Today’s Warm-Up asks students to fully solve some challenging optimization problems. At this time, some students will likely need more scaffolding to do this, so the Optimization Problems Organizer is provided as an optional resource.
The rest of the warm-up is focused on more abstract problems and includes a preview of the day’s lesson focus on quadratic data tables. To help students use their time effectively, I tell them that they should definitely be able to solve problems (1) and (2), and that problems (3) and (4) relate to content that they have not fully mastered yet. This gives them the chance to think about the content in advance and possibly make sense of it on their own (MP1).
It might be useful to ask one or two students to create a reference poster to show their strategies on problem (2), or if no students are interested in doing this, I complete a poster myself to show a few examples. I don’t want students to memorize any algorithms, but it is nice for them to have a reference when they get stuck.
As problem (3) and (4) are previews, I ask students to find a new partner and try to figure out as much as they can about these problems after they complete problem (1) and (2). Often being asked to think about a problem rather than fully solve it gives students a little more space to think and motivates them to make sense of their work.
The fourth problem can be used as a transition to the new learning of the day, so ask students to write down all their thinking about this problem specifically. Students can work ahead if necessary, but everyone should come back to this problem for the transition.
This is a challenging lesson for many reasons. First, students bring several misconceptions to this lesson. They think that any time a data table shows a “vertex” or has symmetry, that it must be quadratic. For this reason, the data table of an absolute value function and a quartic function are both included. Students may think that these are quadratic functions until they examine the differences and the second differences of the y-terms.
Students may or may not discover this pattern in the second differences. The purpose of including the quadratic data tables on the warm-up is to ask them to think about these data tables before explicitly teaching them anything. Once they have made some predictions here, ask them to justify their predictions (MP3). Use the four data tables from the warm-up to facilitate a brief whole class discussion. You can ask students to look at the differences to see what they notice—likely this is enough of a hint for them to realize that the second differences are constant. After this brief discussion, they can get started on the problem set.
The problem set (Quadratic Data Tables.docx) asks students to determine whether or not the data tables show quadratic functions and also to find rules to fit these functions. This is a time to let students choose how much instruction they way. I tell them that they can definitely figure it out themselves, though they might get stuck (MP1). Giving them a chance to figure things out for themselves, and also explicitly telling them that this is possible is a way to encourage them to try thinking about the problems on their own.
As students struggle to find rules, you can ask them to think about whether or not they have the x-intercepts or the vertex, and whether they could use those points to set up a function in one of the three forms. They may or not realize that they can use other data points to solve for unknown parameters. Additionally, sometimes it is helpful to extend the data table in both directions to see more of the points. Students may struggle to figure out how to use the patterns to extend the table, so this is another interesting challenge.
Students may choose to use graphing technology to plot points and try to find a curve that fits the data (MP5). As students work, circulate and ask them how they could check their answers. It can also be helpful to ask a few students to display their work on whiteboards so that students who are struggling can refer to other students’ work for reference.