Lesson Directory
Foundational Structures introduces the language, objects, laws, representations, and reusable tools needed throughout Malone Math. Lessons are grouped below by section.
Sections
1.1 Mathematical Language and Objects
- What Is Mathematics?
- Mathematical Objects
- Definitions, Conventions, and Classification
- Symbols and Notation
- Variables, Constants, Parameters, and Coefficients
- Terms, Expressions, Equations, Formulas, and Identities
- Statements and Open Sentences
- Equality, Equivalence, and Identity
- Examples, Nonexamples, and Counterexamples
- Finite, Infinite, Concrete, and Abstract Objects
1.2 Logic and Mathematical Reasoning
- Truth, Propositions, and Statements
- Logical Connectives
- Conditional Statements
- Converse, Inverse, and Contrapositive
- Necessary and Sufficient Conditions
- Predicates and Quantifiers
- Definitions, Axioms, Postulates, and Theorems
- Deduction, Induction, and Conjecture
- Counterexamples and Contradiction
- Informal Proof and Direct Verification
- Valid and Invalid Reasoning
1.3 Numbers and Quantities
- Number Systems Overview
- Counting, Natural, and Whole Numbers
- Integers and Signed Quantities
- Fractions, Decimals, Ratios, and Percentages
- Rational, Irrational, and Real Numbers
- Imaginary and Complex Numbers
- Number-System Inclusion, Closure, and Extension
- Place Value and Magnitude
- Exact, Approximate, and Measured Values
- Units, Dimensions, and Scalars
1.4 Operations, Properties, and Laws
- Operations: The Six Operations
- Addition
- Subtraction
- Multiplication
- Division
- Exponentiation
- Radicalization
- Opposites, Inverses, and Reciprocals
- Identity, Zero, and Cancellation
- Substitution, Evaluation, and Composition
- PEMDAS and the Order of Operations
- Commutative, Associative, and Distributive Properties
- Equality and Inequality Properties
- Exponent, Radical, Sign, and Equivalence Laws
1.5 Order, Position, and the Number Line
- Anatomy of the Number Line
- Position, Direction, and Origin
- Ordering and Comparing Numbers
- Distance and Absolute Value
- Intervals, Endpoints, and Rays
- Bounds and Interval Notation
- Addition and Subtraction on the Number Line
- Multiplication and Division on the Number Line
- Translation, Scaling, and Reflection
- Density, Discreteness, Midpoints, and Approximation
- Floor and Ceiling Functions
1.6 Sets, Relations, and Functions
- Set Language and Membership
- Subsets, Equality, and Power Sets
- Union, Intersection, Difference, and Complement
- Finite and Infinite Sets and Cardinality
- Cartesian Products
- Venn Diagrams
- Ordered Pairs and Relations
- Relation Properties and Equivalence Relations
- Domain and Range
- Functions as Mappings
- Inputs, Outputs, and Function Notation
- Injective, Surjective, and Bijective Functions
- Identity, Inverse, and Composite Functions
- Representations of Functions
1.7 Representations and Mathematical Spaces
- Multiple Representations
- Natural Language, Numbers, and Symbols
- Tables, Lists, and Sequences
- Diagrams, Number Lines, and Coordinate Graphs
- Algorithms and Data Displays
- Coordinates, Axes, Origins, and Scales
- Cartesian Coordinates and the Real Plane
- The Complex Plane
- Polar and Higher-Dimensional Representation
- Tuples, Vectors, Matrices, and Arrays
- Networks, Graphs, and Trees
- Translating Among Representations
- Choosing an Efficient Representation
1.8 Measurement, Scale, and Precision
- Measurable Quantities and Units
- Unit Systems and Unit Conversion
- Dimensional Consistency
- Rates and Derived Units
- Scale and Reference Quantities
- Exact and Measured Quantities
- Rounding and Significant Figures
- Precision and Accuracy
- Absolute, Relative, and Percentage Error
- Tolerance and Measurement Bounds
- Scientific Notation and Orders of Magnitude
1.9 Algorithms and Computational Thinking
- Procedures and Algorithms
- Problems, Inputs, Processes, and Outputs
- Decision Rules and Branching
- Repetition and Iteration
- Introductory Recursion
- Tables and Pseudocode
- Calculator and Spreadsheet Methods
- Symbolic and Numerical Computation
- Exact and Approximate Methods
- Termination and Stopping Conditions
- Introductory Efficiency
- Testing, Debugging, and Error Detection
- Communicating Algorithms Clearly
1.10 Problem Solving and Verification
- The Malone Math Problem-Solving Framework
- Known and Unknown Information
- Objects, Conditions, Restrictions, and Units
- Selecting a Representation
- Choosing an Operation, Law, Model, or Algorithm
- Carrying Out a Solution
- Estimation and Reasonableness
- Substitution and Inverse-Operation Checks
- Boundary Tests, Special Cases, and Counterexamples
- Dimensional and Graphical Checks
- Independent Solution Methods
- Interpreting and Communicating Results