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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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## Happy Families, Continued

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

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Readers comment on published articles or offer their own ideas.

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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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## 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.

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## Generating Student Interest with Mathematical Stories

Research of enacted curriculum supports the role of sequence in framing lessons that are both coherent and interesting for students.

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## Solving and Graphing Quadratics with Symmetry and Transformations

One of the central components of high school algebra is the study of quadratic functions and equations. The Common Core State Standards (CCSSI 2010) for Mathematics states that students should learn to solve quadratic equations through a variety of methods (CCSSM A-REI.4b) and use the information learned from those methods to sketch the graphs of quadratic (and other polynomial) functions (CCSSM A-APR.3). More specifically, students learn to graph a quadratic function by doing some combination of the following:

• Locating its zeros (x-intercepts)

• Locating its y-intercept

• Locating its vertex and axis of symmetry

• Plotting additional points, as needed

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## Reader Reflections – March 2013

Readers comment on published articles or offer their own mathematical ideas.

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