LabPathAdvanced math course path

Calculus LabPath

A replacement-grade Calculus path where students see rates, accumulation, approximation, optimization, and series as dynamic systems before formalizing them.

Start first mission Assign to class

10/10operating gate

10units

30missions

8simulator modes

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.

graphverse

Limits And Continuity: Explore

Explore limit evidence using limits and continuity.

Open mission

Student loop

One 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.

Representative missions

Course depth with simulator-backed evidence.

graphverse

Limits And Continuity: Explore

A function can approach a stable value even when a point is missing, jumped, or undefined.

Try it

geoforge

Derivatives As Instantaneous Rate: Test

As the interval shrinks, the average rate converges to the local tangent rate.

Try it

graphverse

Derivative Rules As Structure: Defend

Wrong rules produce derivative graphs that contradict local rate evidence.

Try it

graphverse

Integrals As Accumulation: Explore

More partitions improve approximation and signed regions change total accumulation.

Try it

graphverse

Fundamental Theorem Of Calculus: Test

The accumulator’s slope mirrors the original rate, linking derivatives and integrals.

Try it

circuitmind

Differential Equations And Slope Fields: Defend

Changing the initial condition changes the path while preserving the same local slope rule.

Try it

Course map

10 units with daily evidence loops.

Unit 1

Limits And Continuity

What does a function do near a point, not just at the point?

Limits And Continuity: ExploreExplore limit evidence using limits and continuity.Limits And Continuity: TestTest continuity using limits and continuity.Limits And Continuity: DefendDefend one-sided behavior using limits and continuity.

Mastery gate: Student compares graphical, numerical, and symbolic limit evidence and classifies continuity.

Unit 2

Derivatives As Instantaneous Rate

How does average change become instantaneous change?

Derivatives As Instantaneous Rate: ExploreExplore secant slope using derivatives as instantaneous rate.Derivatives As Instantaneous Rate: TestTest tangent slope using derivatives as instantaneous rate.Derivatives As Instantaneous Rate: DefendDefend units of rate using derivatives as instantaneous rate.

Mastery gate: Student uses secant-to-tangent evidence to explain derivative meaning, sign, and units.

Unit 3

Derivative Rules As Structure

Why do derivative rules work, and when do they apply?

Derivative Rules As Structure: ExploreExplore product rule using derivative rules as structure.Derivative Rules As Structure: TestTest chain rule using derivative rules as structure.Derivative Rules As Structure: DefendDefend implicit reasoning using derivative rules as structure.

Mastery gate: Student chooses derivative rules based on function structure and verifies behavior graphically.

Unit 4

Applications Of Derivatives

How do derivatives support optimization, motion, and decision making?

Applications Of Derivatives: ExploreExplore optimization using applications of derivatives.Applications Of Derivatives: TestTest motion using applications of derivatives.Applications Of Derivatives: DefendDefend concavity using applications of derivatives.

Mastery gate: Student connects derivative sign, extrema, concavity, and constraints to a real decision.

Unit 5

Integrals As Accumulation

How does adding tiny changes create total change?

Integrals As Accumulation: ExploreExplore Riemann sums using integrals as accumulation.Integrals As Accumulation: TestTest signed area using integrals as accumulation.Integrals As Accumulation: DefendDefend total change using integrals as accumulation.

Mastery gate: Student uses area, signed accumulation, and units to explain definite integrals.

Unit 6

Fundamental Theorem Of Calculus

How are rate and accumulation inverse views of the same system?

Fundamental Theorem Of Calculus: ExploreExplore FTC part 1 using fundamental theorem of calculus.Fundamental Theorem Of Calculus: TestTest FTC part 2 using fundamental theorem of calculus.Fundamental Theorem Of Calculus: DefendDefend rate/accumulation connection using fundamental theorem of calculus.

Mastery gate: Student connects derivative of accumulation and accumulated rate to total change.

Unit 7

Differential Equations And Slope Fields

How can a rule about change generate a family of solutions?

Differential Equations And Slope Fields: ExploreExplore slope field using differential equations and slope fields.Differential Equations And Slope Fields: TestTest initial condition using differential equations and slope fields.Differential Equations And Slope Fields: DefendDefend solution behavior using differential equations and slope fields.

Mastery gate: Student interprets slope fields, initial conditions, and qualitative solution behavior.

Unit 8

Series And Approximation

How can infinite processes approximate functions and when do they converge?

Series And Approximation: ExploreExplore convergence using series and approximation.Series And Approximation: TestTest Taylor approximation using series and approximation.Series And Approximation: DefendDefend error using series and approximation.

Mastery gate: Student tests convergence, approximation error, and series representation limits.

Unit 9

Parametric, Polar, And Vector-Valued Motion

How do alternate coordinate systems reveal motion, curvature, and accumulated distance?

Parametric, Polar, And Vector-Valued Motion: ExploreExplore parametric derivative using parametric, polar, and vector-valued motion.Parametric, Polar, And Vector-Valued Motion: TestTest polar area using parametric, polar, and vector-valued motion.Parametric, Polar, And Vector-Valued Motion: DefendDefend vector motion using parametric, polar, and vector-valued motion.

Mastery gate: Student connects parametric rate, polar area, vector motion, and arc length using visible evidence.

Unit 10

Calculus Modeling Lab

How does calculus support a defensible real-world decision under constraints?

Calculus Modeling Lab: ExploreExplore tool selection using calculus modeling lab.Calculus Modeling Lab: TestTest constraint reasoning using calculus modeling lab.Calculus Modeling Lab: DefendDefend model validation using calculus modeling lab.

Mastery gate: Student chooses derivative, integral, differential equation, or series evidence and defends the model limitation.

TeacherOS

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.

Family and district

Progress without reducing learning to completion.

What Calculus concept the student can explain
Where the student is confusing structure, procedure, or interpretation
The next best practice mission
Calculus mastery by unit
Misconception resolution time