I will let students work individually on exponential_growth2_warmup for the first 3-4 minutes of class. Students will first need to understand the problem. Then, they will need to show why the the taxes would now be the same or different (MP3). I have students turn-and-talk with a partner to share their justification. While students are engaged in discussion, I listen in to find students who are explain their ideas in a correct and clear way. These students can then share out with the class to help all members of the class understand why the tax bills would no longer be the same.
While this Warm Up deals strictly with the concept of percent, it hits on a common misconception and an important difference between linear and exponential relationships. Many students will think that the tax bill will return to the original amount. My intention here is for students to realize that 50% is being taken of two different numbers. This observation helps to improve their perception of percent change (MP2). Thinking in this way to start will help students be more successful in the lesson.
I give students about 10 minutes to work through exponential_growth2_direct with their partner. First, students will need to read and understand the details of the problem and make a plan for solving it. My students are not likely to use the formula for exponential growth initially. I expect them to think about the salaries based in terms of percent change.
Once students have had time to put together a first response, I will use a Non-Verbal Cue to determine where the class stands in terms of which job they would take. Then, I will have a pair of students from each perspective share their work. I plan to be selective; I will look for students to share who have organized their work.
Year 1: Salary
Year 2: Salary, etc.
Then, I will help these students guide the class towards understanding why the company that starts out at 30K will actually be better during the first five years.
Next, I will display a student's work that shows the first company as being better. I plan to have the students in the class divide each of the consecutive salaries and see what they notice (MP8). I expect all of my students will see that the common ratio is approximately 1.06. I'll ask, "Why this is the case?" Then, I will have them complete a think-pair-share to determine where this number is coming from. When students share out, they should notice that the .06 is connected to the 6% increase each year.
This is a perfect segway to Slide 2. In this slide, I show students the formula for exponential growth and have them verify the work they already did by substituting the values in for a, r and x. The values for x should be 0-4 with zero being the salary in the first year.
This question will help students see the role of negative x-values when graphing exponential functions. Students will write a function using the exponential growth model and make a table. Some students may need assistance setting up their table with the x-values of [-3,4].
Once students have their table constructed they can being answering the questions on this slide. After students have had a chance to answer all of the questions, bring them back together to discuss the solutions as a class and ensure all students are understanding the concepts.
1) In this case the domain would be [-3,4] and the range would be [1.45, 5.18]. While this coin happened to go up by 20% for 7 years it does not mean that it will continue to go up by that much. Connect this idea to the zombie problem or the rumor problem from a previous lesson. Sometimes we just want to focus on a restricted portion of the function.
2) The coordinate (0, 2.50) would be in the middle of the table. Since this value correlates to the year 2005, the value in 2003 would correlate to an x-value of -2. Students can use a calculator to determine these output values.
3) The y-intercept represents the year Sue bought the coin (time zero for her owning it). It can also represent the year 2005.
This is a very straight forward Ticket out the Door. It will help me to see if students are able to apply the exponential growth model from class. This is also a good closure task because students can begin to realize the power of compound interest over time. Some students may be surprised at the value of the money after 14 years. It leaves them thinking as they depart.
As they go, I will remind students of how the paper being folded over and over would get incredibly thick (see the lesson on exponential growth). While the money is not doubling (it is growing at a slower rate) over time the value will get fairly large.