Rather than “doing” variation theory, I am choosing to attend to the variation in the task whilst designing (or considering the design of someone else’s task). Below is a task to illustrate this point. It is worth noting that this task was written as part of a lesson for a mixed attainment year 7 class. The whole lesson can be found here.
Questions 1 to 5 are chosen so that a previous answer could be used to find the prime factorisation. Although it is expected that many students will work out each question from scratch, this then provides an opportunity for students to reflect on the link or be pointed out explicitly. Question 6 draws out an inherent difficulty with prime factorisation – it is common for students to believe 91 to be prime and this provides opportunity to ask questions such as how do you know. Questions 7 to 12 are again chosen so that the previous answer could be used, but again may be done from scratch and again provides opportunity for noticing something interesting.
Questions 13 to 18 are about getting students to work with the prime factored form of a composite number. Question 13 requires introducing a 3 to the prime factored form, whereas 14 also has a simplification, then 15 uses a composite number (one seen before so that this is already known). Then 16 to 18 adds in the consideration of raising a number to a power in prime factored form. This idea is then extended in the final question.