Getting Colder Quicker: Newton’s Law of Cooling In Action!
Overview (Days 1-3)
Grade Level:
Duration:
11th – 12th
3 x 50 min
periods
Subject:
Calculus: Applications of Rates
of Change/Derivative
Prepared By:
Mike Borowczak
Materials Needed
Minimum reqs. Thermometer & Watch; See individual days for full listing with optional equip.
Analyze Learners
Overview & Purpose (STEMcinnati theme):
Students will learn about Newton’s Law of (Heating and) Cooling, specifically they
will learn about real world applications of Newton’s Law of Cooling. By learning
about one specific application of rates of change (Newton’s Heating & Cooling
Law) students will be able to understand how “rates of change” (derivatives) play a
part in their daily lives.
Applications:
Cooling of Nuclear Reactors down to the cooling of an Integrated Chip in a Cell
Phone. Maintaining warmth from buildings to your favorite winter beverage.
Careers:
Materials & Chemical Engineering, Mechanical Engineering, Civil and
Environmental Engineering, Electrical Engineering, Physicist, Chemist, Biologist,
Forensic Pathologists (See Appendix A)
Society:
Keeping us safe from Nuclear Disaster, down to allowing our everyday technology
to exist without exploding.
Science: Heat Transfer between two materials of different temperature
Technology: Use of temperature probes, calculators, and computers 1 to collect
and visualize data.
Engineering: Creating containers and casings that allow rapid/slow heat transfer
to their environment.
Mathematics: Computation of Slopes at various points to derive non-trivial
derivative; taking the derivative of given functions; primer for the concept of
differential equations.
1
Education Standards Addressed
Ohio Benchmarks 11th/12th Grade:
Measurement
D: Solve problem situations involving derived
measurements
Patterns, Functions and Algebra
A. Analyze functions by investigating rates of
change, intercepts, zeros, asymptotes, and local
and global behavior.
Data Analysis and Probability
A. Create and analyze tabular and graphical
displays of data using appropriate tools, including
spreadsheets and graphing calculators.
B.
Use descriptive statistics to analyze and
summarize data, including measures of center,
dispersion, correlation and variability.
C.
Design and perform a statistical
experiment, simulation or study; collect and
interpret data; and use descriptive statistics to
communicate and support predictions and
conclusions.
Scientific Inquiry & Scientific Ways of Knowing
A for both.
Lesson can be adapted for use provided access to: thermometer and watch
Select Goals and Objectives
Goals and
Objectives
(Specify skills/information that will
be learned.)
Teacher Guide
Goals:
Students will understand the fundamentals of
Newton’s Law of Cooling, giving them the insight
to understand the importance of the derivative
(rates of change) in their everyday life.
Objectives:
1. Students will be able to recognize the how
the temperature of an object changes over
time (during cooling).
2. Students will analyze how changing an
object’s material composition or Size
affects the “rate of change” of heat
transfer.
3. Students will be able to explain some of
the factors involved in designing a
container to have specific heat transfer
characteristics.
4. Students will be able to approximate the
derivative of an unknown/non-trivial
function.
Student Guide
While this lesson is geared
towards students in an
Introductory High School
calculus class with a strong
grasp of pre-calculus, this
lesson can be adapted to any
level which has fundamental
knowledge of slopes (High
School Geometry).
The ultimate goal is for
students to grasp that things
that change over time have
measurable “Rates of
Change.”
Assessment
Getting Colder Quicker!
Day One
Materials Needed
Thermometer, Watch, Ruler, Containers (See Attached), and Hot H 2O (Sig. Above Room Temp 80 ºC+);
Optional/Recommended: TI:84+ (or better), CBL Temperature Probe
Analyze Learners
Overview & Purpose (STEMcinnati theme):
Day One covers all four STEM areas!
Select Goals and Objectives
Teacher Guide
Goals and
Objectives
Goals:
1. Students will be able to recognize the how
the temperature of an object changes over
time (during cooling).
2. Students will analyze how changing an
object’s material composition or Size
affects the “rate of change” of heat
transfer.
4. Students will be able to approximate the
derivative of an unknown/non-trivial function.
(Specify skills/information that will
be learned.)
