堵塞的数学语言描述
“在完成这堂课之前，我倾向认为高速路车距提示行驶中的前后两车距离，满足后车刹车距离，避免追尾。但通过下面这节课学习发现，不发生追尾事故的前提下，前后两车的距离与堵塞有直接的数量关系”
需求背景：
任务式教学的概念大行其道，但相当多的STEM课程设计没有摆脱传统的课程思路的束缚，遵循自上而下
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堵塞的数学语言描述
“在完成这堂课之前，我倾向认为高速路车距提示行驶中的前后两车距离，满足后车刹车距离，避免追尾。但通过下面这节课学习发现，不发生追尾事故的前提下，前后两车的距离与堵塞有直接的数量关系”
需求背景：
任务式教学的概念大行其道，但相当多的STEM课程设计没有摆脱传统的课程思路的束缚，遵循自上而下
Top
down形式设定一个任务目标，这样做的风险是目标任务的设定先入为主，有陷入先射箭再画靶心的危险。
运算能力极大释放，数据丰富的现实条件具备了数据中找规律的客观条件。数据支持的结论，透过数据模型找规律是更优解，更接近客观事实。
任务式教学关注点在完成任务的过程，而非目标。过程需要援引真实的数据，正确运用的科学原理，选择简单事实和基本逻辑，整个逻辑推导的过程建立在坚实的事实基础上，结论出乎意料但却可信。
形象的说高楼的基石要结实，尽量运用“元知识”和第一原理”（此处与特斯拉的Elon Mask一再强调的第一性原理不谋而合），觉得抽象的同学，脑补一下几何学中的公理概念。掌握计算工具包含数学模型，计算模型，最后推导出复杂情景下的规律。与前者的过程比较用“差之分毫，谬之千里” 来形容不为过。
澄清以上两种教学不同点，最好办法恰好也是关注上课过程。真实再现一堂美国STEM课程过程，彻底理解真正的任务式教学良苦用心。
课程内容：描述堵车的数学模型，分析触发堵车的因素
课程内容以前后衔接的9个任务为主线，课程开始不预设任何结论，完成9个子任务，结论自然由客观的数据模型推出。
学习收获：
小学3-6年级完成任务1-6，理解数学语言描述堵车和疏散状态
初中1-3年级引入物理中的相变，理解液体变气态临界特征现象
高中引入人工智能snap.berkeley.edu建模仿真(预备中)
PS: 欢迎父母按照任务顺序交给孩子完成，每个任务只有一个正确选项。课程只需小学3年级数学，只有结论部分用到六年级坐标概念。答案在线下课程辅导中公布。
课程特点：
Likewise同理做比较两种教学方式的差别时，我也避免先下结论，再论证的我的观点。先忠实再现课程过程
课程开始：TRAFFIC 交通
任务步骤1：
Two observers are measuring the flow of eastbound vehicles on a highway 15 kilometers apart. Observer A is west of observer B, and there are no on-ramps or off-ramps between the observers. Suppose B measures a flow rate of 15 cars per minute, and A measures 10 cars per minute.
两两名观察员正在测量相距15公里的一条公路上向东行驶的车辆的流量。观察者A的位置在观察者B的位置以西，并且观察者之间没有入口或出口匝道。假设B测量每分钟15辆车的经过，而A测量每分钟10辆车的经过。
If there is a traffic jam on a section of the highway between A and B, is the jam likely growing or shrinking in the long run?
如果在A到B之间的一段高速公路上发生交通堵塞，那么从长远来看，这种堵塞会增大还是缩小？ 请选1还是2
1、Growing 增大
2、Shrinking 缩小
Details and Assumptions:
细节和假设：
No alternate routes are available for the cars on this road.
这条路上的汽车没有可供选择的路线。
No cars make an illegal U-turn to avoid the jam.
没有汽车为了避开堵车而非法转弯。
喵言：数学和算术的区别在这表现出来。让孩子试试看！
问题继续如下：
任任务步骤 2：A large tree falls across the road blocking traffic completely. After some time, a stationary line of cars is waiting for the tree to be removed, forming a traffic jam.
