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.

reverse_factory · math · transformer.inverse

Concept: Inverse functions

Player does: Run the machine backward to recover the input

Breaks: f⁻¹(x) means 1/f(x)

The inverse function f⁻¹ undoes f by swapping inputs and outputs; it is not the reciprocal 1/f(x).

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.

input_filter · math · transformer.domain_filter

Concept: Domain & range

Player does: Choose which crates the √ or 1/x machine accepts without jamming

Breaks: Every real number is a valid input

A function's domain excludes inputs that break it, like negatives under a square root or zero in a denominator.

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.

transform_key · math · transformer.geometric

Concept: Rigid transformations & symmetry

Player does: Rotate, reflect, translate the key until it fits the slot

Breaks: A line reflection equals a 180° rotation

A reflection flips a shape's orientation while a 180° rotation preserves it, so the two are not the same transformation.

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.