LabPathCourse replacement path

Algebra II LabPath

A replacement-grade Algebra II path where students learn functions, structure, modeling, uncertainty, and transfer through interactive evidence, not worksheet mimicry.

Start first mission Assign to class

10/10replacement gate

11units

33missions

12simulator modes

Day 1

Start by making functions behave, not by memorizing notation.

Algebra II begins with input-output structure, then connects graphs, equations, tables, context, uncertainty, and transfer. The point is model judgment, not symbol pushing.

algebralab

Function Machine Commissioning

Interpret function notation as an input-output process across equations, tables, and graphs.

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Student loop

Every mission links structure, action, consequence, and explanation.

Teach the key structure visually before symbols.
Predict what a parameter or operation will change.
Run one model-native action and inspect the consequence.
Explain the same idea in graph, equation, table, and context.
Revise, defend, and transfer the model to a new situation.

Representative missions

Functions, quadratics, polynomials, logs, trig, probability, and modeling.

algebralab

Function Machine Commissioning

Changing the rule changes every output and exposes when notation is being read backward.

Try it

proportionlab

Piecewise Policy Simulator

A bad breakpoint or rule causes visible jumps, gaps, or context violations.

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algebralab

Multiplicity Bounce Lab

Even multiplicity bounces while odd multiplicity crosses, with higher powers flattening near the root.

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algebralab

Radical Domain Gate

Squaring can create candidate solutions that fail the original radical equation.

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statsphere

Exponential Model Evidence Brief

The wrong model family leaves a visible residual pattern that weakens the claim.

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waveworld

Wave Parameter Studio

Each parameter changes a specific visible feature of the wave without changing the others.

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Course map

Ten units from functions to transfer-ready modeling.

Unit 1

Functions As Machines And Models

How does a function turn inputs into predictable, testable outputs?

Function Machine CommissioningInterpret function notation as an input-output process across equations, tables, and graphs.Domain And Range Boundary LabUse graph behavior and context constraints to define domain and range.Representation Transfer StudioBuild a function from context and verify it across table, graph, and equation.

Mastery gate: Student connects a function rule, table, graph, and context and defends whether a relation is a function.

Unit 2

Linear, Absolute Value, And Piecewise Systems

How do different rule structures describe different kinds of change?

Slope As A Control SystemInterpret slope and intercept as controls that shape a linear model.Absolute Value Distance FieldModel absolute value as distance from a center and solve by symmetry.Piecewise Policy SimulatorUse intervals and rule changes to construct and interpret a piecewise function.

Mastery gate: Student selects and defends the right model family for constant, distance-from-center, and changing-rule behavior.

Unit 3

Quadratics And Complex Numbers

How do shape, roots, vertex, and complex solutions reveal a quadratic system?

Parabola Launch RangeConnect quadratic parameters to vertex, axis of symmetry, and intercepts.Factoring And Zero Product LabUse structure and zero product reasoning to solve quadratic equations.Complex Roots RadarInterpret complex roots when a quadratic has no real x-intercepts.

Mastery gate: Student moves between factoring, vertex form, graph behavior, and complex roots with evidence.

Unit 4

Polynomial Structure And End Behavior

How do degree, leading coefficient, roots, and multiplicity shape a polynomial?

End Behavior Wind TunnelPredict polynomial end behavior from degree and leading coefficient.Multiplicity Bounce LabConnect root multiplicity to crossing, touching, or flattening behavior.Polynomial Model FitChoose a polynomial model that fits data without overfitting noise.

Mastery gate: Student predicts end behavior, root behavior, and graph shape from algebraic structure.

Unit 5

Rational Functions And Asymptotes

How do ratios of functions create restrictions, holes, and long-run behavior?

Rational Restriction DetectorFind excluded values and distinguish holes from vertical asymptotes.Asymptote NavigationUse graph and algebraic structure to explain vertical and horizontal asymptotes.Rational Model Safety CaseUse rational functions to model rate, concentration, or efficiency constraints.

Mastery gate: Student identifies domain restrictions, holes, vertical asymptotes, and horizontal/end behavior from rational structure.

Unit 6

Radicals And Inverses

How do inverse operations undo functions, and when do they introduce restrictions?

Radical Domain GateUse radicand constraints to determine valid inputs and solve radical equations.Inverse Function MirrorConstruct and interpret inverse functions using input-output reversal and graph reflection.Operation Undo ChainDefend each inverse operation used to solve or invert an equation.

Mastery gate: Student solves radical equations, checks extraneous solutions, and interprets inverse function restrictions.

Unit 7

Exponential And Logarithmic Models

How do multiplicative change and logarithms explain growth, decay, and scale?

Growth Factor ReactorDistinguish additive and multiplicative change in exponential models.Logarithm Scale DecoderUse logarithms to solve for an exponent and interpret scale.Exponential Model Evidence BriefFit exponential models to data and defend whether the model is appropriate.

Mastery gate: Student models exponential change, transforms logarithmic form, and interprets parameters in context.

Unit 8

Sequences, Series, And Recursion

How do repeated rules create patterns, totals, and long-term behavior?

Recursive Signal BuilderBuild recursive rules and compare them to explicit descriptions.Arithmetic Or Geometric DetectorClassify sequences by constant difference or constant ratio and justify the rule.Series AccumulatorInterpret finite and geometric series as accumulated repeated quantities.

Mastery gate: Student connects explicit, recursive, arithmetic, geometric, and series representations.

Unit 9

Trigonometric Functions And Periodic Models

How do circular motion and wave behavior create periodic functions?

Unit Circle Function GeneratorGenerate sine and cosine values from motion around the unit circle.Wave Parameter StudioUse amplitude, period, and phase shift to model periodic behavior.Periodic Model DefenseFit and defend a trigonometric model for periodic data.

Mastery gate: Student connects unit circle, amplitude, period, phase, and contextual periodic models.

Unit 10

Probability, Combinatorics, And Decision Evidence

How can counting and probability quantify uncertainty without pretending to remove it?

Sample Space SimulatorBuild a sample space and use it to calculate probability.Conditional Probability FilterUse filters and two-way tables to reason about conditional probability.Counting Strategy LabChoose permutations, combinations, or organized lists based on the structure of the count.

Mastery gate: Student models probability, conditional probability, and counting with clear assumptions and evidence.

Unit 11

Integrated Modeling And Transfer

How does a mathematically mature student choose, test, and defend a model?

Model Family TriageChoose between linear, quadratic, exponential, rational, and periodic models using evidence.Constraint Optimization BriefUse constraints to optimize a design or decision and explain the tradeoff.Algebra II Capstone TransferTransfer Algebra II tools to a new system and defend the evidence chain.

Mastery gate: Student selects a model family, validates it against evidence, explains limitations, and transfers the reasoning.

TeacherOS

Intervene by structure, not just score.

Assign the next LabPath mission to the class or a misconception cluster.
Inspect TeacherOS clusters by representation, structure, and transfer errors.
Launch a TeachProof rehearsal for the hardest misconception before reteaching.
Send parent and district evidence without reducing math to completion.

Parent and district

Readable evidence of modeling maturity.

Which algebraic structures the student can explain
Where the student confuses procedures with meaning
What model or representation should be practiced next
Function readiness by course and teacher
Modeling transfer across algebra, statistics, science, and CS