Higher Level · maths-aa-hl · Interactive simulations
IB Mathematics: Analysis and Approaches HL Simulation Lab
73 unique simulations43 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 (74 cards in all). Back to the IB Mathematics: Analysis and Approaches HL course →
1 · Number and algebra
- GeoGebraIn the lesson
Sequence terms and partial sums
Choose Geometric, tick Show series, then drag r: the partial sums level off at a limit only when |r| < 1.
- GeoGebraIB AA 1.1
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.
1.2Exponents and logarithms
1 sim- GeoGebraIn the lesson
The three laws of logarithms
Slide p and q and check each line: log(pq) = log p + log q, log(p/q) = log p − log q, log(p^q) = q log p.
1.3The binomial theorem
3 sims- GeoGebraIn the lesson
Binomial expansion, n = 1 to 6
Slide n from 1 to 6, then change a and b: match each coefficient to a row of Pascal's triangle times powers of a and b.
- GeoGebraIB AA 1.3
Pascal's triangle and nCr
Sliders n (row) and m (position in the row) pick out one entry of Pascal's triangle; a button resets both to 0.
- GeoGebraIB AA 1.3
Binomial approximation near x = 0
Vary a, n and the number of terms kept in the expansion of (1 + ax)^n and compare the cut-off polynomial with the true curve.
- GeoGebraIn the lesson
Permutations versus combinations
Set n and r: nPr is always r! times nCr, because each choice of r objects can be arranged in r! orders.
- GeoGebraIn the lesson
Induction as a chain of steps
Place the blue arrow (the inductive step) at each number in turn: truth passes from k to k + 1 only after the base case holds.
- GeoGebraIn the lesson
Cartesian, polar and Euler form
Drag the point: its distance from O is the modulus and the angle from the positive real axis is the argument.
- GeoGebraIB AA 1.6
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.
- GeoGebraIn the lesson
The nth roots of a complex number
Slide n and drag α: the n roots lie on a circle of radius |α|^(1/n), spaced 2π/n apart, a regular polygon.
- GeoGebraIB AA 1.7
Powers of a complex number
Drag z (or type it in the input box) and move the slider n, or press start/stop, to see w = z^n.
1.8Systems of linear equations
1 sim- GeoGebraIn the lesson
Three planes and unique solutions
Move slider a until the common point disappears (no unique solution), then slider b until the three planes share a line.
2 · Functions
- PhETIn the lesson
Graphing Lines
On Point-Slope, drag the point and the slope handle and watch y − y1 = m(x − x1) update; then try Slope-Intercept.
- GeoGebraIB AA 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
Restrict the domain, get an inverse
Slide left and right to shrink the domain until the green reflection in y = x passes the vertical line test.
- GeoGebraIB AA 2.2
Building the graph of g(h(x))
Move x0 along the x-axis: the construction carries it through h and then g, and the red point D marks g(h(x0)).
2.3Quadratic functions and equations
2 sims- GeoGebraIn the lesson
Three forms of a quadratic
Tick a form and move its sliders: p and q are the x-intercepts, (h, k) is the vertex, c is the y-intercept.
- GeoGebraIB AA 2.3
Discriminant and number of roots
Sliders a, b and c reshape the parabola so you can count its x-intercepts.
2.4Transformations of graphs
1 sim- GeoGebraIn the lesson
Transforming y = f(x) with sliders
In g(x) = a f(b(x − h)) + k, a stretches vertically, b scales x by 1/b, h and k translate. Try a = −1.
- GeoGebraIn the lesson
Asymptotes of y = (ax + b)/(cx + d)
Slide c and d: the vertical asymptote sits at x = −d/c. Slide a: the horizontal asymptote moves to y = a/c.
- GeoGebraIn the lesson
Exponential and log as mirror images
Slide a: (0, 1) and (1, a) on y = a^x become (1, 0) and (a, 1) on y = log_a x, a reflection in y = x.
- GeoGebraIn the lesson
Remainder and factor theorems
Drag the slider to a value a: the vertical segment is the remainder f(a), and it vanishes when (x − a) is a factor.
- GeoGebraIn the lesson
Solving |ax + b| against c
Tick <, > or = and slide c: |ax + b| < c gives one interval between the crossings, > c gives the two outer parts.
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 AA 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.
3.3The unit circle and radian measure
2 sims- GeoGebraIn the lesson
Unit circle exact values
Drag the point round the circle; at π/6, π/4 and π/3 its coordinates (cos θ, sin θ) appear in exact form.
- GeoGebraIB AA 3.3
Radians and arc length
Drag the black points to change the angle and the radius and compare the arc lengths.
- GeoGebraIn the lesson
Amplitude, period and shift of sine
Slide A and ω: the amplitude is A and the period is 2π/ω; B lifts the midline. Tick Show properties to check.
- PhETIB AA 3.4
Trig Tour
Drag the point round the unit circle; choose sin, cos or tan to see its value and its graph drawn against the angle.
- GeoGebraIn the lesson
Every solution in the interval
Set a and the start and end of the interval, count where y = a cuts the curve, then tick Show solutions.
