Be Prepared prompts, Try It exercises, and media/online resources.
Organization: 11 chapters outline a progression from Whole Numbers to Graphs, Polynomials, and introductory Algebra; later chapters cover language of algebra, expressions, factoring, and basics of integers.
Chapter 1: Whole Numbers
1.1 Introduction to Whole Numbers
Goal: Introduce counting numbers and whole numbers, place value concepts, and notation.
Key concepts:
Counting numbers (natural numbers) start at 1: 1, 2, 3, …; ellipsis “…” indicates continuation.
Whole numbers: counting numbers plus 0:
ext{whole numbers} = {0,1,2,3,box{…}
}
Place value system: value of a digit depends on its position; base-10 blocks model hundreds, tens, ones.
Visual tools: number line and place‑value charts; origin (0) on the line.
Naming and writing numbers: naming in words by periods (ones, thousands, millions, etc.) with commas to separate periods. The ones period is not named.
Example highlights:
Example 1.1 identifies counting vs. whole numbers.
Example 1.2–1.4 use place‑value notation to model numbers with blocks and to name numbers in words.
Key terms:
Counting numbers, natural numbers, whole numbers, place value, period, origin, ellipsis.
Be Prepared/TRY IT prompts guide prerequisite checks and quick practice on recognizing and naming numbers.
1.2 Add Whole Numbers
Objective: Use addition notation; model addition of whole numbers; translate word phrases to math notation; apply to applications.
Notation and vocabulary:
Addends: numbers being added.
Sum: result of addition.
Expression: a numerical sentence with an operation; e.g., for 3 + 4, the expression is 3 + 4.
Properties and concepts:
Identity Property of Addition: a+0=a
Commutative Property of Addition: a+b=b+a
Modeling addition:
Use base-10 blocks (ones and rods) to model sums where each addend is less than 10 for simple cases.
When sums reach 10 or more, exchange 10 ones for 1 ten (carry).
Procedure (How to Add Whole Numbers):
Step 1: Align place values vertically.
Step 2: Add digits from right to left; carry if a sum in a column is > 9.
Step 3: Continue for all columns.
Example pattern (non-exhaustive):
2 + 6 model with blocks; total blocks show the sum; 2 + 6 = 8.
Two-digit addition demonstrated with carrying (e.g., 17 + 26 = 43 after carrying as needed).
Be Prepared/TRY IT prompts guide verification of readiness and practice problems involving addition notation and modeling.
1.3 Subtract Whole Numbers
Objective: Use subtraction notation; model subtraction of whole numbers; translate word phrases; apply to applications.
Subtraction notation and vocabulary:
Subtrahend: number being subtracted.
Difference: result.
Expression and equation concepts mirror addition: e.g., 7 − 3 = 4; check via addition: 4 + 3 = 7.
Modeling subtraction:
Use base-10 blocks to model subtraction; subtract and circle or remove blocks to show the difference.
Subtraction properties and checks:
Subtraction is inverse to addition; check by adding the difference to the subtrahend to recover the minuend.
Process for subtracting multi-digit numbers:
Align digits; subtract from right to left; borrow when needed; and verify by addition.
Key ideas:
Borrowing (regrouping) when needed; carrying not typically used in subtraction but appropriate for regrouping.
Try It/Be Prepared prompts provide practice on translating and modeling subtraction.
1.4 Multiply Whole Numbers
Objective: Use multiplication notation; model multiplication; translate word phrases; apply to applications.
Conceptual view:
Multiplication as repeated addition; factors and products; product meaning (e.g., a imes b is the product of a and b).
Notation and symbols:
Multiplication symbols: ×, •, \, and sometimes juxtaposition.
Product: result of multiplication.
Key properties:
Multiplication by zero: a imes 0 = 0 (Multiplication Property of Zero).
Identity Property of Multiplication: a imes 1 = a.
Commutative Property of Multiplication: a imes b = b imes a.
Modeling and methods:
Use counters (or tiles) to build a rectangular array representing the product; show that multiple configurations yield the same product (e.g., 3×8 vs 8×3).
Long multiplication: multiply by each digit of the bottom factor; write partial products; align by place value; carry as needed; add partial products.
