Guidance Material for Number Sequences and Fibonacci Sequence

Introduction

The purpose of this document is to provide support to teachers and researchers who wish to use worksheets and tools provided in Weblabs project in their classes with their students. The underlying idea behind the proposed activities is to inform prospective users about the produced materials (worksheets and tools) on the basis of the main principles underlying the designing and implementation of Weblabs materials.

The extensive form of the material can be found on project’s website, in Plone section (http://www.weblabs.org.uk/wlplone). In this section, the prospective users of the project’s produced environment can find some guidance materials that might effectively introduce them in working with activity worksheets and tools.

The guidance material is divided into two parts. The first part is linked to related activities in school mathematics textbooks and the national mathematics curriculum. The second part of guidance material is related to teaching tips on how prospective users can teach effectively the project’s materials and the tools provided in ToonTalk environment. 

Number Sequences

Examples of activities on number sequences and especially number patterns appear extensively in the school textbooks of many courtiers; the students are required to find rules in number patterns and predict the following and the previous numbers in given number sequences. In Cyprus, such activities are included in several grades, from the fourth to the sixth grade mathematics textbook. The aim of the provided number sequences activities is to encourage students to play with numbers, identify patterns and inductively justify the generating rule in selected number sequences. Using the provided tools, that students have already constructed, they have the opportunity to “break apart” the sequences and identify ways to generate them. This leads to better understanding, since students can construct their own representations for linking the abstract form (first term of the sequence, differences) with concrete forms of representations, like the robots, the birds and the nests.

Number Sequence Challenge

In this activity, students will train a robot to generate the sequence of natural numbers (Add1 robot). The first task is well defined and the use of many pictures will help students train their first robot. One of the main ideas is to help students understand that the robots can be trained “by example”. That means that they only have to show the robot something once (e.g., working in their thought bubble) and the robot will execute that as long as they want to. In the following task students are given some number patterns, and asked to predict which of them can be generated by the robot, after they have trained it. The teacher can orchestrate a discussion here, focused on some contradictions that may arise, since some sequences contain negative numbers. Students can work out these examples and discuss their findings with their peers.

In the following task students will add up the numbers generated by Add1 robot (natural numbers). Students can follow the instructions with the related pictures, to train the AddUp robot, which will get all numbers and add them up. Students may need additional help to chain the AddUp robot with the Add1 robot. The AddUp robot can add two numbers together, but the purpose of this activity is to add the natural numbers. Thus, students have to get the output of the Add1 robot (nest with numbers) and put it as an input into the AddUp robot. Doing this will let students chain the two robots and the AddUp robot will work since there are numbers in the nest.

In the next task, students will train a robot to generate a stream with the same number (Constant). This task is quite easy; the teacher can orchestrate a discussion on what sequences this robot can generate and what is the purpose of generating such sequences (e.g. link them with other robots). In the following task, students will chain the constant robot with the AddUp robot. Students have only to follow the pictorial instructions. The activity is quite easy.

In the last task of the activity students will work with all the three robots. They will think of some of them, or all of them, or even use one for more than one times, in order to generate a clever number sequence. Students can work in pairs to chain the necessary robots and predict the result in their worksheets. Teacher can write their predictions in the whiteboard and let students use their robots to test their predictions in ToonTalk. A discussion will follow, where students can announce their results, think of possible reasons why their predictions did not come true and make suggestions to improve their solutions. In a final discussion, students can use other students’ solutions and try to predict what will happen if they make simple changes, like changing the first number or chaining the robots in a different way.

Working with more Advanced Sequences

In this activity, students work with more complicated sequences and try to understand how a sequence can be constructed by connecting a number of ‘simple’ robots or procedures. Another aspect of this activity is to help students view sequences both as a process and a product i.e. to see that one sequence may be composed of another sequences. Students use the same robots like the ones in the previous activity and they will also make some minor changes in these robots, in order to complete the tasks in this activity.

In the first task, the students are asked to predict the next two terms of the following sequence [1, 2, 4, 7, 11…] and find out the rule behind the sequence. The sequence is within student reach, so the students are expected to find both the next terms and the rule for generating the sequence. According to the group of students, the teacher may have to lead a discussion about the relationship between decomposing the sequence into differences, and composing the sequence using sums (e.g. the AddUp robot). For example, in the given sequence, the students can observe that the differences are increased by 1. If they write down the differences, they will observe that they form another simple sequence [1, 2, 3 ...]. Students can easily understand that they can construct that sequence, using the Add1 robot (from the previous activity). The role of the teacher is crucial in helping them realize that since they have a way to construct the differences (Add1 robot), they only need to add these numbers to each sum, to get the first sequence (and they can do this, using the AddUp robot). If students experience difficulties, teacher can give them a simple sequence to practice, like the [3, 5, 7, 9, 11 …] and ask them to construct it, using only the Constant and AddUp robots.

