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.

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.

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.

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.

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.

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.

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.

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.