Number of factors inquiry
Mathematical inquiry processes: Identify and extend patterns; conjecture; generate examples and counter-examples; reason. Conceptual field of inquiry: Factors; geometric sequences.
The design of the prompt invites students to identify and extend a pattern. The numbers of factors of the first four terms of the geometric sequence 1, 2, 4, 8 are 1, 2, 3, and 4 respectively. The curious pattern invites questions: Will the pattern continue for 16 and 32? Is there a pattern for other geometric sequences, such as 1, 3, 9, ...? What about other types of sequences?
Through the inquiry students generate conjectures about different cases, which they go on to test. Even if they are ultimately unsuccessful in showing their conjectures are true, they will be involved in purposeful and meaningful practice. Students, in general, are far more motivated to find factors in the context of a mathematical inquiry than they are when required to complete a textbook exercise.
Starting the inquiry
In the initial phase of the inquiry, students attempt to make meaning of the prompt (including the concept of a factor if it is new to them), identify the pattern and speculate about how it might continue. They pose questions and notice properties:
The numbers in the top row are doubling.
What is a factor?
Why do different numbers have different numbers of factors?
Most factors are in pairs.
The number of factors goes up in ones.
Will the pattern continue like this?
The pattern continues for the next case because 16 has five factors: 1, 2, 4, 8, and 16.
As in all inquiries, it is important to navigate slowly through this part of the inquiry. The teacher must ensure that the class has an understanding of a factor and how to find the factors of a number by co-constructing an explanation. During the orientation phase, she is able to identify the extent of each students' current understanding and, through questioning and prompting in a class discussion, can attempt to construct a deeper understanding.
As students become more fluent at finding factors, they should be encouraged to make connections between the factors of terms in the same geometric sequence. For example, 16 has one more factor than eight, which is 16 itself. One way of thinking about this is to double each factor (just as eight has been doubled) and then include one.
Intriguingly, the number of factors of other geometric sequences form different patterns. When the common ratio is four (with one as the first term), the number of factors are odd numbers because the terms in the sequence are all square numbers. They are also consecutive odd numbers because each term has two new factors, itself and half of itself.
When the common ratio is six, the number of factors are all themselves square numbers.
Lines of inquiry
(1) First term greater than one
What difference does it make if the geometric sequence starts with a number greater than one? Let's try two with a common ratio of three: 2, 6, 18, 54, ... . The numbers of factors are 2, 4, 6, and 8 respectively, which are double the numbers of factors when the first term is one for the same common ratio (see below).
Would the number of factors be 3, 6, 9, and 12 if the first term was three? Let's try. The geometric sequence starts 3, 9, 27, 81. The numbers of factors are 2, 3, 4, and 5. So the rule breaks down, but, intriguingly, it works if we start the sequence with four. The number of factors for 4, 12, 36, 108 are 3, 6, 9 and 12 respectively.
We can make a new conjecture: For a geometric sequence with a common ratio of 3 and a first term that is a power of two, the numbers of factors are consecutive multiples of one more than the power of two. For example, when the sequence starts with four (22), the numbers of factors are multiples of three (2 + 1). Does that mean that when the first term is eight (23), the numbers of factors are multiples of four?
(2) Arithmetic sequences
Is it possible for the factors of the terms in an arithmetic sequence to form a pattern? By creating their own examples, students soon realise that this is unlikely. If we try 3, 7, 11, 15, 19, for example, the numbers of factors are 2, 2, 2, 4. and 2 respectively. The greater frequency of prime numbers between 1 and 10 and 11 and 20 and their irregular pattern proves to be an obstacle to creating an arithmetic sequence with a pattern for the numbers of factors.
What if we use only even numbers? If we try 2, 6, 10, 14, 18, ...., for example, then the numbers of factors are 2, 4, 4, 4, and 6 respectively. This could form the start of a pattern, so let's extend the sequence: 22 and 26 have 4 factors each and 30 has 6 factors. So now we have 2, 4, 4, 4, 6, 4, 4, 6. The next three terms have 4, 4, and 6 factors as well. Will the pattern continue 4, 4, 6?
(3) Factors and prime factors
For older students, the inquiry could lead into the relationship between factors and prime factors. In a mixed attainment grouping, some students could move quickly onto this line of inquiry (or the one below) after exploring and explaining cases arising from the number of factors prompt.
(4) Different types of sequences
Another line of inquiry for older students is to compare and contrast arithmetic, geometric and quadratic sequences and the general form for the nth term of each type of sequence. Can they identify each type of sequence, generate a sequence from the nth term and find the nth term from a sequence?