Lesson Directory
Algebraic Relationships develops symbolic, numerical, graphical, and algorithmic methods for representing and solving mathematical relationships. Lessons are grouped below by section.
Sections
2.1 Algebraic Language and Translation
- Algebraic Language and Notation
- Variables, Constants, Parameters, and Coefficients
- Terms, Factors, Bases, and Exponents
- Expressions, Equations, Inequalities, Formulas, and Identities
- Known, Unknown, Dependent, and Independent Quantities
- Translating Verbal Phrases into Symbols
- Translating Symbols into Natural Language
- Translating Tables, Diagrams, and Patterns
- Choosing Variables and Defining Parameters
- Conditions, Restrictions, Assumptions, and Units
- Constructing and Interpreting Algebraic Relationships
2.2 Equivalence and Symbolic Transformation
- Equality, Equivalence, and Replacement
- Properties of Equality
- Combining Like Terms
- Distribution, Collection, and Rearrangement
- Expansion and Factoring
- Rearranging Formulas
- Common Denominators and Rationalization
- Rewriting Powers and Radicals
- Completing the Square
- Algebraic and Conditional Identities
- Domain Restrictions and Extraneous Solutions
- Reversible Steps and Solution Preservation
2.3 Equations, Inequalities, and Solution Sets
- One-Step and Multistep Equations
- Equations with Variables on Both Sides
- Literal Equations and Formula Rearrangement
- Absolute-Value Equations
- Polynomial, Rational, and Radical Equations
- Exponential and Logarithmic Equations
- Equations with Parameters and Multiple Variables
- Linear Inequalities
- Compound and Absolute-Value Inequalities
- Polynomial and Rational Inequalities
- Sign Analysis and Boundary Points
- Interval, Set-Builder, and Graphical Notation
- Feasible Regions
- Solution Counts, Identities, and Conditional Equations
- Verifying Candidate Solutions
2.4 Ratios, Rates, Proportions, and Variation
- Ratios and Proportions
- Rates and Unit Rates
- The Proportion Principle
- Understanding Proportions
- Solving Proportions
- Constants of Proportionality
- Direct Proportion
- Inverse Proportion
- Joint and Combined Variation
- Percentages and Percent Change
- Scale and Similarity
- Dimensional Analysis and Unit Conversion
2.5 Functions and Dependency
- Functions as Relationships
- Domain, Codomain, and Range
- Function Notation and Evaluation
- Tables, Graphs, Formulas, and Algorithms
- Independent Variables, Dependent Variables, and Parameters
- Zeros, Intercepts, and Fixed Points
- Increasing, Decreasing, and Constant Behavior
- Maximum and Minimum Values
- Injective, Surjective, and Bijective Functions
- Function Composition
- Inverse Functions and Restricted Domains
- Piecewise and Implicitly Defined Functions
- Parametric Representations
- Transformations and Function Families
2.6 Linear and Affine Relationships
- Constant, Linear, and Affine Functions
- Slope as Rate of Change
- Intercepts
- Slope-Intercept Form
- Point-Slope and Two-Point Forms
- Standard Form
- Horizontal, Vertical, Parallel, and Perpendicular Lines
- Graphing Lines
- Constructing Lines from Data and Conditions
- Linear Equations and Inequalities
- Interpolation, Extrapolation, and Trends
- Piecewise-Linear and Absolute-Value Graphs
- Linear Modeling
2.7 Systems and Matrix Methods
- Systems of Equations
- Systems of Inequalities
- Classifying Systems and Solutions
- Graphical Solution by Intersection
- Substitution, Comparison, and Elimination
- Nonlinear Systems and Parameters
- Matrices as Structured Representations
- Augmented Matrices and Row Operations
- Row-Echelon and Reduced Row-Echelon Form
- Gaussian and Gauss–Jordan Elimination
- Matrix Arithmetic and Multiplication
- Identity, Inverse, and Determinant Methods
- Feasible Regions and Constraint Systems
2.8 Polynomial and Power Relationships
- Polynomial Language and Standard Form
- Polynomial Addition, Subtraction, and Multiplication
- Distribution and Special Products
- Common Factors and Factoring by Grouping
- Factoring Trinomials and Special Forms
- Polynomial and Synthetic Division
- Remainder and Factor Theorems
- Power Functions
- Quadratic Functions and Equations
- Completing the Square and the Quadratic Formula
- Discriminants and Complex Roots
- Higher-Degree Polynomial Equations
