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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, Gina Webber asks her students to think about patterns. She began the lesson by reading from the book How Many Feet in the Bed? Students made a chart of the number of people and feet in the bed. Watch them discuss the relationship between people and feet. **Note 3**

You can find this segment on the session video, approximately 18 minutes and 36 seconds after the Annenberg Media logo.

**Problem A1**

Reflect on the questions (a), (b), and (c) above.

**Problem A2**

How does Ms. Webber encourage students to use patterns to predict the number of feet and people in the bed?

**Problem A3**

How does Ms. Webber incorporate the theme of “doing and undoing” we explored in Session 3?

**Problem A4**

How does Ms. Webber use recursive thinking in this lesson?

**Note 2**

Before examining specific problems at this 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 first-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 ideas (content) in this video are patterns that lead to functions and proportional reasoning. |

b. |
The skills that the teacher expects students to demonstrate involve representing patterns with tallies, pictures, or charts to find relationships between numbers of people and numbers of feet or toes. |

c. |
Students demonstrate their knowledge of the intended content by drawing pictures or tallies, or counting by twos or fives to answer questions about the relationships. |

**Problem A2**

Ms. Webber encourages students to use patterns to predict when she asks students to look at the table to answer questions such as the numbers of feet for six or 10 people.

**Problem A3**

Ms. Webber uses “doing and undoing” when she asks students to work backwards to find the number of people for 70 toes.

**Problem A4**

Ms. Webber uses recursive thinking when she encourages students to count by twos or fives to answer the questions.