Lektionen — 12. Klasse

Diese Lektionen sind auf Englisch, erzählt von Koydo-Tutoren.

Lumi solves ax²+bx+c=0 with the quadratic formula and discriminant — not parabola x-intercepts as a graph job

Math · mit Atlas · 40 Min.

A modern classroom with a large digital board displaying a labeled parabola graph, a graphing calculator on a student desk, a notebook open to quadratic equations, and geometric shapes on nearby shelves while Atlas points toward the axis of symmetry on the board.

Lumi names a sequence as a list and a series as a sum — not a convergence test

Math · mit Atlas · 30 Min.

A library with tall bookshelves lining the walls and a large scroll displaying sequence notation, while Atlas writes formulas on a parchment scroll at a reading table beside an interactive sequence generator and a series sum calculator.

Lumi computes P(A) from counting — combinatorics in a sample space, not nPr versus nCr as the whole job

Math · mit Atlas · 40 Min.

A colorful game board with dice, cards, and balls in different colors. The guiding character, education/atlas, holds a probability spinner and points to a chart showing permutations and combinations formulas.

Lumi uses a determinant as Cramer's numerator and denominator — not Av=λv

Math · mit Atlas · 35 Min.

In a high-tech math lab, glowing 3D parallelogram models float beside matrix equations projected on wall screens, while Atlas holds a tablet displaying cofactor expansion steps and determinant calculations updating in real time.

Lumi plots (r, θ) and converts to (x, y) — polar graphing, not polar conics and not Argand

Math · mit Atlas · 35 Min.

A large circular grid drawn on the floor of a science hall. Atlas demonstrates plotting points using radius and angle with a laser pointer. Surrounding walls show images of polar roses and spirals.

Lumi writes a conic as r = ed / (1 + e cos θ) — polar eccentricity, not cartesian standard form

Math · mit Atlas · 30 Min.

A polar coordinate graph showing an ellipse, parabola, and hyperbola traced in different colors. Atlas points to the focus and the directrix lines, with eccentricity values displayed.

Lumi raises cis θ to an integer power with De Moivre — not the polar writing of z

Math · mit Atlas · 40 Min.

A vibrant scene depicting a large Argand diagram projected on a screen, showing points representing complex numbers and vectors illustrating magnitude and argument. The guiding character 'education/atlas' stands next to the diagram holding a compass and protractor. Surrounding the scene are mathematical notes highlighting Euler's formula, polar coordinates, and roots of unity.

Lumi treats z = f(x,y) as two inputs, one output — a surface, not |x| pieces

Math · mit Atlas · 45 Min.

Inside a science lab, towering 3D coordinate axes stand beside contour maps spread across workbenches, while Atlas reaches into a shimmering holographic 3D surface plot that rotates and updates dynamically as he adjusts input variable sliders on a nearby panel.

Lumi moves a figure with rigid motions — translate, rotate, reflect — not stretching topology

Math · mit Atlas · 40 Min.

A colorful geometric art gallery with various polygons displayed, some showing lines of symmetry, others illustrating rotations and reflections with arrows and angles.

Lumi names injective, surjective, and bijective as mapping types — not even/odd tests

Math · mit Atlas · 50 Min.

A futuristic digital laboratory with holographic screens displaying function mappings between sets, arrows illustrating correspondences, and a guiding character pointing at injective and surjective examples.

Lumi lists domain as allowed inputs and range as produced outputs — not the fork-test for relations

Math · mit Atlas · 30 Min.

In a high-tech classroom bathed in cool blue light, digital whiteboards display the labeled graph of f(x) = 2x + 3 with color-coded domain and range brackets, while Atlas traces input-output arrows on a diagram and function notation examples pulse on the interactive board behind him.

Lumi compares set size by bijection — countable versus not, not Venn unions

Math · mit Atlas · 40 Min.

A cosmic-themed room with floating number sets, infinite sequences spiraling into the distance, and a large digital display contrasting finite and infinite set sizes.

Lumi translates a situation into variables and a relation — not solving the model

Math · mit Atlas · 45 Min.

A detailed classroom with a large interactive whiteboard displaying graphs, equations, and real-world images such as traffic flow, population growth charts, and environmental data.

Lumi extends sine to any angle on the unit circle — not right-triangle SOH-CAH-TOA as the whole job

Math · mit Atlas · 40 Min.

A high school classroom with a large interactive whiteboard displaying a unit circle, right triangles with labeled sides and angles, and graphs of sine and cosine functions. The guiding character, education/atlas, stands beside the board holding a protractor and pointing to the unit circle.

Lumi splits rationals from irrationals inside R — repeating versus not, not the N⊂Z⊂Q nest as a poster

Math · mit Atlas · 30 Min.