一棵大树横倒在路上，完全阻塞了交通。一段时间后，排成一队的汽车在等待树木挪走，形成了交通堵塞。
If the average flow rate along this section of road is a constant 5 cars per minute, how many cars would you expect in the backup 6 minutes after the tree falls?
如果沿着这条路段的平均流速是每分钟5辆车，那么在树倒下6分钟后，估计会有多少辆车堵塞？
任务步骤 3：
As the cars accumulate behind the tree, each driver parks his vehicle a safe distance from the car preceding him in line.
当汽车被树挡住，每个司机都把车停在离他前面排队的车安全距离的地方。
At what rate is the jam getting longer? We will call this the propagation speed of the jam's trailing edge.
堵塞队伍变长了，增长的速度是多少？将其称为阻塞队尾的传播速度。
Details and Assumptions:
细节和假设：
The average length of each car, plus a small safety buffer between cars, is L0 = 15feet.
每辆车的平均长度，加上车之间的一个小安全缓冲区，是。
The flow rate into the jam remains 5 cars per minute.
进入堵塞物的流量保持在每分钟5辆车。
30 feet per minute5 feet per minute75 feet per minute
89% of people got this right.
喵言：引入波动的形象描述为物理的相变埋下伏笔，问题继续如下：
任务步骤 4：
After some time, the tree is removed and traffic starts to flow again. One by one drivers begin to accelerate back up to the speed limit.
一段时间后，树被移除，车辆再次开始流动。一个接一个的司机开始加速恢复到限速。
Which direction does the leading boundary of the jam—the road segment with stopped vehicles—appear to move?
堵车的队伍的最前一排应该朝哪个方向移动？
How far does the leading edge of the jam move after 1 minute? We will call this the leading-edge propagation speed.
1分钟后阻塞队伍的前缘移动多远？我们将称之为前沿传播速度。
Details and Assumptions:
细节和假设：
Assume each driver takes t0=10s to react to the movement of the car directly in front of him.
假设每个驾驶员都花费t0=10s来对正前方的汽车运动作出反应。
The average length of a car is again L0= 15feet
汽车的平均长度也是L0=15英尺
任务步骤 6：
Now that the tree is cleared, both the trailing edge and the leading edge of the jam are moving, and the jam appears to propagate along the road like a wave.
现在树被清除了，堵塞车队后缘和前缘都在移动，看起来像波浪一样沿着道路传播。
Using the leading and trailing-edge propagating speeds we have calculated, determine how far from the tree the jam advances before it completely clears.
使用我们计算的前沿和后沿传播速度，确定阻塞在完全清除之前前进到离树多远。
Assume the length of the jam was 5280 feet (1 mile) at the moment the tree was cleared.
假设堵塞的长度是5280英尺（1英里），此时树木被清除。
任务步骤 7：
Suppose the eastbound flow rate entering the road behind the jam increased to 8 cars per minute before the jam dissipated completely. How would this affect the lifetime of the jam?
假设在堵塞完全消散之前，进入堵塞后面道路的向东流速增加到每分钟8辆车。这将如何影响堵塞的时间？
相变概念的引入：A phase transition is a qualitative change in the behavior of a system made of many interacting parts. The onset of a phase transition can be sudden and can be accompanied by fluctuating instability. This might make you think of earthquakes or avalanches—rightly so, these phenomena are best understood with the physics of phase transitions.
相变是由许多相互作用的部分组成的系统的行为的定性变化。相变的开始可以是突然的，并且可以伴有波动不稳定性。这可能会让你想到地震或雪崩——没错，这些现象最好用相变的物理来理解。
Phase transitions are characterized by vast and sweeping transformations, precipitated by small changes in the system. For example, the trigger of a major traffic jam could be nothing more than a driver tapping his brake when traffic is near a critical density.
相变的特征是巨大的和广泛的转变，由系统中的小变化沉淀。例如，当交通接近临界密度时，主要交通堵塞的触发器可能仅仅是司机踩刹车。
Gas is less dense than liquid.
