A connected introduction of angle measure and the sine function entails quantitative reasoning.

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### 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.

### Lorraine A. Jacques

Teachers have what they need-students, a data projector or an interactive whiteboard, and connection to the Internet. Teachers know what they want-students observing mathematics in action, making conjectures, and supporting their conjectures with solid reasoning. However, when using applets, teachers quickly encounter two difficulties: how to choose them and how to use them.

### Heather Lynn Johnson

This article explores quantitative reasoning used by students working on a bottle- filling task. Two forms of reasoning are highlighted: simultaneous-independent reasoning and change-dependent reasoning.

### Helen M. Doerr, Donna J. Meehan, and AnnMarie H. O'Neil

Building on prior knowledge of slope, this approach helps students develop the ability to approximate and interpret rates of change and lays a conceptual foundation for calculus.

### Colin Foster

Exploring even something as simple as a straight-line graph leads to various mathematical possibilities that students can uncover through their own questions.

### Christopher J. Bucher and Michael Todd Edwards

In the introductory geometry courses that we teach, students spend significant time proving geometric results. Students who conclude that angles are congruent because “they look that way” are reminded that visual information fails to provide conclusive mathematical evidence. Likewise, numerous examples suggesting a particular result should be viewed with skepticism. After all, unfore–seen counterexamples render seemingly valid conclusions false. Inductive reasoning, although useful for generating conjectures, does not replace proof as a means of verification.

## Connecting Research to Teaching: Why Did You Do That?

### Reasoning in Algebra Classrooms

### Daniel Chazan and Dara Sandow

Secondary school mathematics teachers are often exhorted to incorporate reasoning into all mathematics courses. However, many feel that a focus on reasoning is easier to develop in geometry than in other courses. This article explores ways in which reasoning might naturally arise when solving equations in algebra courses.