Steps for multiply two whole numbers (high level):
Step 1: Align digits by place value.
Step 2: Multiply the digit in the ones place of the bottom number by the top number; write the partial product; carry if needed.
Step 3: Move to the next digit of the bottom number; multiply and write the next partial product; shift left appropriately; repeat for all digits.
Step 4: Sum all partial products.
Important concepts:
Zero padding and alignment; partial products; place-value shifting.
Practical tips:
Memorize one-digit multiplication facts; use modeling when stuck; verify with distributive intuition.
Be Prepared/TRY IT prompts provide practice with single-digit multiplication and expanding to two-digit multiplication.
1.5 Divide Whole Numbers
Objective: Use division notation; model division; translate word phrases; apply to applications.
Language and roles:
Dividend: number being divided.
Divisor: number dividing the dividend.
Quotient: result of division.
Division notation and reading:
Examples: rac{12}{4}=3 read as “twelve divided by four equals three.”
Long division basics:
Divide the first digit(s) of the dividend by the divisor; write the quotient; multiply and subtract; bring down the next digit; repeat.
When no more digits remain, the process ends; the remainder (if any) is written as R followed by the remainder.
Special cases:
Division by zero is undefined.
Zero divided by any nonzero number is 0.
Any number divided by 1 is the number itself; any number divided by itself is 1.
Key properties:
Division by zero: undefined.
Division by one and by itself are special cases (Division Properties of One and Zero).
Applications: Solve word problems by translating to division expressions and solving; use long division when necessary.
Chapter 2: The Language of Algebra
(Overview for this chapter focuses on the symbolic language of algebra, translation between words and symbols, exponents, order of operations, and basic algebraic manipulations.)
2.1 Use the Language of Algebra
Core ideas:
Distinguish between variables and constants; understand expressions vs equations.
Common symbols: +, −, ×, ÷, =, ≠, <, >, ≤, ≥, and grouping symbols: parentheses (), brackets [], braces {}.
A variable represents a number that can change; a constant has a fixed value.
Use of exponents to indicate repeated multiplication: a^n (where n is a positive integer; a is the base).
Exponential notation:
Definitions and vocabulary: base, exponent; example: a^n = ext{the base } a ext{ multiplied by itself } n ext{ times}. For example, 2^3 = 2 imes 2 imes 2 = 8.
Order of Operations (PEMDAS):
P: Parentheses (grouping symbols)
E: Exponents
MD: Multiplication and Division (from left to right)
AS: Addition and Subtraction (from left to right)
Mnemonic: Please Excuse My Dear Aunt Sally
Equations and expressions:
An expression is a single mathematical object (no equal sign).
An equation has two expressions separated by an equal sign, e.g., 2x+3=7.
Examples and practice prompts illustrate translating word phrases to algebra, and vice versa; grouping symbols help manage order of operations.
2.2 Evaluate, Simplify, and Translate Expressions
Evaluate: substitute a given value for a variable and simplify according to the order of operations.
Example pattern: evaluate 2a+3 at a=4: 2(4)+3=11.
Like terms and coefficients:
A term is a constant or a product of a coefficient and variables.
The coefficient is the factor multiplying the variable(s).
Like terms share the same variable factors and exponents.
Simplify by combining like terms:
Add/subtract coefficients of like terms; keep the same variable parts intact.
Translate word phrases to algebraic expressions:
Words like "sum", "difference", "product", and "quotient" signal operations; watch for order words like "of" and parentheses.
Example templates:
Addition: the sum of a and b → a+b
Subtraction: the difference of a and b → a-b
Multiplication: the product of a and b → a imes b or ab
Division: the quotient of a and b → rac{a}{b}
Exponent rules and order-of-operations examples illustrate evaluation and simplification.
2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
Central ideas:
A solution to an equation is a value for the variable that makes the equation true.
Subtraction Property of Equality: If a=b, then a-c=b-c for any c.
Addition Property of Equality: If a=b, then a+c=b+c for any c.
Process to solve one-step equations:
Identify the operation; apply appropriate property to isolate the variable; verify by substitution.
Translating word phrases to algebraic equations:
Use phrases like "the sum of x and 5 is 12" → x+5=12; then solve for x.
