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## Complete the What?

Solving quadratic functions is a cornerstone of first year algebra, but students struggle to gain a conceptual understanding of the process of completing the square. With the help of a historical perspective, students can gain both a deep geometric and algebraic understanding of the algorithm.

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## Galileo, Gauss, and the Green Monster

Using ideas of Galileo and Gauss but avoiding calculus, students create a model that predicts whether a fly ball will clear the famous left-field wall at Fenway Park.

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## The Joy of Paper Folding

A paper-folding problem is easy to understand and model, yet its solution involves rich mathematical thinking in the areas of geometry and algebra.

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## Happy Families, Continued

Explore the relationship between families of quadratic expressions factorable over the integers and Pythagorean triples.

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## Technology Tips: Building Interactive Demonstrations with Sage

Sage is an open-source software package that can be used in many different areas of mathematics, ranging from algebra to calculus and beyond. One of the most exciting pedagogical features of Sage (http://www.sagemath.org) is its ability to create interacts—interactive examples that can be used in a classroom demonstration or by students in a computer laboratory. By accessing a simple Web-based Sage Notebook interface, we can quickly compute a diverse range of examples, such as finding the prime factorization of a positive integer or graphing transformations of functions. Graphing calculator explorations translate nicely into Sage, and the interact feature makes them much more dynamic.

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## Connecting Research to Teaching: Linked Representations in Algebra: Developing Symbolic Meaning

Algebra involves negotiating meaning between the worlds of mathematical ideas and the symbols that represent them. Here we examine classroom interactions and explorations as they relate to the connection of these worlds through the use of dynamically connected representations in a technology-rich environment.

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## Exploring Quadratic Functions with Logger Pro

Using technology, students in an extended second year algebra class engage in an activity that introduces them to quadratic functions.

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## Using Secant Parabolas to Find Roots

Almost all root finding methods use linear functions to calculate the next approximation to a real root. The authors introduce a method based on using parabolas through three points and use one of the two roots via the quadratic formula to identify the following approximation.

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## Tasks That Make Connections through Representations

Students think resiliently about using the quadratic formula, analyzing factors graphically, finding the shortest distance between two points, and finding margin of error.