- GeoGebraIn the lesson
Where sin(α + β) comes from
Step the slider to 4: the left edge sin(α + β) matches the right edge, sin α cos β plus cos α sin β. Drag A or B to vary the angles.
- GeoGebraIn the lesson
Adding, subtracting, scaling vectors
Drag A and B and tick addition: b is redrawn from the tip of a, so a + b closes the triangle. Try k < 0 for ka.
3.8Vector equations of lines
1 sim- GeoGebraIn the lesson
The line r = a + λb in 3D
Slide λ: P moves along the line through A in the direction AB; λ = 0 gives A and λ = 1 gives B.
3.9Scalar and vector products; planes
2 sims- GeoGebraIn the lesson
Vector product in 3D
Change the components of u and v and turn the 3D view: u × v stays at right angles to both vectors.
- GeoGebraIB AA 3.9
Normal form of a plane
Move the sliders for λ and μ to carry point R around the plane; the vector AR and the normal n are drawn.
4 · Statistics and probability
4.1Descriptive 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 AA 4.1
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.
- GeoGebraIB AA 4.1
Frequency chart to cumulative frequency
Drag the blue points to change the class intervals and frequencies; the cumulative frequency graph and box plot redraw.
- GeoGebraIn the lesson
Why it is called least squares
Drag the two Drag Me points to shrink the sum of squares, then tick Show LSRL and Show Centroid.
- GeoGebraIB AA 4.2
Getting a feel for r
Choose the number of points with the slider, drag them anywhere, guess the correlation coefficient and then reveal it.
- GeoGebraIn the lesson
Venn diagram drawn to scale
Slide P(A ∩ B) and check P(A ∪ B) = P(A) + P(B) − P(A ∩ B); the overlap shrinks as the events separate.
- GeoGebraIB AA 4.3
Probability tree with rolling balls
A tree diagram with an adjustable probability on each branch; 100 balls roll through and are counted at each exit.
- GeoGebraIn the lesson
Build a distribution and find E(X)
Give each value a probability so the total is 1, run the simulation and watch the sample mean settle towards E(X).
4.5The binomial distribution
2 sims- GeoGebraIn the lesson
Binomial probabilities for n = 20
Slide p and watch the bars centre on the mean 20p; slide r for P(X = r), then tick cumulative for P(X ≤ r).
- PhETIB AA 4.5
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.
4.6The normal distribution
2 sims- GeoGebraIn the lesson
Normal probability as an area
Set a = μ − σ and b = μ + σ: the shaded area P(a < X < b) is about 0.68 whatever μ and σ are.
- GeoGebraIB AA 4.6
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
Bayes' theorem: testing for a disease
Set the incidence to 1% and the false positive rate to 5%: the pie chart shows how small P(infected | positive) is.
- GeoGebraIB AA 4.7
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
From bars to a density function
Increase n to narrow the bars, then change m: P(0 ≤ X ≤ m) is the total area of the bars up to m.
- GeoGebraIB AA 4.8
Density and cumulative distribution
Shows a probability function or density next to its cumulative distribution function for discrete and continuous examples.
5 · Calculus
- GeoGebraIn the lesson
From chord to tangent as h → 0
Tick secant line and show slopes, then slide h towards 0: (f(x+h) − f(x))/h closes in on the tangent's slope f'(x).
- PhETIB AA 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
The product rule as a growing area
Slide t and tick rate of change: the area uv grows by two strips, u·dv/dt and v·du/dt, which is the product rule.
- GeoGebraIB AA 5.2
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).
- GeoGebraIn the lesson
Reading f, f′ and f″ together
Drag the point on the curve and tick Show f ' and Show f '': the tangent is flat where f ' = 0; f '' changes sign at the inflection.
5.4Optimization and kinematics
2 sims- GeoGebraIn the lesson
Open box: volume against cut size
Slide c to change the cut, then switch Function ON: V = c(l − 2c)(w − 2c) peaks where dV/dc = 0.
- GeoGebraIB AA 5.4
Particle motion: s, v and a
Press RESET then START: a particle moves along the vertical axis while position (blue), velocity (green) and acceleration (red) are graphed.
- GeoGebraIn the lesson
The family of antiderivatives F(x) + C
Slide C: every curve F(x) + C is the same shape shifted vertically, so all have the same gradient f(x) at each x.
- 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
u-substitution keeps the area
Drag the limit points in the left picture: the region under the original integrand and the one under g(u) keep the same area.
- GeoGebraIB AA 5.8
Choosing the parts
Type two factors f(x) and g(x) of an integrand and see the first step of integration by parts set out for that choice.
- GeoGebraIn the lesson
Slope field plotter
Type dy/dx as a function of x and y, then drag point A: each solution curve follows the little slope lines through its starting point.
5.10Maclaurin series
2 sims- GeoGebraIn the lesson
Maclaurin polynomials for e^x
Raise the slider n to add terms, then move point A away from x = 0 to see the error between e^x (red) and the polynomial (blue).
- GeoGebraIB AA 5.10
Maclaurin: the standard functions
Plots e^x, sin x, cos x, ln(1 + x) and other functions together with their Maclaurin polynomials up to order 10.
Each simulation stays the property of its source and is credited on the card inside the lesson. Back to the IB Mathematics: Analysis and Approaches HL course →