Using basic derivatives to calculate the gradient function of a hill $y=-e^x+b\ln\left(x\right)+c$, and then substituting values to find the gradient at specific distance from the sea. 

A normal curve is shown, with the mean and +/- 1, 2 and 3 standard deviations marked. An area of the curve is shaded and students are asked to estimate the shaded area using the heuristic rule. There are 6 possible areas: (-infty, mu-2sd), (mu-2sd, mu-1sd), (mu-1sd,mu), (mu, mu+1sd), (mu+1sd, mu+2sd), (mu+2sd, infty). The mean and standard deviation are randomised.

Find and compare means and standard deviation using EXCEL (downloadable randomised dataset)

Unit circle definition of sin, cos, tan using degrees

Unit circle definition of sin, cos, tan using radians

The student has to enter `diff(y,x,2)`, equivalent to $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$, as their answer. It's marked by pattern matching, using a custom marking algorithm.

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

Solving a system of simultaneous congruences using the Chinese Remainder Theorem.

An explore mode question which allows you to write a solution straight away, or break it down into steps.

After solving a system of three congruences, you can ask for a new problem with more congruences.

Solve a problem using a linear equation.

Using the chain rule within product rule problems.

An introduction to using the quotient rule

Using the chain rule with polynomials.

Introduction to using the product rule

Using BODMAS within mathematical expressions

An introduction to using the quotient rule

Using the chain rule with polynomials

Factorising polynomials using the highest common factor

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Using BODMAS within mathematical expressions

Using the chain rule with polynomials

Using the chain rule within product rule problems

An introduction to using the quotient rule

Using the chain rule with polynomials and negative powers.

Introduction to using the product rule

Using the chain rule with polynomials and negative powers

Introduction to using the product rule

Using the chain rule within product rule problems

An introduction to using the quotient rule