The relationship between the frequency of an allele A, $x$, at a genetic locus in a diploid population and the fitness of a population with this frequency of allele A, $w$, is described by the function $w=ax^2+x(b-x)+c(b-x)^2$ . The aims are (a) ti simplify the algebraic expression, (b) calculate the fitness of a population with a given allele A frequency, and (c) calculate the allele A frequency when the fitness of the population is given.

The proportion of the sodium carbonate, $p$, which has dissolved by time $t$ seconds is given by the formula $ p=\frac{bt-at^2}{c}$. The aim is to calculate the proportion of sodium carbonate in a solution at a given time and vice versa.

Edit the Python code to make a 3D plot of a surface defined parametrically.

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The number of patients arriving at a dentist’s surgery each afternoon follows
a Poisson distribution, with a mean of four patients per hour.
Calculate the probability that in a particular one-hour period

Asks students to apply laws of logical equivalence to prove the equivalence between two logical statements. The quiz should accept any correct answer (as long as each step is included, with one law per step), and provides detailed feedback on mistakes.

Partial differentiation question with customised feedback to catch some common errors.

Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions. 

Add, subtract, multiply of divide two mixed numbers. This question has no advice at this stage.

Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions. 

Match prefixes with their abbreviations, from nano to giga. Match prefixes with equivalent scientific notation.

The subtraction algortihm using the borrow and pay back method with integers.

Recall of common units, along with understanding multiplication.

Multiplication algorithm with integers

Multiplication algorithm with integers

Fractions already have a common denominator. Addition and subtraction 50:50 split, when subtracting, the answer is negative half the time. Students shouldn't have to worry about reducing fractions by design.

Subtracting a decimal with 3 decimal places from a decimal with 2 or 3 decimal places. borrowing is necessary. This was modified from a subtraction question using integers with each number divided by 1000 so the variables have names referring to ones, tens, hundreds etc.

a) Multiplying decimals with a single non-zero digit. Students are told to preserve the number of decimal places (from the question to the answer). 

b) Multiplying decimals requiring the multiplication algorithm. 

Simple probability question. Counting number of occurrences of an event in a sample space with given size and finding the probability of the event.

rebelmaths

Shows how to use JSXGraph to make a sine graph with amplitude, frequency and phase controlled by sliders.

This question shows how to generate a random set of $(x,y)$ samples, where $y = mx + c + \mathrm{noise}$.

The JSXGraph extension is used to show a scatter plot of the data. This isn't necessary if you just want to generate the data.

A mathematical expression part with a pattern restriction to ensure that the student has extracted the highest common factor of two terms.

The answer must be of the form $a(b+cx)$, where $b$ and $c$ are coprime.

The student is asked to give the roots of a quadratic equation. They should be able to enter the numbers in any order, and each correct number should earn a mark.

When there's only one root, the student can only fill in one of the answer fields.

This is implemented with a gap-fill with two number entry gaps. The gaps have a custom marking algorithm to allow an empty answer. The gap-fill considers the student's two answers as a set, and compares with the set of correct answers.

The marking corresponds to this table:

A random proper fraction $a/b$ with denominator in the range 2 to 30 is picked, and the student must write $\frac{a}{b} \pi$.

The point of this question is to demonstrate that the correct answer is shown as a multiple of $\pi$ rather than a decimal.

evaluate a function with randomised alphanumeric expressions as inputs.

A custom marking algorithm picks out the names of the constants of integration that the student has used in their answer, and tries mapping them to every permutation of the constants used in the expected answer. The version that agrees the most with the expected answer is used for testing equivalence.

If the student uses fewer constants of integration, it still works (but they must be wrong), and if they use too many, it's still marked correct if the other variables have no impact on the result. For example, adding $+0t$ to an expression which otherwise doesn't use $t$ would have no impact.

Given a function definition in words, evaluate the function with various variable and numeric inputs

Rearrange a formula with a square root to make a variable under the root the subject.

Rearrange a complex formula involving squares, square roots, fractions and additions. This is a fixed question with no randomisation.

Multiplication algorithm with integers