A Kern High School District Task Shipwreck and Murder Hand out Shipwreck and Murder reading piece. A Kern High School District Task

You’ve all been shipwrecked on a tropical island - a wonderful place, with bananas, coconuts, fish, and a pretty constant daytime temperature of 75°and a constant nighttime temperature of 65°. Go through the reading and ask the students to identify the useful information that we might need for later.

Your classmate, John Doe, has been murdered Your classmate, John Doe, has been murdered! You know there’s no one else on the island - it was one of your classmates that did it! None of you will sleep peacefully in your flimsy grass huts until the murderer is discovered. Go through the reading and ask the students to identify the useful information that we might need for later.

You have watches, thermometers, and other simple tools, but no experts on murder investigations. The day John was murdered, everyone was walking around the island noting its features, in hopes that would help you all figure out where you are. Go through the reading and ask the students to identify the useful information that we might need for later.

It turns out that everyone walked by the spot where John's body was found, and recorded the time when they were at a nearby spot (where they could see a volcano on the next island). Figuring out the time of death will likely narrow down the suspects. Go through the reading and ask the students to identify the useful information that we might need for later.

MaryAnn had a hunch that knowing the body temperature (normal body temperature is 98.6°) would help determine the time of death. So, at 1 pm, she checked the temperature of John’s corpse. It was 91.3°. Then at 1:40 pm, it was 87.5°. Go through the reading and ask the students to identify the useful information that we might need for later.

One more clue… this comes from Daniel, “I read a lot of murder mysteries. In one of them, this detective says, ‘A dead body cools off just like a hot cup of joe.’ I don’t know if that helps or not…”

We want to think about how a hot cup of coffee cools off. 1 We want to think about how a hot cup of coffee cools off. 1. What would be a reasonable starting temperature? Have the students guess to get a class high and low to develop a range.

We want to think about how a hot cup of coffee cools off. 2 We want to think about how a hot cup of coffee cools off. 2. About what temperature is it when it’s cold? (Why?) We are trying to get the students think about the room/ambient temperature to help develop the asymptote.

We want to think about how a hot cup of coffee cools off. 3 We want to think about how a hot cup of coffee cools off. 3. How long would it take to reach the “cold” temperature? This is to help students develop the concept of a cooling rate.

Let’s Investigate

Pass-out the Shipwreck and Murder – cooling coffee worksheet

Time Temperature ℉ 10:00pm 154 10:02pm 142 10:03pm 138 10:05pm 131 10:10pm 110 10:15pm 97 10:20pm 85 10:30pm 75 11:00pm 66 The following data represents the temperature of a cup of coffee as it cools over time. This is the same data as in the handout. Students should be plotting the data on the graph provided in the student handout.

Graph the data on the Shipwreck and Murder worksheet. Based on the graph of the data, do you think the coffee will ever reach a temperature of 60℉? Explain your answer. The graph appears to be leveling off around 65 ℉ so the coffee will never reach 60℉.

What is the lowest temperature the coffee will reach? How did you determine this? Discuss the idea of an asymptote. The asymptote will be about y = 65. Remind the students that a non-shifted exponential decay graph has a horizontal asymptote on the x-axis. However, our graph has been shifted so the asymptote has also been shifted. If your class is struggling have them re-read the Shipwreck and Murder for clues to the asymptote.

What type of function could be used to model the data? What would be the general form of the equation for this type of function? Since we have spent the prior days graphing exponentials the students will hopefully recognize the graph as an exponential decay. The equation will be 𝑦=𝑎 𝑏 𝑥 +𝑐 . This would be a good time to review the fact that for exponential decay we have 0<𝑏<1.

How do we find that value? 𝑦=𝑎 𝑏 𝑥 +𝑐 Which unknown value in the equation would be the easiest to find first? How do we find that value? Since asymptotes have already been discussed, students should easily be able to estimate the value of c. If your class is struggling have them re-read the Shipwreck and Murder for clues to the asymptote.

𝑦=𝑎 𝑏 𝑥 +𝑐 To find the values for 𝑎 and 𝑏 choose any two points from the data set and plug them into the equation. You will now have a system of equations. If the student used the first data point and one of the later points in the set, their work should look like below. Using the first point (0, 154) the students should be able to the value of a. 154=𝑎 𝑏 0 +65 to find an a value of 89. Using (10, 110) : 110=89 𝑏 10 +65 −−→ 45=89 𝑏 10 −−→ 45 89 = 𝑏 3 −−→ 𝑏= 10 45 89 ≈0.93407608 −−→𝑦=89 0.93407608 𝑥 +65 If the students have a hard time understanding the why the a value is 89 which is much lower than the first point (0, 154). Remember that the 89 is 89 degrees above the asymptote.

Does your answer seem reasonable based on your graph and table? Use your equation to approximate the temperature of the coffee after 25 minutes. Does your answer seem reasonable based on your graph and table? Let students pick their own 2 points. They will get different values for 𝑎 and 𝑏 but the values should be close. Remind the students that we are looking for a model for the data and the models are close approximations so getting close but differing values is ok. Example: Let c = 65 and use points (0, 154) and (10, 110). 𝑦=89 0.93407608 𝑥 +65 −−→𝑦=89 0.93407608 25 +65 −−→𝑦=81.1787 ͦ Have the students verify the solution on their graph.