Math as a living lab.
A complete interactive path from counting to undergraduate modeling: every concept becomes a visible action, a consequence, an explanation, a revision, and a portable Learning Evidence Record.
Replacement standard
Not worksheets. Not videos. Mathematical action.
Teach the idea visually before controls appear.
Let students act on a model and see the consequence.
Require explanation, revision, defense, and transfer.
Cluster misconceptions for TeacherOS intervention.
Generate TeachProof practice for the teacher move.
Show parents and districts evidence of demonstrated capability.
Daily path engine
Assign once. The next mission adapts.
Mastery is strong enough for adjacent transfer into Ratios And Proportional Reasoning.
unit rate: compare unit rates
Use unit rate to make, test, and defend a mathematical claim.
TeacherOSMiddle School Math LabPath: assign unit rate: compare unit rates. Watch unit-rate-concept-clarity, ratios-and-proportional-reasoning-misconception-cluster, revision-quality.TeachProofAsk the student what unit rate means before solving.ParentYour student connected unit rate across a model, graph, words, and evidence.
Open adaptive next missionCourse spine
Ten paths. Sixty units. One evidence loop.
K-2 Number LabPath
Young learners build number sense through concrete objects, stories, and one-action-at-a-time visual math.
Grades 3-5 Math LabPath
Students move from whole-number fluency to fractions, decimals, geometry, and data with visible models and evidence.
Middle School Math LabPath
Students build the bridge from arithmetic to algebra through ratios, proportionality, expressions, equations, functions, geometry, and data.
Algebra I LabPath
Students learn algebra as relationships: variables, equivalence, functions, systems, quadratics, and modeling through evidence-rich missions.
Geometry LabPath
Geometry becomes construction, invariant testing, proof dependency, measurement, and spatial reasoning.
Algebra II LabPath
Students connect families of functions, transformations, algebraic structure, and modeling decisions.
Precalculus LabPath
Precalculus becomes a bridge from function behavior to calculus through modeling, trigonometry, vectors, and limits.
Calculus LabPath
Calculus is taught as rate, accumulation, approximation, optimization, and differential modeling before symbolic fluency is faded in.
Statistics LabPath
Statistics becomes data design, uncertainty, inference, simulation, and honest claims instead of formula selection.
Undergraduate Math Foundations LabPath
Early undergraduate mathematics becomes proof, linear algebra, differential equations, discrete structures, optimization, and modeling through industrial-grade simulations.
K-2
K-2 Number LabPath
Counting And Cardinality
How do we know how many?
Student counts, groups, compares, and explains quantity with objects and words.Base Ten Foundations
How do tens and ones make numbers easier to understand?
Student composes and decomposes two-digit numbers with tens and ones.Addition And Subtraction Stories
What changes when things join, separate, or compare?
Student represents a story with objects, equation, and explanation.Shapes And Space
How can we describe and build shapes?
Student identifies attributes, composes shapes, and explains spatial relationships.Measurement Beginnings
How do units help us compare length, time, and data?
Student measures with consistent units and explains why units matter.Fluency With Meaning
How do strategies make math faster without hiding meaning?
Student chooses a strategy, shows it, and explains why it works.3-5
Grades 3-5 Math LabPath
Multiplication And Division Structure
How do equal groups and arrays explain operations?
Student models multiplication and division as groups, arrays, area, and equations.Fractions As Numbers
How can fractions be numbers, measures, and operators?
Student locates, compares, and operates on fractions with visual justification.unit fraction: partition wholeUse unit fraction to make, test, and defend a mathematical claim.equivalent fractions: resize unitUse equivalent fractions to make, test, and defend a mathematical claim.fraction comparison: place on number lineUse fraction comparison to make, test, and defend a mathematical claim.
Decimals And Place Value
How do place-value units extend beyond whole numbers?
Student connects decimals to fractions, base-ten models, and magnitude.tenths and hundredths: zoom place valueUse tenths and hundredths to make, test, and defend a mathematical claim.decimal comparison: trade unitsUse decimal comparison to make, test, and defend a mathematical claim.decimal operations: align quantitiesUse decimal operations to make, test, and defend a mathematical claim.
Geometry And Measurement
How do shape attributes and units explain space?
Student measures area, volume, angle, and attributes with justified units.Data And Early Statistics
How do displays help us reason about variation?
Student creates displays, compares distributions, and explains variability.Multi-Step Problem Solving
How do we plan when a problem has more than one step?
Student represents, solves, checks, and explains a multi-step situation.problem representation: draw modelUse problem representation to make, test, and defend a mathematical claim.operation choice: choose operationsUse operation choice to make, test, and defend a mathematical claim.reasonableness: test answer sizeUse reasonableness to make, test, and defend a mathematical claim.
6-8
Middle School Math LabPath
Ratios And Proportional Reasoning
How do two quantities scale together?
Student identifies proportional relationships from tables, graphs, equations, and contexts.Rational Numbers
How do positive and negative numbers describe direction and change?
