Daniel Mansfield

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How to find solutions to a second order recurrence relation.

Introductory exercise about set equality

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$I$ compact interval, $g:I\rightarrow I,\;g(x)=ax^3+bx^2+cx+d$. Find stationary points, local and global maxima and minima of $g$ on $I$

Intorduction to proof and existence statements.

Gentle intro to modular arithmetic through quotients and remainders

Gives students an introduction to the injective property of functions

Introduces students to the floor and ceiling functions

English sentences are given and for each the appropriate proposition involving quantifiers is to be chosen. Also choose whether the propositions are true or false.

Introduces the GCD and a simple form of the Euclidian Algorithm

Introduction to modular arithmetic using a multiplication table and lookup.

Full worked solution using the Extended Euclidian Algorithm

Introduction to modular arithmetic in $\mathbb Z_b$ using a multiplication table and lookup with coprime $b \in \mathbb N$.

Simple counting exercise, with combinations

Introduction to counting with permutations and combinations

Introduction to first order recurrence relations with a simple example, including homogenous and non-homogenous solutions.

A few simple functions are provided of the form ax, x+b and cx+d. Values of the functions, inverses and compositions are asked for. Most are numerical but the last few questions are algebraic.

Quadratic factorisation that does not rely upon pattern matching.

Introductory exercise about power sets.

Introductory exercise about subsets using custom grading code.

Couting application where repitition is allowed and order does not matter.

Applicaiton of the pigeonhole principle

Definition of the pigeonhole principle, and some examples

Simple counting exercise. Students are encouraged to look for the smart way of counting.

English sentences which are propositions are given and for each the appropriate proposition  involving quantifiers is to be chosen. 

English sentences which are propositions are given and the appropriate logical expression chosen for the negation of the sentence.

Intorduces strong induction and uses it to verify the solutions of a second order linear recurrence.

Intorduction to using the defnition to prove simple stamtements.

Simple intro to mod arithmetic