## Too Hot To Handle, Too Cold To Enjoy

- Lesson

Predicting the right time to take that first sip of any hot beverage is difficult. Unfortunately, the temperature of hot coffee does not decrease steadily (linearly) over time. If so, it would be easy to predict when to take that first sip. Which function best represents the rate at which coffee cools: linear, quadratic, square root, absolute value, exponential or logarithmic?

Have you ever taken a sip of coffee and burned your lip or tongue? Then after you set the coffee aside to cool, it gets too cold. This lesson is designed for students to analyze why this happens and to help predict when to take that first sip.

In order to achieve this goal, students will take measurements of hot coffee (actually,*simulated*hot coffee) at various times, and then graph the results. Before they can determine the perfect time for the first sip, however, they must conduct some research to determine what temperature is hot enough to burn.

Lead a class discussion about *Liebeck v. McDonald's Restaurant*,
better known as the "McDonald's Coffee Case," a lawsuit regarding a
fast food restaurant and a person who was scalded by their coffee.
(Note that you may want to exclude the name of the restaurant when
discussing the case in class.) The lawyer for the defendant said that
McDonald's provided him its operations and training manual, which says
its coffee must be brewed at 195–205° and held at 180–190° for optimal
taste.

Pass out the Too Hot to Handle Activity Sheet. Tell students that they are going to determine the best time to take the first sip of a hot drink by using a function to model how the temperature of the drink changes over time. Ask them to predict what kind of function will represent this relationship. If they struggle, remind them of the kinds of functions they have studied — linear, absolute value, quadratic, square root, exponential and logarithmic. Do not draw the graphs, however, because they will do this later.

Too Hot to Handle Activity Sheet

If possible, conduct the following experiment in class. Boil a pot of water. Then, place a thermometer in a cup and fill it with hot water. Record the temperature about every 3 minutes. It will take approximately 30 minutes for the coffee to cool to 120°, which is a reasonable temperature at which the first sip might be taken.

As a time-saving alternative, you can show students the following video that shows a mug of hot water cooling, while a thermometer and timer indicate the temperature and elapsed time. Showing the video gives several advantages: the data collection occurs more quickly; all students are able to watch the thermometer throughout the entire process; and, all students will be using the same set of data. One interesting strategy is to require all students to collect at least ten data points, without restriction on which data points they must choose. Although students will collect different sets of data, the resulting graphs should be very similar.

Using either method, students should collect data until the
temperature gets to the point where clearly no scalding will occur.
Because scalding results partially due to temperature and partially due
to the amount of time exposed to the heat, some research should be done
to find the scalding temperature. If you are doing the experiment, wait
until the temperature gets down to 120° and then gently touch the
surface of the water with your finger. If you cannot hold your finger
there for very long, then it is still too hot. **Do NOT have a student do this.**
If you are using the video, research can be done in class as a
teacher-led web exploration or in a lab as individual or group
research. Students should determine what temperature would be good for
a first sip. A reasonable estimate is approximately 120°.

At this point, students should graph their data. Some students will need guidance through the setup of the axes, scales and intervals. After plotting the data, students should choose the type of function that best models this cooling temperatures. Finally, students should use their scalding temperature research and their graph to determine the best time to take the first sip.

- Computer projector
- Graph paper
- Ruler or straight edge
- Too Hot To Handle Activity Sheet
- Too Hot To Handle Answer Key

**Assessment Options**

- Collect
the activity sheets and evaluate students' understanding by looking at
their answers to the questions. Answers to the activity sheet can be
found in this answer key.

Too Hot To Handle Answer Key

Note that the answers to all questions will vary, but if the video is used, the graph in Question 6 will be an exponential function that passes through the point (30, 120) and is asymptotic to the line*y*= 70. Consequently, the answer to Question 10 will be "approximately 30 minutes." - Lead a classroom discussion regarding the various mathematical models and how they relate to this lesson. Ask whether anyone thought that the cooling process would be linear and why? Did anyone think that another mathematical model would apply and why?
- Randomly select some students to present to the class and the class can participate in reviewing and critiquing the presentations. Of particular interest is the time (and temperature) that students decided would be appropriate for taking the first sip.

