## Learning Math: Patterns, Functions, and Algebra

# Algebraic Structure

## Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.

**In This Session:**

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

In Session 8, we concluded our study of functions by looking at cyclic and inverse functions. You learned that cyclic functions have repeating outputs and that in an inverse proportion, the outputs get smaller as the inputs get larger, and vice versa. Finally, you found that many different kinds of functions can be drawn through the same two points.

Throughout the past several sessions, we’ve seen that functions are relationships between inputs and outputs. We have explored properties of different kinds of functions, situations where they arise, and how they can be expressed in tables, equations, and graphs. This session explores algebra as structure by taking a closer look at the properties of functions.** Note 1**

### Learning Objectives

The main goal for Session 9 is to find out what is meant by “algebraic structure,” a relatively modern branch of mathematics. To that end, in this session you will:

- Explore important concepts in the study of algebraic structure
- Explore a new algebraic structure
- Solve equations in the new structure

### Key Terms

**Previously Introduced:**

algorithm

set

solution set

modular arithmetic

**New in This Session:**

algebraic structure

binary operation

commutativity

associativity

distributivity

closure

identity element

inverse

group

field

### Notes

**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.

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.