气体比液体密度小
说明：STEM教学的优势跨学科。巧妙的引入物理学中相变现象，为孩子理解水的液体、气态的转变打下基础。继续课程，要放大招了
运算能力极大释放，数据丰富的现实条件具备了数据中找规律的客观条件。数据支持的结论，透过数据模型找规律是更优解，更接近客观事实。
任务式教学关注点在完成任务的过程，而非目标。过程需要援引真实的数据，正确运用的科学原理，选择简单事实和基本逻辑，整个逻辑推导的过程建立在坚实的事实基础上，结论出乎意料但却可信。
形象的说高楼的基石要结实，尽量运用“元知识”和第一原理”（此处与特斯拉的Elon Mask一再强调的第一性原理不谋而合），觉得抽象的同学，脑补一下几何学中的公理概念。掌握计算工具包含数学模型，计算模型，最后推导出复杂情景下的规律。与前者的过程比较用“差之分毫，谬之千里” 来形容不为过。
澄清以上两种教学不同点，最好办法恰好也是关注上课过程。真实再现一堂美国STEM课程过程，彻底理解真正的任务式教学良苦用心。
课程内容：描述堵车的数学模型，分析触发堵车的因素
课程内容以前后衔接的9个任务为主线，课程开始不预设任何结论，完成9个子任务，结论自然由客观的数据模型推出。
学习收获：
小学3-6年级完成任务1-6，理解数学语言描述堵车和疏散状态
初中1-3年级引入物理中的相变，理解液体变气态临界特征现象
高中引入人工智能snap.berkeley.edu建模仿真(预备中)
PS: 欢迎父母按照任务顺序交给孩子完成，每个任务只有一个正确选项。课程只需小学3年级数学，只有结论部分用到六年级坐标概念。答案在线下课程辅导中公布。
课程特点：
- 挑战和解决现实问题的趣味为扳机
- 堵车的形成和消失过程逻辑分析扎实 条理清晰
- 分解为9个子任务循循善诱，步步深入
Likewise同理做比较两种教学方式的差别时，我也避免先下结论，再论证的我的观点。先忠实再现课程过程
课程开始：TRAFFIC 交通
任务步骤1：
Two observers are measuring the flow of eastbound vehicles on a highway 15 kilometers apart. Observer A is west of observer B, and there are no on-ramps or off-ramps between the observers. Suppose B measures a flow rate of 15 cars per minute, and A measures 10 cars per minute.
两两名观察员正在测量相距15公里的一条公路上向东行驶的车辆的流量。观察者A的位置在观察者B的位置以西，并且观察者之间没有入口或出口匝道。假设B测量每分钟15辆车的经过，而A测量每分钟10辆车的经过。
If there is a traffic jam on a section of the highway between A and B, is the jam likely growing or shrinking in the long run?
如果在A到B之间的一段高速公路上发生交通堵塞，那么从长远来看，这种堵塞会增大还是缩小？ 请选1还是2
1、Growing 增大
2、Shrinking 缩小
Details and Assumptions:
细节和假设：
No alternate routes are available for the cars on this road.
这条路上的汽车没有可供选择的路线。
No cars make an illegal U-turn to avoid the jam.
没有汽车为了避开堵车而非法转弯。
喵言：数学和算术的区别在这表现出来。让孩子试试看！
问题继续如下：
任任务步骤 2：A large tree falls across the road blocking traffic completely. After some time, a stationary line of cars is waiting for the tree to be removed, forming a traffic jam.
一棵大树横倒在路上，完全阻塞了交通。一段时间后，排成一队的汽车在等待树木挪走，形成了交通堵塞。
If the average flow rate along this section of road is a constant 5 cars per minute, how many cars would you expect in the backup 6 minutes after the tree falls?
如果沿着这条路段的平均流速是每分钟5辆车，那么在树倒下6分钟后，估计会有多少辆车堵塞？
- 30 cars
- 12 cars
- 83 cars
喵言：堵塞队伍延长的速度
任务步骤 3：
As the cars accumulate behind the tree, each driver parks his vehicle a safe distance from the car preceding him in line.
当汽车被树挡住，每个司机都把车停在离他前面排队的车安全距离的地方。
At what rate is the jam getting longer? We will call this the propagation speed of the jam's trailing edge.
