I begin this lesson with a story problem on the board that can be modeled by an exponential function instead of linear or quadratic. This builds on the previous lesson, giving my students a chance to solidify their understanding of algebraic models.
In 1980 wind turbines in Europe generated about 5 gigawatt-hours of energy. Over the next 15 years the amount of energy generated increased by about 59% per year. Approximately what year did the energy generated top 80 gigawatt-hours.
I ask my students to figure out the answer in any way they choose and to be ready to share their answer and math. (MP1, MP4) While they're working I walk around offering encouragement and redirection as necessary. When everyone is done I ask for volunteers to put their work and answers on the board. I have my students review the board and check the equations/answers for themselves to see which if any seem to work. I talk about what I do if none of the answers are correct in my video. Usually at least one student recognizes that an annual percentage increase means an exponential equation and has written a fairly good solution. I write the word "exponential" next to that problem then ask about any other functions we've looked at this year and serve as scribe to write them down. When everyone has had a chance to make suggestions, I tell my students they will be working with a variety of functions today.
You will need copies of the Story Problems for this section of the lesson. For this part of the lesson I tell my students they will be working in teams to attack and solve real-world problems. I explain that they will be responsible for setting up the equations, finding answers to the questions asked, and checking their solutions for reasonableness. I suggest that there might be some easy questions and some that will be more challenging, but assure my students that they can solve them all. I tell them today they will work with their back-partner but that they must each write out their own work for each problem. For those who complain about duplicating each other's work, I advise them to try the problems separately first, then compare their work so it will be different. (MP1, MP2, MP4) I distribute the Story Problems and while my students are working I walk around offering encouragement and redirection as necessary.
After about 30 minutes or when everyone is done, I randomly select teams to post their equation and solution for a problem on the board. My board fits three to four students at a time, so it doesn't take long to get all the problems posted. I then ask my students to compare their work to what is posted and offer any questions or comments. (MP3) This gives everyone a chance to check their work and to see several different ways to solve the problems.
To close this lesson I ask my students to reflect silently for a moment about one problem that was solved in a way they found particularly interesting or that they didn't expect and be ready to discuss it with the class. After a minute or so, I randomly select a student to share his/her thoughts, reminding my students that this is not a time for critiquing, but rather an opportunity to hear alternative thinking. I continue until everyone has an chance to share, then make my own observations about what they've just said, summarizing any patterns I heard.