I have 15-years experience as a classroom teacher at a middle school (grades 6-8) in downtown Boulder, CO. My teaching/learning lessons are designed with student's individual learning abilities as the focus to provide each student with an opportunity to gain a greater understanding of math concepts through guidance and perseverance. The format of each lesson is specific to the material being presented, as well as to the student's preference of learning (practice, repetition, lecture, note-taking, problem solving, online applications, investigation, etc.). However, within each lesson I do try to implement some other form of game/fun, technology, word association, common mistakes, and assessment strategies in conjunction to the content/material being studied.
For instance, when teaching about linear functions, I would start the student(s) off with an activity involving a puzzle with number patterns. I would help to guide them in describing those patterns with more algebraic terminology to subtly direct them towards developing number sentences. After developing a proficient level of understanding of number patterns and their descriptions, we would move towards modifying those number sentences into something more visual on a coordinate-plane. In conjunction with an input-output table, where we would again see those patterns represented, we would then start to generate linear forms of equations. Analyzing these patterns through graphs (visual), equations (algebraic), and tables (pattern recognition), we would solidify an advanced level of understanding and connection of interactions with number operations and variables. The application of this specific standard (linear functions) could be represented and practice in numerous formats (pencil/paper, online applications, graphing calculators, etc.).