Manipulate geometric, trigonometric, statistical, and financial models in real time with 60fps interactive SVG canvases.
Visualize how fixed monthly payments shift from interest-heavy early installments to principal reduction, with interactive extra-payment acceleration.
Compare standard 15-year and 30-year mortgage payoff trajectories and inspect how additional monthly principal curtailments compress total interest cost.
Analyze 36-, 48-, 60-, 72-, and 84-month vehicle financing structures and see total interest overhead vs. monthly payment affordability.
Plot student debt balance decay under standard 10-year and extended 20-year schedules and quantify interest saved per $50/month extra payment.
Separate your future portfolio balance into initial principal, cumulative monthly contributions, and pure exponential compound interest earnings.
Model long-horizon retirement portfolio accumulation and compute sustainable monthly passive income under the 4% safe withdrawal rule.
Track disciplined monthly SIP installments and inspect the wealth-multiplier ratio between total invested capital and terminal maturity value.
Compare nominal savings growth against inflation-adjusted purchasing power and visualize how compounding CPI erodes uninvested cash.
Sweep any angle around the unit circle ($R = 1$) to inspect live $\sin\theta$, $\cos\theta$, and $\tan\theta$ geometric projections alongside exact quadrant signs.
Watch how uniform circular motion unrolls into continuous sinusoidal waveforms $y = A\sin(\omega\theta)$ and phase-shifted cosine curves.
Convert seamlessly between degrees ($\theta^\circ$), exact $\pi$ radians, and subtended arc length $s = R\theta$ on an interactive polar dial.
Scale base $a$ and altitude $b$ to dynamically compute hypotenuse $c$, acute angles $\alpha$ and $\beta$, altitude to hypotenuse $h_c$, and inscribed/circumscribed radii.
Inspect the Euclidean geometric area proof of $a^2 + b^2 = c^2$ with live square projections on all three sides of a dynamic right triangle.
Calculate surveyor sightline distance, target height, and inclination grade (%) from horizontal ground run and vertical rise.
Compare two fractions $a/b$ and $c/d$ side by side using proportional tape bars and circular sector slices, with live addition, subtraction, multiplication, and GCD reduction.
Line up two fraction bars and circular dials to visually test equivalence ($ad = bc$), cross-multiplication inequality ($<, =, >$), and LCD partitioning.
Convert improper fractions ($a > b$) into whole unit bars plus fractional remainders $W\frac{r}{b}$ and see how multiple fractions combine past $1.0$.
Translate any fraction $a/b$ into its exact or repeating decimal expansion, percentage of 100%, and simplified part-to-whole ratio.
See percentages come alive on a precision $10 \times 10$ hundredths matrix alongside proportional value bars for $P\%$ of $V$, part-of-whole, and modified totals.
Visualize positive growth ($+P_1\%$) and negative drawdown ($-P_2\%$) on a 100-block grid and see why a $50\%$ loss requires a $+100\%$ recovery gain.
Decompose unit price, promotional discounts, sales tax, cost of goods sold (COGS), gross margin %, and markup % in one unified visual block grid.
Place masses $w_1, w_2, w_3$ at positions $x_1, x_2, x_3$ along a physical balance beam and watch the equilibrium fulcrum track the exact weighted mean.
Drag extreme outlier values along a number-line beam to see why the arithmetic mean shifts by torque while the median remains robust.
Balance Homework, Midterm, and Final Exam syllabus weights to see your exact overall course grade and how much the final exam shifts your standing.
Decompose any composite integer into its recursive binary factor tree down to highlighted prime leaf nodes, canonical exponent form, and divisor count.
Compare the prime factor trees of two integers $A$ and $B$ to extract their Greatest Common Divisor ($\gcd$) intersection and Least Common Multiple ($\text{lcm}$) union.
Compare ordered arrangements $P(n,r)$ against unordered subsets $C(n,r)$ with Pascal's triangle highlighting and factorial cancellation trees.
