在前所述大陸課綱中提到了十項理念、美國更提出五大新目標、而新加坡的五邊形 框架中，提出了以概念、技能、態度、元認知、過程為五邊，數學問題解決為核心的想 法，在這五種規範、學習中，所有的精神皆指向希望能夠運用數學來達到解決生活週遭 的現象。另在德國的教育標準中，提出了十年級畢業生應具備的六大能力，其中更直接
地談到「數學建模」一詞，將建模的概念引入了教學當中。此外，因本身於普通高中任
二
而方能將數學思維進行連結、整合，甚至透過活動產生對生活週遭現況的理解，進而分 析並深入探索；同時，亦需要進行「數學表達」的訓練：藉由不同的活動來增進其上述 每種能力的不同層次，否則未來訓練出來的學生只會一昧地運用所學得的數學公式、解 題技巧來解題，但卻無法將數學的精神真正融入於日常生活當中！
參考文獻
陳揚盛（2000） 。吳鐵雄：需在關鍵時刻融入資訊教育。階梯雜誌，第 59 期。
梁鑑添（1980） 。評論近二十年來中學數學課程改革。抖摟雙月刊，第 38 期，54－57。
黃毅英（1995b） 。香港數學課程改革：迷失於十字路口？香港數學教育：轉變的時機？
專題研討發表論文。香港：香港中文大學。
黃毅英、黃家鳴（1997） 。十地區數學教育課程標準。數學傳播季刊，第 21 卷，第 2 期。
趙東方（2007） 。21 世紀大學數學精品教材－數學模型與計算。北京：科學出版社。
維基百科。阿基米德浮力論。取自：http://zh.wikipedia.org。
蕭文強（1983） 。數學、數學史、數學教師。抖摟雙月刊，第 53 期，67-72。
英文部份(依作者姓氏筆劃順序排列)
KMK. Bildungsstandards im Fach Mathematik fuer den Mittleren Schulabschluss [D]. 2003.
P15.
National Council of Teachers of Mathematics (1970), A History of Mathematics Education in the United States and Canada (32nd Yearbook), Washington, DC: NCTM.
National Research Council (1990), Reshaping School Mathematics, Washington, DC:
National Academy Press.
Press Releases, President Establishes National Mathematics Advisory Pand, April 18, 2006, from the World Wide Web:
http://www.ed.gov/news/pressreleases/2006/04/04182006a.html.
Secondary Mathematics Syllabuses, from the World Wide Web:
http://www.moe.gov.sg/education/syllabuses/sciences/files/maths-secondary.pdf.
Siu, F. K., Siu, M. K. & Wong, N. Y. (1993), The changing times in mathematics education:
The need of a scholar-teacher, In C. C. Lam & Y. W. Fung (Eds.), Proceedings of the International Symposium on Curriculum Changes for Chinese Communities in Southeast
Asia: Challenge of 21st Century, 223-226, Hong Kong: Department of Curriculum and Instruction, The Chinese University of Hong Kong.
The High School Mathematics Contest in Modeling, from the World Wide Web:
http://www.comap.com/highschool/contests/himcm/index.html.
The National Council of Teachers of Mathematics, from the World Wide Web:
http://standards.nctm.org.
W. Blum & C. Drueke-Neo. Bildungsstandards Mathematik: konkret. Cornelsen: Berlin. 2006.
P33-52.
附錄一 附錄一 附錄一
附錄一：：：：2005-2009 全美高中學生數學建模競賽全美高中學生數學建模競賽全美高中學生數學建模競賽全美高中學生數學建模競賽(HIMCM)
2005 Problem A
Modeling Ocean Bottom Topography
Background:
A marine survey ship maps ocean depth by using sonar to reflect a sound pulse off the ocean floor. Figure A shows the ship’s location at B on the surface of the ocean. The sonar apparatus aboard the ship is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. In two dimensions this arc is shown within Figure A by
∠ ABC
, and the emanating sound pulses are displayed by the dashed lines and the solid lines BA and BC.When a sonar sound pulse hits the bottom of the ocean, the pulse is reflected off the ocean bottom the same way a billiard ball is reflected off a pool table; that is, the angle of incidence
α
equals the angle of reflectionβ
, as illustrated in Figure B. Since the ship is moving when the sound pulse is emitted, it will pick up a reflected sound pulse at location F in this picture. The actual depth of the water is the length of BD in Figure AFigure A Figure B Useful Information:
Oceanography vessels usually travel at a speed of 2m/s while Navy vessels travel at 20m/s. The sonar apparatus aboard these ships is capable of emitting sound pulses in an arc
measuring from 2 to 30 degrees. The typical speed at which a sonar sound pulse is emitted is 1500m/s.
Devise a model for mapping the topography of the ocean bottom. Write a letter to the science editor of your local paper summarizing your findings.
2005 Problem B
Gas Prices, Inventory, National Disasters, and the Mighty Dollar
It appears from the economic reports that the world uses gasoline on a very short supply and demand scale. The impact of any storm, let alone Hurricane Katrina, affects the costs at the pumps too quickly. Let’s restrict our study to the continental United States.
