1. Spatial Reasoning and Geometric Visualization[list][*]Ability to visualize how equations correspond to geometric shapes in the coordinate plane.[br][/*][*]Comfort with recognizing symmetry (about axes, origin, or focus-directrix structures).[br][/*][*]Experience interpreting transformations such as translations, reflections, dilations, and rotations of curves.[br][/*][/list]2. Conceptual Understanding of Algebra and Coordinate Systems[list][*]Fluency in manipulating quadratic and higher-degree equations, including completing the square.[br][/*][*]Understanding how equations define loci of points (e.g., “all points equidistant from a focus and directrix” → parabola).[br][/*][*]Comfort switching among algebraic, graphical, and geometric representations of curves.[br][/*][/list]3. Proportional, Quantitative, and Functional Reasoning[list][*]Ability to work with ratios, distances, and midpoints (for definitions of ellipses and hyperbolas).[br][/*][*]Understanding the role of parameters in equations and how varying them changes the curve.[br][/*][*]Fluency in function transformations and interpreting domain/range restrictions.[br][/*][/list]4. Mathematical Reasoning and Proof[list][*]Experience testing conjectures (e.g., “what happens if [i]a = b[/i] in an ellipse?”).[br][/*][*]Ability to justify geometric properties (such as why a parabola reflects rays through its focus).[br][/*][*]Familiarity with derivations (from geometric definitions to algebraic equations) and logical argumentation.[br][/*][/list]5. Productive Disposition and Inquiry Mindset[list][*]Willingness to experiment with technology (GeoGebra, graphing calculator) to explore curve behavior dynamically.[br][/*][*]Curiosity about real-world applications (e.g., satellite orbits, optics, architecture).[br][/*][*]Growth mindset toward engaging with higher-degree curves that may at first seem unfamiliar or complex.[br][/*][/list]