Applications: Word problems translated to equations; solve and check.
2.4 Find Multiples and Factors
Multiples: A number is a multiple of n if it can be written as n imes k for some integer k.
Divisibility tests (common rules): quick checks to see if a number is divisible by small integers (2–10).
Factors: Numbers a and b are factors of N if N=a imes b.
Prime vs composite:
Prime: has exactly two distinct positive factors: 1 and itself.
Composite: more than two positive factors.
1 is neither prime nor composite.
Factorization approach (trees and ladders): representing a number as a product of primes; use to prepare for LCM calculations.
Inverse relationships:
Multiples/divisibility connect to factoring and greatest common divisors.
2.5 Prime Factorization and the Least Common Multiple
Prime factorization: express a composite number as a product of primes (unique up to order).
Factor tree method and ladder method enable systematic factoring.
Example: 84 = 2^2 imes 3 imes 7 (via factorization tree or ladder).
Least Common Multiple (LCM): smallest positive integer that is a multiple of two numbers.
Methods:
Listing multiples (find common multiples; the smallest is LCM).
Prime-factorization method: use the prime factorization of both numbers, then take the highest power of all primes involved.
Applications include adding and subtracting fractions with different denominators.
Chapter 3: Integers
3.1 Introduction to Integers
Integers extend the integers to negative numbers and zero.
Positive vs negative numbers on the number line; zero is neither positive nor negative.
Opposites: numbers equidistant from zero on opposite sides; e.g., the opposite of a is -a.
Absolute value: distance from zero; defined as |a|=egin{cases}a,&a\ge 0-a,&a<0
eq ext{(if }a<0)
extrm{.}
ight.
Number line basics; plotting points; ordering positive and negative numbers using inequality notation: a<b means a is to the left of b on the number line.
Example concepts:
Plot coordinates, identify opposites, and use absolute value to compare magnitudes.
3.2 Add Integers
Add integers using a model (blue for positive, red for negative counters) to illustrate rules.
Key rules from modeling:
Same signs: add magnitudes; keep sign.
Different signs: subtract smaller magnitude from larger; sign of the result is the sign of the larger magnitude.
Neutral pairs cancel to zero.
Examples with counters show addition of like signs, then mixed signs via pairing and cancellation.
Evaluate expressions by combining integers and using the absolute value as needed.
Applications include arithmetic with temperatures, elevations, financial balances, and other real-world quantities.
3.3 Subtract Integers
Subtraction as adding the opposite: a-b=a+(-b).
Subtraction Property of Equality in the integer context: subtract same integer from both sides to preserve equality.
Examples illustrate subtracting positive and negative numbers, including cases where you must add neutral pairs to facilitate subtraction.
Practice includes translating word phrases to subtraction expressions and solving.
3.4 Multiply and Divide Integers
Rules for signs:
Product of two integers: positive if signs are the same; negative if signs differ.
Quotient of two integers: same sign rule as multiplication.
Zero property: any number multiplied by 0 is 0; division by zero is undefined; zero divided by a nonzero is 0.
Modeling and rules help to handle larger expressions and to evaluate variable expressions with integer arithmetic.
3.5 Solve Equations Using Integers; The Division Property of Equality
Solve equations in which integers appear; apply multiplication/division properties as appropriate to isolate the variable.
Division property of equality mirrors the multiplication property: if ac=bc and c
eq0 then a=b after division by c.Word problems translate to integer equations; solve and verify with substitution.
Key Equations and Concepts (LaTeX formatted)
Addition and related properties:
Identity: a+0=a
Commutative: a+b=b+a
Multiplication and related properties:
Identity: a\cdot1=a
Zero: a\cdot0=0
Commutative: a\cdot b=b\cdot a
Exponents:
a^n = \underbrace{a\cdot a\cdots a}_{n\text{ times}}
Order of Operations (PEMDAS):
ext{Parentheses} \to \text{Exponents} \to \text{Multiplication/Division} \to \text{Addition/Subtraction}
Absolute value:
|a|=a \text{ if } a\ge 0, \quad |a|=-a \text{ if } a<0
Integer addition rules (quick summary):
If signs match: add magnitudes, keep sign.
