Higher Level · maths-ai-hl · Interactive simulations
IB Mathematics: Applications and Interpretation HL Simulation Lab
70 unique simulations40 lessons2 sources
Every simulation here runs the real model — the applets a teacher would put on the board, from PhET and GeoGebra, not a video of one. Pick a topic, open a card inside its lesson to run it next to the notes, or jump straight to the source. A simulation that fits several lessons is listed under each of them (71 cards in all). Back to the IB Mathematics: Applications and Interpretation HL course →
1 · Number and algebra
- GeoGebraIB AI 1.1
Rounding on a number line
A number is placed on a scale; the green slider changes how many significant figures the scale is marked to.
- GeoGebraIn the lesson
Geometric sequence and its sum
Type a first term and a ratio r, then slide n: each term is the last one times r, and the sum S_n levels off when |r| < 1.
- GeoGebraIB AI 1.2
Arithmetic series as an area
Enter a first term and common difference (1 to 5), drag the slider to add terms, then use the buttons to reflect and translate the staircase.
- GeoGebraIn the lesson
Compounding more often
Set the rate with the top slider, tick the blue box and raise M (compoundings a year): the step graph nears a smooth curve.
- GeoGebraIB AI 1.3
Paying off a loan
Drag the green blob to set the total term of a mortgage; the monthly repayment is calculated and shown on the graph.
1.4Exponents and logarithms
1 sim- GeoGebraIn the lesson
Exponential and log as inverses
Tick both curves and the axis of symmetry, then slide a: every point (x, y) on y = a^x mirrors to (y, x) on y = log_a x.
- GeoGebraIn the lesson
Two equations, one intersection
Use the slider bars to change each equation; the point where the two lines cross is the solution of the system.
- GeoGebraIn the lesson
Determinant as an area scale factor
Type a 2×2 matrix and press Apply Matrix: the square's area is multiplied by |det|, and det = 0 squashes it onto a line.
1.7Eigenvalues and eigenvectors
1 sim- GeoGebraIn the lesson
Eigenvectors: when Av is parallel to v
Set the matrix entries, then turn the unit vector v until Av lies along v: that direction is an eigenvector, the stretch is λ.
- GeoGebraIn the lesson
Modulus and argument of z = a + bi
Slide a and b to move z = a + bi, then tick the boxes: r = √(a² + b²), and θ depends on the quadrant, not only on tan⁻¹(b/a).
- GeoGebraIB AI 1.8
Multiplying complex numbers
Drag Z or W on the Argand diagram and watch the product WZ; the points on the sliders reveal the scaling and rotation steps.
2 · Functions
- GeoGebraIn the lesson
Gradient and intercept of a line
Drag the two red points: the gradient is the rise over the run between them, and c is where the line crosses the y-axis.
- PhETIB AI 2.1
Graphing Lines
Drag a point and a slope handle to build a line; the Point-Slope and Slope-Intercept screens write its equation live.
- GeoGebraIB AI 2.1
Gradients of perpendicular lines
Move the points that set the gradient of your line; a perpendicular line is drawn and you compare the two gradients.
- GeoGebraIn the lesson
A function and its inverse
Type f(x), press Reflect and run the horizontal line test: the mirror image in y = x is a function only if each line cuts f once.
- GeoGebraIn the lesson
Key features of a quadratic model
Slide a, b and c: watch the vertex, axis of symmetry and intercepts move, and the parabola flip when a changes sign.
- PhETIB AI 2.3
Curve Fitting
Drag data points onto the graph and fit a linear, quadratic or cubic curve; the fit statistics update as the points move.
- GeoGebraIn the lesson
Exponential model with an asymptote
Slide C, A₀ and k in f(t) = C + A₀e^(kt): C moves the horizontal asymptote, and the sign of k switches growth to decay.
- GeoGebraIB AI 2.4
Half-life decay
Type the initial amount (green box) and the half-life (blue box), then drag the red point along the decay curve.
2.5Modelling with sinusoidal functions
2 sims- GeoGebraIn the lesson
Ferris wheel height as a sine model
Change the radius, centre height and seconds per revolution: they set the amplitude, principal axis and period of the height graph.
- GeoGebraIB AI 2.5
Amplitude, period and shift of sine
Sliders change the amplitude, period and shifts of a sine curve.
