Teaching-mechanic library

sea_level_roots · math · function_world.roots

Concept: Zeros / roots

Player does: Mark every point where the terrain meets sea level

Breaks: Every quadratic has two real roots

A quadratic can have two, one, or zero real roots, depending on whether its graph crosses, touches, or misses the x-axis.

approach_target · math · function_world.limit

Concept: Limit as x → a

Player does: Walk toward x = a while the f(x) marker approaches its destination

Breaks: A limit is the value at the point

A limit describes where f(x) approaches as x nears a, which can differ from f(a) or exist even when f(a) does not.

two_path_convergence · math · function_world.limit

Concept: Two-sided limits

Player does: Approach by left and right routes; the gate opens only if both reach the same height

Breaks: One side is enough

A two-sided limit exists only when the left-hand and right-hand limits agree.

directional_scanner · math · function_world.limit

Concept: One-sided limits

Player does: Scan the machine from one side only; predict where the output heads

Breaks: x → a⁻ means x is negative

x → a⁻ means x approaches a from values less than a, not that x itself is negative.

broken_tile · math · function_world.limit

Concept: Limit exists at a hole

Player does: The tile at x = a is missing, but both paths converge on one spot

Breaks: No f(a), no limit

A limit can exist at a point even when the function itself is undefined there, as long as both sides converge to the same value.

decoy_destination · math · function_world.limit

Concept: Limit ≠ function value

Player does: Nearby paths converge on one platform, but the tile at a teleports elsewhere

Breaks: The limit always equals f(a)

A limit describes where nearby values head, which can differ from the function's actual defined value at that point.

split_gate · math · function_world.limit

Concept: Jump discontinuity (DNE)

Player does: Approaching from the left raises one gate half; from the right, a different height

Breaks: Every function has a limit at every point

When the left and right limits disagree, as at a jump, the two-sided limit does not exist.

runaway_elevator · math · function_world.limit

Concept: Infinite limits

Player does: Near the forbidden floor the elevator climbs faster and never settles

Breaks: An infinite limit is a very large number

An infinite limit means the function grows without bound, describing behavior, not a specific large numeric value.

long_horizon · math · function_world.asymptote

Concept: Limits at infinity

Player does: Travel far; the terrain levels toward a fixed altitude

Breaks: A curve can never cross its asymptote

A function can cross a horizontal asymptote near the middle of its domain; the asymptote only describes its far-away behavior.

closing_walls · math · function_world.squeeze

Concept: Squeeze theorem

Player does: Upper and lower walls close in; predict where the trapped orb ends

Breaks: The trapped function can escape between the bounds

If two bounding functions converge to the same limit, the squeezed function trapped between them must converge there too.

precision_zone · math · function_world.epsilon_delta

Concept: ε-δ definition

Player does: The target band shrinks; choose how close x must stay to a

Breaks: You pick δ first, then ε

The ε-δ definition works the other way: for any tolerance ε chosen first, a matching δ must be found.

unbroken_track · math · function_world.continuity

Concept: Continuity

Player does: A cart rides the track; mark where holes, jumps, and blow-ups stop it

Breaks: Continuous just means defined everywhere

Continuity requires the limit to exist and equal the function's value at every point, not merely being defined there.

repair_the_track · math · function_world.continuity

Concept: Removable discontinuity

Player does: Place the missing piece where the surrounding track says it belongs

Breaks: Any discontinuity can be patched with one point

Only a removable discontinuity, where the limit exists but the function value doesn't match, can be fixed by redefining one point.

slope_scanner · math · function_world.slope

Concept: Slope depends on position

Player does: Cross curved terrain with a tangent scanner; mark the steepest spot

Breaks: A curve has one slope

A curve's slope, given by its derivative, changes from point to point rather than staying fixed.

speed_snapshot · math · function_world.secant

Concept: Instantaneous rate

Player does: Freeze time over shorter windows to read the vehicle's speed

Breaks: Instantaneous rate = average rate over the trip

Instantaneous rate is the limit of the average rate as the time window shrinks to zero, not the average over the whole trip.

closing_gates · math · function_world.secant

Concept: Difference quotient → derivative

Player does: Two checkpoints slide together while Δy/Δx updates

Breaks: The secant slope is the derivative

The derivative is the limit of the secant slope as the two points merge, not the secant slope at any fixed separation.

momentum_direction · math · function_world.slope

Concept: Sign of f′

Player does: Mark where the terrain pushes you forward vs back

Breaks: f′ < 0 means f < 0

The sign of f′ tells whether f is increasing or decreasing, which is unrelated to whether f itself is positive or negative.

zero_force_zones · math · function_world.slope

Concept: Critical points

Player does: Find every spot where the horizontal push vanishes

Breaks: f′ = 0 always means a max or min

A critical point where f′ = 0 can be a local max, local min, or neither, like a saddle point.

curvature_gravity · math · function_world.slope

Concept: Concavity

Player does: Steering bends with how the slope itself changes

Breaks: Concave down means decreasing

Concavity describes how the slope itself is changing, which is independent of whether the function is increasing or decreasing.

gravity_flip · math · function_world.slope

Concept: Inflection points

Player does: Mark where the world's curvature flips

Breaks: An inflection point is where f′ = 0

An inflection point is where concavity changes, marked by f″ = 0, not necessarily where f′ = 0.