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To begin the exploration of what algebraic thinking looks like in a classroom at your grade level, watch a video segment of a teacher who has taken the “Patterns, Functions, and Algebra” course and has adapted the mathematics to her own teaching situation. When viewing the video, keep the following three questions in mind:

**Note 2**

a. |
What fundamental algebraic ideas (content) is the teacher trying to teach? Think back to the big ideas of the previous sessions: patterns, functions, linearity, proportional reasoning, nonlinear functions, and algebraic structure. |

b. |
What mathematical thinking tools (process) does the teacher expect students to demonstrate? Think back to the processes you identified in the first session: problem-solving skills, representation skills, and reasoning skills. |

c. |
How do students demonstrate their knowledge of the intended content? What does the teacher do to elicit student thinking? |

**Video Segment**

In this video segment, Lolita Mattos introduces her students to the process of backtracking. She begins by giving her students an algorithm. She then asks them to undo the algorithm by reversing operations.

You can find this segment on the session video, approximately 46 minutes and 3 seconds after the Annenberg Media logo.

**Problem A1**

Reflect on questions (a), (b), and (c) above. **Note 3**

**Problem A2**

How does Ms. Mattos incorporate the concept of doing and undoing into the solving of equations?

**Problem A3**

How do students think about operations as a result of backtracking?

**Problem A4**

How do students think about solving equations when using backtracking?

**Note 2**

Before examining specific problems at your grade level with an eye toward algebraic thinking, we’ll watch another teacher — one who has also taken the course — teaching in her classroom. The purpose is not to be critical of the teacher’s methods or teaching style. Instead, look closely at how the teacher brings out algebraic ideas in teaching the topic at hand, as well as how the teacher extends the lesson and asks questions that elicit algebraic thinking.

Review the meaning of algebraic ideas (the content of algebra) and mathematical thinking tools (the processes used in analyzing problems). Keep in mind questions (a), (b), and (c) as you watch the video.

**Note 3**

**Groups:** Work in small groups on Problems A1-A4. Share answers to Problem A3 especially, since backtracking is not a common method of solving equations.

**Problem A1**

Reflection on the three questions should include the ideas described below.

a. |
The fundamental algebraic idea (content) in this video is the notion of inverse function as reversing processes and using inverse operations. |

b. |
The teacher expects students to write a function rule to start, with a relationship shown as an equation, and then backtrack to find a rule for the inverse function. |

c. |
Students demonstrate their knowledge of the intended content by drawing a flow chart for a multi-step function, drawing the backtracking flow chart, and then writing an inverse function. |

**Problem A2**

Ms. Mattos asks students to focus on properties of operations, not properties of numbers.

**Problem A3**

The students discuss operations and inverse operations when they backtrack.

**Problem A4**

The students describe backtracking as reversing a process and using inverse operations. This process helps them decide what to do when they solve equations.