Enhancing Calculus
with Modeling and Technology
TCM Conference 2017
Cheryl Gann
[email protected]
What is Mathematical Modeling?
At NCSSM, we consider a mathematical
experience in which students make choices
about how to use mathematics to create
representations of a real-world process to
be a form of mathematical modeling.
Modeling is the process of creating
representations (models) that help us
understand a phenomenon while using
mathematical concepts and the principles
and language of mathematical symbolism.
What is Mathematical Modeling?
“Mathematical Modeling is when you use
mathematics to understand a situation in
the real world, and then perhaps use it to
take action or even to predict the future,
and where both the real world situation and
the ensuing mathematics are taken
seriously.”
SIAM Math Modeling Handbook:
https://m3challenge.siam.org/resources/modeling-handbook
Ease in to Modeling
Incorporating smaller problems that require
students to think and struggle is beneficial
in its own right.
This also helps prime the students for more
extensive modeling in the future.
One way this can be done is by giving
students problems to work on before
showing them a new technique.
Another approach can be to take away
some details of a problem so that they must
make decisions about what information
they need.
Benefits of Tech in Calculus
• Technology reaches visual learners. It
can also help students with algebra
difficulties see and understand important
course topics.
• Technology can help engage students in
the problem solving process.
• Rather than using technology to replace
thinking, it is most effective when used
to allow us to solve problems that we
otherwise could not.
Quick Survey
I will be demonstrating activities using
Desmos, LoggerPro, GeoGebra, and
Spreadsheets (for Euler’s method).
Which of the following describes your
experience with these?
I have experience with:
A) Spreadsheets to implement Euler’s method
B) Not spreadsheets but one other
C) Two of the mentioned apps
D) Three or more of the mentioned apps
E) None of the above
Enter letter choice at b.socrative.com with
Room Code: NCSSMGann
Warm Up Examples
Warm Up Examples
Landing a Plane Using Calculus
• This activity can be done as soon as students can take the
derivative of cubic functions.
• Perhaps best used after the ideas of continuity and
differentiability have been discussed.
• To implement this activity, I used the Desmos Activity
Builder.
https://teacher.desmos.com/activitybuilder/custom/5810e9de139131c209144577
Landing a Plane Using Calculus
𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
2𝐻
𝑎= 3
𝐷
3𝐻
𝑏= 2
𝐷
𝑐=𝑑=0
Note that from here, the
class could consider what
non-cubic functions could
work.
Related Rates and LoggerPro
• This activity can be used to introduce related rates or as a
follow-up activity after students have worked through
several related rates examples.
• If used as an intro, likely less analysis would be appropriate
initially. The class could then return to the problem after
more work on the topic has been done.
• Using LoggerPro (used in many science classrooms), data
can be collected from a video.
Related Rates and LoggerPro
In this activity, students will explore how the volume of a liquid
in cone-like container changes as the container is filled.
After collecting some data from a video clip, students use
calculus knowledge to explore the rate of change of the volume
of the liquid in the container.
Activity Handout & LoggerPro file
Related Rates and LoggerPro
𝜋 2
𝑉= 𝑟 ℎ
3
𝑑𝑉 𝜋 2 𝑑ℎ 2𝜋
𝑑𝑟
= 𝑟 ⋅
+
ℎ⋅𝑟⋅
𝑑𝑡 3
𝑑𝑡
3
𝑑𝑡
Note that if the glass were a perfect cone,
we could also use that
ℎ
𝑟
=
3.25
2.34375
⇒ ℎ ≈ 1.387𝑟:
1.387𝜋 3
𝑉=
𝑟
3
𝑑𝑉
𝑑𝑟
2
= 1.387𝜋𝑟 ⋅
𝑑𝑡
𝑑𝑡
Excel calculations
Skydiving with Euler
• This activity can be done once students know Euler’s
Method.
• Before doing this activity, I strongly recommend that
students have practiced using Euler’s Method and have used
Excel to do Euler’s Method.
Euler and Excel
• It is helpful to think about Euler’s Method as:
𝑛𝑒𝑤 𝑥 = 𝑜𝑙𝑑 𝑥 + 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
𝑛𝑒𝑤 𝑦 = 𝑜𝑙𝑑 𝑦 + 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 = 𝑜𝑙𝑑 𝑦 + 𝑜𝑙𝑑 𝑠𝑙𝑜𝑝𝑒 ⋅ (𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥)
• Excel and Euler example: Newton’s Law of Cooling
𝑑𝑇
= −0.25 𝑇 − 70
𝑑𝑡
𝑇 0 = 120
Skydiving with Euler
Extreme skydivers try to set records for very high altitude
jumps. These jumps are typically made not from airplanes,
but from hot-air balloons.
The record for extreme jumping is 39 kilometers (about 24.2
miles) set by Felix Baumgautner during which he set a record
for fastest free-fall velocity of 1357.64 km/hr (843.6 mi/hr).
Skydiving with Euler
Skydiving with Euler
In this activity, students will determine how fast such a
skydiver is going at his fastest and how long it takes him to
reach the ground.
𝑛𝑒𝑤 𝑡 = 𝑜𝑙𝑑 𝑡 + Δ𝑡
𝑛𝑒𝑤 𝑣 = 𝑜𝑙𝑑 𝑣 + 𝑜𝑙𝑑 𝑎 ⋅ Δ𝑡
𝑘=
𝐶𝜌𝑆
2𝑚
𝐶 = 0.57
𝑛𝑒𝑤 ℎ = 𝑜𝑙𝑑 ℎ + 𝑜𝑙𝑑 𝑣 ⋅ Δ𝑡
𝑎 = −9.8 + 𝑘 ⋅ 𝑣 2
𝑚 = 75 𝑆 =
0.7 𝑖𝑓 ℎ > 2610
25 𝑖𝑓 ℎ ≤ 2610
Activity Handout & Solution
𝜌 = 1.3 0.9999
ℎ
An AP Example
Geogebra setup
Thank you for attending!
Please contact me with any
questions.
Cheryl Gann
[email protected]