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Linear and quadratic sequences

The prompt links the concepts of linear and quadratic sequences. It was designed to follow on from the intersecting sequences prompt as a bridge to quadratic sequences later in a scheme of learning. The prompt contains similar nth terms to draw out features that are the same and different and to address misconceptions, such as 2n = n2. The terms in the quadratic sequence appear in the linear sequence with an increasing gap between them - one number between the first two terms, then two between the second and third and so on.

Prompt sheet
Notesheet for a structured inquiry

Classroom inquiry
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 n
3. 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 notesheet. As the inquiry developed, students began to make generalisations about sequences that, for example, only generated even or odd numbers.
A second line of inquiry started when the teacher introduced a procedure for finding the nth term of a 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.