The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) propose a new vision for the mathematics classroom with updated content standards and Standards for Mathematical Practice (SMP). These practices are founded on NCTM processes (Problem Solving, Reasoning and Proof, Communication, Representation, and Connections) and abilities (adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition) that have always been important in mathematics and have gained significant relevance in the mathematics curriculum during the last thirty years. As teachers are called on to implement these mathematical practices in their classrooms, they need support in identifying, designing, and modifying tasks that elicit student engagement, reasoning, and participation, and embrace equity. This article describes the authors efforts to identify, design, and modify mathematical tasks to promote and support student engagement and learning by using different approaches to problem solving: arithmetic, algebraic, and geometric. Specifically, they start with a Diophantine equation and show different approaches to solve it, as a way to enrich teachers with ideas to better understand and experience "what mathematicians do: solve and pose problems." During the discussion, we touch on some mathematical practices (such as SMP 1: Make sense of problems and persevere in solving them, SMP 2: Reason abstractly and quantitatively, and SMP 5: Use appropriate tools strategically) that are recommended in current reforms. We have used several of these activities in classes and professional development workshops for future and current secondary mathematics teachers. The presentation offers some suggestions on how to discuss the ideas in the classroom. However, teachers will make their own decisions on the basis of the level and needs of their students.