函数模型及其应用教案.doc

Hangzhou Dongfang QCE：Providing a Passport for the Future

Modeling and Problem Solving
——函数模型及其应用教案
中澳课程部 王晓叶
学情分析：澳方MathB每次的Paper Test都分为两部分，其中Knowledge and Procedures（知识与过程）这个和普通高中数学相似，学生A/B率比较高，但是另外一部分Modeling and Problem Solving（建模与实际问题的解决）学生的A／B率不高。这一部分内容题目普遍很长、生词量较多，并且都是将数学知识应用于实际生活中，所以大多数学生遇到此类题目都是放弃不做。MathB这门课又特别注重实际生活问题的解决，而我们的学生这方面意识比较薄弱，抽象概括能力较弱。所以，我们的教学任务是提高学生的考试成绩等级，提高OP成绩。但是另一方面，12年级的学生大多数能灵活的使用图形计算器，具有一定的英语语言基础。
教学目标：1.了解函数模型在现实生活中的运用。
2.能够建立恰当的函数模型，并对函数模型进行简单的分析。
3.利用所得函数模型解释有关现象，对某些发展趋势进行预测。
教学重难点：1.建立合适的函数模型
2.利用得到的函数模型解决实际问题
教学过程
一、 引入案例、探索新知（如何确定最合适的函数模型）（18分钟）
案例：根据《Daily Mail》报道，上个月一名中国留学生将自己车速飙到180公里／小时的录像传到了Instagram个人网页上，并以配以中文：“从Albany开回Perth，一路180公里／小时，将4.5小时的车程缩短到3.5小时。”
目前，他正在接受警方调查。
警察表示，视频显示这名男子在限速110公里／小时的高速公路开到了180公里／小时，他将面临巨额罚款、吊销驾照以及拘留。

Example1:The table below shows the relationship between the velocity of a car and the distance after it braking.
Velocity
10
20
30
40
50
60
70
80
90
Distance
2
10
15
20
27
38
47
60
75
a. Use the calculator to find the relationship between the velocity of a car and the distance after it braking.
b. What’s the minimum safe following distance for a car travelling at 110 km/h on the motor way?
澳洲法律常识
项目
罚款
扣分
超速少于10km/h
163澳元
扣2分
超速10km/h－20km/h
357澳元
扣3分
超速20km/h－30km/h
726澳元
扣5分
超速30km/h－40km/h
866澳元
扣7分
未系安全带
341澳元
扣3分
闯红灯
437澳元
扣3分
开车使用手机
315澳元
扣3分

（设计意图：从生活案例引入新知，激发学生的学习兴趣。从简单题目入手，目的是让学生掌握图形计算器的使用，能够利用图形计算器建立合适的函数模型，为解决函数模型的应用做铺垫。同时在课堂中渗透德育内容，让学生知法懂法守法。）

小结：如何建立合适的函数模型？
• To Draw the Scatterplot

Choose the most suitable Model

Solve the practical problem

Exercise．Some Chemistry students measured the concentration of chlorine remaining in a swimming pool over a period of 8 hours on a hot summer day. Chlorine had been placed in the pool at 8 am. Their results were as follows.

Morning
Afternoon
Time
9
10
11
12
1
2
3
4
Chlorine concentration(ppm)
5.0
3.8
2.9
2.2
1.6
1.2
0.9
0.7

a. Develop a model for the data。

b. What’s the concentration of chlorine at 8 am?

（设计意图：通过练习，巩固加强掌握图形计算器的使用，为下一个例题的讲解做好铺垫。）

二、 例题精讲(函数模型的应用)（12分钟）
A/B Standard-Modeling and Problem Solving
Example2．Some Chemistry students measured the concentration of chlorine remaining in a swimming pool over a period of 8 hours on a hot summer day. Chlorine had been placed in the pool at 8 am. Their results were as follows.

