Precalculus LabPath
A bridge from Algebra II to Calculus where students reason about advanced functions, trigonometry, limits, vectors, polar systems, and model choice through visible consequences.
Day 1
Teach the idea, run the model, defend the evidence.
Students start with a visual walkthrough, then use one clean control change to produce evidence they can explain, revise, defend, and transfer.
Function Families And Transformations: Explore
Explore parent function recognition using function families and transformations.
Open missionOne visible mathematical consequence per mission.
- Learn the structure with a visual walkthrough.
- Predict which control will change the model.
- Run one clean test and inspect the consequence.
- Explain the evidence in words, symbols, and context.
- Revise, defend, and transfer to a new situation.
Course depth with simulator-backed evidence.
Function Families And Transformations: Explore
Every transformation changes a visible feature while preserving the underlying function family.
Try itPolynomial And Rational Behavior: Test
Hidden factors create visible crossings, bounces, holes, walls, and long-run behavior.
Try itExponential And Logarithmic Systems: Defend
Small factor changes compound into large long-run differences and logs recover hidden exponents.
Try itAnalytic Trigonometry And Identities: Explore
Invalid transformations fail at test angles or outside allowed domains.
Try itVectors, Parametric Motion, And Polar Systems: Test
The same path can look simpler or harder depending on the coordinate system.
Try itLimits Before Calculus: Defend
Approach behavior can converge even when the function value is missing or undefined.
Try it10 units with daily evidence loops.
Unit 1
Function Families And Transformations
How do parent functions and transformations create predictable families of behavior?
Function Families And Transformations: ExploreExplore parent function recognition using function families and transformations.Function Families And Transformations: TestTest transformation order using function families and transformations.Function Families And Transformations: DefendDefend representation transfer using function families and transformations.
Mastery gate: Student identifies the parent function, predicts transformation effects, and defends the graph/equation/context match.Unit 2
Polynomial And Rational Behavior
How do roots, asymptotes, and end behavior reveal hidden algebraic structure?
Polynomial And Rational Behavior: ExploreExplore root behavior using polynomial and rational behavior.Polynomial And Rational Behavior: TestTest asymptote reasoning using polynomial and rational behavior.Polynomial And Rational Behavior: DefendDefend end behavior using polynomial and rational behavior.
Mastery gate: Student predicts roots, holes, asymptotes, and end behavior from symbolic structure and validates them visually.Unit 3
Exponential And Logarithmic Systems
How do multiplicative change and inverse exponent reasoning model scale?
Exponential And Logarithmic Systems: ExploreExplore growth factor using exponential and logarithmic systems.Exponential And Logarithmic Systems: TestTest log inverse using exponential and logarithmic systems.Exponential And Logarithmic Systems: DefendDefend scale interpretation using exponential and logarithmic systems.
Mastery gate: Student solves exponential/logarithmic models and explains base, factor, scale, and limits.Unit 4
Trigonometry As Circular And Wave Motion
How does circular motion become sine, cosine, and periodic modeling?
Trigonometry As Circular And Wave Motion: ExploreExplore radian measure using trigonometry as circular and wave motion.Trigonometry As Circular And Wave Motion: TestTest unit circle projections using trigonometry as circular and wave motion.Trigonometry As Circular And Wave Motion: DefendDefend periodic parameters using trigonometry as circular and wave motion.
Mastery gate: Student connects unit circle projections, radian measure, amplitude, period, and phase to a real periodic system.Unit 5
Analytic Trigonometry And Identities
When is an identity a structural truth rather than an algebra trick?
Analytic Trigonometry And Identities: ExploreExplore identity verification using analytic trigonometry and identities.Analytic Trigonometry And Identities: TestTest equivalence using analytic trigonometry and identities.Analytic Trigonometry And Identities: DefendDefend counterexample reasoning using analytic trigonometry and identities.
Mastery gate: Student verifies, transforms, and defends trig identities using structure and counterexample checks.Unit 6
Vectors, Parametric Motion, And Polar Systems
How do direction, magnitude, and alternate coordinate systems describe motion?
Vectors, Parametric Motion, And Polar Systems: ExploreExplore vector components using vectors, parametric motion, and polar systems.Vectors, Parametric Motion, And Polar Systems: TestTest parametric time using vectors, parametric motion, and polar systems.Vectors, Parametric Motion, And Polar Systems: DefendDefend polar coordinates using vectors, parametric motion, and polar systems.
Mastery gate: Student models motion with vectors, parametric equations, and polar coordinates and explains coordinate choice.Unit 7
Limits Before Calculus
How can a value be approached even when direct substitution breaks?
Mastery gate: Student uses numerical, graphical, and symbolic evidence to reason about limits and discontinuity.Unit 8
Sequences, Series, And Discrete Growth
How do recursive and explicit rules describe long-run behavior before calculus?
Sequences, Series, And Discrete Growth: ExploreExplore recursive reasoning using sequences, series, and discrete growth.Sequences, Series, And Discrete Growth: TestTest geometric growth using sequences, series, and discrete growth.Sequences, Series, And Discrete Growth: DefendDefend series convergence using sequences, series, and discrete growth.
Mastery gate: Student compares arithmetic, geometric, recursive, and series representations and defends convergence or divergence.Unit 9
Conics, Implicit Relations, And System Geometry
How do equations describe geometric constraints that are not always functions?
Conics, Implicit Relations, And System Geometry: ExploreExplore conic structure using conics, implicit relations, and system geometry.Conics, Implicit Relations, And System Geometry: TestTest implicit relation using conics, implicit relations, and system geometry.Conics, Implicit Relations, And System Geometry: DefendDefend system intersection using conics, implicit relations, and system geometry.
Mastery gate: Student identifies conic structure, transforms equations, and defends when relation, function, or system language applies.Unit 10
Precalculus Modeling Capstone
How does a mature modeler choose the right function family and defend limitations?
Mastery gate: Student selects, tests, revises, and transfers a model family using residuals, constraints, and domain evidence.Evidence becomes action.
- Assign the next course mission to a class or misconception cluster.
- Inspect TeacherOS evidence by concept, representation, and transfer.
- Launch TeachProof practice for the hardest teaching move.
- Share readable parent/district evidence without reducing learning to completion.
Progress without reducing learning to completion.
- What Precalculus concept the student can explain
- Where the student is confusing structure, procedure, or interpretation
- The next best practice mission
- Precalculus mastery by unit
- Misconception resolution time