- GeoGebraIn the lesson
Logistic growth and carrying capacity
Slide c to move the level the curve flattens at (the carrying capacity L); a sets the start value c/(1 + a), b the growth rate.
- GeoGebraIn the lesson
Straightening an exponential with logs
Type a growth factor and an initial value: the points (x, log y) taken from the curve line up, with gradient log b and intercept log a.
3 · Geometry and trigonometry
- GeoGebraIn the lesson
Cone: volume, surface area and net
Move the sliders for h and r and follow the worked volume and total surface area; find the slant height on the net.
- GeoGebraIB AI 3.1
Angles inside a tetrahedron
A tetrahedron with base sides 8 cm and height 10 cm: drag the sliders to turn the view and tick the HINTS boxes.
3.2Trigonometry: sine and cosine rules
3 sims- GeoGebraIn the lesson
The sine rule in any triangle
Drag a vertex: the ratios a/sin A, b/sin B and c/sin C stay equal however the triangle changes.
- GeoGebraIB AA 3.2
Cosine rule in any triangle
Drag any vertex; the six parts of the triangle are shown together with c², a² + b² and a² + b² − 2ab cos γ.
- GeoGebraIB AA 3.2
The ambiguous case (SSA)
With angle A fixed, move point X to set side b and use the slider for side a; the possible triangles appear in red and green.
- GeoGebraIn the lesson
Bearings measured from north
Slide B to turn the arrow: the true bearing is measured clockwise from north (316°T); the compass form gives the acute angle (N44°W).
- GeoGebraIn the lesson
Voronoi cells for three sites
Drag sites A, B and C and watch the boundaries move; untick v to hide them and predict the edges before showing them.
- GeoGebraIB AI 3.4
Placing post offices
Drag the coloured dots to place 3 to 6 post offices; the applet draws the region each one should serve.
- GeoGebraIB AI 3.4
Growing circles build the diagram
Move the points, press Go and equal circles grow from every site until they meet.
- GeoGebraIn the lesson
Adding, subtracting and scaling vectors
Drag A and B and tick each box: a + b joins tip to tail, a − b adds the reverse of b, and the slider k stretches or flips a.
- GeoGebraIn the lesson
Relative position and closest approach
Tick the boxes in turn: B relative to A is r_B − r_A, and the two are closest when that is perpendicular to the relative velocity.
- GeoGebraIn the lesson
A graph and its adjacency matrix
Set the number of nodes, generate a random graph and tick Show Matrix: an entry is non-zero only where an edge joins two vertices.
- GeoGebraIn the lesson
Kruskal and Prim, step by step
Pick Kruskal's or Prim's with the slider, then move the vertical slider one step at a time and read why each edge is added.
- GeoGebraIB AI 3.8
Travelling salesman: five cities
Sliders set the weight of each edge between 5 cities; tick boxes choose edges and the total weight of your circuit is shown.
- GeoGebraIB AI 3.8
Shortest tour versus spanning tree
Drag the points or change how many there are; the travelling salesman path and the minimum spanning path are both drawn.
4 · Statistics and probability
4.2Descriptive statistics
3 sims- PhETIn the lesson
Center and Variability
On Mean & Median, kick several balls, then drag one far out: the mean chases the outlier while the median barely moves.
- GeoGebraIB AI 4.2
Frequency chart to cumulative frequency
Drag the blue points to change the class intervals and frequencies; the cumulative frequency graph and box plot redraw.
- GeoGebraIB AI 4.2
Mean and SD under X → aX + b
Drag the data points, then use sliders a and b to transform every value by X → aX + b; mean and SD are shown for both sets.
- PhETIn the lesson
Least-Squares Regression
Drag points onto the graph, tick Best-Fit Line and read r; then drag one point away and watch the line and r change.
- GeoGebraIn the lesson
Tree diagram without replacement
Set the total and the number of blue balls, then move the step slider: the second-draw fractions change because the first ball is not replaced.
- GeoGebraIB AI 4.4
Venn diagram drawn to scale
Set P(A), P(B) and P(A ∩ B) directly; the two circles and their overlap are drawn to scale.
- GeoGebraIn the lesson
Build a distribution and find E(X)
Enter values and probabilities that sum to 1, then simulate: the sample mean x̄ settles towards μ = Σ x·P(X = x) as n grows.
4.6Binomial and Poisson distributions
2 sims- GeoGebraIn the lesson
Binomial and Poisson side by side
Tick Binomial and slide n and p: the bars centre on np. Tick Poisson and raise λ: the peak follows λ and the spread widens.
