Khaemenes Academy · Open-Age Mathematics
Printable Teacher Answer Keys
Ninety-two lesson-aligned keys. Keep evaluator materials separate from learner-facing folders.
Function Language, Relations & Multiple Representations
Unit 01 · Lesson 01
Domain, Range, Intercepts & End Behavior
Unit 01 · Lesson 02
Average Rate of Change & Difference Quotients
Unit 01 · Lesson 03
Transformations and Families of Functions
Unit 01 · Lesson 04
Composition of Functions & Process Chains
Unit 01 · Lesson 05
Inverse Functions and Restricted Domains
Unit 01 · Lesson 06
Choosing, Fitting & Validating Models
Unit 01 · Lesson 07
Power Functions and Polynomial End Behavior
Unit 02 · Lesson 01
Zeros, Factors & Multiplicity
Unit 02 · Lesson 02
Polynomial Graphs from Structure
Unit 02 · Lesson 03
Polynomial Division & the Remainder Theorem
Unit 02 · Lesson 04
Rational Functions: Domain, Holes & Asymptotes
Unit 02 · Lesson 05
Rational Equations, Inequalities & Sign Analysis
Unit 02 · Lesson 06
Polynomial and Rational Modelling
Unit 02 · Lesson 07
Optimization and Technology-Assisted Validation
Unit 02 · Lesson 08
Exponential Growth, Decay & Percent Change
Unit 03 · Lesson 01
The Number e and Continuous Change
Unit 03 · Lesson 02
Logarithms as Inverses and Exponents
Unit 03 · Lesson 03
Properties of Logarithms
Unit 03 · Lesson 04
Exponential and Logarithmic Equations
Unit 03 · Lesson 05
Logarithmic Scales and Data Interpretation
Unit 03 · Lesson 06
Exponential Regression and Model Validation
Unit 03 · Lesson 07
Angles in Standard Position & Coterminal Angles
Unit 04 · Lesson 01
Degrees, Radians & Conversion
Unit 04 · Lesson 02
Arc Length, Sector Area & Angular Speed
Unit 04 · Lesson 03
Unit-Circle Coordinates and Reference Angles
Unit 04 · Lesson 04
Exact Sine and Cosine Values
Unit 04 · Lesson 05
Tangent and Reciprocal Functions on the Circle
Unit 04 · Lesson 06
Circular Motion and Periodic Position
Unit 04 · Lesson 07
Graphing Sine and Cosine from the Unit Circle
Unit 05 · Lesson 01
Amplitude, Period, Midline & Phase Shift
Unit 05 · Lesson 02
Building Sinusoidal Equations from Graphs
Unit 05 · Lesson 03
Tangent, Cotangent & Their Asymptotes
Unit 05 · Lesson 04
Secant and Cosecant Graphs
Unit 05 · Lesson 05
Inverse Trigonometric Functions & Principal Values
Unit 05 · Lesson 06
Sinusoidal Regression and Periodic Data
Unit 05 · Lesson 07
Harmonic Comparisons and Model Limitations
Unit 05 · Lesson 08
Reciprocal, Quotient & Pythagorean Identities
Unit 06 · Lesson 01
Strategies for Verifying Identities
Unit 06 · Lesson 02
Sum and Difference Formulas
Unit 06 · Lesson 03
Double-Angle and Half-Angle Formulas
Unit 06 · Lesson 04
Product-to-Sum and Sum-to-Product Connections
Unit 06 · Lesson 05
Solving Basic Trigonometric Equations
Unit 06 · Lesson 06
Solving Multi-Step Equations on Intervals
Unit 06 · Lesson 07
General Solutions, Technology & Extraneous Results
Unit 06 · Lesson 08
Right-Triangle Trigonometry Review at Precalculus Depth
Unit 07 · Lesson 01
Law of Sines and the Ambiguous Case
Unit 07 · Lesson 02
Law of Cosines
Unit 07 · Lesson 03
Area Formulas for Oblique Triangles
Unit 07 · Lesson 04
Bearings, Navigation & Triangulation
Unit 07 · Lesson 05
Periodic and Triangle Models in Context
Unit 07 · Lesson 06
Measurement Error, Precision & Model Defense
Unit 07 · Lesson 07
Polar Coordinates and Multiple Representations
Unit 08 · Lesson 01
Converting Between Polar and Cartesian Forms
Unit 08 · Lesson 02
Graphing Polar Curves and Symmetry
Unit 08 · Lesson 03
Complex Numbers in Rectangular and Polar Form
Unit 08 · Lesson 04
Multiplication, Division & Rotation in the Complex Plane
Unit 08 · Lesson 05
De Moivre's Theorem and Powers
Unit 08 · Lesson 06
Roots of Complex Numbers & Geometric Structure
Unit 08 · Lesson 07
Vectors: Magnitude, Direction & Notation
Unit 09 · Lesson 01
Vector Components and Resultants
Unit 09 · Lesson 02
Vector Operations and Linear Combinations
Unit 09 · Lesson 03
Dot Product, Angle & Orthogonality
Unit 09 · Lesson 04
Parametric Equations and Eliminating the Parameter
Unit 09 · Lesson 05
Velocity, Direction and Projectile Models
Unit 09 · Lesson 06
Vector and Parametric Design Applications
Unit 09 · Lesson 07
Sequences, Recursive Definitions & Explicit Rules
Unit 10 · Lesson 01
Arithmetic Sequences and Finite Series
Unit 10 · Lesson 02
Geometric Sequences and Series
Unit 10 · Lesson 03
Infinite Geometric Series and Convergence
Unit 10 · Lesson 04
Sigma Notation and Series Manipulation
Unit 10 · Lesson 05
Binomial Theorem and Pascal Structure
Unit 10 · Lesson 06
Discrete Modelling and Financial Applications
Unit 10 · Lesson 07
Matrices, Dimensions & Operations
Unit 11 · Lesson 01
Matrix Transformations and Composition
Unit 11 · Lesson 02
Systems of Equations and Matrix Methods
Unit 11 · Lesson 03
Parabolas in Standard and Translated Forms
Unit 11 · Lesson 04
Ellipses and Their Geometry
Unit 11 · Lesson 05
Hyperbolas and Asymptotic Structure
Unit 11 · Lesson 06
Conic Identification, Rotation & Applications
Unit 11 · Lesson 07
Average Rate of Change Revisited
Unit 12 · Lesson 01
Difference Quotients and Secant Lines
Unit 12 · Lesson 02
Intuitive Limits from Tables and Graphs
Unit 12 · Lesson 03
Limit Laws and Algebraic Evaluation
Unit 12 · Lesson 04
Continuity, Discontinuity & End Behavior
Unit 12 · Lesson 05
Instantaneous Change and Accumulation Preview
Unit 12 · Lesson 06
Defining a Researchable Modelling Question
Unit 13 · Lesson 01
Selecting Variables, Assumptions & Function Families
Unit 13 · Lesson 02
Collecting, Cleaning & Representing Data
Unit 13 · Lesson 03
Fitting, Comparing & Validating Models
Unit 13 · Lesson 04
Communicating Mathematics and Limitations
Unit 13 · Lesson 05
Oral Defense, Revision & Comprehensive Synthesis
Unit 13 · Lesson 06