The prompt gives students the opportunity to explore the differences and connections between factors and prime factors. It is aimed at classes who have learnt about factors and prime numbers, but have yet to meet prime factorisation. In the initial orientation to the prompt, these questions have arisen in classroom inquiry:

• What does prime factorisation mean? Is it connected to prime numbers?

• Is an integer a normal number?

• Are a and b supposed to be indices?

• What numbers do p, q, a and b stand for?

• I know that 8 has four factors: 1, 2, 4 and 8.

• Why do you add one to a and b?

The statement in the prompt is always true. For example, the prime factorisation of 12 is 2² x 3¹ from which a = 2 and b = 1. It follows that 12 has (2 + 1)(1 + 1) = 6 factors. Before the inquiry starts, the teacher should prepare to explain how to express an integer as the product of its prime factors or, perhaps, flip learning by asking students to study a video or written explanation before the lesson.

Developing the inquiry

Once students are familiar with the concept of prime factorisation, they often use the regulatory cards to decide to create more examples. Integers such as 20 and 24 follow the general statement in the prompt because the prime factorisations are 2² x 5¹ and 2³ x 3¹ respectively. However, students soon realise that some integers are expressed by the product of more or less than two different prime numbers. For example, the prime factorisation for 8 is 2³ and that for 30 is 2¹ x 3¹ x 5¹ . Nevertheless, the rule still holds. Thus, 8 has four factors (3 + 1) and 30 has 8 factors (1 + 1)(1 + 1)(1 +1). The inquiry might end with students trying to amend or re-write the statement in the prompt to take account of the new finding.

February 2021