- PhETIB AI 4.6
Plinko Probability
Drop balls through rows of pegs; set the number of rows and the chance of bouncing right, and compare the histogram with the ideal one.
4.7The normal distribution
2 sims- GeoGebraIn the lesson
Area under the normal curve
Type μ and σ, then drag the two red points: the shaded area is P(a < X < b), about 0.68 for μ ± σ and 0.95 for μ ± 2σ.
- GeoGebraIB AI 4.7
One standard deviation either side
100 random normal data points: randomise them, drag the movable lines to guess mean ± 1 SD, then tick Show Statistics.
- GeoGebraIn the lesson
χ² test: reject or not?
Slide the purple point along the axis: reject H0 once the purple p-value area is smaller than the blue α area.
- GeoGebraIB AI 4.8
Goodness-of-fit test in steps
Sliders choose a data set, a confidence level and a step number; each step of a χ² goodness-of-fit test is shown in turn.
- GeoGebraIn the lesson
What changes a confidence interval
Raise the confidence level and the red interval widens; raise n and it narrows, because the margin is z·σ/√n. Compare with the fixed blue one.
- GeoGebraIB AI 4.9
Confidence intervals from new samples
Each new sample draws a new confidence interval while the population mean stays fixed; change the confidence level and sample size.
5 · Calculus
- GeoGebraIn the lesson
From secant slope to tangent slope
Drag the red point along the curve towards the purple one: the secant's slope closes in on the tangent's slope, the derivative there.
- PhETIB AI 5.1
Calculus Grapher
Drag on the graph of f(x) to reshape it and watch the derivative graph redraw beneath; a tangent tool shows the slope at a point.
- GeoGebraIn the lesson
Tangent and normal at a point
Slide a to move P along y = x² and tick both boxes: the tangent has gradient 2a, the normal −1/(2a), so the two gradients multiply to −1.
- GeoGebraIB AI 5.2
Open box: volume against cut size
Sliders set the size of the cardboard and the cut size c; the box is drawn and its volume is plotted against c.
- GeoGebraIn the lesson
Definite integral as area
Type f(x) and slide the limits a and b: the shaded area equals F(b) − F(a), with the antiderivative F shown in the working.
- GeoGebraIB AI 5.3
Area so far is an antiderivative
g(x) is the area under f from a to x: tick Show f(t) and Show FTC, then drag the points x and a.
- GeoGebraIn the lesson
Trapezoidal rule strip by strip
Raise the number of trapezia and watch the sum close in on the exact area under the curve.
- GeoGebraIn the lesson
Reading f, f′ and f″ together
Trace the tangent along f and line up features with the cursor: f″ > 0 where f bends upward and f″ = 0 at an inflexion.
- GeoGebraIB AI 5.5
Derivatives of sin(ax) and cos(ax)
Move the red point along the curve to see its gradient; one slider swaps sine and cosine, another sets a in f(x) = sin(ax).
- GeoGebraIB AI 5.5
The product rule as a growing area
A rectangle whose sides u and v both change with t; follow how its area uv grows.
- GeoGebraIn the lesson
Volume of revolution about the x-axis
Keep f(x) = sqrt(x) and slide b: the solid grows and the volume is π∫ y² dx from a to b, here π·b²/2 when a = 0.
- GeoGebraIB AA 5.6
Area between two curves
Drag the red points along the x-axis to set the limits; type your own two functions to see the area between them calculated.
- GeoGebraIB AA 5.6
Area so far is an antiderivative
g(x) is the area under f from a to x: tick Show f(t) and Show FTC, then drag the points x and a.
- GeoGebraIn the lesson
Euler's method on a slope field
Drag A to set the start point and shrink Δx: each step adds Δy = (dy/dx)·Δx, and the path follows the slope field more closely.
- GeoGebraIB AI 5.8
Slope field plotter
Type dy/dx = f(x, y) in the input box; tick the Solution boxes and drag points A to D to move the solution curves.
- GeoGebraIn the lesson
Phase portrait of a coupled system
Type x' and y' for a system and drag the green start points: here the paths spiral outwards because the eigenvalues 1 ± i are complex.
Each simulation stays the property of its source and is credited on the card inside the lesson. Back to the IB Mathematics: Applications and Interpretation HL course →