Introduction

This activity focuses on generating and analyzing numeric sequences. It is a foundation activity, in the sense that it establishes tools, practices and base knowledge useful for a wide range of activities such as Convergence and Divergence, Coding and Fibonacci.

When investigating sequences, students will create mechanisms to generate sequences, and then observe the real-time execution of the process. The advantage of ToonTalk is that the sequences can be viewed not only as a product (a series of terms), but also as a process that can be observed and altered. This is in contrast with traditional algebraic representations that are 'static'. These alternative representations may lead to different understandings, for example, the sequence {0, 2, 5, 7, 10, 12…} has a relatively simple rule (+2, +3) that is easily programmed in ToonTalk, but not easily described by conventional algebra.

When generating more complex sequences, students will 'chain' (connect using bird-nests) a number of robots together. By altering the parameters and number of chained robots, any polynomial sequence can in theory be generated. Students will explore the relationships between the number of robots and the complexity of the corresponding sequence (the order of the polynomial). They will also investigate how rules for sequences are determined, and build robots to help them with this task.

Aims

• To develop a non-algebraic language for describing, discussing and reasoning about polynomial (and maybe some non-polynomial) sequences.
• To develop an understanding of the generation of number sequences, the rules that sequences rely on and how sequence generation relates to the ToonTalk environment (robots, birds, etc).
• To gain some insight into the relationship between the number of chained robots and the type and complexity of the corresponding sequence.
• To develop students' ability to make conjectures, suggest more than one solution to a problem, evaluate arguments and reason.

Prerequisites

This activity is fairly self-contained, both in terms of mathematical content and programming skills. It has been tried successfully with ToonTalk beginners, after a few sessions playing the puzzle game.
However, we strongly recommend that students go through the basic sequences activities before starting the game.


 

Activity plan and resources

Motivation

A possible way to start the activity is by playing Guess my Sequence in a group discussion. 

• Write several sequences on the whiteboard and ask students to provide the next terms.
• Ask the students to explain how they worked out the rules of these sequences.
• Let them propose sequences for you to work out, and let other students respond to those.
• Ask a student who proposed a challenge how she would train a robot to generate that sequence. If possible, train a robot together with the students.

At this point, you can play the paper version of Guess My Sequence (use GuessMySequenceGame.doc) or turn directly to the web-based game.

Game

In most cases, the best way to get started on the game is by responding to an existing challenge. Direct students to the Guess my Robot page, and ask them to choose a challenge and respond to it. Explain that they should respond by adding a comment with their robot (the full rules are listed here).

After they have responded to someone else's challenge, ask students to pose their own challenge, by using the Guess my Robot Template. Once some of the students have done so, you can encourage them to respond to each others' challenges.

You should allocate at least 3 sessions to playing the game, so that students have a chance to build up a rich interaction with remote peers.

Here are various tools which can be used for constructing sequences.

 

Reflection

Ask students to use the Reflections on Guess my Robot template to summarize their experiences of playing the game. Allow students sufficient time for thoughtful writing. After at least half of the group have published their personal reports, conduct a group discussion and publish a group report using the same template.

 

Related activities

• Cryptography
• Number Sequences and Fibonacci
• Convergence and Divergence
   
  

Things to look out for

Past experience with this activity has shown several classes of canonical sequences to be popular among students. Read about the mathematical structure of these sequence types.

 

References and further reading

• Sequences, differences of terms, applications : General mathematical background.
   
• Pedagogical advice on number sequences