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Faculty Work: Comprehensive List
1-7-2017
Weighing Fog: Hands on Modeling for Day 1 of
Differential Equations
Tom Clark
Dordt College, [email protected]
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Clark, Tom, "Weighing Fog: Hands on Modeling for Day 1 of Differential Equations" (2017). Faculty Work: Comprehensive List. 639.
http://digitalcollections.dordt.edu/faculty_work/639
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Weighing Fog: Hands on Modeling for Day 1 of Differential Equations
Abstract
The first day of many mathematics classes contains formalities and very little mathematics. Here an alternative
is presented where modeling is placed as the centerpiece to orient students to the real work of differential
equations. Namely, to capture as beautifully and compactly as possible through the process of conjecture and
investigation, the deep and interesting aspects of the physical world. A demonstration of the sublimation of
dry ice sits at the center of the lesson. Students collaborate in groups to design an experiment that could
measure the change in mass of a piece of dry ice that is dropped into water, the experiment is then carried out,
and finally the students (with some guidance) build and solve a model for the phenomenon. I will also present
my reflections on the lesson as well as guidance and resources for use in the differential equations classroom.
Keywords
mathematical models, differential equations, physical science, dry ice
Disciplines
Mathematics
Comments
Presented at the 2017 Joint Mathematics Meetings held January 4-7, 2017, in Atlanta, Georgia.
This conference presentation is available at Digital Collections @ Dordt: http://digitalcollections.dordt.edu/faculty_work/639
Weighing Fog: Hands on Modeling for Day 1 of
Differential Equations
Thomas J. Clark
January 7, 2017
Dordt College
1
in my beginning is my end
The Goal
Mathematics begins in wonder and ends in understanding.
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The Goal
Mathematics begins in wonder and ends in understanding.
1. Capture attention of students.
2. Ask interesting questions.
3. Start with modeling, let the differential equations follow from
need.
4. Students do the thinking, generate ideas.
5. Let day 1 be a microcosm of the semester.
6. Don’t discuss syllabus on day 1; they can read it, or I can
make a screencast and they watch later.
7. The more I think about it, this could probably fill the first two
days of class, especially if it is done via IBL.
3
The Question
Question
What will happen if I drop a piece of dry ice into water?
4
The Question
Question
What will happen if I drop a piece of dry ice into water?
Question (Follow Up)
What is the graph of the mass over time? How do you find the
equation of that function?
4
5
The Outline
1. Pose dry ice question.
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The Outline
1. Pose dry ice question.
2. Students design experiment that could measure mass of dry
ice over time.
6
The Outline
1. Pose dry ice question.
2. Students design experiment that could measure mass of dry
ice over time.
3. Run experiment (just borrow the scale and dry ice from the
chemistry department).
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The Outline
1. Pose dry ice question.
2. Students design experiment that could measure mass of dry
ice over time.
3. Run experiment (just borrow the scale and dry ice from the
chemistry department).
4. Make conjectures about mass/time function.
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The Outline
1. Pose dry ice question.
2. Students design experiment that could measure mass of dry
ice over time.
3. Run experiment (just borrow the scale and dry ice from the
chemistry department).
4. Make conjectures about mass/time function.
5. Model with differential equations. Discuss what a
mathematical model is.
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The Outline
1. Pose dry ice question.
2. Students design experiment that could measure mass of dry
ice over time.
3. Run experiment (just borrow the scale and dry ice from the
chemistry department).
4. Make conjectures about mass/time function.
5. Model with differential equations. Discuss what a
mathematical model is.
6. Compare “solution” to data, revise/improve model.
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The Model
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The Model
1. Students suggest “exponential decay”.
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The Model
1. Students suggest “exponential decay”.
2. Clarify to y (t) = Ce kt .
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The Model
1. Students suggest “exponential decay”.
2. Clarify to y (t) = Ce kt .
3. Move to y 0 = ky , what this means...change in mass is
proportional to mass. Does that seem right?
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The Model
1. Students suggest “exponential decay”.
2. Clarify to y (t) = Ce kt .
3. Move to y 0 = ky , what this means...change in mass is
proportional to mass. Does that seem right?
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The Model: Take 2
1. How does the dry ice sublimate? Where?
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The Model: Take 2
1. How does the dry ice sublimate? Where? On the surface.
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The Model: Take 2
1. How does the dry ice sublimate? Where? On the surface.
2. So y 0 = k · Surface Area.
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The Model: Take 2
1. How does the dry ice sublimate? Where? On the surface.
2. So y 0 = k · Surface Area.
3. What is mass proportional to?
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The Model: Take 2
1. How does the dry ice sublimate? Where? On the surface.
2. So y 0 = k · Surface Area.
3. What is mass proportional to? Volume.
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The Model: Take 2
1.
2.
3.
4.
How does the dry ice sublimate? Where? On the surface.
So y 0 = k · Surface Area.
What is mass proportional to? Volume.
How do you relate volume and surface area for a cube?
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The Model: Take 2
1.
2.
3.
4.
5.
How does the dry ice sublimate? Where? On the surface.
So y 0 = k · Surface Area.
What is mass proportional to? Volume.
How do you relate volume and surface area for a cube?
Propose y 0 = ky 2/3 . But what is the solution?
9
The Model: Take 2
1.
2.
3.
4.
5.
How does the dry ice sublimate? Where? On the surface.
So y 0 = k · Surface Area.
What is mass proportional to? Volume.
How do you relate volume and surface area for a cube?
Propose y 0 = ky 2/3 . But what is the solution?
9
The Extensions
1. Vary mass to surface area ratio: y 0 = ky r .
dy
= kdt
yr
1
y 1−r = kt + C
1−r
1
y (t) = (1 − r )(kt + C ) 1−r .
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The Extensions
1. Vary mass to surface area ratio: y 0 = ky r .
2. Assume k = k(t), changes with time. (Plot ‘k(t)‘ = y 0 /y 2/3 )
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The Extensions
1. Vary mass to surface area ratio: y 0 = ky r .
2. Assume k = k(t), changes with time. (Plot ‘k(t)‘ = y 0 /y 2/3 )
3. Attempt to model water mass loss from “fog”.
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The Resources
Worksheet:
http://homepages.dordt.edu/~tclark/DryIce/DryIce.pdf
Run 1: 4.4 grams of dry ice sublimating:
https://youtu.be/1qyiyBRpj-8
Run 1: Screen capture of mass/time data:
https://youtu.be/_4DhfnH02a8
Run 2: 11.42 grams of dry ice sublimating:
https://youtu.be/EogDv4yXBJA
Run 2: Screen capture of mass/time data:
https://youtu.be/Xa9wmxzgxGM
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the end of all our travels
Thanks For Your Attention!
If you would like to see any of my materials contact me at
[email protected].
References
1. Clark, Thomas J, “Weighing Fog: Modeling on Day 1 of
Differential Equations,” (pre-print).
2. Winkel, Brian (2015), “1-12-T-SublimationCarbonDioxide,”
https://simiode.org/resources/435.
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