We are back for Week 2!

#### Walk the Graph

We began the morning with walking the graph.

#### FA-1: Processes

Andrew led discussion on the informal concept of “processes.” This is like input-output machines that many teachers are already familiar with. Our discussion took us many places, but some of the highlights were:

- Some processes are invertible, that is, they can be undone to return to the original input for any input. Inverse operations generalize the idea of how addition and subtraction are inverses of each other, and how multiplication and division are inverses of each other.
- Some processes are not invertible, such as multiplication by 0. Knowing that a number produced 0 as the output is not enough to know what the original number is.
- We can break more complicated procedures into smaller processes.
- When chaining together procedures, the order matters. For example, “multiply by 2 then add 3” is different from “add 3 then multiply by 2.”
- We indicate the order of procedures by using parentheses in an algebraic expression.
- Two procedures are equivalent if they always produce the same output given the same input. For example, “add 3 then multiply by 2” is a different process than “multiply by 2 then add six,” but they are equivalent in that they always produce the same output given the same input. In other words, they represent different journeys to the same destination.

#### Role-playing helping a student

We divided into pairs and took turns role-playing a teacher helping a student with a mathematical misunderstanding. We audio-recorded these so that we can use them for tonight’s homework.

#### FA-2: Proportions

Matt led discussion on Proportions. We explored the relationship between the number of feet and the corresponding number of inches, and contrasted it with a length and the area of a square with that side length. We will conclude the discussion and work on related problems Tuesday.

### Homework

#### Math

Complete problems 1, 2, 3 on page FA1.9–10.

#### Pedagogy

Read Principles to Actions: “Pose Purposeful Questions” (p. 35-41) and “Elicit and Use Evidence of Student Thinking” (p. 53-57)

In your notebook, write responses to these three prompts:

- In questioning small groups of students working on a problem, a teacher noticed that when she asked a “focusing” question, the students continued to look at their work and continued to engage in their own dialogue. When she asked a “funneling” question, the students looked up at the teacher. Comment on these observations.
- Listen to your audiorecording from today. Use fig. 14 on p. 36 and fig. 16 on p. 39 to write a description of your question patterns.
- How might you change your questioning to elicit and then use evidence of your students’ thinking to move the student forward to the mathematical goal of the problem?