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**In This Session**

**Part A:** What Do You See?

**Part B:** Patterns in Situations

**Part C:** Different Uses of Variables

**Part D:** Counting Stairs

**Part E:** Summing Up

**Homework**

In Session 1, we began to develop a definition of algebraic thinking. We used mathematical thinking tools to reason about situations, and we used pictures, charts, and graphs to describe how different quantities are related. Although we were thinking algebraically, we did not use what many people think of when they think of algebra: variables. This session explores the uses of variables in describing patterns and relationships.

**Note 1**

In this session, we will explore patterns in different situations. We will:

- Find, describe, explain, and predict using patterns
- Determine whether or not patterns in tables are uniquely described
- Distinguish between closed and recursive descriptions of patterns
- Understand that a table of data associated with a specific situation determines a unique pattern
- Understand that there are different conceptions of algebra

**New in This Session:**

**Variable: **The term variable can have different meanings:

- An indeterminate, a symbol used to represent generalized arithmetic (such as 0 x a = 0)
- An unknown, used to represent a particular number (such as a + 3 = 7)
- A relationship between quantities (such as a + b = 10)
- A term used to help in understanding the study of algebraic structures (such as a + b = b + a)

**Input: **Almost all rules and algorithms start with an input and end with an output. For example, if a rule says to add three to a number, the input of 8 leads to the output of 11.

**Output: **Almost all rules and algorithms start with an input and end with an **output**. For example, if a rule says to add three to a number, the input of 8 leads to the **output** of 11.

**Recursive Description: **A recursive description of a pattern tells you how to proceed from one step to the next. For example, a recursive description might be, “Add two to the value of the output each time the input goes up by one.” The Fibonacci sequence, where each output is the sum of the two numbers before it, is a recursive description of a pattern.

**Closed-Form description: **A closed-form description of a pattern tells how to get from any input to its output, without having to know any previous outputs. A rule such as “take the input, triple it, and add two” is a closed-form description of a pattern.

**Note 1**

This session introduces a framework for thinking about patterns. We will explore patterns in different situations by finding, describing, explaining, and using patterns to make predictions. An important goal for this session is to progress beyond simply identifying patterns to being able to use patterns to make predictions in situations that are not easily calculated or identified.

Often, patterns given in tables are assumed to have a unique description. This session makes the point that patterns in tables, separate from contexts or situations, can be described in multiple ways. If, however, the pattern is associated with a specific context (such as the toothpick problem in Part B), the description is more uniquely defined.

Another goal of this session is to use variables to describe situations both in and out of context. The staircase and toothpick problems motivate us to think about using variables to describe what happens “down the line” in stages where it would be difficult to extend the table or the drawings.

**Materials Needed:**

Thirty-five blocks or cubes for individuals working alone or per group if not doing the Counting Stairs interactive activity

**Review Previous Session**

Begin by reviewing the main idea of Session 1: Reasoning about situations and then describing how quantities in those situations are related is one of the big ideas in algebraic thinking. In Session 1, although we were thinking algebraically, we may not have been using variables. In Session 2, we will explore situations in which variables may be more useful in describing patterns and relationships.

**Homework Review**

Groups: Discuss any questions about the homework. Consider discussing everyone’s reactions to the “Algebra for Everyone” readings, particularly Bob Moses’s assertion that algebra is part of a civil rights agenda. Also, react to Richard Riley’s argument about algebra for all in the eighth grade.

Groups: Consider discussing Problem H3 from the homework. It is challenging to make the connection between the sloped sides of the container and the curve of the graph. This picture may help explain the situation better.