For today's opener, I post this graph, which is a screenshot from Desmos. As students arrive, I tell them that their task is to write an equation for the curve. After they think about it for a few moments, a lot of kids are frustrated, and really not sure where to start. I tell them to try to write the coordinates of the points they can see on the graph, and this is enough to get them talking to each other and trying some possibilities.
When three or four minutes have passed, I start by asking everyone what they've been able to figure out. Usually, students will volunteer the coordinates of a few points they can see. I record these in a table as I hear them: (0,500), (1,700), and then something along the lines of (2, "almost 1000"). I use whatever words kids provide for that third point. Making an estimation like that - and being confident about that the level of precision is appropriate - is a great mathematical habit.
As we examine the table, I ask, "What kind of function is this?" By now, kids are quick and confident enough in their response that I can build on that. "What makes this an exponential function?" I continue. The shape of the graph is one obvious response. But then we investigate the table of values. The output column is not growing at a constant rate: from 500 to 700 is a difference of 200, but then from 700 to "almost 1000" is a change of almost 300. We all recognize (hopefully because students raise the point, but if they don't, I do) that we need to figure out the common ratio for the geometric sequence in the output column (we still move back and forth between the language of functions and the language of sequences).
Finding that multiplier, which is 1.4, most often involves a process of guess and check. Students notice that it must be less than 2 but greater than 1, so they often try 1.5 first. When they see that the product of 500 and 1.5 is a little too high, they keep guessing. Now, in the best scenario (which happens about half the time, in my experience), students will insist that there must be a better way than guess and check to find the common ratio here. "Of course there is!" I say, and I show them the equation
500x = 700
As I write it, I say, "Starting from 500, what do I have to multiply by to get 700?" Guess and check, and the efficiency gains that algebra offers will be a focus next week, as we start the Systems of Equations unit.
Whatever method we apply, once we figure out that we must multiply successively by 1.4, we test it out. "Does 700 times 1.4 give us almost 1000?" I ask. Kids are satisfied to see that it does. Then we review the y-intercept, and its role in an exponential function. Starting from 500, we repeatedly multiply by 1.4, which is a reading of the rule
f(x) = 500 * 1.4^x
As we move on to today's variety of review activities, the work of this opener will come in handy.
While students got started today, I returned this check in quiz from yesterday's class. For many students, the quiz was easy: at minimum, they are able to distinguish between linear and exponential functions. So in a week where they're submitting a project and taking a unit exam, this means they've already demonstrated mastery on the first of the four learning targets for the unit. There are students who struggled with this quiz, and now I've identified them.
For both sets of students, this quiz serves as I prep tool for the exam. Yesterday, as an extension, I asked students to try to write equations to represent each of these situations. Today, I tell them that that's a great place to start studying for the exam. As the lesson transitions from the opener to a choice of work time, students have that option.
Of the problem here, #2 and #7 were the most common for kids to mis-identify, so I'm ready to go over those today.
With the opener complete and quizzes returned, I take a few minutes to outline some options for today's class. Throughout the year, I've taken some time a day or two before our exams to talk about how to study. I always emphasize that effective studying means actively doing something. When I ask, many of my students will tell me that studying means "looking through your notes." I explain that notes are just a starting place: you have to be solving problems and applying the knowledge that you're expected to have.
As outlined on today's agenda, here are some options that I lay out:
With options laid out, I give students the rest of class to use the time as they will. Wonderful chaos ensues, and three thoughts go through my mind: first, that it's stressful to try to keep with what every one of my kids needs all at once. Second, that this is exactly what school should be: kids knowing what they know and advocating for themselves to learn what they don't. At this point in the year, they understand the idea of learning targets, and they can ask specific questions in those terms. Finally, at the end of one unit, I always look forward to what we're doing next. A project and an exam tie a bow on the unit, and I get see all the varying levels of mastery my students have achieved.