Education Standards Addressed
See Overview
Student Guide
Use pre-assessment to
address major
misconceptions:
e.g. temperature of a
liquid can not drop below
room temp (given no
change in pressure etc).
Also use pre-assessment
to determine biggest area
of concern e.g graphing
the derivative
Objectives:
Focus on Main Objectives 1,2, & 4
Select Instructional
Strategies –
Information
(Catch, give and/or demonstrate
necessary information,
misconceptions, etc…)
Utilize Technology
1. Pre-assessment (10 minutes)
2. Catch (10 minutes)
a. Teacher will place the same temperature
liquid (hot water) in two similar containers
(e.g. two different ceramic mugs) – students
will touch the exterior and experience and
deduce that different objects transfer heat
differently.
Note: You may want to look for videos online
showing rapid heat transfer and its effects (e.g.
rapid heating / cooling of glass & ceramics)
Use of data collection tools
Basic thermometer & watch  CBL system
Assessment
During this time students will
be introduced to the idea of
heat transfer and exposed to
the wide breadth of possible
applications of the derivative.
See Assessment
Questions Document
(GCQ_assess.[docx|pdf])
Other Resources
Require Learner
Participation
Activity
(Describe the independent activity to
reinforce this lesson)
Evaluate (Assessment)
(Steps to check for student
understanding) – See Objectives
above
Inquiry based activity (50 minutes)
Use Cool_it_down_lab.[docx|pdf] as Guide
1. Quickly Review Procedure w/Students [2
minutes]
2. Create groups for each Container Material
Type & Allow students to get materials &
setup [3 minutes]
3. Students
will
now
measure
the
temperature of each of their 3 containers
once every 30 seconds to 1 minute
a. Optional Section
So that students understand the
difficulties in real world data
collections students will collect
data round-robin
For each data point a new student
will measure the temperature and
will not share their measurement
with the others
4. While student are measuring others are
plotting and computing the slope of the
tangent line at several points for each of
the containers
Through observation the
students should see the
cooling curve (an exponential
decay)
Reflect & Assess (5-10 minutes)
1. Bring Students back and ask them what
they have observed
a. How do they think what they’ve
done happens in “the real world”
2. Ask students to plot the tangent line
slopes for homework (finish computing
them)
In the reflection phase
students will confirm the
relationship, and will begin
understanding objective 4
(approximation of the
derivative)
Students should also see that
size matters – in particular
volume/surface area.
+ The lower the number the
faster the heat transfer occurs
Students will calculate the
rate of change by hand
At this point the students will
not see the difference
between different material
types
Students will also not yet
realize the importance of
effective data collection
Additional Notes
Getting Colder Quicker!
Day Two (Optional – Replace with Discussion of Homework)
Materials Needed
Computing Resource, Excel or OpenOffice (free!)
Analyze Learners
Overview & Purpose (STEMcinnati theme):
Education Standards Addressed
See Overview
In Class: Use of Technology and Mathematics prevail in use of Excel & Calculus
Homework: Science and Engineering to develop the idea for their aparatus
Select Goals and Objectives
Goals and
Objectives
(Specify skills/information that will
be learned.)
Select Instructional
Strategies –
Information
Teacher Guide
Assessment
Goals: Be able to plot data and observe overall
difference between various characteristics.
Objectives:
1. Students will analyze how changing an
object’s material composition or Size
affects the “rate of change” of heat
transfer.
Ask students how accurate they think their
measurements are – how would they represent it
and model it. Point out how
(Catch, give and/or demonstrate
necessary information,
misconceptions, etc…)
Utilize Technology
Student Guide
Computing Resource, Excel or OpenOffice (free!)
Students will see why it’s
important to clearly represent
the data that has been
collected.
Other Resources
Require Learner
Participation
Activity
(Describe the independent activity to
reinforce this lesson)
Evaluate (Assessment)
(Steps to check for student
understanding) – See Objectives
above
Taking the data collected the previous day
students will input and plot the following:
1. The combined group data for each
container with trend lines & Equations
2. Optional: Individual data and trend lines +
equations
Students will then present their findings to the
class in a 1 minute “blurb.”
Students will have first hand
experience using excel a tool
for rapid plotting and
evaluation of data.
