Before class I split my board in half and post an inequality on one side and an equation on the other (see Equations for class opener). Generally my students will begin trying to factor these as they come in to class so while they're working and talking I walk around listening. This gives me a chance to determine which students are on track and which might need additional support during today's lesson. I encourage them to identify as many key points as they can for each of these problems using whatever methods and tools they choose then ask them to post their work and solutions. (MP1, MP5) I ask for volunteers to critique the each side of the board then ask if anyone has any other ideas about possible methods for better understanding these problems. (MP3) After checking for understanding with open (I got it!) or closed (I'm lost!) hands and working additional examples if necessary, I ask my students to brainstorm with their back partner about real-world situations that might be modeled by these equations and what the key points might represent. (MP4) I allow two to three minutes then randomly select students to share what they discussed with their partner. This sets up the next section where they will be solving real-world problems involving both inequalities and equations.
For this section I have my students to work in teams to graph real-world problems and discuss meaning of their solution. I allow them to choose their partner because it is not critical to control ability groupings for this activity and because it increases their sense of ownership of the process. We briefly discuss using graphing calculators, tables or some other algebraic method of their choosing to solve these problems and generally determine that the calculators would be most efficient. I distribute the problems and ask if there are any questions, then walk around offering assistance and encouragement as needed. (MP1, MP2, MP4) For example, for students who still struggle with interpreting their results I ask questions like "What are the axes representing for this problem?" and "What kind of function is this?" to help them focus their thinking. I anticipate still having some students who struggle to use the graphing calculator effectively, so if those students are not getting help from their partner, I might suggest they come in for extra help outside of class. As I'm walking I also keep an eye out for teams that have good solutions or an interesting approach to a specific problem so that I can have them share with the class. When everyone is done, I have each team briefly present their solutions and explanations to the problems I noted earlier. They either use the document camera or the whiteboard to share so that their classmates have the opportunity to question and critique each problem. (MP3)
To help students really solidify their understanding of graphing as a method for solving problems I ask them to pair-share what kinds of problems graphing works well for and why, and what kind it isn't the best option for. I randomly select students to share what they discussed with the class then ask them to summarize our discussion in their notes so that they have a reference for when and how to best use graphing to solve problems because, as I discuss in my Graphing the Big Picture video, this is an important part of working with technology. (MP5)