**Extensions**

- Conduct another experiment by putting a thermometer in a beaker of water and recording how the temperature rises as the water is heated. A bunsen burner from a chemistry lab or a sterno can could be used to heat the water.
- Have students graph their data on semi-log paper. (You and your students can create semi-log paper using Illuminations' Dynamic Paper interactive. Click on the Graph Paper tab, and then choose the Semi-Log
option.) When you graph an exponential function on semi-log paper, you
get a straight line which you usually only see when graphing a linear
function. Semi-log paper is used because graphs of exponential
functions drop and rise so quickly that it is difficult to measure the
change in time accurately and the change in temperature when too much
time has passed. With semi-log paper, these changes become magnified.

Dynamic Paper AppletA similar result occurs when looking at the graph of the Dow Jones stock index since the Great Depression. It is an exponential function, but many stock sites make it look linear by using a logarithmic scale. Use

*Yahoo! Finance*to change the scale from logarithmic to linear and see the differences in the graphs. - Ask students to perform an exponential regression using a graphing
calculator or Microsoft Excel. But be careful! Because the asymptote is
*y*= 70, students will have to do a transformation first by subtracting 70 from each value in the range. Also note that the regression given by most graphing calculators takes the form*y*=*ab*, whereas the form used by Excel is^{x}*y*=*ae*, where^{bx}*e*is the base of the natural logarithms. - Have a class discussion where students attempt to relate what they
observed with Newton's Law of Cooling, which states that the rate of
change of the temperature of an object is proportional to the
difference between its own temperature and the ambient temperature (the
temperature of its surroundings). The formula is:
*T*=*A*+*Se*^{‑kt}*T*= temperature at any instant*A*= temperature of surroundings (ambient temperature)*S*= initial temperature of the liquid minus the ambient temperature*t*= elapsed time*k*= constant determined by the conditions

**Questions for Students**

1. Now that you've graphed the data, can you explain why when you set the coffee aside for a few minutes, many times it is too cool when you finally take a sip?

[Because of the exponential decay, the temperature drops quickly from the starting temperature.]

2. If the coffee was left to cool for much longer than 30 minutes, what temperature would it reach? Why? Will the temperature continue to decrease until it reaches 0°?

[Eventually, the temperature of the coffee will reach an ambient temperature that we call

room temperature. The asymptote will be somewhere neary= 70, since room temperature is about 70°.]

3. What is the name for the concept that describes the limit that the temperature will never get to?

[Exponential decay functions have horizontal asymptotes. In this experiment, there is a final temperature and it is the ambient temperature, however, the temperature just below the ambient temperature will never be reached and that can be considered the asymptote]

4. What is the inverse of exponential decay? If you were to heat a liquid and measure the temperature as the liquid gets hotter over time, would the graph look like the inverse of the exponential function?

[The inverse of exponential decay is logarithmic growth. And, yes, heating a liquid will have logarithmic growth.]

5. What factors might affect the cooling rate of hot coffee?

[Factors that affect cooling include: type of liquid, shape of container, container material, outside air temperature, etc.]

6. If you were to design a coffee mug which could help people to avoid scalding themselves, what shape would you use? What shape would you use to keep the coffee the hot the longest? Can a mug be made that prevents scalding and keeps coffee hot or are these conflicting goals?

[Answers will vary.]

7. What other natural phenomena might be represented by exponential decay?

[An example is depreciation.]

**Teacher Reflection**

- If you chose to do the experiment in class, could anything have been done to improve safety?
- Was the experiment part of the activity inclusive of all students? What could be done to include more students?
- Did all students set up the graph correctly? What could have been done to prepare students for that part of the lesson?
- Were students able to relate to the lesson? Did they feel that this was practical knowledge that would be useful?

### Learning Objectives

Students will:

- Collect, graph and analyze real world data.
- Choose an appropriate mathematical model for a particular situation.
- Make predictions based on their analysis.

### NCTM Standards and Expectations

- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

- Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.

- Draw reasonable conclusions about a situation being modeled.

- Analyze precision, accuracy, and approximate error in measurement situations.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.