LabPathDepth expansion

Mathematics LabPath

A full mathematics path where students learn by manipulating quantities, structures, graphs, proofs, and models until reasoning is visible.

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10/10depth gate

8course units

24mission labs

8domain engines

What must be true

The course must teach, simulate, assess, and adapt.

Start with the concrete meaning before notation.
Predict what a manipulation will preserve or change.
Act on a mathematical object and see the invariant, graph, proof, or data consequence.
Explain why the move was legal or why the model changed.
Transfer the structure to a new representation or context.

Mode-native engines

Each domain gets its own controls, models, and evidence.

Number Sense Manipulative Lab

Make place value, fractions, ratios, and operations tactile.

Controls

compose unitssplit wholesdrag ratio blocksregroupcompare magnitude

rational-number and base-ten invariant checks

Algebra Transformation Engine

Show equations as balanced systems with legal transformations.

Controls

move termscale both sidesfactorexpandtest solution

symbolic equivalence, inverse operations, domain checks

Function Graph Dynamics Lab

Connect parameters, graphs, tables, stories, and rates.

Controls

drag parametermove pointanimate inputcompare functionstrace rate

function evaluation, transformations, finite differences, asymptote/intercept checks

Geometry Proof Studio

Make conjecture, construction, invariants, and proof dependencies visible.

Controls

drag vertexconstruct linemark congruenttest invariantbuild proof chain

constraint geometry, congruence/similarity invariants, proof graph validation

Statistics Simulation Lab

Turn sampling, variation, bias, inference, and causation into repeated simulations.

Controls

sample populationchange biasbootstraprandomizeinspect interval

sampling distributions, bootstrap/randomization approximation, correlation checks

Calculus Rate And Area Engine

Make derivative, integral, accumulation, and differential models visible.

Controls

move tangentshrink intervalaccumulate areachange rate lawsolve flow

finite-difference derivative, Riemann accumulation, Euler approximation

Discrete Network Lab

Represent counting, logic, graph theory, algorithms, and finite systems.

Controls

connect nodescount pathstoggle conditionrun algorithmfind counterexample

graph traversal, combinatorics counting, truth-table checks, recurrence simulation

Modeling And Optimization Lab

Turn messy situations into variables, constraints, objectives, and defended decisions.

Controls

choose variablesset constraintchange objectiverun scenariocompare tradeoff

linear/nonlinear objective scoring, sensitivity and constraint checks

Try the shape

Representative labs that show the new depth standard.

base-ten construction mat

Place Value Regrouping Lab

How does changing units change the way a number is written?

Open mission

equation balance scale

Equals Sign Balance Lab

What makes an equation stay true?

Open mission

input-output simulator

Function Machine And Domain Lab

What makes a relationship a function?

Open mission

dynamic geometry board

Invariant Construction Lab

What stays true when the figure moves?

Open mission

population sampling lab

Sampling Variability Simulator

Why do samples differ even when the method is fair?

Open mission

tangent microscope

Derivative Zoom Lab

How does average rate become instantaneous rate?

Open mission

combinatorics construction board

Counting Without Double Counting Lab

How do you count all possibilities exactly once?

Open mission

model design studio

Model Variables And Assumptions Lab

What should the model include and what should it ignore?

Open mission

Full course map

Eight units. Twenty-four labs. Evidence every week.

Unit 1

Number, Quantity, And Units

What does a number mean in a structure or situation?

Place Value Regrouping Labplace value, regrouping, and base-ten structure Fraction Magnitude Explorerfractions as numbers and equivalent units Ratio Table Missionratios, rates, and proportional scaling

Mastery gate: Represent, compare, and operate on quantities while preserving unit meaning.

Unit 2

Expressions And Equations

Which transformations preserve meaning?

Equals Sign Balance Labequals sign as relationship Linear Equation Transformation Lablinear equations and inverse operations Factoring Structure Labfactoring, expansion, and expression equivalence

Mastery gate: Solve and transform expressions/equations while defending equivalence and domain.

Unit 3

Functions And Graphs

How does a relationship behave across representations?

Function Machine And Domain Labfunction, domain, range, and mapping Transformation Playgroundfunction transformations and parameters Rate Of Change Motion Graph Labslope, average rate, and changing rate

Mastery gate: Analyze functions through graphs, tables, parameters, and context.

Unit 4

Geometry, Measurement, And Proof

What remains true when a figure changes?

Invariant Construction Labconstructions and invariants Congruence And Similarity Courtcongruence, similarity, and transformations Proof Chain Builderdeductive proof and dependency

Mastery gate: Build, test, and defend geometric claims with invariants and proof chains.

Unit 5

Statistics And Probability

How do data and chance support or weaken claims?

Sampling Variability Simulatorsampling variation and representativeness Confidence Interval Builderconfidence intervals and uncertainty Correlation And Causation Courtassociation, confounding, experiment, and causation

Mastery gate: Use simulation, sampling, and inference evidence to defend claims with uncertainty.

Unit 6

Calculus And Change

How do rates and accumulation describe changing systems?

Derivative Zoom Labinstantaneous rate and derivative Accumulation Area Labintegral as accumulation Differential Model Missiondifferential equations and Euler approximation

Mastery gate: Use derivative/integral evidence to model, approximate, and defend change.

Unit 7

Discrete Mathematics, Logic, And Algorithms

How do finite structures support rigorous reasoning and computation?

Counting Without Double Counting Labcounting principles and overcounting Logic Counterexample Lablogical implication, converse, and counterexample Graph Path Algorithm Labgraphs, paths, and algorithm traces

Mastery gate: Use logic, counting, graph, and algorithm evidence to defend finite-system claims.

Unit 8

Mathematical Modeling And Optimization

How can mathematics make a decision under constraints?

Model Variables And Assumptions Labvariables, assumptions, and model structure Optimization Constraint Missionobjective functions and constraints Sensitivity And Transfer Defensesensitivity, robustness, and transfer

Mastery gate: Build, test, revise, and defend a model with assumptions and limits.

TeacherOS

Daily intervention CTA

Group students by structural misconception rather than wrong answer.
Recommend one targeted manipulative/proof/modeling task.
Create a Studio assessment that requires reasoning evidence.
Generate TeachProof practice for explaining the misconception Socratically.

District / university

Evidence rollup

Math growth by concept adjacency and prerequisite repair.
Reasoning evidence across representations.
Misconception resolution in algebra, functions, geometry, statistics, and calculus.
College readiness through modeling, proof, and transfer.