A serene library with shelves shaped like a number line; 'education/atlas' holding a magnifying glass over decimal expansions; holograms show rational and irrational numbers side by side.

Lumi tests even, odd, and monotonic as properties — not the one-output definition

Math · mit Atlas · 40 Min.

A modern classroom with a large digital whiteboard displaying graphs of various functions, including linear, quadratic, and exponential curves. The guiding character, education/atlas, stands beside the board holding a pointer, with mathematical symbols floating around.

Lumi wraps a Caesar shift as addition mod 26 — not a public-key protocol

Math · mit Atlas · 25 Min.

A high-tech futuristic lab with holographic encrypted messages floating in the air; 'education/atlas' holds a transparent tablet displaying a Caesar cipher wheel and encrypted text examples.

Lumi optimizes a linear objective on a polygonal feasible region — not a derivative critical point

Math · mit Atlas · 45 Min.

A bright workspace with large coordinate planes displayed on interactive whiteboards, 3D models of feasible regions, and 'education/atlas' guiding students through step-by-step optimization problems involving resource allocation.

Lumi reads a vector as magnitude plus direction — not a parametric line

Math · mit Atlas · 30 Min.

A physics lab with a large whiteboard showing a vector arrow drawn from the origin to point (3, 4) on a grid. Atlas stands beside a protractor and ruler, demonstrating how to measure vector magnitude and angle.

Lumi treats a differential equation as an unknown function — y' = f, not the meaning of f' as slope

Math · mit Atlas · 35 Min.

A laboratory-style classroom with a large screen showing animated slope fields and solution curves. The guiding character 'education/atlas' gestures to a simple differential equation on a digital board, while students use tablets to explore solutions interactively.

Lumi treats a matrix as an m-by-n array you add entrywise — not an inverse

Math · mit Atlas · 30 Min.

A high school classroom with a blackboard displaying colorful matrix grids and numeric arrays, while a teacher character points at a large three-by-three matrix on a digital screen, with students working on graph paper at their desks.

Lumi isolates a radical, squares, then checks extras — not rational polynomial fractions

Math · mit Atlas · 30 Min.

A science lab classroom with geometric shapes and radical symbols floating around a large screen, while a teacher character holds a square root symbol and a calculator next to an interactive prime factorization tree projected on the wall.

Lumi restricts arcsin to [-π/2, π/2] so the inverse is a function — not a sine graph's amplitude

Math · mit Atlas · 25 Min.

A futuristic study space with holographic displays showing graphs of inverse sine, cosine, and tangent functions. Atlas stands beside a digital tablet highlighting the restricted domains and ranges on the unit circle.

Lumi plots x+yi as the point (x,y) on an Argand diagram — not r cis θ and not e^{iθ}

Math · mit Atlas · 45 Min.

A vibrant classroom with a large digital whiteboard displaying an Argand diagram. The background includes geometric shapes and coordinate axes with complex numbers plotted as points; education/atlas stands pointing at the diagram with a stylus.

Lumi writes e^{iθ} = cos θ + i sin θ — Euler, not integer-power De Moivre

Math · mit Atlas · 25 Min.

A high-tech digital board displaying the famous Euler’s formula e^{iθ} = cos θ + i sin θ, with education/atlas demonstrating how it connects exponential and trigonometric forms. Complex plane diagrams show roots of unity.

Lumi treats topology as what stretching cannot erase — holes, not Euclidean distance

Math · mit Atlas · 30 Min.

A vibrant classroom filled with 3D models of doughnuts, Möbius strips, and coffee cups, with a large digital screen displaying colorful diagrams of open sets and continuous deformations.

Lumi uses the Fundamental Theorem: derivative of accumulation, and net change F(b)−F(a)

Math · mit Atlas · 30 Min.

A bright classroom with a large chalkboard displaying graphs of functions and their integrals, a student attentively watching the guiding character 'education/atlas' who stands next to a digital tablet illustrating the area under a curve.

Lumi nests N ⊂ Z ⊂ Q ⊂ R ⊂ C — which system a number lives in, not interval marks

Math · mit Atlas · 30 Min.

A futuristic classroom with holographic floating numbers and symbols representing various number sets; a digital blackboard displaying Venn diagrams of number systems; 'education/atlas' pointing to the diagrams.

Lumi reads a discrete random variable through its pmf — not binomial-versus-normal families

Math · mit Atlas · 35 Min.

A digital graph display showing bar charts of probabilities for dice rolls and coin toss outcomes with a character pointing at the graphs.

Lumi simplifies a polynomial fraction and excludes zeros of the denominator — not radicals

Math · mit Atlas · 30 Min.

