Teaching-mechanic library

number_line_leap · math · mapper.number_line

Concept: Magnitude, fractions, negatives

Player does: Land on 3/8 or −2.5 with only coarse landmarks labeled

Breaks: A bigger denominator means a bigger fraction

A fraction's size comes from how far along the line it lands, and a bigger denominator makes each step smaller.

equivalent_landing · math · mapper.number_line

Concept: Equivalent fractions

Player does: Two different-looking fractions must land on the same stone

Breaks: Different numerals mean different amounts

Equivalent fractions name the same point on the line, no matter how they are written.

log_scope · math · mapper.number_line

Concept: Scientific notation, orders of magnitude

Player does: Place 10⁻⁹ m, 10³ m, 10⁶ m on a zoomable log line

Breaks: 10⁻⁹ is a large negative number

A negative exponent means a tiny positive number; each step on a log line multiplies by ten.

integer_lift · math · mapper.number_line

Concept: Signed-number arithmetic

Player does: Ride the elevator through basement floors: −3 − (−5)

Breaks: Subtracting a negative makes a number smaller

Subtracting a negative moves you up the line, because it undoes a move down.

gear_ratio_gate · math · tuner.formula

Concept: Ratios & proportion

Player does: Set gear tooth counts so the portcullis rises at the target speed

Breaks: Ratios scale by adding the same amount

A ratio scales by multiplying both parts by the same factor, never by adding to them.

percent_shop · math · tuner.formula

Concept: Percent change

Player does: Apply successive markups and discounts to hit an exact price

Breaks: −20% then +20% returns to the start

Successive percent changes multiply, so a 20% drop and a 20% rise do not cancel.

unit_pipeline · math · transformer.composition

Concept: Dimensional analysis

Player does: Chain conversion-factor pipes; joints connect only when units cancel

Breaks: Units are labels you can drop

Units multiply and cancel like numbers, which is exactly how a chain of conversions works.

order_of_ops_forge · math · transformer.trace

Concept: Order of operations

Player does: Predict what the calculator machine outputs for a nested expression

Breaks: Operations always go left to right

Operations follow precedence and grouping, not reading order.

balance_chamber · math · balance.equation

Concept: Solving linear equations

Player does: Apply the same operation to both pans until x stands alone

Breaks: You can move a term across without changing its sign

Moving a term to the other side is really applying the same operation to both sides, which flips its sign.

intersection_hunt · math · mapper.plane

Concept: Systems of equations

Player does: Mark the one point both laser paths pass through

Breaks: A solution only has to satisfy one equation

A system's solution must satisfy every equation at once, which is exactly where the lines intersect.

allowed_region · math · mapper.plane

Concept: Inequalities

Player does: Only tiles where y < 2x + 1 are solid; mark the safe region

Breaks: Multiplying by a negative keeps the inequality direction

Multiplying or dividing an inequality by a negative number flips the inequality's direction.

distance_field · math · mapper.number_line

Concept: Absolute value

Player does: Beacons light where |x − 3| < 2; mark the whole lit stretch

Breaks: |x| just deletes a minus sign, so |x| = x

Absolute value is distance from zero, so |x| equals x only when x is nonnegative; otherwise it flips the sign.

graph_terrain · math · tuner.curve

Concept: Slope & intercept

Player does: Tune m and b so the ramp meets the ledge

Breaks: Slope is run over rise

Slope is rise over run, the vertical change divided by the horizontal change, not the other way around.

function_factory · math · transformer.function_machine

Concept: What a function is

Player does: Feed inputs, watch outputs, predict the output for a new input

Breaks: A function must be a formula

A function is any rule pairing each input to exactly one output, whether given by a formula, table, or graph.

machine_pipeline · math · transformer.composition

Concept: Composition

Player does: Order two machines so the crate exits as the target

Breaks: f(g(x)) = g(f(x))

Function composition is generally not commutative, so f(g(x)) and g(f(x)) usually differ.

projectile_architect · math · tuner.curve

Concept: Quadratics (standard form)

Player does: Tune a, b, c so the arc threads the rings

Breaks: c changes how wide the parabola is

In standard form the leading coefficient a controls the parabola's width, while c only shifts the y-intercept.

move_the_arc · math · tuner.curve

Concept: Vertex form

Player does: Drag h, k, a to move the parabola onto its target

Breaks: (x − 3)² shifts the graph left

In vertex form, (x − h)² shifts the graph right by h, so (x − 3)² shifts right, not left.

