A-Level · 9709 · Interactive simulations
68 unique simulations37 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 (78 cards in all). Back to the A Level Mathematics course →
Tick each form in turn and move its sliders: the factored form shows the roots p and q, the vertex form the turning point (h, k).
Adjust A, B and C in y = Ax² + Bx + C and watch the discriminant value alongside the number of real roots.
Change a, b, c in standard form or a, h, k in vertex form and watch the parabola, its vertex and its roots move.
Move the sliders to restrict the domain until f is one-one, so that an inverse function can be defined.
Drag the blue point along the x-axis and follow how its image is built in two steps, one function after the other.
Move the Domain Shadow and Range Shadow sliders to squash the graph onto the x-axis and onto the y-axis.
Two perpendicular lines with their gradient triangles: drag the points to tilt one line and the other follows.
Slider r changes the radius of the circle; drag P round it to move the point of contact and its tangent.
Drag two points to set a line and read its gradient, or build a line in point-slope and slope-intercept form.
A sector with a radius slider and a point you drag round the arc; tick boxes show the angle in degrees or radians.
Drag the point round the circle and tick Show π/6, π/4 or π/3: its coordinates at those angles are the exact values of cos θ and sin θ.
Drag the point round the unit circle and watch sin, cos or tan trace out its graph, in degrees or radians.
Each of four sliders controls one transformation of the sine wave: amplitude, frequency, horizontal shift and vertical shift.
Bars for the partial sums of a geometric series, with sliders for the first term a and the ratio r.
Sliders set the function and the number of terms; the blue polynomial is the truncated binomial expansion of the red curve.
Drag a point along a curve: the gradient of its tangent is plotted underneath as you go, tracing the gradient function.
Step through three reveals: the point P on y = f(x), the tangent at P, then the normal at P.
Sculpt a curve f(x) with the drawing tools and see the graph of its derivative update underneath.
Type a function and slide the two limits: the shaded area matches the worked integral, F(b) − F(a).
A region under a curve spun about the x-axis: set the limits a and b and see the solid it sweeps out.
Shape f(x) and watch the graph of its integral build up from the signed area under the curve.
Change a, b and c: the solutions of |ax + b| < c are the x-values where the V-shaped graph lies below the line y = c.
The graph of y = log_b x with a slider for the base b.
Change the base a of f(x) = aˣ: the left panel shows f and its derivative, the right panel shows their difference.
Four sliders reshape f(x) = c·a^(x − h) + k, changing the graph and its equation together.
Drag D round the circle: the length DE read two ways gives sin 2α = 2 sin α cos α, and OE gives cos 2α = 2cos²α − 1.
A diagram for a sin θ + b cos θ with sliders for a, b and θ.
Move the slider to plot the points (t, sec t) against the fixed graph of cos t.
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.
Drag the red point on the bank where river cable meets land cable, and change how much dearer river cable is.
A cubic with its first and second derivatives evaluated at a point you drag along the x-axis.
Raise the number of trapezia and watch the sum close in on the exact area under the curve.
Drag the red points on the x-axis to set the limits, or type in your own pair of functions.
Type a rearrangement g(x) and step the iterations: a staircase or cobweb closing on y = x means the sequence converges to the root.
Drag the curve to move its roots; the decimal search homes in on the interval where f(x) changes sign.
Tick <, > or = and slide c: |ax + b| < c gives one interval between the crossings, > c gives the two regions outside them.
Move A along y = eˣ: its gradient equals its y-value, while the mirror point A′ on y = ln x has gradient 1/x.
Change the base a of f(x) = aˣ: the left panel shows f and its derivative, the right panel shows their difference.
Change the starting value a and the rate r, then move the point along the curve to read the amount at time t.
Drag D round the circle: the length DE read two ways gives sin 2α = 2 sin α cos α, and OE gives cos 2α = 2cos²α − 1.
A diagram for a sin θ + b cos θ with sliders for a, b and θ.
Move the green slider to trace the parametric curve, then the blue slider to see the gradient, tangent and normal.
Type a polynomial equation in x and y, then drag the red point C round the curve; P plots the gradient at C.
Drag the red points on the x-axis to set the limits, or type in your own pair of functions.
Type a rearrangement g(x) and step the iterations: a staircase or cobweb closing on y = x means the sequence converges to the root.
Drag the curve to move its roots; the decimal search homes in on the interval where f(x) changes sign.
Slide λ and watch P travel along the line: its position vector is always a + λb.
Move the slider to slide B along the line l and watch the length AB; a check box reveals the coordinates.
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.
Drag A round the Argand diagram and see where the modulus r and argument θ in the highlighted polar form come from.
Drag two complex numbers on the Argand diagram and compare their moduli and arguments with those of the product.
Drag z₁ while keeping the length |z₁ − w| equal to k, and see the path it is forced to follow.
On Friction, push the crate a little harder each time: it stays in equilibrium while friction can match your push, and accelerates once it cannot.
Drag the top of the slope to change its angle; the applet reports when the friction needed reaches its maximum.
Drag vectors onto the grid, show their components and build the resultant by adding them.
Slide the points on the velocity–time graph up or down: the position and acceleration graphs redraw to match.
Set two masses and velocities and run the collision: total momentum before equals total momentum after, whatever the elasticity.
Set the incline angle and both masses, press Run, then tick the box to check the acceleration and the tension.
Push a crate with a chosen applied force and watch the resultant force, speed and acceleration, with or without friction.
Show the bar graph and release the skater: kinetic and potential energy trade places while the total stays level; add friction and it drains to thermal energy.
Drag the blue points to change the class intervals and frequencies: the cumulative frequency graph and box plot update.
Drag the data points, then use the sliders for a and b to transform every value to aX + b.
Kick balls to build a data set, then show its median, mean, range and interquartile range.
Set P(A), P(B) and P(A∩B) and watch the regions resize. Make P(A∩B) equal P(A) × P(B), then make it 0.
Change the probability on each branch of the tree, then roll 100 balls through it and count where they come out.
Slide n and p: the bars stay centred near np, symmetric at p = 0.5 and skewed either side of it.
Type a probability for each value, make them total 1, then simulate and compare the frequencies.
Drop balls through rows of pegs and watch the bins fill into a binomial histogram.
Set μ and σ, then slide a and b: the shaded area under the curve is P(a < X < b).
Shows B(n, p) with P(X ≤ r); tick boxes overlay N(np, npq) and then the continuity-corrected area up to r + 0.5.
Change the parameter μ and watch the bars shift and spread; set the interval to read P(X ≤ k) straight off the chart.
A binomial distribution B(n, p) drawn over the Poisson with the same mean, with sliders for n and p.
Slide a and b: the mean follows aμ + b, but the spread is scaled by a alone — b never changes it.
Raise n for narrower bars, then change m: P(0 ≤ X ≤ m) is the total bar area, which becomes the area under f(x).
Let the sample means pile up, then stop, clear the data and change the sample size: larger samples give a narrower spread.
Take new samples and watch the interval move while the population mean stays fixed; vary the % level and the sample size.
Sliders set the mean, standard deviation and sample size n; the blue curve is X and the red curve is the mean of n values.
Change α and the true mean μ: the Type I area sits under H₀, and the Type II area shrinks as μ moves away from it.
Each simulation stays the property of its source and is credited on the card inside the lesson. Back to the A Level Mathematics course →