By this day of the unit, students should be choosing higher levels of problems to work on. They should also be making more connections on the web of multiple representations for distance functions. My plan is to give them a lot of time and various resources (their previous warm-ups, computers with graphing technology, each other) to make further progress in the depth and connectedness of their learning.
I display a poster with all the multiple representations spread out in a web:
As I circulate to ask them questions about their work, I refer to the poster and ask them to make more connections. This is how I push students to think about identifying new connections or to identify shortcuts.
At first, the shortcut from the verbal description to the graph will seem like a big cognitive leap. As you coach students to look for connections, highlight opportunities for students to make this connection. Doing the same problems for several days gives students a lot of time to build on their previous understanding to make this connection, but it is often helpful to push.
I will encourage students to attempt higher level problems by circulating around the room and briefly asking each student, "What is your plan to progress from what you accomplished yesterday?" I find that it helps to ask students to refer to their warm-up from yesterday, so they can see the progress they are making.
If students give answers that are more about work product and output ("Yesterday I only did Problem 1 and today I am doing Problem 3"), encourage them to think about their progress in terms of the conceptual understanding and skill development ("Yesterday I didn't really know how to choose the inputs carefully and I now I can choose the inputs really easily to make sure I see the whole graph.") These kinds of goal setting conversations help students hold themselves accountable for their own learning.
I like to ask students to create their own notes based on the investigation of the day. I tell them:
In a textbook or in a traditional math class, the teacher would write a reference for the students that explains how the verbal descriptions transform the graphs. Can you create this reference for yourselves?
I ask them to look at the specific parts of the verbal description. I display an example match-up on the board. I ask students to explain how each part of the verbal description affected the graph:
After a brief demonstration, I ask students to make their own reference describing the graph transformations and to use this reference to defend or to verify some of their match-ups. This can be done in writing, or verbally between pairs of students, depending upon the amount of accountability you want to set up for this activity and how much you can rely on students to actually have the conversation with their partner if you don't ask them to write it down.