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.

factor_forge · math · builder.tiles

Concept: Factoring

Player does: Arrange algebra tiles into a rectangle whose sides are the factors

Breaks: x² + 5x + 6 = (x + 5)(x + 1)

Factoring requires two numbers that multiply to the constant and add to the middle coefficient, here 2 and 3, giving (x + 2)(x + 3).

replication_chamber · math · simulator.reach_state

Concept: Exponential growth

Player does: Set the growth factor so the colony hits the target on day n

Breaks: Exponential growth is fast linear growth

Exponential growth multiplies by a constant factor each step, which eventually outpaces any linear growth no matter how fast.

reverse_growth · math · tuner.formula

Concept: Logarithms

Player does: Choose how many doublings reach the target population

Breaks: A logarithm is division

A logarithm answers how many times to multiply by the base to reach a value; it is the inverse of exponentiation, not division.

pattern_path · math · transformer.function_machine

Concept: Arithmetic & geometric sequences

Player does: Jump to the stone holding the next term

Breaks: Every sequence adds a constant

An arithmetic sequence adds a constant difference, but a geometric sequence multiplies by a constant ratio instead.

inheritance_machine · math · simulator.reach_state

Concept: Recursive sequences

Player does: Set a₁ and the rule so generation 5 matches

Breaks: Recursive and explicit rules can't describe the same sequence

A recursive rule and an explicit formula can generate exactly the same sequence, just computed differently.

exponent_mimic · math · truth_finder.mimic

Concept: Exponent rules

Player does: Expose the chest claiming x²·x³ = x⁶

Breaks: Multiply exponents when multiplying powers

When multiplying powers with the same base, exponents add, so x²·x³ = x⁵, not x⁶.

mirror_angles · math · tuner.formula

Concept: Reflection angles

Player does: Set mirror angles so the beam hits the crystal

Breaks: Light bounces straight back off any mirror

Light reflects so the angle of incidence equals the angle of reflection, not straight back unless it hit head-on.

area_tiler · math · builder.tiles

Concept: Area vs perimeter

Player does: Tile the room, then fence it; compare both counts

Breaks: More area always means more perimeter

Area and perimeter can change independently, so a shape can have more area with less perimeter, and vice versa.

pythagoras_bridge · math · tuner.formula

Concept: Pythagorean theorem

Player does: Cut the plank that spans the diagonal gap

Breaks: The hypotenuse is a + b

The hypotenuse's length is the square root of a² + b², not the simple sum of the legs.

scale_shifter · math · tuner.formula

Concept: Similarity; square-cube law

Player does: Grow ray: double your size, 8× your weight, too big for the gap

Breaks: Doubling length doubles volume

Volume scales with the cube of linear size, so doubling length multiplies volume by eight, not two.

congruence_stamp · math · sorter.bins

Concept: Congruence criteria

Player does: Sort triangle pairs into SSS / SAS / ASA / not enough info

Breaks: SSA proves congruence

Side-Side-Angle does not guarantee congruent triangles; it can produce two different triangles from the same measurements.

proof_ladder · math · sequencer.linear

Concept: Two-column proofs

Player does: Order the rungs of a proof; one decoy reason doesn't follow

Breaks: If it looks true in the diagram, it's proven

A geometric proof needs a chain of justified statements, since a diagram can look true without the relationship being proven.

volume_pour · math · tuner.formula

Concept: Volume

Player does: Fill differently shaped tanks to the same volume

Breaks: The taller container holds more

Volume depends on the full shape, not just height, so a shorter but wider container can hold more.

coordinate_treasure · math · mapper.plane

Concept: Distance & midpoint

Player does: Mark the midpoint between two torches on the grid

Breaks: The midpoint is the difference of the coordinates

The midpoint is the average of the two points' coordinates, not their difference.

circular_navigator · math · mapper.plane

Concept: Unit circle

Player does: Walk the unit circle to where the x-shadow reads the given cos θ

Breaks: sin θ is the x-coordinate

On the unit circle cos θ is the x-coordinate and sin θ is the y-coordinate.

projection_machine · math · mapper.number_line

Concept: sin & cos as projections

Player does: Predict where the rotating point's shadow lands at θ

Breaks: sin and cos are unrelated functions

Sine and cosine are the two shadows of one rotating point, which is why they share a period and trade peaks.

