Number Treasury3

Investigations, Facts and Conjectures about More than 100 Number Families

Margaret J Kenney, Stanley J Bezuszka

• Publisher: WSPC
• Year: 2015
• Edition: 3
• eBook ISBN:
  9789814603713
• Topic: Mathematik
• Subtopic: Mathematik unterrichten

This resource volume is an enlargement as well as an update of the previous edition. The book aims to introduce the reader to over 100 different families of positive integers. A brief historical note accompanies the descriptions and examples of several of the families together with a mix of routine exercises and problems as well as some thought provokers to solve. Number Treasury 3 especially aims to stimulate further study beyond the scope of the introductory treatment given in the book. The emphasis in Number Treasury 3 is on doing not proving. However, the reader is directed to think critically about situations, to provide explanations, to make generalizations, and to formulate conjectures. To engage the reader from the start, the book begins with a set of rich Investigations. These are standalone activities that represent each of the chapters of the book.

Contents:

• A Perfect Number of Investigations 28 = 1 + 2 + 4 + 7 + 14
• Numbers Based on Divisors and Proper Divisors
• Plane Figurate Numbers
• Solid Figurate Numbers
• More Prime Connections
• Digital Patterns and Noteworthy Numbers
• More Patterns and Other Interesting Numbers

Readership: Secondary and intermediate classroom teacher and tertiary mathematics education instructor; undergraduates whose interest is in teaching mathematics at the pre-tertiary level and the segment of the general public for whom mathematics might be a hobby.
Key Features:

• Differs from other books that treat numbers (formally or informally) because it contains numerous exercises and problems, detailed examples for the reader to follow, and the narrative is kept to a minimum
• Has broad appeal for different audiences
• Is a gateway to families of positive integers which leads and encourages the reader to go beyond the book to deeper study of the topics presented

Chapter 1

A Perfect Number of Investigations 28 = 1 + 2 + 4 + 7 + 14

A GREAT discovery solves a great problem but there is a grain of discovery in any problem.

George Polya (1887–1985)

What are They?

The Investigations that follow are a set of stand-alone activities. Each Investigation focuses on at least one number family or topic relating to numbers. All but six of the Investigations are described on one page. Share with students that the problems in an Investigation are intended to be challenging in many ways:

• The time needed to complete an Investigation may vary and exceed the time required to finish a typical homework assignment.

• The computation necessary to bring closure may be lengthy and demanding — even with the use of technology.

• The amount of writing, discussing, explaining and illustrating may be more than anticipated.

Teacher Tips

The Investigations are listed in ascending order of difficulty in the table on the next two pages. There are three levels of difficulty represented in the 28 Investigations that can be assigned individually or adapted for group work. The lowest level consists of the first eight Investigations that have the least prerequisites. The middle level consists of the next nine Investigations and requires more use of abstract reasoning and familiarity with algebraic expressions. The final 11 Investigations challenge the student to persist and probe more deeply in order to complete the work.

There is also a column in the table naming the most significant prerequisite(s) needed by the student to understand and carry out the work in each Investigation. The teacher may choose to provide additional content background for some specific Investigations prior to assigning them. Assign the Investigations as extended homework or as in-class work. Some Investigations call for the preparation of reports. Thus, students may need further directions, especially about the kind of resources available for them to use. Students should know the Internet is an excellent resource, and that it should be used appropriately as they compile their reports.

Finally, pages noted in the Prerequisites column refer to related material contained in Chapters 2 through 7.

Page	Investigation	Prerequisites & Text reference
4	Footsteps of Lagrange	Square numbers, p. 85
5,6	Trying Trapezoids	Triangular numbers & trapezoids, p. 74
7	Hexagons in Black & White	Familiarity with recursive & explicit formulas, p. 107
8,9	Marble Art	Figurate numbers, following directions, pp. 74–111
10	Honest Number Hunt	Counting in a language, searching resources, p. 197
11	Seeking Honesty in Numbers	Groups work together to organize their data, p. 197
12	Geoboard Journeys	Trial/Error pursuit, link Catalan & Pascal, p. 200
13	Mysterious Mountains & Binary Trees	Doing & arranging sketches, p. 200
14	Fermat Factorings	Prime factorization, p. 42
15,16	Factor Lattices	LCM, prime factorization, p. 46
17	A Juggling Act	Describing systematically, p. 46
18	The Super Sum	Reasoning with patterns, p. 74
19,20	Conjecturing with Pascal	Articulating patterns, p. 79
21	Pythagorean Triple Pursuits	Evaluating expressions, p. 90
22	Pentagonal Play	Squares & triangles within, p. 101
23	Triangular Number Turnarounds	Visualizing triangles, p. 80
24	Centered Triangular Numbers	Making algebraic generalizations, p. 74
25,26	Catalan Capers	Connect algebra & geometry, p. 201
27	Highly Composite Numbers	Counting divisors, p. 45
28,29	Tower of Hanoi & the Reve’s Puzzle	Recursive actions & thinking, p. 56
30	Perfect Number Patterns	Using logs to count, p. 59
31	Crisscross Cubes	Perimeter, are...

Table of contents

• Cover page
• Title page
• Copyright page
• Dedication
• Contents
• Foreword
• 1. A Perfect Number of Investigations 28 = 1 + 2 + 4 + 7 + 14
• 2. Numbers Based on Divisors and Proper Divisors
• 3. Plane Figurate Numbers
• 4. Solid Figurate Numbers
• 5. More Prime Connections
• 6. Digital Patterns and Noteworthy Numbers
• 7. More Patterns and Other Interesting Numbers
• Recommended Readings
• Glossary of Numbers
• Solutions to Investigations
• Solutions to Exercises
• Index