We introduce the Into Math Graph tool, which students use to graph how “into" mathematics they are over time. Using this tool can help teachers foster conversations with students and design experiences that focus on engagement from the student’s perspective.

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### Amanda K. Riske, Catherine E. Cullicott, Amanda Mohammad Mirzaei, Amanda Jansen, and James Middleton

### Samuel Otten and Andrew Otten

Students make strategic choices–and justify them–to solve a system of two linear equations.

### Michael Weiss

Core content provides opportunities to focus on the structure of mathematical theory, proof, and anticipation of subsequent topics.

A set of problems of many types.

### Victoria Miles

Performance tasks provide effective ways to differentiate mathematics instruction while allowing students to be creative and to incorporate mathematics content that is suitable for their interest and readiness. A project that I have enjoyed assigning to my high school students, Algebra about Me, is designed to introduce, reinforce, and review equation-solving concepts and skills (for a customizable activity sheet, go to www.nctm.org/mt).

Readers comment on published articles or offer their own ideas.

### Shiv Karunakaran, Ben Freeburn, Nursen Konuk, and Fran Arbaugh

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

### Darla R. Berks and Amber N. Vlasnik

Two teachers discuss the planning and observed results of an introductory problem to help students nail a conceptual approach to solving systems of equations.

### Connie Colton and Wendy M. Smith

A multiphase lesson on selling T-shirts guides students from single-variable equations to linear relationships.

### Rachel Levy

The mathematical concept of slope can be made real through a set of simple, inexpensive, and safe experiments that can be conducted in the classroom or at home. The experiments help connect the idea of slope with physical phenomena related to surface tension. In the experiments, changes in surface tension across the surface of the water, which correspond to greater slopes on the graph, lead to increased motion of the fluid. The mathematical content, targeted to middle school and high school students, can be used in a classroom or workshop setting and can be tailored to a single session of thirty to ninety minutes.