• 6-8: 4E/M4. Energy appears in different forms and can be transformed within a system. Motion energy is associated with the speed of an object. Thermal energy is associated with the temperature of an object. Gravitational energy is associated with the height of an object above a reference point. Elastic energy is associated with the stretching or compressing of an elastic object. Chemical energy is associated with the composition of a substance. Electrical energy is associated with an electric current in a circuit. Light energy is associated with the frequency of electromagnetic waves.
• 9-12: 4E/H9. Many forms of energy can be considered to be either kinetic energy, which is the energy of motion, or potential energy, which depends on the separation between mutually attracting or repelling objects.
• 9-12: 4E/H10. If no energy is transferred into or out of a system, the total energy of all the different forms in the system will not change, no matter what gradual or violent changes actually occur within the system.

• 6-8: 4F/M3a. An unbalanced force acting on an object changes its speed or direction of motion, or both.
• 9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.

• 9-12: 11A/H1. A system usually has some properties that are different from those of its parts, but appear because of the interaction of those parts.

• 9-12: 12B/H1. Use appropriate ratios and proportions, including constant rates, when needed to make calculations for solving real-world problems.
• 9-12: 12B/H2. Find answers to real-world problems by substituting numerical values in simple algebraic formulas and check the answer by reviewing the steps of the calculation and by judging whether the answer is reasonable.

• N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

• A-SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
• A-SSE.2 Use the structure of an expression to identify ways to rewrite it.

• A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

We highly recommend this resource to promote deeper understanding of the interactions that influence the motion of a mass hanging from a spring. The author anticipates student conceptual roadblocks and helps them recognize how to use alternative problem-solving methods when the kinematic equations would become overly tedious. Meets numerous Math Common Core standards and AAAS Benchmarks and would be a good supplement for the PhET simulation "Masses & Springs" (above under Activities).

• Conceptual Physics Conservation of Energy and Forces on a Spring Unit
• Algebra-Based Physics Conservation of Energy and Forces on a Spring Unit
• AP/Calculus-Based Physics Conservation of Energy and Forces on a Spring Unit