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.

Reasoning

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.

Other cases

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.