The warm-up for this lesson involves students doing some skills that they have had practice with in a way that is organized for them to work towards some generalizations. They should have some realizations when looking at the first four functions and the two graphs provided (if they have not made this connection already) that replacing the multiplier with its reciprocal has the same effect as making the exponent negative, so the 4 functions given actually only create two different graphs. The actual matching will take them little time, but the important thing to talk about with students is the generalization. This is why the flexibility of the warm-up time is really nice: if students easily complete the match-ups, they might just skip over the generalization altogether and get to work finding the function rules, which is totally fine, as long as when you circulate you ask them to come back to the generalization. Many students will have a much easier time explaining their ideas aloud and might need help organizing their ideas. If this is the case, guide them to use “If-then” statements and in some cases I tell them, “I think you really need 4 different if-then statements here.” They should then be able to look at all 4 cases to formulate an articulate generalization. (MPS3.)
As with all warm-ups, part of the purpose is for students to practice the skills, so if students are struggling to even do the problems, make sure that they find a partner to help them or they find a reference page. This is a time where it can be smart to have one student save their work on a whiteboard so you can just hand that whiteboard to a struggling student. The goal is for each student to have the least possible amount of resources and help provided to them so that they are still able to make progress. Once all students understand the skill, it is important to make sure to have some conversation about the ideas of the problems with each student or group of students. If students do skip the conceptual questions, ask them to return to these questions. I never get mad at students for skipping questions, because I train them to always be working so if they get stuck it is fine to skip something, but I also want them to have a question ready to go when I get to their table, so that they can make progress. There will be some students who have skipped something and don’t want to go back to it, so it is important to explicitly discuss with them the role of feedback and failure in the learning process.
Some students will whiz through this lesson, while other students will struggle to deal with the fractions. The goal for everyone is to articulate the generalization clearly and apply it to these match-ups. This is an important time for you to quickly formatively assess each student so that you can assign students the more challenge problems that ask them to write equations for transformed graphs. I find it helpful to set a clear expectation for these students early on to make sure that they don’t think they are done once they have done the match-ups. This is where being explicit about the purpose of differentiation helps. Even if you think you have said it a thousand times, I always take the time to restate: “My job is to push each and every person beyond their comfort zone, so for you this means taking on this extra challenge.”
Students who are working on the match-ups can be coached to test data points or, if this is to easy for them, they can be coached to use generalizations about whether functions are increasing or decreasing and to use these generalizations to make the match-ups. Likely, students will finish this quickly. If they do, they can either continue working on the data tables or the two data points. If they are ready for more of a challenge, they can start looking at writing functions for the transformed graphs.
If students take on this challenge, some of them might quickly realize that the horizontal asymptote reveals the d value (though they likely won’t articulate it this way.) In that case, they might breeze through these problems. They can then tackle the next level, which has graphs where it is not easy to see the exact horizontal asymptote and the data tables have points with higher exponents so the equations become quadratic instead of linear.
Students who are ready for more of a challenge can begin work on Changing Bases of Exponential Functions.