A modern classroom where education/atlas stands at a large digital blackboard pointing to a towering algebraic fraction, while manipulable algebra tiles and an interactive factoring tree diagram fill the display beside the character.

Lumi plots an ordered pair as an address on two axes — not a distance formula

Math · mit Atlas · 25 Min.

An open field with a large transparent grid overlay, guiding character 'education/atlas' pointing to plotted points and labeled axes with arrows and grids.

Lumi finds where a line meets a circle by substitution — not writing the two equations

Math · mit Atlas · 45 Min.

A vibrant graphing environment showing coordinate axes with multiple lines and circles plotted. The guiding character, education/atlas, is holding a large compass and ruler, pointing at a circle with center at (3, -2) and radius 5, alongside a line intersecting the circle.

Lumi lists a sample space and uses 0≤P≤1 with complements — not counting-formula probability

Math · mit Atlas · 30 Min.

A vibrant classroom with colorful charts showing sample spaces, Venn diagrams, and factorial notation. The guiding character 'education/atlas' interacts with a large touchscreen displaying card decks and dice to illustrate probability scenarios.

Lumi writes a line as r(t) = r0 + t d — a vector path, not |v| of one arrow

Math · mit Atlas · 45 Min.

An outdoor setting featuring a three-dimensional coordinate grid with arrows representing vectors originating from the origin. Atlas stands beside a digital screen displaying parametric curve plots, with a drone illustrating motion along a parametric path. The environment includes labeled axes and vector components highlighted in bright colors.

Lumi writes a composite as a unique product of primes — not nested number systems

Math · mit Atlas · 30 Min.

A vibrant ancient Greek marketplace scene with prime numbers displayed on scrolls; 'education/atlas' holding a factor tree diagram; animated numbers splitting into prime factors.

Lumi hunts maxima by critical points where f'=0 or f' fails — not what f' means as slope

Math · mit Atlas · 40 Min.

An outdoor scene showing a hilly landscape with peaks and valleys superimposed with graphs highlighting maxima and minima points. Education/atlas holds a tablet displaying the first and second derivative graphs alongside the original function graph.

Lumi turns a word problem into an objective function — setup, not f'=0 and not LP vertices

Math · mit Atlas · 50 Min.

A modern laboratory with graphs of functions on screens, a 3D model of a box, and a chalkboard showing derivative calculations and constraint equations.

Lumi evaluates AND, OR, and NOT on 0-1 bits — not predicate quantifiers

Math · mit Atlas · 35 Min.

A modern laboratory setting showing circuit boards with blinking LEDs, schematic diagrams of AND, OR, and NOT gates on a large touchscreen, and a digital assistant character 'education/atlas' guiding students through circuit simulations.

Lumi writes a×10^k and counts significant figures — not nested number systems

Math · mit Atlas · 25 Min.

A modern science lab setting with microscopes, measuring instruments, and a large digital screen displaying large and small numbers in scientific notation. The guiding character 'education/atlas' holds a calculator and points at a chart illustrating significant figures rules.

Lumi inverts a square matrix so AA^{-1}=I — not a stretch-line eigenvalue

Math · mit Atlas · 40 Min.

A modern classroom with large touchscreen displays showing each matrix inversion step in color, a smartboard where Atlas demonstrates the determinant formula with labeled elements, and printed example problem sheets spread across student desks.

Lumi writes the equation of a line and of a circle — not their intersection

Math · mit Atlas · 30 Min.

A vibrant classroom with a large coordinate plane on a smartboard displaying a blue line and a red circle, with gridlines and axis labels. The guiding character, Atlas, stands next to the board holding a geometric compass and ruler.

Lumi splits center (mean, median, mode) from spread (range, IQR, SD) — not H0 tests and not 68-95-99.7

Math · mit Atlas · 30 Min.

A classroom setting with a data table on a projector, histograms illustrating data spread, and the character education/atlas explaining the concepts.

Lumi contrasts binomial counts with a continuous normal model — not 68-95-99.7 and not a generic pmf

Math · mit Atlas · 40 Min.

A split-screen image showing a binomial tree diagram alongside a smooth bell curve graph, with the character education/atlas explaining the relationship.

Lumi uses sin²θ + cos²θ = 1 to solve a trig equation — not amplitude and not inverse ranges

Math · mit Atlas · 45 Min.

A high-tech math lab with holographic projections of unit circles and sine wave graphs. The guiding character, education/atlas, gestures toward the projections while holding a tablet displaying formulas.

Lumi reads f'(x) as instantaneous rate and slope — not the f'=0 max/min test

Math · mit Atlas · 55 Min.

A futuristic lab with a digital graph displaying a curve and its tangent line. The guiding character, education/atlas, uses a stylus to mark points of inflection and maxima on the graph.

