I introduce this lesson with a video on Exponential Functions from Math Dude of Montgomery County Public Schools. The video begins with an introduction of the following problem:
Which is more: being given $100,000, or one penny the first day, double that penny the next day, then double the previous day's pennies, and so on for a month?
I stop the video when Math Dude says, "Which would you choose?" as he holds up 2 pennies.
I ask students to use their own paper to write down their decision and support their decision mathematically. Students initially want to say that the best answer is the $100,000 because the value of a penny is so small. However, they think it is a trick question, and this suspicion gives them the motivation to start working the math to support their decision.
I will present the end of the video to the students at the end of the lesson.
While students are working on their decision and the math to support it, I walk around the room to identify learners whom I will ask to share their findings. I try to select a variety of students. I am looking for common mistakes, correct answers, and wrong answers.
By presenting a variety of methods or answers, it helps me better question and guide all students towards the lesson's objective. Again, by the end of this lesson, students should be able to recognize the parts of an exponential equation, understand the common ratio, and graph it.
This lesson was fun for students and for me. Watching them think that they would take the $100,000, but then wondering if it was a trick or not.
This suspicion made the students drive forward to find a solution. Students were writing out the entire table of values instead of deriving a formula from the table. It takes a little more time than I had originally thought, but it is worth it. To see them finally solve the problem and then realize the doubling the pennies every day was a better deal.
The Math Dude shows enthusiasm to set up the problem well, and when I pause it at the end of the question, students begin to think and work. Students then want me to show the remainder of the video to verify the solution.
A teaching strategy that I use often is comparison. Comparison allows students to see differences for themselves when observing and interacting with other students. Note-taking is another strategy that I think helps improve student achievement, and I have students complete the notes for this lesson using a Frayer model graphic organizer.
Comparison of student work:
I present 4 examples of student work below. Students learn early in the year that my class is a safe place to make mistakes and that I expect students to share their work. When I ask students to make comparisons, I do not share the correct answer to the problem. Most students realize the correct answer as we discuss and share work, but I do not state it. I post all of the different answers on the board.
I compare the work of student 1 with student 2. When comparing the work of these two students, I want my class to recognize the difference in the initial value on day 0. In the video, day 0 was given as 1 penny. Several students made the mistake of writing the initial value as 0 on day 1. As a result, there final answer on the 30th day was incorrect. However, students were still able to observe the doubling of the output each day, and that the best choice was the penny.
I then compare the work of student 2 to student 3. Student 2 has the correct answer of 1,073,741,824 but states it as dollars. Student 2 has not converted the value on day 30 from amount of pennies to dollars. Therefore, there are over 1 billion pennies, but the correct dollar amount is stated by student 3 of $10,737,418.24. Student 3 states an opinion that he blows money quickly in his reasoning, and I remind students to answer the question given.
Finally, I have student 4 share his/her work. Some students write the output values in dollars as shown in the work of student 4. The conversion to dollars is not necessary if the unit in the table is already dollars.
Some of the students tried to write the equation instead of writing out the t-table for 30 days. Some students had written 2x instead of 2^x. I collect the student work before presenting the correct answer. I use this example to lead into writing the equation from the table.
I have students hand in their choice, the math, and the reasoning of their decision before I present the lesson. I have provided a blank T-table and graph to project on the screen. I also have students take notes on a Frayer Model graphic organizer.
I demonstrate creating the equation from the table in the video below:
After I present the table, I let the students view the end of the video on Exponential Functions to see the outcome presented by Math Dude. This demonstration reinforces what we have done in class. I often find that it helps my students to see another teacher present a problem. I have provided a copy of a completed Frayer Model in the video below with a brief explanation.
It was helpful for students to have the example of the penny and money to refer to write the exponential equations. Students should be able to write the exponential equations using growth factor and decay factor. Recognizing that b (the common ratio) is equivalent to the growth or decay factor is an aha! moment for students when they recognize it.
For some students, it takes longer to realize how the parts of the equation are formed. However, the Frayer Model was definitely beneficial to student understanding in this lesson.