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## Deciphering Multiplication Algorithms with the Area Model

Prospective teachers explore Egyptian multiplication and Russian peasant multiplication to address the quantitative and qualitative relationships underlying these algorithms.

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## Factoring for Roots

Justifications, methods, and results compare two classes of students who used a new technique that ties together procedural fluency and conceptual understanding in a manner unlike other current strategies.

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## Problem Solving + Problem Posing = Mathematical Practices

A Diophantine equation engages mathematics students as they problem solve.

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## When Foil Falls Apart

A critique of FOIL provides an alternate method of multiplying polynomials.

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## The No-to-Yes Project: Conquering Common Algebraic Mistakes That Drive Us and Our Students Crazy

Help students stop making typical, persistent errors related to misconceptions about exponents, distribution, fraction simplification, and more.

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## Solving Equations: Exploring Instructional Exchanges as Lenses to Understand Teaching and Its Resistance to Reform

Many research studies have sought to explain why NCTM's vision for mathematics classrooms has not had greater impact on everyday instruction, with teacher beliefs often identified as an explanatory variable. Using instructional exchanges as a theoretical construct, this study explores the influence of teachers' institutional positions on the solving of equations in algebra classrooms. The experimental design uses surveys with embedded rich-media representations of classroom interaction to surface how teachers appraise correct solutions to linear equations where some solutions follow suggested textbook procedures for solving linear equations and others do not. This paper illustrates the feasibility of studying teaching with rich-media surveys and suggests new ways to support changes in everyday mathematics teaching.

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## Number Talks: Gateway to Sense Making

Support your high school students in the challenging transition from memorization to understanding.