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9 articles on explainer: the maths or physics explained, ready-to-teach lesson plans, and how to build each slide in Tracé.

Three trajectories at the same speed: 30° and 60° land together at 35.3 m, 45° goes furthest at 40.8 mphysics · mechanics · high school · explainerWhy 45° Is the Best Launch Angle (and When It Isn’t)The proof that 45° gives the longest throw, why 30° and 60° land in the same place, and why real throws use less: launch height and air resistance.Two balls racing from the same start: the one on a cycloid curve is ahead of the one on a straight rampphysics · mechanics · calculus · explainerThe Brachistochrone: Why the Fastest Path Is a CurveWhy a ball on a cycloid beats one on a straight ramp (3.17 s against 3.76 s for a 20 m drop), the 1696 challenge behind it, and how to animate the race.The Taylor polynomial 1 + x + x²/2 + x³/6 + x⁴/24 closing in on the curve of eˣcalculus · university · explainerTaylor Series: Approximating Functions with PolynomialsHow Taylor series build eˣ and sin x from polynomials term by term, how accurate a few terms are, the small-angle approximation, and a click-by-click slide.Newton's method on x² − 2: tangents from x₀ = 3 down to the root √2, with the values in a tablecalculus · numerical methods · university · explainerNewton's Method, Animated: One Tangent per ClickTeach Newton's method visually: follow each tangent down to the x-axis, compare it with bisection, and see why it converges so fast, or fails.Euler's method with step 0.5 for y′ = y, below the exact curve eˣ, over the direction fielddifferential equations · calculus · university · explainerEuler's Method and Direction Fields, VisuallyA visual lesson on differential equations: draw the direction field, follow solution curves, then watch Euler’s method step away from the exact solution.A 16-QAM constellation in noise with decision boundaries and Gray-coded bit labelssignals · engineering · university · explainerQAM Constellation Diagrams, Explained (BPSK to 256-QAM)What a constellation diagram shows, why 256-QAM needs a cleaner channel than QPSK, Gray coding, noise and phase errors, with animated examples.Sequence diagram of the TCP three-way handshake: SYN, SYN-ACK and ACK between client and servernetworking · engineering · university · explainerHow to Explain the TCP Three-Way HandshakeTeach the TCP handshake with a sequence diagram that plays one message per click: SYN, SYN-ACK, ACK, sequence numbers, and what happens when a packet is lost.The Koch snowflake at depths 0, 2 and 5geometry · sequences · high school · explainerFractals in the Classroom: Koch, Sierpinski, MandelbrotThree fractals that hook a class: the Koch snowflake step by step, the Sierpinski triangle and the Mandelbrot set, with the maths behind each.Bifurcation diagram of the logistic map from r = 2.5 to 4, with a line at r = 3.56sequences · dynamical systems · university · explainerThe Logistic Map: Teaching the Road to ChaosOne simple rule, x → r x (1 − x), goes from a steady state to cycles to chaos as r grows. Cobweb diagrams and the bifurcation diagram, explained.