Student operates with rational numbers using number-line and context evidence.signed numbers: walk number lineUse signed numbers to make, test, and defend a mathematical claim.absolute value: reflect distanceUse absolute value to make, test, and defend a mathematical claim.rational operations: combine signed changesUse rational operations to make, test, and defend a mathematical claim.
Expressions And Equations
How do symbols preserve relationships?
Student writes, transforms, and solves expressions/equations with legal moves.expression structure: combine termsUse expression structure to make, test, and defend a mathematical claim.one-step equations: balance sidesUse one-step equations to make, test, and defend a mathematical claim.multi-step equations: undo operationsUse multi-step equations to make, test, and defend a mathematical claim.
Functions And Linear Relationships
How does one quantity determine another?
Student connects functions across rule, table, graph, and story.Geometry And Transformations
What stays the same when figures move or scale?
Student proves congruence, similarity, and measurement claims through transformations.Statistics And Probability
How do data and chance support careful claims?
Student uses samples, distributions, and simulations to make qualified claims.sampling: sample populationUse sampling to make, test, and defend a mathematical claim.center and spread: compare distributionsUse center and spread to make, test, and defend a mathematical claim.probability simulation: run chance modelUse probability simulation to make, test, and defend a mathematical claim.
6-8 / 9-12
Algebra I LabPath
Variables And Patterns
How does a changing situation become a rule?
Student translates a relationship across words, table, graph, and equation.variable meaning: choose variableUse variable meaning to make, test, and defend a mathematical claim.pattern rule: build ruleUse pattern rule to make, test, and defend a mathematical claim.representation transfer: sync representationsUse representation transfer to make, test, and defend a mathematical claim.
Expressions And Equivalence
When are two forms really the same?
Student proves equivalence by structure and substitution.Equations And Inequalities
What moves preserve a solution set?
Student solves, checks, graphs, and explains legal moves.balance equations: balance sidesUse balance equations to make, test, and defend a mathematical claim.multi-step equations: undo operationsUse multi-step equations to make, test, and defend a mathematical claim.inequality boundary: test boundaryUse inequality boundary to make, test, and defend a mathematical claim.
Linear Functions
How do rate and starting value shape a relationship?
Student interprets slope, intercept, and domain from context.Systems And Quadratics
How do multiple relationships or changing rates create decisions?
Student solves systems and quadratics from graphs, equations, and contexts.Algebraic Modeling
How does algebra make claims about the real world?
Student builds, tests, revises, and defends a model with limits.9-12
Geometry LabPath
Foundations Of Proof
What makes a mathematical argument convincing?
Student builds a proof chain from definitions, conjectures, and counterexamples.Transformations And Congruence
What stays invariant under motion?
Student proves congruence through rigid transformations.Similarity And Trigonometry
How does scale preserve shape and create ratios?
Student uses similarity to justify trigonometric ratios and indirect measurement.Circles And Coordinate Geometry
How do algebra and geometry describe the same objects?
Student connects geometric relationships to equations and coordinates.circle theorems: move point on circleUse circle theorems to make, test, and defend a mathematical claim.distance formula: measure distanceUse distance formula to make, test, and defend a mathematical claim.coordinate proof: trace coordinate proofUse coordinate proof to make, test, and defend a mathematical claim.
Area, Volume, And Modeling
How do measurements change when dimensions change?
Student models area, volume, density, and optimization with units.Capstone Design Proof
How can geometry justify a design decision?
Student creates a design, tests constraints, proves a property, and defends limitations.design constraints: set constraintsUse design constraints to make, test, and defend a mathematical claim.proof choice: choose theoremUse proof choice to make, test, and defend a mathematical claim.spatial optimization: stress-test designUse spatial optimization to make, test, and defend a mathematical claim.
9-12
Algebra II LabPath
Function Families
How do different functions behave and transform?
Student compares function families from features, transformations, and contexts.parent functions: shift graphUse parent functions to make, test, and defend a mathematical claim.transformations: stretch graphUse transformations to make, test, and defend a mathematical claim.feature comparison: compare featuresUse feature comparison to make, test, and defend a mathematical claim.
Quadratics And Polynomials
How do factors, zeros, and end behavior reveal structure?
Student connects symbolic, graphical, and contextual polynomial evidence.Exponentials And Logarithms
How do repeated growth and inverse reasoning work?
Student models exponential change and uses logs as inverse evidence.Rational And Radical Functions
How do restrictions and inverse operations shape functions?
Student explains domain restrictions, asymptotes, and radical constraints.domain restrictions: move denominator zeroUse domain restrictions to make, test, and defend a mathematical claim.asymptotes: trace asymptoteUse asymptotes to make, test, and defend a mathematical claim.radical equations: test extraneous solutionUse radical equations to make, test, and defend a mathematical claim.
Sequences, Series, And Complex Numbers
How do patterns extend beyond real-number intuition?
