These are the questions and observations of a year 9 mixed attainment class. They range from a request to define n to noticing features of the sequences. One pair of students has suggested a change to the nth term to include n3. Their speculation that the sequence increases in intervals of three suggests that they could hold a misconception about the sequences generated from 3n and n3. 

The class used the six regulatory cards to decide to find more examples in which the nth terms (one linear and one quadratic) generated common terms. Those that wanted more structure chose to use the structured inquiry sheet. As the inquiry developed, students began to make generalisations about sequences that, for example, only generated even or odd numbers.

The teacher's contribution

A second line of inquiry started when the teacher introduced a method for finding the nth term of a quadratic sequence. 

To consolidate the connection between linear and quadratic sequences, the teacher used the following method:

(a) Find a as half of the second difference (justified using the first and second differences between the general terms, a + b + c, 4a + 2b + c, 9a + 3b + c, 16a + 4b + c)

(b) Subtract an2  from the quadratic sequence to leave a linear sequence.

(c) Find the nth term of the linear sequence (bn + c).

(d) Sum an2  and bn + c to give the complete expression for the nth term of the quadratic sequence.

Now the students made up their own linear and quadratic sequences with common terms, found the nth terms and compared them to draw more conclusions. 

The inquiry ended with whole-class presentations. The pair of students who had suggested the change to the prompt to include n3 described how the third difference was the same, which led on to further speculation about the fourth difference of the sequence generated from n4.