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## Puddle Play!

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## Examining Errors and Framing Feedback

Engaging mathematics students with peers in analyzing errors and formulating feedback improves disposition, increases understanding, and helps students uncover and correct misconceptions while informing opportunities for targeted instruction.

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## Triangles from Three Points

Using technology to solve triangle construction problems, students apply their knowledge of points of concurrency, coordinate geometry, and transformational geometry.

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## Graphing Inequalities, Connecting Meaning

tudents often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and my students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an inequality, students may lack a deep understanding of the relationship between the inequality and its graph. Hiebert and Carpenter (1992) stated that mathematics is understood “if its mental representation is part of a network of representations” and that the “degree of understanding is determined by the number and strength of the connections” (p. 67). I therefore developed an activity that allows students to explore the graphs of inequalities not presented as lines in slope-intercept form, thereby making connections between pairs of expressions, ordered pairs, and the points on a graph representing equations and inequalities.

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## Working the System

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

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## Using Tiered Explorations to Promote Reasoning

Address the needs of diverse learners with a class structure that is designed around a crime scene theme and based on student choice and perceptions of the math being studied.

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## A Natural Approach to the Number e

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.

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## Student-Generated Questions in Mathematics Teaching

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

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

### Reasoning in Algebra Classrooms

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.