Finding composite functions of a linear function and a logarithmic function.

Finding composite functions of a linear function and a function of the form $x^n+a$.

Finding composite functions of 2 linear functions.

Finding the value of a variable

Extra practice on some basic algebra skills, including solving linear equations. You can try as many times as you like and also generate new versions of the questions for extra practice.

Drag points on an axis to plot a linear graph (integer gradient and intercept only)

Calculate the length of a line segment given two points

Find the gradient of the line that passes through the points 

Find the equation from the image of graph

Drag points on an axis to plot a linear graph (rational gradient and intercept)

Identifying which graph shows the function $y=\frac{x-a}{x+b}$

Example of an explore mode question. Student is given a 2x2 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 eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

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

Given a cubic expression and a linear factor, calculate the remaining quadratic factor.

Calculating two unknown coefficients of a cubic expression when given two of the expression's linear factors.

Solving a system of three linear equations in 3 unknowns using Gaussian Elimination (or Gauss-Jordan algorithm) in 5 stages. Solutions are all integers. Introductory question where the numbers come out quite nice with not much dividing. Set-up is meant for formative assessment. Adapated from a question copied from Newcastle.

Solving a system of three linear equations in 3 unknowns using Gaussian Elimination (or Gauss-Jordan algorithm) in 5 stages. Solutions are all integers. Introductory question where the numbers come out quite nice with not much dividing. Set-up is meant for formative assessment. Adapated from a question copied from Newcastle.

Solve simple two step linear equations with feedback.

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Construct a line through two points in a GeoGebra worksheet. Change the line by setting the positions of the two points when the worksheet is embedded into the question.

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

This particular example has a positive gradient.

A quadratic and a graph of it is given. A tangent is also sketched. The equation of the tangent line is asked for.

Find the equation of a straight line which has a given gradient $m$ and passes through the given point $(a,b)$.

Solve for $x$: $\displaystyle ax+b = cx+d$

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Dealing with expressions in online tests and in Matlab.

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".

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

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Educational calculation tool rather than "question".

This allows the student to input a linear system in augmented matrix form (max rows 5, but any number of variables). Then the student can decide to swap some rows, or multiply some rows, or add multiples of one row to other rows. The student only has to input what operation should be performed, and this is automatically applied to the system. This question has no marks and no feedback as it's just meant as a "calculator".

It has some rounding to 13 decimal places, as otherwise some fraction calculations become incorrectly displayed as a very small number instead of 0.

It would be possible to extend to more than 5 rows, one just has to put in a lot more variables and so on. I just had to choose some place to stop.