Algebra Connections / Edition 1

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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.

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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 Z n

4.6 Multiplicative inverses in Z n

4.7 Elementary applications of modular arithmetic

4.8 Fermat’s Theorem and Wilson’s Theorem ii

4.9 Linear equations defined over Z n

4.10 Extended studies: The Chinese Remainder Theorem

4.11 Extended studies: Quadratic equations defined over Z n

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 x 2 + y 2 = z 2

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


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