Studies in Algebraic & Geometric Thinking
A series of explorations in algebra and geometry with commentaries which draw upon mathematical powers, themes and pedagogical choices to illustrate the fact that any task can be seen as merely one example for a rich domain of similar tasks, and putting forward the view that a task which does not permit and encourage learners to use their own powers, to encounter core mathematical themes, is not a pedagogically effective task.
Algebraic Thinking
AS1: Up & Down Sums or

Constructs called upon include

AS2: Sums to One: a study in generalisation
If someone has chosen two numbers which sum to one, which of the following two will be the larger?
the sum of the square of the larger with the smaller or the sum of the square of the smaller with the larger 
Constructs called upon include

S3: Marble Sharing: a study in variation
The tasks begin with a traditional sharing problem:

The problem is simplified and then recomplexified
The notions of Dimensions of Possible Variation and Range of Permissible Change are exploited both to simplify and then generalise the task.