Student reasons with recursive/explicit forms, series, and complex operations.Algebra II Modeling
How do we choose the right function for evidence?
Student fits, compares, critiques, and defends a model family.model selection: fit familiesUse model selection to make, test, and defend a mathematical claim.residuals: inspect residualsUse residuals to make, test, and defend a mathematical claim.parameter interpretation: defend parameterUse parameter interpretation to make, test, and defend a mathematical claim.
9-12 / undergrad
Precalculus LabPath
Advanced Function Analysis
How do features predict behavior?
Student analyzes functions from domain, range, rate, asymptotes, and composition.composition: compose machinesUse composition to make, test, and defend a mathematical claim.inverse functions: reverse mappingUse inverse functions to make, test, and defend a mathematical claim.asymptotic behavior: move asymptoteUse asymptotic behavior to make, test, and defend a mathematical claim.
Trigonometric Functions
How do circular motion and waves create trig functions?
Student connects unit circle, graph, identities, and applications.Vectors And Parametric Motion
How do components describe motion and force?
Student represents and analyzes motion with vectors and parametric equations.vector components: drag vectorUse vector components to make, test, and defend a mathematical claim.parametric path: animate parameterUse parametric path to make, test, and defend a mathematical claim.relative motion: combine velocitiesUse relative motion to make, test, and defend a mathematical claim.
Polar And Complex Models
When is a new coordinate system more useful?
Student chooses polar/complex representations and explains the advantage.polar coordinates: sweep angleUse polar coordinates to make, test, and defend a mathematical claim.complex rotation: multiply complex pointUse complex rotation to make, test, and defend a mathematical claim.coordinate choice: compare gridsUse coordinate choice to make, test, and defend a mathematical claim.
Limits And Continuity Preview
What does a function approach?
Student reasons about limits numerically, graphically, and verbally.Precalculus Modeling Capstone
How do functions model real systems before calculus?
Student builds a multi-function model and defends assumptions and constraints.piecewise model: choose function piecesUse piecewise model to make, test, and defend a mathematical claim.periodic model: fit cycleUse periodic model to make, test, and defend a mathematical claim.constraint analysis: stress constraintsUse constraint analysis to make, test, and defend a mathematical claim.
9-12 / undergrad
Calculus LabPath
Limits And Derivatives
How does average change become instantaneous change?
Student estimates, visualizes, computes, and defends derivative meaning.Derivative Applications
How do rates explain motion, shape, and decisions?
Student applies derivatives to motion, optimization, and related rates.Integrals And Accumulation
How do small pieces add up to a whole?
Student connects Riemann sums, area, accumulation, and antiderivatives.Fundamental Theorem
How are rate and accumulation inverse ideas?
Student explains and applies the connection between derivative and integral.Series And Approximation
How can functions be approximated by simpler pieces?
Student builds, tests, and bounds approximations.Differential Modeling
How do equations describe changing systems?
Student models, solves, simulates, and critiques differential systems.differential equation: draw slope fieldUse differential equation to make, test, and defend a mathematical claim.slope field: simulate trajectoryUse slope field to make, test, and defend a mathematical claim.model limitation: change parameterUse model limitation to make, test, and defend a mathematical claim.
9-12 / undergrad
Statistics LabPath
Data And Distributions
How do displays reveal and hide structure?
Student summarizes distributions with center, spread, shape, and context.Study Design
How does data collection shape what claims are allowed?
Student identifies bias, confounding, sampling method, and experimental design.Probability And Simulation
How can chance be modeled and tested?
Student simulates probability and compares expected with observed variation.Inference For Proportions
How do samples support population claims?
Student builds intervals/tests and explains uncertainty in context.sampling distribution: resampleUse sampling distribution to make, test, and defend a mathematical claim.confidence interval: set confidenceUse confidence interval to make, test, and defend a mathematical claim.hypothesis test: test nullUse hypothesis test to make, test, and defend a mathematical claim.
Inference For Means And Regression
How do models connect variables with uncertainty?
Student analyzes means, slopes, residuals, and conditions.Statistical Communication
How do we make claims that are useful and honest?
Student writes a claim with evidence, uncertainty, limitations, and transfer.undergrad
Undergraduate Math Foundations LabPath
Proof And Mathematical Structures
How do definitions, examples, and logic create certainty?
Student proves, disproves, and repairs arguments across structures.Linear Algebra
How do vectors and transformations organize systems?
Student connects matrices, spaces, eigenvectors, and applications.Differential Equations
How do rates define systems over time?
Student models, simulates, analyzes, and critiques dynamic systems.Discrete Mathematics And Algorithms
How do finite structures support computation and proof?
Student reasons with graphs, counting, recurrence, and algorithm traces.Optimization And Numerical Methods
How do constraints and approximation guide decisions?
Student optimizes under constraints and explains error and sensitivity.Mathematical Modeling Capstone
How does mathematics become accountable evidence in the real world?
Student chooses tools, builds a model, validates evidence, and states transfer limits.