In the following task, students will create their own hard sequences and challenge other students to solve them. They will write down the rule for their sequences and how they constructed it in the ToonTalk environment. Students should chain more than one AddUp robot. They might also try other types of (non-polynomial) rules e.g. +2 +3 or geometric sequences. The next two tasks can be used as assessment tasks for this activity.

The last task in this activity is a part of the “Guess my Robot Game”, where students challenge other students from the same or other sites to solve their hard sequences. After generating a hard sequence, students can post them on the project’s website, where other students will try to train robots to generate these sequences. More guidance on the “Guess my Robot Game” can be found at:

http://www.weblabs.org.uk/wlplone/Members/yish/my_reports/Report.2004-03-08.1130/index_html.

Fibonacci Sequences

The task on Fibonacci sequences involves two activities. In the first one the students are expected to construct the Fibonacci sequence, using only the AddUp robot, after doing some changes on it. In the second activity, students explore divisibility in Fibonacci sequences.

Constructing the basic Fibonacci sequence

In this activity students will understand the basic rule in generating the Fibonacci sequence and how they can construct this sequence by connecting a number of ‘simple’ robots (or procedures) and modeling the robots’ action in ToonTalk’s environment. In the first task students have to work in pairs and try to predict the next two terms in the sequence [1, 1, 2, 3, 5, 8, 13…] and the rule for generating this sequence. Teacher will orchestrate a discussion on the findings of the first task. Since most of the students have worked before with number patterns, it is expected that they will reach easily the answers in that task.

In the following task, which can be used for assessment purposes, students will work with a sequence such as [… 21, 34, 55…], where they have to apply the rule they found in the previous task, to find the next two terms and the two previous terms. In the following discussion, students will compare their results with their classmates’ results, write and discuss any differences. 

In the next task students will model the Fibonacci sequence in ToonTalk’s environment. The great challenge of that task is that students will try to model the sequence, using only the AddUp, Add1 and Constant robots (of course they can do it, using only the AddUp robot). First, students will draw a model in paper, showing how they can make the necessary changes in the ready-made robots, to generate the Fibonacci sequence. Teacher may need to help students how to chain different robots, or how to make a copy of the AddUp nest and put it back to robot’s box, so it can work with its own output as an input. After a discussion, students will work in ToonTalk, trying to build up the modified AddUp robot or any other robots chain, to generate a Fibonacci sequence. After completing the task, teacher will start a discussion, where students can share their difficulties in training the robot(s) and try to explain why something went wrong. They will finally publish their results on the project’s website.

Divisibility in Fibonacci Sequences

In this activity students will work on the divisibility patterns in Fibonacci sequences, like F_k | F_nk, i.e F₂ | F_2n, F₃|F_{3n .} F₄|F_{4n .} …. They will also explore the formula (F_m , F_k)= F_(m,k) about the Greatest Common Divisor (GCD) and they will use it effectively in solving problems. They will finally use effectively the tools provided in ToonTalk to solve tasks on Fibonacci sequence’s divisibility. 

In the first task students are asked to generate the sequence [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…] in ToonTalk, using the tools they constructed in the previous activity. Then, they have to find the divisors of 144. The purpose of this activity is to identify that some of its divisors are terms in the same sequence (2, 3 and 8). In the next activity, they will use the rank tool in ToonTalk, to observe the relation between the rank of the first number (the rank of 144 is 12) and the rank of the divisors (2’s rank is 3, 3’s rank is 4 and 8’s rank is 6). Teacher can orchestrate a discussion to help students understand the relation between the rank of these numbers, that is F_k | F_nk, i.e F₃ | F₁₂, F₄ | F_{12 .} F₆ |F_12.  Students can use their results to extend the discussion in a different Fibonacci sequence as well.

In the following task, which can be used for assessment purposes, students will work on a problem, where they have to find which terms of the Fibonacci sequence are divisors of the number 220 (the 10^th term). The teacher can explain to students that they do not have to find the numbers themselves, but only the rank of the numbers in the sequence.

In the following task, students will observe that the greatest common divisor for two numbers in a Fibonacci sequence is itself a Fibonacci number. They will also observe that the rank of the GCD is the GCD for the ranks of the first two numbers. In the first activity, students may work with the sequence such as [3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432…] and they will try to find the GCD for numbers 432 and 102. Students can use the tools in ToonTalk, for generating the sequence and finding the GCD. Teacher can lead a discussion where students will announce and compare their results with their peers’. The results of the discussion can be posted on the project’s website, under a group report. The following activity is for assessment purposes. Students have to find the rank of a number in a Fibonacci sequence, if they know that the rank of a second number is 25 and the rank of the GCD is 5. Students will also explain their answers and post their results on project’s website.