- Zeros, Roots, Factors, and Multiplicity
- End Behavior and Turning Points
- Polynomial Identities and Interpolation
2.9 Rational and Radical Relationships
- Rational Expressions and Domain Restrictions
- Simplifying and Canceling Rational Expressions
- Operations with Rational Expressions
- Complex Fractions
- Rational Equations and Inequalities
- Reciprocal Relationships and Inverse Variation
- Zeros, Holes, and Asymptotes
- Roots, Radicals, and Radicalization
- Rational Exponents
- Simplifying and Operating with Radicals
- Conjugates and Rationalizing Denominators
- Radical Equations and Extraneous Solutions
2.10 Exponential and Logarithmic Relationships
- Exponential Functions
- Initial Values, Growth Factors, and Repeated Multiplication
- Exponential Growth and Decay
- Compound Interest and Continuous Compounding
- Doubling Time and Half-Life
- Exponent Laws
- Logarithms as Inverses
- Common and Natural Logarithms
- Logarithm Laws and Changing Bases
- Solving Exponential and Logarithmic Equations
- Linear versus Exponential Change
- Exponential, Logarithmic, and Logistic Models
2.11 Sequences, Recurrences, and Algebraic Patterns
- Sequences, Terms, and Indices
- Explicit and Recursive Definitions
- Arithmetic Sequences
- Geometric Sequences
- Polynomial Sequences and Finite Differences
- Fibonacci-Type Recurrences
- Sequence Tables and Discrete Graphs
- Sigma and Product Notation
- Finite Series
- Arithmetic and Geometric Sums
- Iteration and Recurrence Algorithms
- Discrete Functions, Patterns, and Introductory Convergence
2.12 Graphical Methods
- Coordinate Systems, Scale, and Plotting
- Graphs of Equations, Inequalities, Functions, and Relations
- Intercepts, Zeros, and Intersections
- Domain and Range from Graphs
- Increasing, Decreasing, and Extreme Behavior
- Symmetry and Function Families
- Translations, Reflections, Stretches, and Compressions
- Piecewise Graphs and Discontinuities
- Asymptotes and End Behavior
- Parametric, Polar, and Discrete Graphs
- Graphical Estimation and Solution Methods
- Graphing Technology, Scale, and Distortion
2.13 Numerical and Algorithmic Methods
- Exact and Approximate Solutions
- Numerical Tables and Successive Approximation
- Iteration, Recursion, and Search Intervals
- Bracketing and Bisection
- Fixed-Point Iteration
- Newton's Method
- Secant Methods
- Numerical Solutions of Equations and Systems
- Polynomial Evaluation and Synthetic Division Algorithms
- Calculator, Spreadsheet, and Computer-Algebra Methods
- Pseudocode, Branches, Loops, and Stopping Conditions
- Convergence, Error, Tolerance, and Verification
2.14 Introductory Analysis of Curves and Change
- Constant, Changing, and Average Rates
- Secant Lines, Tangent Lines, and Local Rate
- Difference Quotients and Finite Differences
- Derivatives of Simple Polynomials
- Slope Functions and Critical Points
- Increasing, Decreasing, Extrema, and Concavity
- Introductory Optimization
- Area and Accumulated Change
- Rectangular Approximation and Numerical Area
- Simple Antiderivatives and the Rate–Accumulation Connection
2.15 Generalized Algebraic Structures
- Sets with Operations and Binary Operations
- Closure, Associativity, and Commutativity
- Identity and Inverse Elements
- Operation Tables
- Semigroups, Monoids, and Groups
- Subgroups and Symmetry Groups
- Rings, Integral Domains, and Fields
- Modular Arithmetic and Boolean Algebra
- Scalars, Vectors, and Vector Spaces
- Linear Combinations, Span, and Independence
- Bases and Dimension
- Linear Transformations
- Homomorphisms, Isomorphisms, and Invariants
2.16 Applied Algebraic Problem Families
- Ratio, Proportion, and Percent Problems
- Scale, Conversion, and Variation Problems
- Distance, Rate, and Time
- Relative and Opposing Motion
- Work, Rate, and Time
- Single, Combined, and Opposing Work Rates
- Staggered, Interrupted, and Unknown Work Rates
- Mixture Problems
- Concentration, Dilution, and Replacement
- Alloy, Composition, Price, and Value Mixtures
- Cost, Revenue, Profit, and Break-Even
- Simple and Compound Interest
- Growth, Depreciation, and Population Models
- Supply, Demand, and Market Equilibrium
- Taxes, Discounts, Commissions, and Markups
- Scientific, Engineering, and Scaling Models
- Network, Flow, and Constraint Systems
- The Algebraic Modeling Process