If signs differ: subtract smaller magnitude from larger, sign of the larger magnitude.
Prime factorization and LCM (summary):
Prime factorization: express N as a product of primes: N=p1^{e1}p2^{e2}\cdots
LCM: smallest multiple common to two numbers; can be found via listing multiples or using prime factorizations.
Connections and Relevance
The structure of whole numbers builds a foundation for algebra: understanding place value, basic operations, and arithmetic properties forms the basis for solving equations, simplifying expressions, and translating word problems.
The Language of Algebra links everyday language to symbols, enabling translation between verbal descriptions and algebraic models, essential for modeling real-world problems.
Integers extend the number system to negative quantities, modeling many real-world phenomena (temperature, elevation, debt, etc.). Mastery of absolute value and sign rules is crucial for higher mathematics (linear algebra, analytic geometry, and beyond).
Practice and Study Tips
Regularly practice translating phrases to expressions and vice versa to build fluency in algebraic language.
Use the counter models (ones/tens blocks) for addition, subtraction, and integer operations to build intuition before relying on memorized rules.
When unsure, verify by checking: for equations, substitute the solution back into the original equation; for arithmetic, re‑express results in another form (e.g., use the inverse operation to check).
Memorize key properties (Identity/Zero/Commutative for both addition and multiplication) and common divisor rules to speed problem solving.
Quick Reference: Select Tables and Key Rules (LaTeX)
Identity properties:
a+0=a,\
Chapter 2: The Language of Algebra
(Overview for this chapter focuses on the symbolic language of algebra, translation between words and symbols, exponents, order of operations, and basic algebraic manipulations.)
2.1 Use the Language of Algebra
Core ideas: Algebra introduces symbolic language to represent mathematical relationships. It distinguishes between variables (letters representing numbers that can change) and constants (numbers with fixed values). We also differentiate between an expression (a combination of numbers, variables, and operations, e.g., 2x+3) and an equation (two expressions separated by an equal sign, e.g., 2x+3=7).
Exponential notation: This is used to indicate repeated multiplication. In a^n, a is the base and n is the exponent. It means the base a is multiplied by itself n times. For example, 2^3 = 2 \times 2 \times 2 = 8.
Order of Operations (PEMDAS): To ensure consistent results when simplifying expressions, we follow a specific order:
Parentheses (or any grouping symbols like brackets, braces)
Exponents
Multiplication and Division (performed from left to right as they appear)
Addition and Subtraction (performed from left to right as they appear)
Mnemonic: Please Excuse My Dear Aunt Sally
How to do/Practice Steps:
Work inside grouping symbols first.
Calculate all exponents.
Perform all multiplication and division from left to right.
Perform all addition and subtraction from left to right.
Example 1: Applying PEMDAS
Evaluate (5+3) \times 2^2 - 10 \div 5Parentheses: (8) \times 2^2 - 10 \div 5
Exponents: 8 \times 4 - 10 \div 5
Multiplication/Division (left to right): 32 - 2
Addition/Subtraction (left to right): 30
2.2 Evaluate, Simplify, and Translate Expressions
Evaluate Expressions: This means substituting a given value for a variable and then simplifying the resulting numerical expression using the order of operations.
How to do/Practice Steps:
Replace each variable with its given numerical value.
Follow the Order of Operations (PEMDAS) to simplify the expression.
Like terms and coefficients:
A term is a constant or a product of a coefficient and variables (e.g., 5x, 7, y^2).
The coefficient is the numerical factor multiplying the variable(s) (e.g., in 5x, 5 is the coefficient).
Like terms are terms that have the exact same variable factors raised to the same exponents (e.g., 3x and 7x are like terms; 2y^2 and 5y^2 are like terms; 4x and 4x^2$$ are not).
Simplify by combining like terms: We can add or subtract the coefficients of like terms while keeping the variable parts the same.
How to do/Practice Steps:
Identify all like terms in the expression.
Combine the coefficients of each set of like terms using addition or subtraction.
Write the simplified expression with each unique term.
Translate word phrases to algebraic expressions: Many common words signify mathematical operations:
Addition: