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How to Derive the Quadratic Formula - One Page Proof

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Here’s a step-by-step guide on how to derive the quadratic formula, the formula we use to solve quadratics we can’t factorize. Note that this requires an understanding of how to ‘complete the square’.

Root 2 is Irrational - One Page Proof

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A fully explained proof (by contradiction) that √2 is irrational.

Infinite Primes - One Page Proof

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Here’s a proof, using the method that requires the least prerequisite knowledge, that there are infinitely many prime numbers. One particular part of this proof tends to confuse people, so this PDF also includes a lemma explaining that step in great detail.

A Guide to Set Notation

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An explanation on the various symbols used when talking about sets, each with an example. Also contains definitions for the sets of natural, integer, rational, real and complex numbers.

Arithmetic Sequences and Series

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Contains a worksheet on arithmetic sequences and series, and summations. Also contains eight multi-part questions on the topic each with a fully worked solution. This pack is designed for a tutor to work through and annotate the worksheet with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Proofs

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Contains a worksheet on proof by deduction, disproof by counter-example, proof by exhaustion and proof by contradiction, each with worked examples. Also contains twelve questions on the topic each with a fully worked solution. Note that this does not include inductive proofs. This pack is designed for a student to work through alone, or a tutor to work through and annotate the with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Prime Factorisation & HCF

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Contains a worksheet on prime numbers, the Sieve of Eratosthenes, prime factorisation through factor trees, and the highest common factor (hcf). Also contains five questions (some multi-part) each with a fully worked solution. This pack is designed for a tutor to work through and annotate the worksheet with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Exponentials and Logarithms

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Contains a worksheet with the laws of indices and laws of logs (with proofs) and a couple of graphs to demonstrate the relationship between the graphs of logs and exponentials. Also contains ten tricky questions each with a fully worked solution. This pack is designed for a tutor to work through and annotate the worksheet with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Converting Between Decimals, Percentages and Fractions

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Contains a worksheet on converting between decimals, percentages and fractions. Also contains three ten-part questions covering every combination of the conversions each with a fully worked solution. This pack is designed for a tutor to work through and annotate the worksheet with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Geometric Sequences and Series

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Contains a worksheet on geometric sequences and series. Also contains seven multi-part questions on the topic each with a fully worked solution. This pack is designed for a student to work through alone, or a tutor to work through and annotate the with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Surds - Rationalising the Denominator

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Contains a worksheet on rationalising the denominator with worked examples. Also contains questions on the topic each with a fully worked solution. This pack is designed for a student to work through alone, or a tutor to work through and annotate the with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Fraction Arithmetic

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Contains a worksheet on adding, subtracting, multiplying and dividing fractions, with a focus on addition and subtraction by cross multiplication. Also contains a set of questions on the topics each with a fully worked solution. This pack is designed for a student to work through alone, or a tutor to work through and annotate the with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Binomial Expansion

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Contains a worksheet on Pascal’s Triangle, factorials, and binomial expansion using the Binomial Theorem. Also contains seven questions on the topic each with a fully worked solution. This pack is designed for a tutor to work through and annotate the worksheet with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Simplifying Surds

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Contains a worksheet on simplifying surds with four worked examples. Also contains over twenty questions on the topic each with a fully worked solution. This pack is designed for a tutor to work through and annotate the worksheet with a student, go through a few questions together and leave them the rest to work on on their own. Enjoy!

Sequences and Series

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Contains an information sheet, questions and fully worked solutions on arithmetic sequences and series, and the same again on geometric sequences and series.