Contextualization
Mathematics is a subject that is present in nearly all parts of our lives and understanding the logic behind numbers and operations can help us solve various types of problems. One of the ways to organize and manipulate numbers is through number sequences.
In order to understand the idea of a sequence, we must first understand the meaning of a term in a sequence, which refers to an individual number in a sequence. In a sequence, the terms are usually listed in a specific order, determined by a rule or a pattern.
Number sequences can be simple, such as the sequence of odd numbers (1, 3, 5, 7,....) or complex such as the famous Fibonacci sequence: (1, 1, 2, 3, 5, 8, 13,...). In either case there is a rule, a formula, that enables us to determine the forthcoming number, based solely on the preceding numbers.
There are countless sequences all throughout our everyday lives such as calendar dates; the numbers we use to identify homes on a city street; bus numbers and so forth. Number sequences are also extremely common throughout many intellectual disciplines, including computer science, biology, physics and the social sciences
To provide one specific examples, in computer science, the binary string of (0,1) underlies all computer programming, In physics, sequences might be used to define the movement of a body, according to its location at various different points in time or in biology the sequence of our DNA is essential to understand and identify the diversity of life.
For delving further into this subject, students have the following materials from which to choose
- Khan Academy support materials:
- YouTube video serving as an introduction to numerical sequences:
- Book entitled: " Mathematics : Context & Applications ," written by Luiz Roberto Dante.
# Practical Activity
## Project Title:" A Leap into Infinity : Exploring Number Sequences"
Project Objective
This project is geared towards providing students an opportunity for applying hands on learning with number sequences while solving problems, researching patterns and identifying term inter relationships, Additionally the projects seeks to promote teamwork communication and time organization and management techniques.
Detailed Project Description
The class will divided into teams of between three to five members, with each team assigned and tasked with working one a different number sequence, The different sequences will vary and increase in degrees of complexity thereby enabling the incorporation and examination of more advanced algebraic concepts as necessary. Here are a number suggested number sequences for your consideration: a basic numerical sequence a Fibonacci sequence or the sequence for finding prime numbers amongst others,
Students are challenged to thoroughly research the sequences assigned to their team seeking always to recognize and identify patterns as well as developing algebraic models for the different sequences. Teams will check the accuracy equivalency as it pertains the formulas developed to generate the sequences and are expected predict forthcoming sequence numbers . Students are tasked with applying all their mathematical skills with additional outside independent research necessary for a deeper more comprehensive knowledge of their chosen sequence.
Further, Students will research the real-world applications for the numerical sequence assigned to them; this will include identifying the disciplines where this sequence is utilized in addition to giving at least one specific working example
Required Materials
Students will work in groups on this project, but each student in every group is responsible individually for the completion the following requirements:
- 3-ring binder for notes.
- Research materials (books, access to the Internet and or computer databases).
- Graph paper (for plotting graphs of functions).
- Graphing calculator
- A variety of different colored pencils, and or highlighters.
- Ruler
- Protractor
- Markers for creating posters
Step-by- Step directions
1 - Group Formation : The class is split up or divided into different working teams that consist of exactly 3 or possibly as many as six members each.
2- Sequence distribution : Each team/group is handed a number sequence for further exploration.
3- Investigation : Students will engage in individual as well group research on this topic. Students should actively seek to determine the pattern, write the recursive or explicit equation, identify the domain and the range (if possible), find connections between functions, and explore multiple representations. Students work together to create visual aids as needed for the different function families.
4 -Presentation: The groups are to create group presentations of the function family that was assigned.
Project Submission
This projects is culminated with the submission of one written report per team to be handed in at a time designated by your classroom instructor; there is the further expectation and requirement that each team provide an in class oral presentation of their research. Oral presentations allow for all students in class to share what is learned from exploring the properties and characteristics and real world uses found in the assigned function families.
Report Format:
1 - Introduction: a) State the assigned function family b) Discuss its history and real world connections and applications c) State the objectives to be examined and investigated within your group research
2- The Body/Main Content: a) Explain in detail any and all mathematical concepts which are related to your assigned function families. - Include graphs, equations and or other representations that strengthen the mathematical context. b) Discuss your group’s findings in regards the patterns and or trends within the function families studied.
- Discuss your group’s findings in regards real world connections to a variety of academic disciplines as well as general life applications. 3- Conclusion: a) Restate the main objective for undertaking and completing the assigned project, and summarize your project's major findings. b) Discuss how the function family relates in some way to your own life or to other members in your group.
