Notes for Session 9

Note 1: In this session, we'll explore a primary focus of modern algebra: algebraic structure. Algebraic structures are systems with objects and operations, and the rules or properties governing those operations, that can be used to calculate and solve equations. The objects are often rational numbers and the operations are usually addition, subtraction, division, and multiplication. In other areas of mathematics, however, different objects or operations may be used to solve different kinds of equations.

 Part A Notes: Comparing Operations Part B Notes: Guess My Rule Part C Notes: Algebraic Structures Part D Notes: Working with Algebraic Structures Part E Notes: Summing Up

Important concepts in the study of algebraic structure include comparing processes, doing and undoing, equivalence, and properties of systems.

In Part A, we'll focus on properties of operations as they start to develop "operation sense." The activities in this part revisit the idea of doing and undoing as a way of thinking about the structural relationships between operations.

In Part B, we'll examine the idea of equivalence through the "Guess My Rule" activity. (For example, in what sense is the algorithm "Take a number, add 2, and then multiply the result by 2" the same as "Take a number, double it, and then add 4"?) This section will also reinforce the idea that the same function can be described by different rules.

We'll also look at the units digit of whole numbers in Parts C and D, thinking about an arithmetic structure whose objects are the digits 0 through 9 (essentially, mod 10), and whose operations are addition and multiplication. We'll explore inverses, reciprocals, and the commutative property in this system to expand our conceptual understanding of these properties in a different environment. We'll also have a chance to compare this finite system to our familiar, real number system. Finally, we'll look at cryptography as an application of modular systems.

Review
Groups: Discuss any questions from the homework. You may want to spend some time on the "mod 3" function, Problems H1-H5, because this session covers several "mod" functions. Review the function of "taking the remainder." Note that this is in fact a cyclic function.

Note that "mod 3" was arbitrary. You can work mod on any positive whole number. Take a moment to think about what numbers make sense as inputs (integers), outputs (integers less than the mod number), and remainders (again, integers) before moving on to this session.

 Session 9: Index | Notes | Solutions | Video