GRADE BAND: HS ALGEBRA Domain: Seeing Structure in Expressions Why this domain is a priority for professional development Many teachers will look at the standards and declare they already teach these. In fact, they address similar content procedurally without attending to structure. It is new content for teachers to look at familiar-seeming expressions and equations and address questions that get at structure. Why this domain is a priority for professional development To start working with teachers, one can present familiar-looking problems but ask unfamiliar questions. One might even start with a particular expression and ask, “Tell me all you can about this.” For example, 460 (1.93)t Why this domain is a priority for professional development The content of this strand is the upper-grade version of what we were talking about this morning. Students need to work back and forth between symbols and a variety of representations and/or contexts expressions might be drawn from. They need to be able to look at the components of an expression and think about the behavior of the operations involved to see how the expression changes with the change of a variable. Key standards in this domain that pd should focus on 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (These standards are of a larger grain size than the others in this section.) Key standards in this domain that pd should focus on “Write expressions in equivalent forms to solve problems”—This should be interpreted with a sense of purpose. Why should expressions be written in different forms? What benefits do you get from each form? Once why is addressed, then get at how—that is, develop skills of symbolic manipulation. This cluster needs lots of good examples to get somewhere with groups of students. Simplification is a basic category of this statement, but the statement covers much more. Domains and standards that are in need of new resources In addition to working on the content, PD needs to support teachers with techniques for asking students to do something new. Teachers need help in facilitating purposeful discussion.