This project focuses on the Initial Value Problem (IVP) for ordinary differential equations with the application of multipoint recursion schemes. The effectiveness and convergence of these schemes are explored and subsequently applied to examine the properties of a compound pendulum, specifically the dependence of the oscillation period on energy. The chapter then focuses on Newton’s laws of motion, laying the foundation for understanding the motion equation. The project uses a simple pendulum to illustrate the concept, looking at how changes in amplitude affect the period of harmonic oscillations. Numerical methods, such as recursive methods based on local extrapolation, are then employed to derive formulas. The project concludes by discussing the integration of Runge–Kutta methods and implicit schemes to solve the equations. This project ultimately questions the viability of the pendulum as a standard unit of time, adding value to ongoing discussions in physics and mathematics education.

This chapter focuses on mathematics and computational thinking. In each chapter, the practice is dissected into distinct and clear learning tasks that serve as process goals for learning the practice. These tasks are then examined within the context of a self-regulated learning cycle and coaching strategies for instruction and assessment are emphasized. The instruction and assessment strategies are contextualized for students in grades 9–12 and focus on conducting an investigation on the factors influencing the period of a pendulum. The data practices for the investigation are infused with computational thinking. The tasks are reassembled into two case studies focused on the heating curve of water– one positive and one negative – to demonstrate how the learning tasks can be used by students and how teachers can support students learning how to plan and carry out investigations.

In this chapter, we begin by examining the work due to a torque. We then define the concept of the rotational kinetic energy for a point mass, systems of discrete masses, and continuous rigid bodies. We develop the angular work-kinetic energy theorem and use it to study the conservation of energy and the conservation of mechanical energy in systems involving rotational motion. To develop these theorems, we draw from our understanding of the analogous theorems in linear motion.