Over the past six years, Canada has been the leading foreign supplier of oil to the United States, including both crude and refined oil products. (Petroleum Supply Monthly, Table S3 - Crude Oil and Petroleum Product Imports, 1988-Present. See page 5 for Canadian exports to the United States.)
Canada was the largest foreign supplier of oil to the United States again in 2004, for the sixth year running (from 1999, when the country displaced Venezuela, to 2004
inclusive).
In 2002, Canada supplied the United States with 17 percent of its crude and refined oil imports — more than any other foreign supplier at over 1.9 million barrels per day.
Western Canadian crude oil is imported principally by the U.S. Midwest and the Rocky Mountain states.
Eastern Canada's offshore oil is imported principally by the U.S. East Coast states, and even by some Gulf Coast states.
Many refiners are buying enough to serve motorists' current needs, but not enough to rebuild stocks. "They are looking to buy the oil when they need it,” according to The
Washington Post. "When they are uncertain about the future, they hold back." (The Washington Post: Crude Oil Imports to U.S. Slow With War 3/31/03.)
Build a better model for the oil industry for its use and consumption in the United States that is fair to both the business and the consumer. You can build your model based on a peak day.
Write a letter to the President’s energy advisor summarizing your findings.
2006 ProblemA Inflation of the Parachute
A parachute is made from thin, lightweight fabric, support tapes and suspension lines.
The lines are usually gathered through cloth loops or metal connector links at the ends of several strong straps called risers. The risers in turn are attached to the harness containing the load.
Deployment systems
Freefall deployed parachutes are pulled out of their containers by a smaller parachute called a pilot chute.
A way of deploying a parachute directly after leaving the aircraft is the static line. One end of the static line is attached to the aircraft, and the other to the deployment system of the parachute container.
Types of parachutes Round parachutes
An American paratrooper using an MC1-1C series 'round' parachute
Round parachutes, which are pure drag devices (i.e., they provide no lift like the ram-air types), are used in military, emergency and cargo applications. These have large dome-shaped canopies made from a single layer of cloth. Some skydivers call them "jellyfish 'chutes"
because they look like dome-shaped jellyfish. Rounds are
rarely used by skydivers these days. The first round parachutes were simple, flat circulars, but suffered from instability, so most modern round parachutes are some sort of conical or
parabolic.
Some round parachutes are steerable, but not to the extent of the ram-air parachutes. An example of a steerable round is provided in the picture of the paratrooper's canopy; it is not ripped or torn but has a "T-U cut". This kind of cut allows air to escape from the back of the canopy, providing the parachute with limited forward speed. This gives the jumpers the ability to steer the parachute and to face into the wind to slow down the horizontal speed for the landing.
Annular & pull down apex parachutes
A variation on the round parachute is the pull down apex parachute invented by a
Frenchman named LeMoigne-- referred to as a Para-Commander-type canopy in some circles,
canopy apex that apply load there and pull the apex closer to the load, distorting the round shape into a somewhat flattened or lenticular shape.
Often these designs have the fabric removed from the apex to open a hole through which air can exit, giving the canopy an annular geometry. They also have decreased horizontal drag due to their flatter shape, and when combined with rear-facing vents, can have considerable forward speed around 10 mph (15 km/h).
Ribbon and ring parachutes
Ribbon and ring parachutes have similarities to annular designs and they can be designed to open at speeds as high as Mach 2 (two times the speed of sound). These have a ring-shaped canopy, often with a large hole in the center to release the pressure. Sometimes the ring is broken into ribbons connected by ropes to leak air even more. The large leaks lower the stress on the parachute so it does not burst when it opens.
Often a high speed parachute slows a load down and then pulls out a lower speed parachute. The mechanism to sequence the parachutes is called a "delayed release" or
"pressure detent release" depending on whether it releases based on time, or the reduction in pressure as the load slows down.
Ram-air parachutes
Most modern parachutes are self-inflating "ram-air" airfoils known as a Para foil that provide control of speed and direction similar to Para gliders. Para gliders have much greater lift and range, but parachutes are designed to handle, spread and mitigate the stresses of deployment at terminal velocity. All ram-air Para foils have two layers of fabric; top and bottom, connected by airfoil-shaped fabric ribs. The space between the two fabric layers fills with high pressure air from vents that face forward on the leading edge of the airfoil. The fabric is shaped and the parachute lines trimmed under load such that the ballooning fabric
inflates into an airfoil shape.
A U.S. NAVY display jumper landing a 'square' ram-air parachute
Reserves
Paratroopers and sports parachutists carry two parachutes. The primary parachute is called a main parachute, the second, a reserve parachute. The jumper uses the reserve if the main parachute fails to operate correctly.
Reserve parachutes were introduced in World War II by the US Army paratroopers, and are now almost universal. For human jumpers, only emergency bail-out rigs have a single parachute and these tend to be of round design on older designs, while modern PEPs (i.e P124A/Aviator) contain large, docile ram-air parachutes.
