For today's lesson students will explore two applications of exponential functions:
During the Warm Up students predict the shape of a graph modeling a cup of hot cocoa cooling. In the next section of the lesson, students will compare it to the graph of an actual cup of cooling hot cocoa.
In this Think-Pair-Share activity, the Cocoa story, I provide students five minutes to read it and to work the problem on their own. I have them assigned to a partner in my classroom to work about 3 minutes after the individual time to compare and make changes. The partners are assigned homogeneously in my classroom unless I feel the need to place lower level students in heterogeneous groups in order for them to keep progressing.
I expect that some of my students will struggle with drawing a graph of the story without numbers. I instruct them to search on the internet for the temperature of a hot cup of cocoa and the degree of room temperature to help determine the numbers to scale the graph. I post the following questions while they are working to prompt student thinking:
Student 1 drew 3 line segments to demonstrate the Cocoa cooling quickly, then at a slower rate, and then a horizontal line to represent a constant temperature at room temperature.
Student 2 drew more of a curve and then a constant line at the end.
I have six students sketch their graphs on the Applying Exponential Functions PowerPoint. Students will refer back to these predictions after collecting the actual data from a hot cup of cocoa and draw the actual graph also.
After the students complete their predictions, as a class, we perform an experiment on an actual cup of cooling cocoa. I have provided a copy of our actual data and graph. In the PowerPoint, I provide a page for collecting class data. This could also have been done in small groups for more student engagement if it is set up right with safety precautions. I was concerned about the hot beverage, so we did the experiment as a class.
1 paper cup-material of cup may change the data
data chart to record data
graph paper to graph the data
To collect the data for the actual cup of cooling cocoa:
As is shown, it models a continuous curve or exponential function that does not level out until around 80 degrees, and actually in our experiment it does not reach the room temperature of 70 degrees. It begins cooling rapidly, and cools more slowly as it approaches room temperature. Student two predicts closer to the actual graph than student one. I also model this as an exponential function from the table by finding the common ratio. As a class, we find the average of the ratios between each output value and its previous output value to identify the common ratio.
After the analysis of the actual cup of cooling cocoa, students should recognize it as an exponential function. I build from this example, to introduce Newton's Law of Cooling. I use a PowerPoint to provide students with notes and examples on the importance of e and Newton's Law of Cooling.
The Time of Death Problem provides a second application for students to explore. I review the Time of Death Problem in the video below:
I use this Exit slip as a formative assessment to check student progress and understanding of exponential functions and how to apply Newton's Law of Cooling to problems involving a continuos change of temperature.
The Exit Slip also provides an example that shows when a constant is added or subtracted to the temperature, it affects the data. Some other factors that may affect the data when using Newton's Law of Cooling are listed below:
Some mistakes that students made when completing this problem is mostly making incorrect substitutions or problem using the natural log to solve for t in the exponent. I provide a key to the Exit Slip in the resource section.