Numberama: Recreational Number Theory In The School System

Chapter 1 comprises the nuts and bolts of Numberama, as it includes all the problems that I have included in Recreational Number Theory as part of my Numberama program. Each problem has the designated skill level required, and the problems begin with the Subsets and Circle problem, which utilizes only addition and subtraction, though a knowledge of basic algebra would be required for middle school or high school students to understand the algebraic formulations of this problem. Early in the sequence of problems I go to the Perfect Numbers problem, which admittedly is my favorite problem, and requires a knowledge of multiplication and division, along with once again a knowledge of basic algebra for the same purpose as the Subsets and Circle problem. The learning and teaching strategies that I have included in the Perfect Numbers problem are especially effective in awakening students to the mystery, surprise element, and beauty inherent in our number system, as well as developing an understanding of mathematics as an exciting open-ended field of study with unknown problems that can be worked on with high level computers. Later on in my sequence of problems, one encounters the Sums of Squares problem, which goes back to Fermat, one of the original founders of Number Theory, and once again is an excellent way to develop an appreciation in students of the beauty and mystery involved in our number system. Taken as a whole, these problems from Recreational Number Theory that I have chosen to utilize in my Numberama program can serve as the “magic” needed to demonstrate the inherent joy of mathematics to our students in the school system—and to people of all ages.

Chapter 2 consists of the games that require the skill levels through addition, subtraction, and multiplication. These games consist of The Tri Game, The Fib-Tri Game, The Multiplicative Persistence Game, The Tri-Square Game, The Tri-Squar- -Cube Game, and The Kaprekar Number Game. These games are in general appropriate for children in grades 1, 2, and 3, depending on the children's arithmetic skills level. However, they can be used as enjoyable recreational games for children of all ages, inclusive of mathematically gifted children. See the sections “Introduction to the Games” and “Game Ideas from Teachers at Numberama Teacher Workshops” in the beginning of this book, and some of the teacher and teacher education Numberama workshop participant responses in the Appendix for relevant information about playing these games effectively with children, as well as how these games were developed.

Chapter 3 consists of the games that require the skill levels through multiplication and division. These games consist of The Prime Number Game, The Perfect Number Game, The Semi-Perfect Number Game, The Powerful Number Game, The Divisor Game, The Sum of Squares Game, The Syracuse Algorithm Game, and The Fib-Tri-Prime Game. The games in Chapter 3 are in general appropriate for children in grades 4, 5, and 6, depending on their skills levels, inclusive of gifted children. The Perfect Number Game has been especially popular with children, in particular because of the stimulating open problems about perfect numbers in mathematics; i.e. are there infinitely many perfect numbers and does there exist an odd perfect number (see the Perfect Number problem section in Chapter 1). The games in Chapter 3 are excellent teaching devices to motivate children to practice their multiplication and division skills in an enjoyable way, in addition to enhancing their creative thinking capacities (see some of the teacher and teacher education Numberama workshop participant responses in Appendix).

Chapter 4 consists of the games that require all the arithmetic skill levels through fractions. These games consist of The Clock Arithmetic Game, The Pascal's Triangle Game, The Anomalous Fractions Game, The Farey Fractions Game, and The Numberama Game. The games in Chapter 4 are appropriate in general for children in grades 6, 7, and 8, depending on their skill levels, inclusive of gifted children. These are the most challenging games in the set of Numberama Recreational Number Theory games in this book. These games can be especially stimulating for gifted children, and are an excellent way for middle school children to practice and improve upon their skills with fractions (see some of the teacher and teacher education Numberama workshop participant comments in Appendix). The Numberama Game is an excellent teaching device to practice virtually all of the Numberama Recreational Number Theory problems in Chapter 1 of this book.