I have really liked the idea of etudes ever since I came across this article about a year ago. Embedding practice into a mathematically rich context seems to perfectly balance the need for practice with the desire that students be working productively towards something meaningful - as well as ensuring that mathematics stays centre-stage in the lesson.

My aim in this post is to link to various resources I believe fulfill the requirement of an etude and to find commonalities in the types of over-arching mathematical goal involved in each case - that is to categorise them!

__Category 1__Students follow a rule leading to a pattern of results - can the pattern be justified?

(ie. the teacher taps into a seam of related examples that follow some rule and students investigate this seam.)

Examples:

- At MathsPad, we developed this task on factorising and expanding (this is still in flash at the time of writing), inspired by an interesting book on productively using algebra by Martin Kindt (available as pdf here.)

- Calculations with fractions in this task by Don Steward.

__Category 2__Students are given a number of criteria and examples must be found that fulfil different combinations of these criteria.

Examples:

- Johnny Griffith's Risps are good examples of these - such as Risp 10 on coordinate geometry, where for each section of a Venn Diagram, an example must be found.

(Also, students must combine given elements to form as many examples as possible - such as RISP 3)

__Category 3__Students work on examples where the results converge towards a limit.

Examples:

- Finding areas of successive polygons surrounded by circles, which in the limit will tend towards pi. Embedded practice of using trigonometry and finding areas.

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