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Integers, Fractions and Arithmetic: A Guide for Teachers [Pehme köide]

  • Formaat: Paperback / softback, 208 pages, kaal: 400 g
  • Sari: MSRI Mathematical Circles Library
  • Ilmumisaeg: 30-Jan-2013
  • Kirjastus: American Mathematical Society
  • ISBN-10: 082188798X
  • ISBN-13: 9780821887981
Teised raamatud teemal:
  • Formaat: Paperback / softback, 208 pages, kaal: 400 g
  • Sari: MSRI Mathematical Circles Library
  • Ilmumisaeg: 30-Jan-2013
  • Kirjastus: American Mathematical Society
  • ISBN-10: 082188798X
  • ISBN-13: 9780821887981
Teised raamatud teemal:
This book, which consists of twelve interactive seminars, is a comprehensive and careful study of the fundamental topics of K-8 arithmetic. The guide aims to help teachers understand the mathematical foundations of number theory in order to strengthen and enrich their mathematics classes. Five seminars are dedicated to fractions and decimals because of their importance in the classroom curriculum. The standard topics are covered in detail, but are arranged in an order that is slightly different from the usual one. Multiplication is treated first, and with that in hand, common denominators and equivalent fractions are more readily understood and are available for use when discussing addition. The book is intended for the professional development of teachers. It is appropriate for teacher education programs as well as for enrichment programs such as Mathematical Circles for Teachers. There are numerous activities in each seminar that teachers can bring into their classrooms. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
To The Reader ix
Introduction xi
Seminar 1 Number Systems
1(14)
1 Arithmetic in the Integers, Part I
2(5)
2 Roll Back, A Number Game of Chance
7(1)
3 Other Number Systems
8(1)
4 Arithmetic in the Integers, Part II
9(6)
Seminar 2 Divisibility and Order in the Integers
15(16)
1 Introduction
15(1)
2 Properties of Divisibility
16(4)
3 Divisibility and Arithmetic in Z.
20(1)
4 An Activity on Secret Codes, I
21(2)
5 Order in Z
23(5)
6 Divisibility and Order in Z.
28(3)
Seminar 3 GCD's and The Division Algorithm
31(20)
1 Introduction
31(2)
2 Euclid's Game
33(2)
3 The Division Algorithm
35(3)
4 Return to Euclid's Game
38(2)
5 The Euclidean Algorithm
40(3)
6 GCD's as Linear Combinations
43(8)
Seminar 4 Prime Numbers and Factorization Into Primes
51(14)
1 Prime Numbers
51(7)
2 Factorization Into Primes
58(7)
Seminar 5 Applications of Prime Power Factorization
65(16)
1 Finding All Positive Divisors of n
65(4)
2 The Locker Problem
69(2)
3 The Greatest Common Divisor, the Least Common Multiple and the Least Common Denominator
71(6)
4 Secret Codes, a Game of Integer Divisors, II
77(4)
Seminar 6 Modular Arithmetic and Divisibility Tests
81(18)
1 Preliminaries
81(1)
2 Introduction
81(1)
3 Examples of Congruence
81(4)
4 Congruence
85(7)
5 Congruence and Tests for Divisibility
92(7)
Seminar 7 More Modular Arithmetic
99(16)
1 Congruence Classes
99(4)
2 The Arithmetic of Congruence Classes
103(6)
3 What is a Linear Congruence?
109(1)
4 Solving Linear Congruences
109(4)
5 Inverses mod m
113(2)
Seminar 8 The Arithmetic of Fractions
115(16)
1 Introduction
115(1)
2 Fractions and Why We Need Them
115(2)
3 Multiplication of Fractions and Common Denominators
117(6)
4 Equivalence of Fractions
123(8)
Seminar 9 Properties of Multiplication of Fractions
131(14)
1 Reducing a Fraction to Lowest Terms
131(3)
2 Multiplication and Equivalence
134(2)
3 Properties of Multiplication of Fractions
136(1)
4 Equality and Equivalence
137(1)
5 Fractions and Mixed Numbers
138(1)
6 Division of Fractions
139(2)
7 Word Problems with Fractions
141(4)
Seminar 10 Addition of Fractions
145(16)
1 Addition of Fractions with the Same Denominator
145(1)
2 The Rule for Addition of Fractions
146(3)
3 Comparison of Methods for Addition of Fractions
149(4)
4 The Properties of Addition of Fractions
153(3)
5 The Distributive Property
156(1)
6 Addition and Equivalence
157(4)
Seminar 11 The Decimal Expansion of a Fraction
161(18)
1 Definitions of Decimal Fraction, Decimal Expansion and Decimal
161(3)
2 Constructing the Decimal Expansion of a Fraction
164(6)
3 Fractions with Terminating Decimals
170(4)
4 Fractions with Nonterminating Decimals
174(5)
Seminar 12 Order and the Number Line
179(20)
1 Positive Fractions and Negative Fractions
179(2)
2 Comparison of Fractions
181(3)
3 Properties of Order in the Set of Fractions
184(4)
4 Ordering of Decimals
188(2)
5 Representing Fractions on the Number Line
190(6)
6 Decimals on the Number Line
196(3)
Appendix A The Mysterious Long Division Algorithm 199(6)
Appendix B The Pigeonhole Principle 205(2)
Index 207
Judith D. Sally, Northwestern University, Evanston, IL, USA

Paul J. Sally, Jr., University of Chicago, Chicago, IL, USA