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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, Liza Jones introduces her students to the process of undoing. She begins by asking her students to figure out the “recipe” or formula she is using to turn an In into an Out. She then asks them to undo the “recipe” by finding the In when given the Out.

You can find this segment on the session video, approximately 33 minutes and 48 seconds after the Annenberg Media logo.

**Problem A1**

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

**Problem A2**

How does Ms. Jones incorporate the concept of doing and undoing into the model of function machines?

**Problem A3**

How do the students think about undoing the recipe they constructed?

**Problem A4**

How could Ms. Jones have used recursive thinking in this lesson?

**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 (that is, 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 A4 especially, because recursive thinking is not necessarily something that third-grade teachers often consider in their teaching. The vocabulary (recursive) is not the important part here. The method of thinking recursively, however, is important for teachers to consider as students make sense of patterns.

**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 function and inverse function as doing and undoing. |

b. |
The teacher expects students to write a function rule to represent a relationship shown in a table of values and then find a rule for the inverse function. |

c. |
Students demonstrate their knowledge of the intended content by writing function rules and by describing inverse functions as reversing a recipe by reversing steps and using “opposite” operations. |

**Problem A2**

Ms. Jones asks students to think of a function as a recipe for “doing,” which then leads to inverse functions as “undoing” the same recipe.

**Problem A3**

The students describe undoing as reversing the steps and using opposite operations.

**Problem A4**

Ms. Jones could have used recursive thinking with the function machine that doubles and adds one. This function could be described recursively by saying, “The Out number for In = 1 is 3. The Out number for any other In is the previous Out + 2.”