phase_gate · math · tuner.oscillator

Concept: Period

Player does: Dial the period so the vault rings lock

Breaks: A bigger b makes the period longer

The period of sin(bt) is 2π/|b|, so a bigger b squeezes the wave and shortens the period.

pulse_matcher · math · tuner.oscillator

Concept: Frequency

Player does: Match pulses per second to the beacon

Breaks: Frequency and period are the same number

Frequency is the reciprocal of the period: cycles per unit time, not time per cycle.

oscillation_reach · math · tuner.oscillator

Concept: Amplitude

Player does: Set the swing so the platform just touches each checkpoint

Breaks: Amplitude is peak-to-trough

Amplitude is the distance from the midline to a peak, half of the full swing.

wave_alignment · math · tuner.oscillator

Concept: Phase shift

Player does: Slide one wave in time until the peaks align

Breaks: sin(x + π/2) shifts right

Adding inside the argument shifts the graph left, because each output now happens earlier.

radian_rune_line · math · mapper.number_line

Concept: Radian measure

Player does: Step onto 5π/6 on the unrolled circle

Breaks: π radians is a full turn

A full turn is 2π radians because the unit circle's circumference is 2π; π is half a turn.

arc_builder · math · tuner.formula

Concept: Arc length s = rθ

Player does: Lay rope along the rim; the angle is arc ÷ radius

Breaks: An angle's radian measure depends on the circle's size

Radian measure is arc length divided by radius, so it is the same for any circle.

identity_circuits · math · truth_finder.mimic

Concept: Trig identities

Player does: Several routes transform the signal; expose the non-equivalent one

Breaks: Identities only hold at special angles

An identity holds for every angle; one counterexample is enough to expose an impostor.

trig_solve_bridge · math · sequencer.linear

Concept: Solving sin x = k

Player does: Lay the planks: isolate, reference angle, quadrants, all solutions

Breaks: sin x = ½ has one solution

Solving a trig equation means isolating, finding the reference angle, then using every quadrant where the sign matches.

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.

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.

peak_efficiency · math · tuner.optimize

Concept: Optimization

Player does: Tune the parameter to maximize the machine's output

Breaks: The max is always at an endpoint

An optimum can occur at a critical point in the interior, not just at an endpoint of the domain.

linked_machines · math · tuner.formula

Concept: Related rates

Player does: Set how fast the radius grows so volume rises at the target rate

Breaks: Linked quantities change at the same rate

Related quantities change at rates connected by the chain rule, which usually differ rather than match.

derivative_machine · math · transformer.function_machine

Concept: Differentiation rules

Player does: The machine turns f into f′; predict its output for a new f

Breaks: d/dx x³ = 3x³

The power rule brings the exponent down as a coefficient and reduces it by one, so d/dx x³ = 3x², not 3x³.

tile_the_region · math · accumulator.riemann

Concept: Riemann sums

Player does: Place narrower and narrower blocks under the curved ceiling

Breaks: Left sums always underestimate

Whether a left or right Riemann sum over- or underestimates the true area depends on whether the function is increasing or decreasing.

fill_the_reservoir · math · accumulator.area

Concept: Definite integral as area

Player does: Stop the gate where the reservoir holds the target volume

Breaks: Area under a curve needs a geometric formula

The definite integral gives exact area under a curve even when no simple geometric shape formula applies.

energy_ledger · math · accumulator.signed

Concept: Signed area

Player does: Area above the axis charges the battery; below drains it

Breaks: Area below the axis counts as positive

The definite integral treats area below the x-axis as negative, subtracting from the total rather than adding.

resource_collector · math · accumulator.rate_total

Concept: Accumulation from a rate

Player does: Ride through changing production rates; predict inventory at t

Breaks: Total = rate × time even when the rate changes

When the rate changes, total accumulation is the integral of the rate over time, not a single rate times time.

rate_total_machine · math · accumulator.rate_total

Concept: Fundamental Theorem

Player does: Switch between the rate gauge and the storage tank; predict each

Breaks: Integrals and derivatives are unrelated

The Fundamental Theorem of Calculus links them directly: differentiation and integration are inverse operations.

integration_window · math · accumulator.area

Concept: Changing bounds

Player does: Drag the window's edges to hit the target total

Breaks: Swapping the bounds doesn't matter

Swapping the limits of integration flips the sign of the definite integral's value.

