Number & operations
Quantity, place value, computation, proportional reasoning, number systems, and numerical analysis.
A complete, non-accredited curriculum library · K through graduate study
Join early access ↗∑Patterns · Reasoning · Proof
From number sense to original mathematical thought.
A connected mathematics sequence that builds meaning before shortcuts, fluency without fragility, and the ability to model, explain, prove, and apply ideas from early arithmetic through graduate study.
∑
FIELD OF STUDY
Mathematics
01
K–2Foundations
5 sample Trees02
3–5Elementary
5 sample Trees03
6–8Middle Years
6 sample Trees04
9–12High School
7 sample Trees05
UndergraduateCollege
7 sample Trees06
GraduateAdvanced
7 sample Trees✦
One connected field. Placement follows demonstrated readiness by branch—not age alone.
The field at a glance
Not a scattered list.
A connected discipline.
Every Knowledge Tree sits inside a larger map of concepts, practices, vocabulary, prerequisites, and meaningful applications.
Algebra & functions
Patterns, expressions, equations, functions, structures, abstract algebra, and functional analysis.
Geometry & measurement
Spatial reasoning, proof, transformation, trigonometry, topology, and differential geometry.
Data & probability
Data literacy, statistics, probability, inference, stochastic processes, and experimental design.
Discrete mathematics
Logic, combinatorics, graph theory, algorithms, optimization, and theoretical computer science.
Modeling & proof
Representing real systems, making assumptions visible, constructing arguments, and communicating results.
K through graduate study
Expectations deepen.
The roots stay connected.
These are representative Knowledge Trees, not a fixed or final list. Families can build a focused path, a complete sequence, or advanced independent study.
01
Foundations
K–2
See, touch, compare, and explain quantity.
- Early Number SenseKnowledge Tree
- Addition & SubtractionKnowledge Tree
- Shapes & Spatial ThinkingKnowledge Tree
- Measurement Through PlayKnowledge Tree
- Patterns & Mathematical TalkKnowledge Tree
02
Elementary
3–5
Build flexible strategies and durable fluency.
- Whole-Number OperationsKnowledge Tree
- Fractions & DecimalsKnowledge Tree
- Measurement & DataKnowledge Tree
- Geometry FoundationsKnowledge Tree
- Problem Solving StudioKnowledge Tree
03
Middle Years
6–8
Connect arithmetic to proportional and algebraic reasoning.
- Math 6Knowledge Tree
- Pre-AlgebraKnowledge Tree
- Algebra IKnowledge Tree
- Geometry FoundationsKnowledge Tree
- Statistics & ProbabilityKnowledge Tree
- Competition Math LabKnowledge Tree
04
High School
9–12
Reason confidently through abstraction and application.
- Algebra I & IIKnowledge Tree
- Geometry with ProofKnowledge Tree
- TrigonometryKnowledge Tree
- PrecalculusKnowledge Tree
- Calculus IKnowledge Tree
- StatisticsKnowledge Tree
- Consumer MathematicsKnowledge Tree
05
College
Undergraduate
Develop disciplinary breadth and technical maturity.
- Calculus I–IIIKnowledge Tree
- Linear AlgebraKnowledge Tree
- Discrete MathematicsKnowledge Tree
- Differential EquationsKnowledge Tree
- Probability & StatisticsKnowledge Tree
- Real AnalysisKnowledge Tree
- Abstract AlgebraKnowledge Tree
06
Advanced
Graduate
Study advanced theory, research methods, and specialized applications.
- Measure TheoryKnowledge Tree
- Complex AnalysisKnowledge Tree
- TopologyKnowledge Tree
- Numerical AnalysisKnowledge Tree
- Mathematical ModelingKnowledge Tree
- Advanced StatisticsKnowledge Tree
- Graduate Research SeminarKnowledge Tree
Active microlearning
Study becomes something
the learner actually does.
Most digital lessons take 5–15 minutes and develop one clear objective through listening, reading, watching, trying, moving, making, practicing, feedback, and recall. Narrated instruction, adjustable playback, and transcripts sit alongside visual, read/write, interactive, kinesthetic, and hands-on elements. Parents can prefer or disable optional modes for each learner while sustained reading, labs, projects, writing, discussion, and performance remain wherever the objective calls for deeper work.
Interactive models
Manipulatives, graphs, simulations, constructions, and dynamic representations make abstract ideas inspectable.
Reasoning conferences
Learners explain a method, compare strategies, and defend why a result is reasonable.
Applied investigations
Model budgets, motion, populations, design constraints, risk, data, and other authentic systems.
Proof & research
Advanced learners move from guided arguments to formal proof, mathematical writing, and independent inquiry.
Evidence of learning
Mastery is more than
a completed screen.
✓Concept mastery by prerequisite✓Written and oral reasoning✓Problem sets and cumulative checks✓Modeling investigations✓Proof portfolios✓Projects and advanced research
Transparent worldview
Choose the framing.
See it before assigning.
CHRISTIAN PATHWAY
Highlights order, intelligibility, stewardship, intellectual humility, and the responsible use of quantitative claims without replacing mathematical proof with devotional assertions.
SECULAR / ACADEMIC PATHWAY
Teaches the same mathematical competencies and historical context without confessional framing or devotional integration.
Knowledge Tree pages identify perspective, sources, debated questions, prerequisites, and objectives. Parents retain final control.
Inside every Knowledge Tree
Complete enough to teach.
Clear enough to trust.
Published scope & sequence
Objectives, prerequisites, vocabulary, lesson order, worldview, materials, estimated time, and mastery expectations.
Multimodal micro-lessons
One focused concept through coordinated visual, auditory, read/write, interactive, movement, and hands-on elements—with feedback and retrieval.
Assessment & grading
Checks, quizzes, tests, projects, and writing—with automatic rubric grading, evidence-linked feedback, and parent review.
Adaptive support
Readiness checks trace weak roots, assign proportionate repair, verify retention, and return learners to the goal.
Parent controls
Adjust pace, tutors, worldview, due dates, interventions, lesson modes, and whether suggested writing grades post automatically or require approval.
Permanent records
Lesson history, grades, attempts, feedback, mastery evidence, hours, portfolio artifacts, and unofficial transcripts.