Factors and prime factors inquiry
Mathematical inquiry processes: Explore by testing particular cases; verify and explain. Conceptual field of inquiry: Factors, prime factors and indices.
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.
Explaining the prompt
Why is the prompt always true?
The prime factorisation of 36 is 2 x 2 x 3 x 3. The factors of 36 are the combination of none, some or all of those prime factors. We could use 0, 1 or 2 twos and 0, 1 or 2 threes. So there are nine (3 x 3) possible combinations altogether, given in the form (number of twos, number of threes): (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) and (2,2). For the case of 0 twos and 0 threes, the factor is one.
The same reasoning applies to 48 (2 x 2 x 2 x 2 x 3). We could use 0, 1, 2, 3 or 4 twos and 0 or 1 threes. The ten (5 x 2) possible combinations, given in the form (number of twos, number of threes), are: (0,0) (0,1) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1) (4,0) (4,1). Again (0,0) represents the factor one.
In the case of 8 (2 x 2 x 2) there is only one prime number that appears three times. So the factors of 8 are given by the products of 0, 1, 2 and 3 twos. In this case, (0) represents one.
Finally, in the case of 30 (2 x 3 x 5) we could use 0 or 1 twos, 0 or 1 threes and 0 or 1 fives, which gives 8 possible combinations: (0,0,0) (1,0,0) (0,1,0) (0,0,1) (1,1,0) (1,0,1) (0,1,1) (1,1,1). In this case, (0,0,0) represents one.
The general case
In general, add one to an exponent to find how many times that prime factor could be used in a combination, including zero times. The total number of combinations is given by the product of one more than each exponent.