pure

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

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.

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.

A worksheet on volumes of revolution.

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.

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

Some practice using integration by parts and by substitution, including finding parametric areas. Solutions here.

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.

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.)

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