Conceptual Learning with Interactive Applets is a project to build highquality webbased applets and supporting resources for enhancing conceptual understanding in undergraduate mathematics and statistics. Our applets are built using GeoGebra.
The project is based in the University of Melbourne, School of Mathematics and Statistics, and funded by a University of Melbourne Learning & Teaching Initiatives grant.
Select a subject to view applets for that subject, or browse the full collection below.

Bases and Coordinates in R2
This applet demonstrates the concept of coordinate vectors in $\mathbb R^2$ A basis $\mathcal B$ of a vector space $V$ is a linearly independent spanning set. One useful feature of a basis is that it gives rise to a way of writing coordinates on $V$. Any vector $\mathbf v \in V$ can be written uniquely […]

Visualising the GramSchmidt Algorithm
This applet demonstrates the GramSchmidt algorithm performed in $\mathbb R^3$. The GramSchmidt algorithm converts a basis of an inner product space into an orthonormal basis. It does this by building up the orthonormal basis one vector at a time. For each vector in turn, we remove any component that is parallel to the vectors which […]

Visualising linear transformations in R2
This applet shows the geometric effect of a linear transformation $T: \mathbb R^2 \to \mathbb R^2$. You can type a matrix $M$ on the right hand side of the applet, and then click Play to see how the vertices of a triangle are transformed when multiplying by $M$. Other resources: Geogebratube page for this applet

Visualising linear transformations in R3
This applet shows the geometric effect of a linear transformation $T$ in $\mathbb R^3$. You can type a 3×3 matrix $M$ on the right hand side of the applet, and then click Play to see how the vertices of a cube are transformed when multiplying by $M$. Can you see if the corresponding transformations are […]

This applet helps visualise the surface generated by cylindrical coordinates using r,θ and z. Click and drag on the sliders on the left to adjust the ranges for r,θ and z. Geogebratube page for this applet

This applet visualises surfaces generated by spherical coordinates using r,θ and φ. Click and drag on the sliders on the left to change the values for r,θ and φ. Click and drag on the graph to change/rotate the view. Geogebratube page for this applet

This applet shows a solution of the heat equation, a partial differential equation from MAST20029 Engineering Mathematics.

This applet visualises the span of two vectors in R3 using linear combinations.

This applet shows a line in R2 and the vector form of its equation.

This applet shows a plane in R3 and the vector form of its equation.

This applet shows a line in R3 and the vector form of its equation.

This applet shows how the determinant is unaffected by the elementary row operation of addition of a scalar multiple of a row to another row.

This applet shows how the determinant is unaffected by the elementary row operation of addition of a scalar multiple of a row to another row.

This applet shows the row, column, and solution spaces of a 3×3 matrix M.

This applet shows how the column space, solution space, rank and nullity of a matrix M change as you append additional columns. Initially the matrix M has a single column. You can add extra columns to M by editing the text boxes on the right of the applet, and clicking the ‘Append column’ button. The […]

This applet explores a geometric interpretation of the parameter t in the parameterisation of the standard hyperbola using cosh and sinh.

This applet explores the velocity vector of a parametric curve, and its relationship to the chord r(t+h)r(t) and the difference quotient.

This applet illustrates the connection between a confidence interval, a formal hypothesis test, and the pvalue of a hypothesis test.

This applet gives a visualisation of the concept of statistical power, and helps illustrate the relationship between power, sample size, standard deviation and difference between the means.

This applet illustrates partitioning of variability into explained (fitted) and unexplained (residual) variability.

This applet illustrates the partitioning of variability into explained and unexplained variability, in the context of ANOVA.

This applet illustrates the effect of a linear transformation in R2 on the unit circle/unit disk, and the geometric meaning of eigenvectors, eigenvalues and determinant.

This applet displays the distribution for the order statististics of a sample of size n from an arbitrary population distribution.

This applet shows the maximum likelihood estimator and (log) likelihood function for several statistical models.

This applet illustrates the εδ definitions of the limit and continuity of a function. It can be used to investigate (non)convergence or (dis)continuity of real functions, including the Dirichlet everywhere discontinuous function and variants.

This applet illustrates the definition of derivative as the limit of the gradient of a chord.

This applet illustrates the εM definition of convergence of a sequence.

This applet illustrates upper and lower Riemann sums and refinement of partitions.

This applet shows the relationship between terms of a sequence and the partial sums of a series. It also allows exploration of some important sequences & series including geometric and harmonic sequences.

This applet explores the normal approximation to the binomial distribution.

This applet shows the construction of the inverse of a function, and can be used to explore whether the inverse is a function.

This applet plots and traces a parametric curve, given as a vector function in R2.

This applet plots two parametric curves simultaneously. It can be used to explore whether two particles collide.

This applet explores the relationship between the pmf or density and the cumulative distribution function of a range of discrete and continuous probability distributions.

This applet explores a logistic population growth model with no harvesting. The phase plot is shown alongside the plot of p vs t.

This applet explores a logistic population growth model with constant harvesting.

This applet shows the relationship between a plot of an estimated empirical CDF and a normal probability plot.

This applet explores QQplots for a range of distributions.

This applet simulates a spring acting under gravity, subject to drag and an external driving force.

This applet illustrates the concept of independent identically distributed random variables.

This applet aims to demonstrate visually the projection of a vector u onto a vector v.

This applet illustrates how the distribution of the sample mean converges towards normality as sample size increases.

This applet calculates the zygote and adult allele and genotype frequencies according to the FisherHaldaneWright model of population genetics, and plots the results.

This applet iterates a difference equation (also known as recurrence relation) and displays the resulting sequence both graphically and numerically.

This worksheet performs iteration and produces cobweb diagrams for a firstorder difference equation (AKA recurrence relation, discrete dynamical system).

This applet shows a linear approximation to a nonlinear difference equation close to an equilibrium, using cobwebbing. It can be used to investigate the accuracy of a linear approximation, or to motivate the linear stability criterion for equilibria of a firstorder difference equation.

Guess the correlation of a sample of bivariate data drawn from a linear or nonlinear population.

Repeatedly sample from a bivariate population, and construct a histogram of sample regression line slope.

This applet illustrates a solution of the wave equation, from the MAST20029 Engineering Mathematics lecture notes.

This applet displays the direction field and solutions for an ordinary differential equation (ODE), and calculates approximate solutions using Euler’s method.

This applet investigates the continuity of a 2branch piecewisedefined function.

This applet investigates the continuity of a piecewisedefined function.

This applet displays the direction field and solutions for an ordinary differential equation (ODE).

This applet explores how the rate of change of a composite function y = f(g(x)) depends on the rates of change of both f and g.