Navigate Pascal's Triangle up to row $n = 12$, highlight recurrence pairs $\binom{n-1}{r-1} + \binom{n-1}{r} = \binom{n}{r}$, and expand $(a+b)^n$ polynomial terms.
Adjust $|A|$, $|B|$, intersection $|A \cap B|$, and universe $|U|$ to visualize Union $A \cup B$, Symmetric Difference $A \triangle B$, and De Morgan complements.
See how conditioning on event $B$ shrinks the sample space from $U$ down to circle $B$, making $P(A \mid B) = \frac{|A \cap B|}{|B|}$ visually unmistakable.
Slide observed value $x$, mean $\mu$, and standard deviation $\sigma$ to shade left-tail, right-tail, and central Gaussian cumulative probabilities $\Phi(Z)$.
Inspect symmetric two-tailed confidence regions $\mu \pm z_{\alpha/2}\sigma$ and tail rejection zones $\alpha/2$ for hypothesis testing.
Visualize the three-sigma empirical rule bands ($\mu \pm 1\sigma$, $\mu \pm 2\sigma$, $\mu \pm 3\sigma$) and Six Sigma quality control thresholds.
Manipulate coefficients $a, b, c$ in real time to see concavity, axis of symmetry $x = -b/(2a)$, vertex coordinates, y-intercept, and real or complex roots.
Observe how the discriminant $\Delta = b^2 - 4ac$ governs whether a parabola crosses the x-axis twice ($\Delta > 0$), touches tangentially ($\Delta = 0$), or floats in the complex plane ($\Delta < 0$).
Plot two Cartesian coordinates $(x_1, y_1)$ and $(x_2, y_2)$ to visualize the $\Delta y / \Delta x$ slope triangle, y-intercept $b$, x-intercept, and perpendicular normal line.
Compute exact 2D Euclidean distance $d = \sqrt{\Delta x^2 + \Delta y^2}$, midpoint $M$, and Manhattan taxicab distance on an interactive coordinate grid.
Rotate and scale vectors $\vec{A}$ and $\vec{B}$ to visualize tip-to-tail parallelogram resultant $\vec{R} = \vec{A} + \vec{B}$, scalar dot product $\vec{A}\cdot\vec{B}$, and 2D cross magnitude.
Resolve two concurrent force vectors $\vec{F}_1$ and $\vec{F}_2$ into orthogonal $F_x, F_y$ components, net resultant $\vec{F}_{net}$, and the exact equilibrant vector $- \vec{F}_{net}$ required for static equilibrium.
Plot Total Revenue $TR = P \cdot Q$ against Total Cost $TC = FC + VC \cdot Q$ to pinpoint the exact Break-Even Quantity $Q^*$ where loss turns into net operating profit.
Compare high-fixed-cost (SaaS/manufacturing) vs. high-variable-cost (service/retail) cost structures and quantify Degree of Operating Leverage $\text{DOL} = \frac{CM}{\text{EBIT}}$.
Adjust circle radius $R$ and central angle $\theta$ to visualize and compute sector area, circular segment area, arc length $s$, chord length $c$, apothem $d$, and sagitta height $h$.
Simulate 2D ballistic flight from launch velocity $v_0$, angle $\theta$, initial cliff elevation $h_0$, and gravity $g$ with apex markers and velocity vector decomposition.
Superpose two harmonic waves $y_1(x)$ and $y_2(x)$ with independent frequencies, amplitudes, and phase shift $\Delta\phi$ to observe constructive/destructive interference and acoustic beats.
Trace Prior probability $P(A)$, True Positive sensitivity $P(+|A)$, and False Positive rate $P(+|\neg A)$ through a 2-stage branching tree to see why rare conditions trigger the Base-Rate Fallacy.
Adjust source Voltage ($V$) and load Resistance ($R$) on a live schematic and 4-quadrant Ohm's Power Wheel to inspect Current ($I$), Joule Heating Power ($P = V \cdot I$), and daily kWh energy.