# pure

## Number theory

### Continued Fractions

A Colab notebook on continued fractions, with nice applications to Pell's Equation and to finding certain Pythagorean triples. Could be worked through independently by pupils, but perhaps better as a collaborative, teacher-led exercise.

## Coordinate geometry

### Circle tangents

circle_tangents.pdf

A nice set of problems about circles with tangents at whole numbered coordinates that pass through the origin.

### Vectors

Coordinate Geometry – from 2D to 3D.pdf

A resource to promote discussion of the relationship between coordinate geometry in 2D and in 3D. The problems give motivation for the vector equation of a line and for the dot product. Teacher notes here, and a Desmos graph to illustrate the 2D version of problem 1.

### Matrix transformations

An interactive Desmos graph to investigate matrix transformations.

## Proof

### Proof

A Colab notebook with exercises exploring methods of proof (a work in progress; proof by contradiction and more examples to come...)

## Integration

### Integrals

area.pdf

A set of problems involving integrals, some coordinate geometry and the odd cubic equation... One instruction for each question: find the exact value of the green area.

### Volumes of revolution

Volumes of revolution.pdf

A worksheet on volumes of revolution.

### Reversing the chain and product rules

reversing_chain_and_product_rules.pdf
What do I differentiate to give….pdf

A guided worksheet and a follow up exercise for reversing the chain and product rules without explicitly mentioning substitution or integration by parts. I've used these in the past for getting pupils familiar with the underlying ideas (rather than learning a technique by rote). Also useful for developing fluency in the chain and product rules themselves. Some follow up questions towards the end of the first sheet, with linked video solutions.

See also this Desmos graph to show how graph transformations can be used as a way of understanding integration by substitution.

### Improper integrals

improper_integrals.pdf

A set of problems involving improper integrals. Only one instruction for each question (find the green area) but plenty to think about!

## Trigonometry

### Compound angle formulae

compoundangle_questions.pdf

A set of problems that can be solved using compound angle formula. I hope these are a bit different from the routine examples in text books.

## Complex numbers

### A complex iteration

A complex iteration investigation.pdf

Teacher notes for an investigation into a (complex) iterative sequence. Accompanying Colab notebook. (No programming experience required, but interested pupils can expand and examine the code.)

## Differential Equations

### Integrating factors

diff_eqns_I.pdf

An introduction to the idea of integrating factors to solve differential equations, starting with the idea of finding a product with that differentiates to a given two-term expression.

## Hyperbolic functions

### Hyperbolic functions

Defined from the hyperbola rather than using exponentials.