Context

This task was designed as part of the Number sequences strand of the WebLabs project. It was introduced at the end of a series of activites on number sequence, and aimed to test students' competence at the skills and techniques developed so far. It can be used as an assessment task, as a time-filler, or as a means of coping with diversity of skills. While the previous two activities are foundational and indispensable in terms of subsequent activities, this one is not. In some classes some of the students attempted it and some didn’t, in others it was skipped altogether.

The task is set up as a puzzle game, or riddle – students are challenged to use the Add-up tool they had constructed to generate unexpected sequences: powers of 2 and the Fibonacci sequence. It can be positioned as more or less 'gameful' depending on the mode of use the teacher wants to emphasize. If it is used primarily as a means of keeping advanced students occupied, then it is presented as a recreational activity, and any assessment or evaluation is done by non-intrusive observations. If it used as a formal assessment tool, then adherence to the elaborate response template is expected. To allow this flexibility, the response template is separated from the game page, which only contains the task-in-a-box, and links to the response template and a concealed 'hint'. To retain the game-like style, this hint is itself enigmatic: 'You *ONLY* need Add up! No other robots! Let the serpent eat its tail...'

Whatever the scenario, this activity marks a transition from linear and quadratic sequences to a broader, perhaps more complex, set. It is worth noting that some students 'discover' such sequences during the first two activities. However, these would typically be generated by replacing the additive in Add-a-num with a function. In the case of this challenge, students are drawn to engage not only with richer mathematical structures, but also with richer processes for generating these structures. This point requires elaboration. Let us use the notation [a|b|c...] to denote a ToonTalk robot's box with the inputs a in the first hole, b in the second, c in the third and so on. An Add-a-num robot receives a box of the form:

[a|b|bird]

And computes the sequence:

S₀ = a; S_n = S_n-1 + b

Which in closed form would be written as:

S_n = a*n + b

In ToonTalk this behaviour is achieved by 'dropping' a copy of b over Sn. This is in fact shorthand for applying the unary function +b to Sn. By replacing b with a function f(x), we generalize to a broader set of recursive (or iterative) sequences:

S₀ = a; S_n = f(S_n-1)

This class includes the powers of 2, for example. Set f(x) = 2x (in ToonTalk terms, place ‘*2’ in the second hole). However, the second sequence in the challenge – the Fibonnaci numbers – can not be produced by an unary function, and thus cannot be obtained as an outcome of Add-a-num. Furthermore, the challenge requires the student to use Add-up to generate the sequences. Add-up operates on a sequence as an object. Using Add-a-num engages students with questions of functional relations between numbers, whereas using Add-up elevates the discussion to functional relations between sequences – or, in other words, second order functions. In the case of this challenge, the question they are asked to solve is:

Given the operation S_n = S_n-1 + T_n, What is the sequence T I need in order to produce a sequences S = {2, 4, 8, 16…} and S = {1, 3, 5, 8…}? The solution requires them to extend their recursive intuitions from the domain of numbers to the domain of sequences, identifying T = S and then T = {T_i | T_n= S_n-1}.