Reflect & Assess (5-10 minutes)
1. Bring Students back and ask them what
they have observed
a. Were they able to see the same
graphs they plotted the day before
b. Comparing the graphs with each
other what did they see when
different materials were used?
In this reflection phase
students will now understand
how to quickly plot and find
trendlines for data sets
They will also understand that
different materials cool down
differently – e.g. they have
different “K” values in their
exponent.
Evaluation in reinforcing objectives 1, 2 & 4.
Assignment of problem to solve:
“Look around your house – find a container or set
of containers that would help you cool a liquid
down quickly”
In reflection students will also
realize the importance of
complete and accurate
measurement. Those
measuring at 0 and 30sec will
have much better
approximations of the
temperature/time graph than
students measuring at 1:00
and 1:30.
Additional Notes
Getting Colder Quicker!
Day Three
Materials Needed
Day 1 Materials + Student Brought Materials + Extra Teacher Provided Materials
Analyze Learners
Overview & Purpose (STEMcinnati theme):
Education Standards Addressed
See Overview
Today is all about putting it all together – Engineer a complete solution based off a
problem statement.
Select Goals and Objectives
Goals and
Objectives
(Specify skills/information that will
be learned.)
Teacher Guide
Goals: Students will design and construct a
container that will rapidly cool down a liquid to
room temperature. Students will see several
practical applications for Newton’s laws of
cooling.
Student Guide
Assessment
Students will see that they
are capable of creating and
designing a container which
can rapidly release heat!
Objectives:
1. Students will be able to explain some of
the factors involved in designing a
container to have specific heat transfer
characteristics.
2. They will explain by actually designing a
container to rapidly cool down a liquid.
Select Instructional
Strategies –
Information
(Catch, give and/or demonstrate
necessary information,
misconceptions, etc…)
Utilize Technology
Catch: Students are doing another experiment –
they think they’re cooling down 1) the same liquid
at 2) the same temperature. They’re not – they are
swapping liquids and the temperature of their
liquid is much hotter than on day 1 – surprise –
redesign if needed.
Same technology as Day 1 + any additional items
students bring
Students may need to rethink
their design since the
temperature is going to be
much hotter now.
They also need to use
teamwork to quickly build
their solution so there’s time
to test.
Other Resources
Require Learner
Participation
Activity
(Describe the independent activity to
reinforce this lesson)
Activity is open ended. Stipulations are listed in the
assignment handout
Design a Container/Apparatus to quickly cool 1
Cup (8 fl oz) of Water. Your final design must fit
within a 1 cubic foot of space (e.g. width x height x
length can not exceed 1 ft3)
Students will complete
against each other to build a
container/apparatus capable
of quickly cooling down a
liquid.
Any liquids in your design (besides water) they
must be non-toxic and must be approved before
hand by instructors.
Students will have 10 minutes to construct their
apparatus and then 8 fl oz of 100 C water will be
placed within their device and a counter will be
started – time will end when once their liquid
reaches room temperature.
Evaluate (Assessment)
(Steps to check for student
understanding) – See Objectives
above
Evaluate student ability to cool down a liquid faster
than the liquid being place in a Styrofoam
container.
Students will experience
friendly competition with their
peers.
See Assessment
Questions Document
(GCQ_assess.[docx|pdf])
Post Assessment
Important Attachments:
1. Pre-Post Assessment
2. Worksheets
3. PowerPoint
4. Reflection after lesson
Reflection:
What worked well:
Student’s enjoyed the ability to work together to investigate Newton’s law. Documents we’re laid out in a straightforward manner – easy to
”fill in the important details.” The clear separation of tasks also was a benefit as the students knew what they were doing and why - e.g. day
1/2 collect data, day 2/3 input collected data. Students knowledge was reinforced that the derivative was nothing more than the change in
slope of the tangent line of the cooling curve – thus accomplishing our initial objective.
What didn’t work as well:
Students quickly became agitated measuring the liquid with such high frequency / long duration. Suggest that you remove a variable from
the equation – e.g. groups work on a single container (either type of size), this way the group is measuring one object and they can switch
on/off measurements. Additionally – students thought that entering data was monotonous – unsure how to remedy this aside from again
making the students switch on/off entering data – much like in the data collection phase.