I will ask my students to get started right away on this Warm-Up, which highlights 4 key skills from the week so far and previews the new skill that will be focused on in today’s investigation.
As I circulate, I constantly ask students: “How do you know?” and “What does this mean?” There are many opportunities on this warm-up for students to use multiple representations to look at functions, so if my students get stuck, I will ask them if they can represent the relationship a different way. This is also a good chance to encourage them to use the graphing calculator (www.desmos.com/calculator) if they think this could help them. It helps to always have a few computers available (MP5).
Problems (3), (4) and (5) make room for much more differentiation. At the most basic level, students can identify the relationships as linear or non-linear. Beyond this, they can find function rules that fit the data or the relationship and display these relationships graphically or with data tables. If students breeze through these problems, ask them to justify their claims or explain their process for finding rules.
There are so more abstract and challenging problems on the second page for students who are ready for more abstract problems relating to properties of linear functions.
This investigation is a great opportunity for students to really engage with MP1, MP2 and MP3. To get started, I leave this investigation as open-ended as possible. I start by displaying the two verbal descriptions from the Warm-Up (problem 5) and asking students to vote on whether they think each relationship described is linear. I do not provide any hints or scaffolds here, because in their absence, I hope students will come up with some different ways to justify their answer. Some students may demonstrate a more abstract idea about how the relationships work, while other students may jump to using numbers to verify their claims (MP2). I find that as long as I never tell students whether I agree or disagree with their arguments, they will listen to each other and collectively try to make sense of this problem (MP1 and MP3).
I facilitate a whole group discussion only long enough to make sure that students understand the question. This takes 5-10 minutes. It does help to eventually agree as a class on whether or not these two particular descriptions are linear. If necessary, you can eventually tell students your opinion, but hold off on this as long as possible.
Then students can get started on the investigation. For students who are struggling to start, I work with a small group or pair to model the method of choosing some values for x and setting up a possible data table. Basically, I try to find some numbers that fit this relationship and organize these numbers in a table. This can be a fun time to use an online graphing calculator (www.desmos.com/calculator) (MP5) so that students can see points appearing on the graph as they set up their table. The graph also makes it incredibly easy to see whether the table shows linear data, so perhaps this shouldn’t be made available to all students, only those who need the extra scaffold.
If students are engaged, they can work in small groups the entire time while you circulate to ask them questions about their justifications. This is a good example of a task where getting the yes or no answers is not the most challenging aspect of the task; providing the justification is. So as you circulate, constantly ask students about the quality of their justifications (MP3). Also, they can start to try to formulate a generalization about what kinds of relationships are linear. This is one of the write-up questions at the end of the task, so you can start asking them this question as they work.
If students start to lose focus, I will interrupt their work and facilitate a brief class discussion about any of the more challenging verbal descriptions, or you can ask a group to share some of their conclusions to see if others agree. This is a good way to break up the work time, if necessary. It is not expected that students finish this task during the work time today. They can finish this outside of class so that it can be used in their portfolio for the unit.