Can We Find Good Modeling Problems in Unexpected Places? Julie Graves NCSSM graves@ncssm.edu Teaching Contemporary Mathematics January 2017
The issue We have access to problems in books that are on our bookshelves or those of our colleagues. But these books don’t necessarily contain the kinds of problems that will help students become confident modelers.
We would like problems that provide opportunities to practice mathematical modeling. Modeling is the process of Good modeling problems require that choosing and students using appropriate • Frame questions about the real mathematics and statistics to world in mathematical terms analyze empirical • Make assumptions situations, to • Study the model using known understand them better, and to mathematical tools improve • Interpret the results in context decisions.
These problems are richer, more realistic, and more fun. They start in the real world, move to the mathematical world, and then return to the real world.
Structured versions of a problem: A rancher wants to enclose a rectangular field with a fence. She have a $6000 to spend on fencing material that costs $12 per foot. A building is on The problem one side of the field and so she won’t need any statement fencing on that side. Determine the dimensions of contains exactly the the field that will enclose the largest area. information needed. OR Nothing is A rancher wants to enclose a rectangular field with missing and a fence. Fencing material that costs $12 per foot. A nothing is building is on one side of the field and so she won’t extraneous.. need any fencing on that side. She needs to enclose 800 square feet. Determine the dimensions of the field that will enclose this area at the least cost.
Modeling Steps • Identify the problem • Make assumptions and identify variables • Do the Math • Analyze and assess the solution • Iterate • Implement the model • Studetns can swolve a structured problem without engaging in any of these modeling stpes #suggesthashtag
Image goes here The problem statement is incomplete and imprecise. It leaves t room for students to engage in the modeling process. Unstructured version of the problem: A poulterer wants to enclose a chicken run, using a existing fence as part of the perimeter. How can she do this most economically?
Structured version of a problem: A right circular cylindrical can is to have a volume of 90𝜌 cubic inches. Find the height h and the base radius r such that the cost of the The student can will be a minimum, given that the top and needs to make very bottom cost 5 cents per square inch and the few decisions. lateral surface area costs 8 cents per square The problem inch. statement has done most of the work and made most of the decisions.
Unstructured version of the problem: What is the most economical way to construct a can, like a soup can? Are cans that you see The student on the grocery store shelf constructed in an makes economical way? decisions about what assumptions to make and what additional information may be needed.
Structured version of a problem: Gas station A is on your driving route to work and sells gas for $3.00 a gallon. Station B, which is 5 miles off your route to work, sells gas for $2.85 a gallon. Your car gets 30 mpg and your friend’s car gets only 10 mpg. Should either of you drive to Station B for gas?
What would an unstructured version of this problem look like?
Unstructured version of the problem: Most drivers have a “usual” route along which they drive to and from work. Gas prices vary widely so that gas may be substantially cheaper somewhere other than along that route. Would it be economical to go to a station that is not along your usual route to purchase gas at a price that is lower than what is available along your route?
http://bit.ly/2GRAVESTCM http://bit.ly/2TCMGRAVES Please use this link to go to a Google Doc where you can share problems that you write. If you can, include both a “structured” and an “unstructured” version.
Structured problems A printer need to make a poster that will have a total area of • 200 in 2 and will have 1 inch margins on the sides, a 2 inch margin on the top and a 1.5 inch margin on the bottom. What dimensions will give the largest printed area? We want to construct a box whose base length is 3 times the • base width. The material used to build the top and bottom cost $10/ft 2 and the material used to build the sides cost $6/ft 2 . If the box must have a volume of 50ft 3 determine the dimensions that will minimize the cost to build the box. #suggesthashtag
Structured problems You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time ? #suggesthashtag
Structured problems A trucker driving over a flat Interstate Highway gets 7 miles per gallon if he drives at 55 miles per hour. For each 1 mile per hour increase in speed the truck’s fuel efficiency decreases by one-tenth of a mile per gallon. Fuel costs $2.45 per gallon and the driver is paid $27.50 per hour in wages. The fixed costs for operating the truck amount to $11.33 per hour. What driving speed will result in the least cost for a 260 mile trip? #suggesthashtag
Un-structured problem Audio books typically have a total listening duration of about 30 hours. Each individual chapter in a written book lasts for only 2-3 minutes in a audiobook. As a consequence, consecutive written chapters are often combined to form a single audio chapter, since listeners seem to prefer audio chapters that last about 10 minutes. Audio books are typically divided into volumes whose approximate length is 8 hours each. How should an audiobook be produced to result in optimal listening? #suggesthashtag
http://www.siam.org/reports/gaimme-full_color_for_online_viewing.pdf http://www.nctm.org/Publications/Mathematics- Teacher/2016/Vol110/Issue5/mt2016-12-336a/ Thank You!
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