In order for a lesson like this to work, students need to be committed to understanding the material. A big risk with a lesson like this is that as soon as one student starts to figure out tricks, everyone else just takes their same tricks without figuring anything out themselves. One important strategy I use to help ensure that all students think about how the situation works is to never tell anybody whether anything they say is “right” or not. If a student has an idea and wants feedback, I ask them to draw a picture or write their reasoning on a whiteboard. They grumble about this, but this is the only way a lesson like this helps anybody learn anything. I take the time to listen to their reasoning and ask them questions about it to make sure that it makes sense to them. Taking the time to do this is essential to creating the right environment for this lesson.
The warm-up exposes students to two of the big problems in this unit: the bunny problems and the candy bar problem. Students may want to be told how to solve the first bunny problem, and seem possibly unwilling or unable to “draw” the situation. I ask them to find a way to represent the situation on their own. Also, I purposefully do not provide data tables or any organizer because I want to leave these problems as open as possible. Most students have no idea how to draw the problem, so I start a color-coded diagram on the whiteboard and refer them to it. I ask students, “What happens during the next month? Why are there so many bunnies?”
The candy bar problem is another important problem throughout this unit, and the grid provided is meant to represent the candy bar. It is ideal for students to use different colored pens or pencils to shade what they eat each day. Again, I do not push students for generalizations at this time. They are already eager to find shortcuts, so it is more important that they do things the “long way” in the beginning to make sure that they fully understand them.
The data tables at the bottom are meant to get students thinking about patterns. The repeated data tables are on purpose. I want students to find two different ways to complete the tables. The numbers chosen are suggestive of exponential data, but having only two points doesn’t make them exponential. So I ask students: “Can you think of a different way to complete the table? How did you know that would be the pattern?” The idea is for them to realize that two data points does not tell us what type of growth pattern is being shown.
During the closing of the lesson, I showcase the deeper student thinking that happened during this lesson (see About the Math). For today, I ask one student to present the idea that, in order to find the number of bunnies in the next month, you can multiply the current number of bunnies by the number of pairs that each bunny has per month and then add the current number of bunnies that result to find the new total.
For instance, in a problem with 5 initial pairs and each pair having 3 pairs per month, I say:
Jessica told me that she could find the next month by multiplying 5 by 3 to find the number of new baby pairs, and then adding 5 to that number. Do you agree? Does this always work to find the next month? Can you tell me why this would work?
Then after students think, I say, “Jonai asked me if this is the same as multiplying 5 by 4. What do you think?” By presenting this as student thinking rather than as my thinking, students continue to think about this thinking. If I tell them it is my thinking, they assume I am the expert and they do not question or critique it. So I leave it in the words of students which makes them more critical of the ideas.
I also presented Natalie's Whiteboard which had the differences shown. I say, “Natalie wanted to find a rule for this data so she started looking at differences. What happened? Did this work?” I was excited about Natalie’s attempt and I want all students to notice what she noticed: these differences never seem to become constant. This is a time when students start to say, “Oh, so these are different kinds of rules than the last ones.” This is a good realization for them to have.
After this discussion, I ask students to write about the Exit Ticket questions. These are pretty “easy” questions, but I want to emphasize this idea of exponential growth. The example of Australia is an interesting real world application. Some of my students liked hearing about this situation (you can easily read about it on Wikipedia to get a quick overview.) Though the numbers in the problems we solved today are obviously an oversimplification, the exponential growth is very real and can be problematic. While students realized many things today, the things I want them to learn for sure are how to answer these basic questions: why is this growth not linear, how you can find the next month based on the previous month and the idea that exponential growth is really fast. Obviously this last idea is expressed informally, but I think it is worth discussing informally, just to address the realization students have when trying to draw or build these situations.