Clip 29/29: Standard 1: Making Sense and Perseverance Graphing Quadratics Part 1
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends...Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Carlos Cabana works with his high school English language learning students on algebraic reasoning and multiple representations around parabolas. In these clips, his students build sketches of parabolas and link them to a T-table. His students have had some exposure to factoring and multiplying, but it's not automatic for any of them yet. They see how to find x and y intercepting with lines at a procedural stage. The students have previously made tables for parabolas and they can find the vertex in the table, and are developing capacity to use the vertex to calculate other parabolic properties.
See this video in the context of an entire lesson.
(Parts 1, 2, 3, 4, 5, 7)