Algebra Connections

by Papick, Ira J.; University of Missouri, UMO

• ISBN13: 9780131449282
• ISBN10: 0131449281
• Edition: 1st
• Format: Paperback
• Copyright: 2005-12-02
• Publisher: Pearson

Summary

Strong mathematics performance in the middle grades is more important than ever-and teachers entering the field need to prepare for this endeavor in new and innovative ways. This new approach introduces some basic concepts of number theory and modern algebra that underlie middle grade arithmetic and algebra, with a focus on collaborative learning combined with extensive in-class and out-of-class assignments.Gives both pre-service and in-service teachers a fundamental understanding of the key mathematical ideas that they will be teaching, so that in turn they can help their students learn important mathematics. Directly connects college-level abstract algebra and number theory to standards-based middle grade mathematics curricula. Gives specific examples from middle-grade curricular materials to show readers the direct connections between the mathematics they are learning and the mathematics they will be teaching. Focuses on the mathematics in new reform materials. Offers Classroom Problems and Classroom Discussions that focus on discovery and collaborative learning.A useful reference for teachers of middle-grades mathematics.

Table of Contents

Preface	vii
Patterns	1	(62)
Classroom Connections: Representing Patterns	1	(2)
Reflections on Classroom Connections: Representing Patterns	3	(11)
Arithmetic Sequences	14	(2)
Classroom Connections: A Quadratic Sequence	16	(1)
Reflections on Classroom Connections: A Quadratic Sequence	16	(5)
Finite Arithmetic Sequences	21	(2)
Geometric Sequences	23	(6)
Mathematical Induction	29	(5)
Classroom Connection: Counting Tools	34	(13)
The Binomial Theorem	47	(4)
The Fibonacci Sequence	51	(12)
Arithmetic and Algebra of the Integers	63	(32)
A Few Mathematical Questions Concerning the Periodical Cicadas	64	(1)
Classroom Connections: Multiples and Divisors	64	(1)
Reflections on Classroom Connections: Multiples and Divisors	65	(5)
Multiples and Divisors	70	(3)
Least Common Multiple and Greatest Common Divisor	73	(2)
The Fundamental Theorem of Arithmetic	75	(7)
Revisiting the LCM and GCD	82	(7)
Relations and Results Concerning LCM and GCD	89	(6)
The Division Algorithm and the Euclidean Algorithm	95	(50)
Measuring Integer Lengths and the Division Algorithm	95	(7)
The Euclidean Algorithm	102	(4)
Applications of the Representation GCD (a, b) = ax + by	106	(4)
Place Value	110	(15)
Prime Thoughts	125	(20)
Arithmetic and Algebra of the Integers Modulon	145	(70)
Classroom Connections: Divisibility Tests	146	(1)
Reflections on Classroom Connections: Justifying the Divisibility Tests	146	(2)
Clock Addition	148	(3)
Modular Arithmetic	151	(13)
Comparing Arithmetic Properties of Z and Zn	164	(5)
Multiplicative Inverses in Zn	169	(5)
Elementary Applications of Modular Arithmetic	174	(14)
Fermat's Little Theorem and Wilson's Theorem	188	(5)
Linear Equations Defined over Zn	193	(7)
Extended Studies: The Chinese Remainder Theorem	200	(3)
Extended Studies: Quadratic Equations Defined over Zn	203	(12)
Algebraic Modeling in Geometry: The Pythagorean Theorem and More	215	(38)
The Significance of Daryl's Measurements and Related Geometry	216	(1)
Classroom Connections: The Pythagorean Theorem	217	(1)
Reflections on Classroom Connections: The Pythagorean Theorem and Its Converse	217	(9)
Computing Distance in Two-Dimensional and Three-Dimensional Euclidean Space: The Distance Formula	226	(1)
An Extension of the Pythagorean Theorem: The Law of Cosines	227	(2)
Integer Distances in the Plane	229	(1)
Pythagorean Triples: Positive Integer Solutions to x2 + y2 = z2	230	(7)
Extended Studies: Further Investigations into Integer Distance Point Sets---A Theorem of Erdos	237	(3)
Extended Studies: Additional Questions Concerning Pythagorean Triples	240	(10)
Fermat's Last Theorem	250	(3)
Arithmetic and Algebra of Matrices	253	(58)
Classroom Connections: Systems of Linear Equations	254	(1)
Reflections on Classroom Connections: Systems of Linear Equations	255	(6)
Rational and Irrational Numbers	261	(7)
Systems of Linear Equations	268	(14)
Polynomial Curve Fitting: An Application of Systems of Linear Equations	282	(4)
Matrix Arithmetic and Matrix Algebra	286	(12)
Multiplicative Inverses: Solving the Matrix Equation AX = B	298	(8)
Coding with Matrices	306	(5)
Glossary	311	(10)
References	321	(2)
Answers to (Most) Odd-Numbered Exercises	323	(22)
Photo Credits	345	(2)
Index	347