Linear and quadratic sequences

The prompt

Mathematical inquiry processes: Notice connections; generate more examples with the same properties; analyse structure. Conceptual field of inquiry: Position-to-term rules for linear and quadratic sequences; algebraic expressions.

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 number of terms between them - one number between the first two terms, then two between the second and third, three between the third and the fourth and so on. 

In the prompt, students might speculate that there is a link between the constant in the nth term of the quadratic sequence and the fact that the differences between the terms increase by two each time. By testing different cases, they come to associate the square with differences that increase (or decrease) by the same amount.

One line of inquiry follows the main line of the intersecting sequences prompt. An example comes from a year 9 student who looked at the two sequences below during an online inquiry. She analysed the mathematical structure of the sequence formed from the common terms.

The even numbers in the quadratic sequence form the 'intersecting' sequence. They appear in pairs, each followed by a pair of odd terms. This means that the differences between the even terms increase by four each time because, firstly, the difference between the terms in in the quadratic sequence increase by one each time and, secondly, there are four spaces from the middle of one pair to the middle of the next. This could be represented in the following way (where the terms in the sequence are in bold text):

even (the difference is even) even (the difference is 1 more than the difference before and is, therefore, odd) odd (+2) odd (+3) even (+4) even

The differences between the pairs of even numbers increase by 12 each time. This time there are three spaces from one even number to the next. The differences are made up of three consecutive whole numbers in the following pattern:

+ 3, + 4, + 5     total +12

+ 7, + 8, + 9     total +24

+ 11, + 12, + 13     total +36

As there are two rules to generate the intersecting sequence, there is no single expression for the nth term. 

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 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 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.