The exit ticket questions are based on different students’ questions and comments about setting up these functions. It turns out that these functions provide a particularly rich context in which students can make meaning of negative exponents. The first question is designed to make this learning explicit: at this point, even if students have no idea about what negative exponents mean, they will likely be able to say, or at least understand, that the negative exponent means “2 months ago” and in order to work backwards they need to divide by the multiplier. This is basically enough to define the negative exponent.
The second question relates to problems in which the current amount of bunnies is given and students need to work backwards to find the “original number” of bunnies. Obviously this question is not totally clearly defined, but it brings forth the idea that the function can be written with different starting values. In order to change the starting value but keep the function the same, the exponent needs to change as well. This question is the beginning of a deeper conversation the horizontal shift and the exponent laws, which some students will engage in in future lessons.
The third question is another attempt to get students thinking about the meaning of the negative exponents. If the internet is readily available to students in your classroom, you could easily give them the opportunity to graph the functions on desmos.com/calculator so that they could see for themselves that these functions are the same. Obviously this is just the beginning of the conversation, because then the question is why are these functions the same.
To facilitate the closing, I state the questions clearly so that all students understand exactly what I am asking. Before I ask them to write about the questions, I explain a little bit about the context of each question and refer to specific students who brought up this question during the lesson. For instance, I say, “Jonai asked a brilliant question about the starting values and the exponents so I want you to think about the two functions in the second question to see if we can answer her question.”
Then, I ask them to write on their own. After giving them some individual think time, I ask some students to share their answers and I highlight the key ideas myself (taking notes on the projector.) Then I give them more time to add to their ideas before turning in their exit ticket.