Algebra Connections

Description

Features

• Classroom Connections – Directly connect college-level abstract algebra and number theory to standards-based middle grade mathematics curricula.
  – Students benefit by learning about the mathematics that underlies algebraic concepts in the middle school curriculum.
  – Specific examples from middle-grade curricular materials show students the direct connections between the mathematics they are learning and the mathematics they will be teaching.
• Focus on the mathematics in new reform materials, including Connected Mathematics, Mathematics in Context, MathThematics, MathScape.
  – Students are shown how to effectively utilize these newly developed, innovative materials.
  – Provides many in-depth examples.
• Classroom Problems and Classroom Discussions – Focus on discovery and collaborative learning.
  – Teaches students to make sense of mathematics and learn many habits of discovery.
  – Encourages group work, prompting students to share ideas and understand the ideas of others.
  – Encourages writing solutions with complete justifications.

Table of Contents

1. Patterns

1.1 Classroom connections: Representing patterns

1.2 Reflections on classroom connections: Representing patterns

1.3 Arithmetic sequences

1.4 Geometric sequences

1.5 Mathematical induction

1.6 Classroom connections: counting tools

1.7 The Binomial Theorem

1.8 The Fibonacci sequence

2. Arithmetic and Algebra of the Integers

2.1 A few mathematical questions concerning the periodical cicadas

2.2 Classroom connections: multiples and divisors

2.3 Reflections on classroom connections: multiples and divisors

2.4. Multiples and divisors

2.5 Least common multiple and greatest common divisor

2.6 The Fundamental Theorem of Arithmetic

2.7 Revisiting the lcm and gcd

2.8 Relations and results concerning lcm and gcd

3. The Division Algorithm and the Euclidean Algorithm

3.1 Measuring integer lengths and the Division Algorithm

3.2 The Euclidean Algorithm

3.3 Applications of the representation gcd(a, b) = ax + by

3.4 Place value

3.5 Prime thoughts

4. Arithmetic and Algebra of the Integers Modulo n

4.1 Classroom connections: divisibility tests

4.2 Reflections on classroom connections: Justifying the divisibility tests

4.3 Clock addition

4.4 Modular arithmetic

4.5 Comparing arithmetic properties of Z and Zn

4.6 Multiplicative inverses in Zn

4.7 Elementary applications of modular arithmetic

4.8 Fermat's Theorem and Wilson's Theorem ii

4.9 Linear equations defined over Zn

4.10 Extended studies: The Chinese Remainder Theorem

4.11 Extended studies: Quadratic equations defined over Zn

5. Algebraic Modeling in Geometry: The Pythagorean Theorem and More

5.1 The significance of Daryl's measurements and related geometry

5.2 Classroom connections: The Pythagorean Theorem and its converse

5.3 Reflections on classroom connections: The Pythagorean Theorem

and its converse

5.4 Computing distance in 2-dimensional and 3-dimensional

Euclidean space: The distance formula

5.5 An extension of the Pythagorean Theorem: The law of cosines

5.6 Integer distances in the plane

5.7 Pythagorean triples: Positive integer solutions to x2 + y2 = z2

5.8 Extended studies: Further investigations into integer distance point sets - a Theorem of Erdös.

5.9 Extended studies: Additional questions concerning Pythagorean triples

5.10 Fermat's Last Theorem

6. Arithmetic and Algebra of Matrices

6.1 Classroom Connections: systems of linear equations

6.2 Reflections on classroom connections: systems of linear equations

6.3 Rational and irrational numbers

6.4 Systems of linear equations

6.5 Polynomial curve fitting: an application of systems of linear equations

6.6 Matrix arithmetic and matrix algebra

6.7 Multiplicative inverses: solving the matrix equation AX = B

6.8 Coding with matrices

Glossary