Lumi runs the ratio test on Σ a_n — not naming sequence versus series

Math · mit Atlas · 30 Min.

A chalkboard filled with sequences and series formulas, graphs showing convergent and divergent behavior, with education/atlas guiding students through various convergence tests like the Ratio and Root tests.

Lumi reads a fractal as the same pattern at smaller scales — not sensitive chaos

Math · mit Atlas · 45 Min.

A digital screen displaying zoomed images of the Mandelbrot set, Sierpinski triangle, and natural fractals like fern leaves and snowflakes, with colorful recursive patterns.

Lumi matches derivatives at a with a Taylor polynomial — not the ratio test

Math · mit Atlas · 35 Min.

A futuristic lab setting with holographic graphs of functions and their Taylor polynomial approximations, education/atlas explaining the concept of infinite sums approximating functions.

Lumi sums an arithmetic or geometric progression — S_n, not the nth-term formula

Math · mit Atlas · 50 Min.

An ancient library setting with scrolls and charts showing sequences and series. The guiding character, education/atlas, points to a large parchment displaying an arithmetic sequence and a geometric series formula.

Lumi names chaos as sensitive dependence on the start — not fractal self-similarity

Math · mit Atlas · 50 Min.

A futuristic lab setting with multiple screens showing fractal images and graphs of the logistic map iterations. The guiding character 'education/atlas' stands near a large touchscreen displaying a bifurcation diagram and a time series plot.

Lumi treats a limit as the value f approaches — the start of calculus, not the continuity checklist

Math · mit Atlas · 40 Min.

A vibrant classroom with a large chalkboard filled with graphs of functions approaching specific points. The guiding character, Atlas, stands near the board holding a compass, pointing to a curve approaching a limit point. Around the room, posters illustrate key calculus concepts like slopes of tangent lines and area under curves.

Lumi computes union, intersection, and complement on a Venn picture — not cardinality bijections

Math · mit Atlas · 40 Min.

A bright classroom with a large whiteboard displaying colorful overlapping circles forming Venn diagrams, sticky notes with set elements, and a futuristic tablet showing set operations dynamically.

Lumi treats a relation as a set of pairs that may fail to be a function — not domain/range lists

Math · mit Atlas · 40 Min.

A large touchscreen display showing arrows connecting elements from one set to another, with labeled domains and codomains. Nearby, flowcharts illustrate function properties with examples.

Lumi sorts parabola, ellipse, and hyperbola by cartesian standard form — not polar e

Math · mit Atlas · 40 Min.

An outdoor scene with a large chalkboard displaying colorful graphs of a parabola, ellipse, and hyperbola, set against a backdrop of rolling hills. Atlas points to the graphs with a wooden pointer, illustrating their shapes and axes of symmetry.

Lumi names an eigenvector as a line that A only stretches — Av = λv

Math · mit Atlas · 45 Min.

A modern classroom digital whiteboard displays matrix A = [[4, 1], [2, 3]] with colorful arrows showing how multiplication stretches two-dimensional vectors, while Atlas points to highlighted eigenvector directions that remain aligned along their own span without rotating.

Lumi reads a network as vertices plus edges, then degree — not a payoff matrix

Math · mit Atlas · 45 Min.

A futuristic digital interface displaying colorful graphs with nodes and edges, a network map of a city’s transportation system, and charts showing degrees of vertices.

Lumi drops Euclid's parallel postulate — many lines through a point, or none

Math · mit Atlas · 40 Min.

A classic study room with a large globe representing spherical geometry and a hyperbolic crochet model on a table. The guiding character 'education/atlas' points to diagrams contrasting Euclidean parallel lines with hyperbolic and spherical cases on a chalkboard.

Lumi solves two linear equations by elimination and shades a half-plane — not one-variable rays

Math · mit Atlas · 50 Min.

A high-tech math lab with dual coordinate-plane screens displays the precise intersection point of two graphed lines, while education/atlas points to the crossing point and a nearby notepad shows the substitution and elimination steps written out in sequence.

Lumi solves one-variable ax+b=c and graphs ax+b<c as a ray — not a two-line system

Math · mit Atlas · 40 Min.

An interactive math lab with a large touchscreen displaying number lines and graphs of linear inequalities. The guiding character 'education/atlas' uses a stylus to plot solutions while students observe.

Lumi splits an antiderivative from a definite integral — definition, not a substitution trick

Math · mit Atlas · 45 Min.

A bright science lab featuring a large digital screen plotting the area under a curve of y = x² between two points. Education/atlas uses a pointer to highlight the shaded region representing the definite integral, with formulas and examples displayed around.