Deployment
Reserve parachutes usually have a ripcord deployment system, but most modern main parachutes used by sports parachutists use a form of hand deployed pilot chute. A ripcord system pulls a closing pin (sometimes multiple pins) which releases a spring-loaded pilot chute and opens the container, the pilot chute is propelled into the air stream by its spring then
canopy, to which it is attached via a bridle.
A hand deployed pilot chute, once thrown into the air stream, pulls a closing pin on the pilot chute bridle to open the container, then the same force extracts the
deployment bag. There are variations on hand deployed pilot chutes but the system described is the more common throw-out system. Only the hand deployed pilot chute may be collapsed automatically after deployment by a kill line reducing the in-flight drag of the pilot chute on the main canopy. Reserves on the other hand do not retain their pilot chutes after deployment.
The reserve deployment bag and pilot chute are not connected to the canopy in a reserve system, this is known as a free bag configuration and the components are often lost during a reserve deployment. Occasionally a pilot chute does not generate enough force to either pull the pin or extract the bag, causes may be that the pilot chute is caught in the turbulent wake of the jumper (the "burble"), the closing loop holding the pin is too tight, or the pilot chute is generating insufficient force, this effect is known as "pilot chute hesitation" and if it does not clear it can lead to a total malfunction requiring reserve deployment.
Paratroopers' main parachutes are usually deployed by static lines which release the parachute yet retain the deployment bag which contains the parachute without relying on a pilot chute for deployment. In this configuration the deployment bag is known as a direct bag system and the deployment is rapid, consistent and reliable. This kind of deployment is also used by student skydivers going through a static line progression, a kind of student program.
Using the modeling process, build a mathematical model for the opening of the parachutes discussed above. We are concerned with how the parachute inflates. Use your model to explain how the geometry of the folding of the parachute affects the inflation and then discuss how we might affect the rate of inflation of the parachute.
2006 Problem B A South Sea Island Resort
A South Sea island chain has decided to transform one of their islands into a resort. This roughly circular island, about 5 kilometers across, contains a mountain that covers the entire island. The mountain is approximately conical, is about 1000 meters high at the center, appears to be sandy, and has little vegetation on it. It has been proposed to lease some fire-fighting ships and wash the mountain into the harbor. It is desired to accomplish this as quickly as possible.
Build a mathematical model for washing away the mountain. Use your model to respond to the questions below.
How should the stream of water be directed at the mountain, as a function of time?
How long will it take using a single fire-fighting ship?
Could the use of 2 (or 3, 4, etc.) fire-fighting ships decrease the time by more than a factor of 2 (or 3, 4, etc.)?
Make a recommendation to the resort committee about what do.
2007 Problem A Smoke Alarms
Fire is one of the leading causes of accidental deaths. It is important for everyone to take every preventative measure and precaution possible to be ready to deal with a fire emergency.
More than half of all fatal fires occur between 10 p.m. and 6 a.m. when everyone in the home is usually asleep. Smoke alarms are necessary to alert you to fires when you sleep. Will smoke alarms allow enough time to evacuate safely?
Build a mathematical model to determine the number and locations of smoke alarms to provide the maximum time for evacuation. Also include a model to determine the number and
location of at-home fire extinguishers to have available. Build a mathematical model for evacuation of a family from both one and two story homes.
Prepare an advertisement for your local fire department to pass out to the community that includes the main results of your mathematical models.
One Story Home
Two Story Home Downstairs
Upstairs
2007 Problem B Car Rentals
Some people rent a car when they are going on a long trip. They are convinced they save money. Even if they do not save money, they feel that the knowledge that "if the car breaks down on the trip, the problem is the rental company's" makes the rental worth it. Analyze this situation and determine under what conditions renting a car is a more appropriate option.
Determine mileage limits on one's own car and a break-even value of "ease of mind" for the driver and her family.
2008 Problem A
National Debt and National Crisis
Mathematical modeling involves two equally important steps – building models based on real world situations and interpreting predictions made by those models back in the real world.
This problem places equal emphasis on both steps.
We are at the start of the 2008 U.S. presidential elections, and one important area of debate is sure to be the national debt. As high school students, you have a particular interest in this subject since you are the people who will pay off or at least manage the national debt in the future. The rate at which the national debt changes depends on the difference between federal income (primarily taxes) and federal expenditures. Your first task is to build a model that can be used to help understand the national debt and make forecasts based on different assumptions. As usual, modeling involves a balance between so much complexity that the model may be intractable and so little complexity that it is unrealistic and useless. Your model needs, at the very least, to allow you to consider different tax policies and different
expenditure policies.
As usual, raw numbers don't carry much information. Those numbers must be placed in context. For example, total national debt is less meaningful than national debt per capita. In
addition, you must be careful about inflation. Many analysts look at the ratio between national debt and gross domestic product as a good indicator of the impact of the national debt. Others worry about the cost of servicing the national debt. This cost is affected by both the size of the national debt and the interest rate the government must pay to borrow money. You may want to look at the Wikipedia article
http://en.wikipedia.org/wiki/National_debt_by_U.S._presidential_terms
http://en.wikipedia.org/wiki/National_debt_by_U.S._presidential_terms