Approaching Application
Problems
Ryan Pietropaolo Ph.D.
NCSSM
January 2014
What/Why?
• Application- the act of bringing something to
bear; using it for a particular purpose.
• Practice skills and interpret answers in
context.
Application = Interesting?
• Train A leaves the station at 8:16 AM going 75
MPH. Train B leaves the station at 9:06 AM
going 85 MPH.
• When will train B catch up with train A?
At the time train B catches up with train A,
how many miles have they traveled?
Princesses?
Different Approach
• Treat the problem as a “springboard” to a
larger story.
• Allow the students to take ownership of their
learning by using creativity to generate their
own follow-up questions.
• Encourage students to cross disciplines.
• Make math problems memorable and fun!
Train cont.
• Analyze the ridership of Amtrak over the past
decade.
• Investigate/compare the fuel efficiency of
trains versus planes and cars.
• Research the impact of trains on the Industrial
Revolution.
What did you learn today?
Salmon Run
• Each year in the North Atlantic and Pacific
millions of Salmon make the long migratory
journey upstream to spawn in riverbeds.
Energy vs. Velocity
• Salmon travel hundreds of miles and climb
thousands of feet before reaching their natal
spawning grounds. The fish expend massive
amounts of energy in the process.
• Make a rough sketch of the relationship between
the amount of total energy expended by the
Salmon over a fixed distance and its swimming
velocity. Then determine a function that satisfies
your sketch.
• Assume the river is flowing at a constant rate.
en route mortality
The salmon that don’t make it provide a
valuable resource for the eco-system!
Energy Expended
• There are many factors that impact a model
for energy expenditure including: the variable
rate of river flow, temperature, distance
traveled (including vertical),
physical/metabolic changes in preparation for
spawning.
• Additional student work…
Energy Equation
• Scientists have theorized that Energy can be
represented by the following equation:
𝐷
𝑛
𝑛
𝐸 𝑣 =𝑐∙𝑣 𝑡 =𝑐∙𝑣
𝑣−𝑟
• How does your equation compare to this
general model? How does the exponent
affect the graph?
Swimming Rates
Researchers have observed that salmon swim at
rates about 50% greater than the flow of the
river. How will this impact your function?
Adjust your function to satisfy this observation.
After the Run?
• Salmon do not feed in freshwater so nearly all
of the fish lack the energy to return to sea.
They spend their final days protecting their
spawning grounds, redds, and ultimately
provide valuable nutrients for the riverbed
when they die.
Research Questions…
Follow-Up Assignment
• Research a topic related to the Salmon Run to
present to the class.
• Write a mock newspaper article summarizing
your findings from your research.
• Recommendation Letters…
Student Work
pH and candy
• Since tooth decay occurs when the pH level in
the mouth is lowest, a candy company designs
their candy so that the pH level reaches a
minimum of 5.0 three minutes after a person
eats candy. Assume that the pH level in the
mouth is normally 6.5. Develop a model for
pH level in the mouth after a person
consumes candy.
AP Calculus Exam
• “Calculus AB and Calculus BC are primarily
concerned with developing the students’
understanding of concepts of calculus, and
providing experience with its methods and
applications. The courses emphasize a multirepresentational approach to calculus, with
concepts, results, and problems being
expressed geometrically, numerically,
analytically, and verbally.”
http://www.csun.edu/~vcmth00m/AP.pdf
Common Core
Focus on modeling
• Estimate how much water and food is needed
for emergency relief in a devastated city of 3
million people, and develop a way to
distribute it.
Modeling cont.
• How far should you drive out of your way to
save on the price of gas?
• How should you design a subway system to
minimize the time it takes to commute?
Discussion
• How do you apply mathematics in your
classroom? Examples…
• How do you feel about crossing disciplines
with the applications?
• What is the ultimate goal for our students?
Thank You!
Enjoy the rest of the conference.
pietropaolo@ncssm.edu