堵塞队伍变长了，增长的速度是多少？将其称为阻塞队尾的传播速度。
Details and Assumptions:
细节和假设：
The average length of each car, plus a small safety buffer between cars, is L0 = 15feet.
每辆车的平均长度，加上车之间的一个小安全缓冲区，是。
The flow rate into the jam remains 5 cars per minute.
进入堵塞物的流量保持在每分钟5辆车。
30 feet per minute5 feet per minute75 feet per minute
89% of people got this right.
喵言：引入波动的形象描述为物理的相变埋下伏笔，问题继续如下：
任务步骤 4：
After some time, the tree is removed and traffic starts to flow again. One by one drivers begin to accelerate back up to the speed limit.
一段时间后，树被移除，车辆再次开始流动。一个接一个的司机开始加速恢复到限速。
Which direction does the leading boundary of the jam—the road segment with stopped vehicles—appear to move?
堵车的队伍的最前一排应该朝哪个方向移动？
- Stationary
- East
- West
How far does the leading edge of the jam move after 1 minute? We will call this the leading-edge propagation speed.
1分钟后阻塞队伍的前缘移动多远？我们将称之为前沿传播速度。
Details and Assumptions:
细节和假设：
Assume each driver takes t0=10s to react to the movement of the car directly in front of him.
假设每个驾驶员都花费t0=10s来对正前方的汽车运动作出反应。
The average length of a car is again L0= 15feet
汽车的平均长度也是L0=15英尺
- 90 feet per min
- 150 feet per min
- 15 feet per min
任务步骤 6：
Now that the tree is cleared, both the trailing edge and the leading edge of the jam are moving, and the jam appears to propagate along the road like a wave.
现在树被清除了，堵塞车队后缘和前缘都在移动，看起来像波浪一样沿着道路传播。
Using the leading and trailing-edge propagating speeds we have calculated, determine how far from the tree the jam advances before it completely clears.
使用我们计算的前沿和后沿传播速度，确定阻塞在完全清除之前前进到离树多远。
Assume the length of the jam was 5280 feet (1 mile) at the moment the tree was cleared.
假设堵塞的长度是5280英尺（1英里），此时树木被清除。
- Less than a mile from where the tree fell
- 1.5 miles from where the tree fell
- 6 miles from where the tree fell
任务步骤 7：
Suppose the eastbound flow rate entering the road behind the jam increased to 8 cars per minute before the jam dissipated completely. How would this affect the lifetime of the jam?
假设在堵塞完全消散之前，进入堵塞后面道路的向东流速增加到每分钟8辆车。这将如何影响堵塞的时间？
- Model parameters and remain the same despite the flow rate
increase.
模型参数不变，尽管流量增加。 - The jam would dissipate in significantly less than 352
minutes
阻塞明显减少，并会在352分钟内消散。 - The jam would start propagating eastward, in the same direction
as traffic flow
- 阻塞将开始向东传播，与交通流的方向相同。
- The jam would never dissipate
阻塞永远不会消散。
相变概念的引入：A phase transition is a qualitative change in the behavior of a system made of many interacting parts. The onset of a phase transition can be sudden and can be accompanied by fluctuating instability. This might make you think of earthquakes or avalanches—rightly so, these phenomena are best understood with the physics of phase transitions.
相变是由许多相互作用的部分组成的系统的行为的定性变化。相变的开始可以是突然的，并且可以伴有波动不稳定性。这可能会让你想到地震或雪崩——没错，这些现象最好用相变的物理来理解。
Phase transitions are characterized by vast and sweeping transformations, precipitated by small changes in the system. For example, the trigger of a major traffic jam could be nothing more than a driver tapping his brake when traffic is near a critical density.
相变的特征是巨大的和广泛的转变，由系统中的小变化沉淀。例如，当交通接近临界密度时，主要交通堵塞的触发器可能仅仅是司机踩刹车。
Gas is less dense than liquid.
气体比液体密度小
说明：STEM教学的优势跨学科。巧妙的引入物理学中相变现象，为孩子理解水的液体、气态的转变打下基础。继续课程，要放大招了