Morning
Afternoon
Time
9
10
11
12
1
2
3
4
Chlorine concentration(ppm)
5.0
3.8
2.9
2.2
1.6
1.2
0.9
0.7

1. Use the calculator to find the relationship between the chlorine concentration and the time elapsed since the chlorine was placed in the pool.
2. Use the results to find the concentration that would be needed at 8 am on a similar day to ensure that the chlorine concentration did not fall below 1.5ppm(parts per million) before 3 pm.
3. If two chlorine doses were used , one at 8 am and another at 12 noon, what concentrations would be needed at these times to ensure that the concentration did not fall below 1.5 ppm before 3 pm.

4. What are the implication of your answers for the effective chlorination of pools at lowest cost?

(设计意图：例题为11年级MathB一次大考中的一道原题，是一道A/B等级的题目，大多数学生遇到此类型的题目都是放弃不做，原因是题目太长，生词太多，难度较大。针对这些问题，所以通过前面的练习让学生看懂题目，拟合出合适的函数模型。精解函数模型的应用。首先是引导学生审题，找关键词，读懂题目。又将题目分解成多个小题，层层递进，由易到难，引导学生学会解决此类问题的方法。遇到大的困难时，先一个一个小困难的解决。)

三、课堂检测（14分钟）
1. An oscilloscope is used by students to measure the voltage across a capacitor as it discharges through a large resistance. Observations made by the students were as follows.

Time(s)
4
8
12
20
30
40
Voltage
49
40
33
22
13
8
a. Develop a model for the data
b. Use yours results to predict the voltage after 2 min.

2.The removal of some substances from the blood by the kidneys depends on the concentration in the blood. The following measurements of the concentration of an antibiotic were taken after the antibiotic was intravenously injected to give an initial concentration of 20 ppm.
Time (hours)
0.5
1
1.5
2
2.5
3
Concentration(ppm)
18
16
14
13
11
10
a. Develop a model for the data。
b. For the drug to be effective, it is known that the concentration must be between 5 and 20ppm. How often must a follow-up dose be given?

（设计意图：由于澳方核心能力考试强调学科间的结合，所以例题涉及化学，课堂练习两道题目涉及分别涉及物理、生物医药。目的是为了检测本节课学习成果，检测学生的掌握程度。题目难度比例题要低一些，确保大多数同学都能完成，增强信心，第二题的第二个小问稍微有些难度，题目由易到难。）

四：总结（1分钟）

五、课后练习：
1.People from the country say that the pace of city is too fast. They claim that people in the big cities walk faster than those in smaller towns. Marc and Helen Bornstern collected data on the speed at which people walk. The table on the following is adapted from their data.
Location
Population
Speed(m/s)
Brooklyn,USA
2602000
1.54
Munich,Germany
1340000
1.91
Prague Czechoslovakia
1092759
1.79
Jerusalem,Israel
304500
1.35
New Haven,USA
138000
1.34
Iraklion,Greece
78200
1.17
Netanya,Israel
70700
1.31
Bastia,France
49375
1.49
Dimona,Israel
23700
1.00
Safed,Israel
14000
1.13
Corte,Corsica
5491
1.01
Itea,Greece
2500
0.69
Psychro,Crete
365
0.84
Athens,Greece
867023
1.59
Brno,Czechoslovakia
341948
1.47
a. Develop a model for the data
b. Use your results to predict the speed of walkers in Toowoomba, which has a population of about 100000.

2. In 1798,Thomas Malthus suggested that exponential growth of the human population must eventually outstrip the possible food supply and lead to wars and disease.
a. Investigate the records of word population given above to find whether you agree with the idea that it does grow exponentially.
b. Malthus originally suggested that the world’s food supply would have been outstripped long before the present day. Suggest some reasons to explain the fact that his dire predictions have at least been delayed.
Year
1000
1200
1400
1500
1600
1700
1750
1800
1850
1900
1950
2000
Population
350
435
465
475
490
630
800
900
1300
1600
2500
6000