WebLabs - Sequence Tools

An alternative version of this sequence that relies upon making graphs from within ToonTalk can be found here.

Related worksheets:

General background

In the core of this activity, you explore convergence and divergence of several sequences and series such as the sequences with a general term:

And the corresponding series:

The advantage of using ToonTalk to explore these sequences is twofold. On the one hand, the process of generating them can be made visible, and thus accessible. On the other, ToonTalk has the capability of assisting learners in the reification of concepts regarding the convergence or divergence of these sequences by providing a visual form for key arguments.

Manipulating the sequence while arguing about its sum aims to help the learner to maintain both views at once: that of the process that generates the sequence and that of the sequence as a whole as a mathematical object.

We do not aspire to “correct children's intuitions about infinity”. On the contrary: we will explore these mathematical structures empirically, raise conjectures based on experience and intuition, and scrutinise these conjectures. Our intention is to lead participants to reflect on the use of empirical and intuitive knowledge, to acknowledge its utility and its limitations at the same time.

Aims

To experience surprises arising out of the tension between intuitions of infinity and evidence revealed through activity.
To develop a non-algebraic language for describing, discussing and reasoning about sequences. Specifically, discussing convergence, divergence and limits. This language will be derived from discussing the programming activities, but will be extrapolated to address the underlying mathematical concepts.
To develop students’ ability to evaluate their and their peers' arguments and reasoning.
To begin to appreciate the nature of divergent and convergent sequences.
To discriminate between empirical evidence and formal argumentation, while using both in exploratory activities.

Prerequisites

It is highly recommended that students participate in the Guess my Robot activity before starting this one. If this is not the case, you should allow sufficient time at the beginning for them to get acquainted with the tools and practices.