level_the_reservoir · math · accumulator.average_value

Concept: Average value

Player does: Flatten uneven water into a rectangle of equal area

Breaks: Average value is the average of the endpoints

The average value of a function over an interval is its integral divided by the interval length, not just the average of the endpoints.

rate_runner · math · accumulator.rate_total

Concept: Position from velocity

Player does: Sketch your ghost's velocity; position is the area under it

Breaks: The velocity graph shows the path's shape

A velocity-time graph shows speed and direction over time; its shape is not the shape of the object's actual path.

vector_winds · math · mapper.plane

Concept: Vector addition

Player does: Place fans; mark where their combined push carries you

Breaks: Vectors add like numbers (3 + 4 = 7)

Vectors add component-wise (or tip-to-tail), so their combined magnitude generally is not the simple sum of their lengths.

component_split · math · mapper.plane

Concept: Components

Player does: Split thrust into x and y engines that land you on target

Breaks: Components add up to the magnitude

A vector's magnitude comes from the Pythagorean combination of its components, not their simple sum.

repeat_experiment · math · simulator.sample

Concept: Variability

Player does: Rerun the identical experiment; predict the spread

Breaks: Same setup, same result

Random variation means identical setups can still produce different results from trial to trial.

prediction_stabilizer · math · simulator.sample

Concept: Law of large numbers

Player does: Choose how many trials until the estimate stays in the band

Breaks: After five heads, tails is due

Independent trials have no memory, so past outcomes don't change future odds; the law of large numbers only stabilizes averages over many trials.

long_run_casino · math · simulator.sample

Concept: Expected value

Player does: Pick the table with the best long-run payoff, then watch 1,000 plays

Breaks: The biggest prize is the best bet

Expected value weighs each payoff by its probability, so a rare huge prize can still be a worse bet than a smaller frequent one.

population_collector · math · simulator.sample

Concept: Sampling bias

Player does: Sample only reachable districts; explain why the estimate drifts

Breaks: A bigger biased sample fixes the bias

A biased sampling method stays biased no matter how large the sample gets; only fixing the method removes the bias.

capture_the_parameter · math · simulator.sample

Concept: Confidence intervals

Player does: Throw interval nets; predict how many catch the hidden true value

Breaks: There's a 95% chance this one interval holds the value

A 95% confidence level describes the method's long-run capture rate across many intervals, not the chance for one already-computed interval.

spread_field · math · sequencer.rank

Concept: Standard deviation

Player does: Rank datasets with the same mean by spread

Breaks: Same mean, same data

Two datasets can share the same mean while differing greatly in standard deviation, which measures the spread around it.

scatter_landscape · math · mapper.number_line

Concept: Correlation

Player does: Place each scatter plot's r on the −1…1 line

Breaks: r = 0 means no relationship of any kind

A correlation of r = 0 rules out a linear relationship but not a strong nonlinear one.

hidden_variable_mystery · math · investigator.elimination

Concept: Correlation vs causation

Player does: Find the hidden third system driving both meters

Breaks: Correlation proves causation

A correlation can be explained by a hidden confounding variable driving both, rather than one causing the other.

evidence_updater · math · mapper.number_line

Concept: Bayes' theorem

Player does: After the test result, set the belief meter

Breaks: A 99%-accurate test means a 99% chance you're sick

Bayes' theorem shows the chance of being sick given a positive test also depends on the disease's rarity, which can make it far below the test's accuracy rate.

restricted_world · math · mapper.number_line

Concept: Conditional probability

Player does: Irrelevant outcomes vanish once the condition is known; place P(A|B)

Breaks: P(A|B) = P(B|A)

Conditional probability is not symmetric in general: P(A|B) and P(B|A) can be very different values.

odds_forge · math · simulator.sample

Concept: Probability & counting

Player does: Choose the fork with the best odds of loot

Breaks: Two outcomes means 50/50

Probability depends on how many ways each outcome can occur, not just how many outcomes are listed.

mean_median_seesaw · math · balance.torque

Concept: Mean vs median

Player does: Add an outlier; move the fulcrum (mean) while the median marker stays put

Breaks: Mean and median always move together

The mean is pulled toward outliers while the median resists them, so the two can move very differently as data changes.

z_score_rapids · math · mapper.number_line

Concept: z-scores, normal distribution

Player does: Place the raft at the reading's z-score

Breaks: z = 2 means twice the average

A z-score of 2 means the value is two standard deviations above the mean, not double the mean itself.