折吸管的数学实验来自比利时的数学实验案例

1、号 Texas InstrumentsZ ZlMLgSuoom.Education Technology折吸管实验Koen Stulens 2011 -Texas Instruments- Product Line StrategyItXAS InstrumentsBending Soda Straws 折吸管实验Based on Soda Straws and Other DiversionssCharles Vonder Embse Central Michigan UniversityHow many times have you been sitting sort of mindles

2、sly playing with a soda straw, bending it into various geometric shapes? In this activity we will play a soda straw game of probability. Grab a soda straw at two random places along its length and bend the ends up toward each other.小时候，你是否玩过用各种备样的吸管折成不同儿何形状的游戏？在接卜来的活动中，我们将用 一根吸管來玩i个概率的游戏。拿一根吸管在任意位宜把

3、吸管两端折起。The figures below show two stages of one case where the bends produce a triangle. 如图，图1a和1b显示弯曲过程中两个阶段,进行到第二阶段（1b）形成一个三角形。lalbAnd the next figure shows a case where the ends will never meet to form a triangle.图1c中显示这样的情况：三边无法形成一个三角形。思考并解释为什么会冇这种情况，即随意弯 曲i根吸管的两个端点成为三角形，我们将研究这种情况出现的概率。Bending S

4、oda Straws1电 Texas InstrumentsKoen StulensData Gathering Experiment数据采集实验Lets start with making a conjecture about what you think the actual probability of forming a triangle should be and explain the reasoning behind your conjecture猗测组成三角形的概率并解释你的推理原因。To help refine your conjecture, make two simult

5、aneous bends in your straw 25 times and record how many successes you have修正你的上述概率猜想：随意折吸n 25次，记录这25次中出现二角形的概率，即成功概率。SuccessesConvert the results of this experiment into a probability by computing F(s)=.Trials把实验结果转换成计算公式。P(s)=成功组成三角形的次数/实验次数。Compare these results to you original conjecture.把你的实验结果和

6、你最初的猜想进行对比。Aggregate all of the experiments from your students into a spreadsheet showing a column for the number of the trial 一 event within the experiment 一 and the number of successes. In another column called probability compute the experimental probability Draw a scatterplot - in Data & Statist

7、ics - of the results of the classroom data gathering experiment.在Tl图形计算器中,建立 个“列表与电子表格”程序。把你的每个学生的实验数据记录在表 中。建立儿列分别表示实验序号，成功次数，在第三列计算概率，如卜图所示。再利用TI图形计 算器的“数据与统计”程序做出概率图像，显示收集到的班级实验数据。如卜图所示：Note that the approx command is used to convert the output into a decimal value rather than a fractio n.注意：Appr

8、ox ():分数转换为小数的系统函数。Another way to approximate the real underlying probability is to look at the cumulative probability of all the events in the various trials as one large trial space This is done by creating the mean of the list probability and store this value as the variable cp. The value of cp 一

9、 cell DI 一 changes each time a new trial is entered in the successes column Create 一 in another column 一 the list cumprob to gather the changing values of the variable cp with the Automated Data Capture tool.另一种计算概率的方法是计算累计概率，即用累积成功次数/累积总实验次数，实验结果如卜图所 示。Bending Soda Straws电 Texas InstrumentsKoen Stu

10、lensBending Soda Straws电 Texas InstrumentsKoen StulensEach time a new trial is entered into the list successes, the cumulative probability in cell DI changes and this new value is gathered in a new cell of the list cumprob.每次在表格中输入成功次数，在D1单元格中的累积概率都会发生变化，新的列数据会被记录 到命名为“cumprob”的单元格中。Since each trial

11、 actually represents 25 separate events, the total number of actual events is equal to 25 times the number of trials. Column F shows these values The scatterplots visualize the result of all students: trial versus probability and totaltrials versus cumprob (cumulative probability).既然每个实验实际上都分别有25个密件

12、，实际实验事件的总次数等于25乘于实验的个数。F 列显示的就是总实验事件的值。从下图的散点中看看出所冇学生的实验结果。蒙特卡罗模拟In a Monte Carlo Simulation, the actual probability event is simulated electronically by creating a model that exhibits the same conditions for success and failure as the physical event itself. These models are generally created using

13、a random number generator built into computers or calculators. 在蒙特卡洛模拟里，真实的概率事件是指用计算机创建个模型，展示在与物理事件本身相同 的条件下的失败和成功的概率。这些模型通常是用同化在计算机或il算器内部的随机数发生器 来创建的。For the Soda Straw Problem, we make the following initial assumption: the length of the the soda straw is one unit 在吸管问题中，我们做下列初始假设：吸管的氏度为-个单位长度。The

14、 model of the straw has three segments 一 left, middle, and right 一 representing the three segments of the real straw as it is bent.吸管被折成三段，左、中、右分别代农吸管上的三条线段。The plan is to assign random numbers as measurements to portions of the 1 unit long straw We only need to choose two segments, say the left and

15、 the middle, and the right segment will be known since the entire length is 1 unit我们先建立下图的数学模型，定义吸管氏度为单位长度2,其中两边分别为X”则第三边为x-y。Bending Soda Straws电 Texas InstrumentsKoen Stulens1 cmTo begin, we create two lists of both 100 random numbers between 0 and 1. 首先，我们创建两个从0到1的随机数列表。2 12.1 |3.l1第吸営问题QQlirstse

16、condAand(IOO)=r3nd(100j980.5300740.5328910.0927970.6691421C0081712|0.850438)101I_I1C2h. *B1C0 |-0.8S043840163186Before we can use these random numbers, representing the length of the segments, we need to rework them as follows.在我们使用这些随机数农示线段的氏度之就，我们需耍进行定的数学上的处理。Without loss of generalization, we can

17、 suppose that the smaller of the two values in each row of columnsA and B (i.e first and second) will be taken for the left most segment of the soda straw.This can be accomplished by creating a new list with the command leftseg:= min(first,second). 在通常情况下，我们通过系统函数min(first,second)lU列随机数组屮的放小值作为弯曲的吸

18、管最左边线段的氏度，同时把所取的值存储给变星leftsegoThe middle segment of the bent soda straw cannot be represented by the other of the pair of random numbers since each of the random numbers in a pair are between 0 and 2. Another way to visualize this is to assume that both random numbers in each pair are measured from

19、the same left 尊 Tkas Instruments卜创隔sts Koen Stulensend of the straw This means that the middle segment is actually the absolute value of the difference of the two random numbers因为每一对中的任意随机数都是在0到1之间取值，另外一列中的随机数不能川来表示弯曲吸 管的中间部分线段。这意味着中间部分线段的值是两个随机数之间差的绝对值。The list midseg, computed with the command mid

20、seg:=abs(second-first), represents the middle segments 生成一个新的数据列,命名为midseg”，用系统函数*abs(second-first)*命令取绝对值,表 示中间部分线段。Once the left and middle segments of the bent soda straw have been determined, the remaining segment, the right side, is determined since the entire length is 1 unit.一H左边和中间部分线段被确定，那么

21、右边部分线段即为l-(leftseg+midseg)oThe command rightseg:=l(leftseg+midseg) creates the list with the values for the right segments. 生成一个新的数据列，命名为*rightsegM ,* 1(leftseg+midseg)取值作为右边线段的值。firstsecondrand(100)randdO)nninfl and dl + fl el and el + fl dl, 1, 0) compares the 3 values in the first row of lists le

22、ftseg, middleseg, and rightseg. Fill down the conditional check up to the 100th cell.这个条件命令when (dl + el fl and dl + fl el and el + fl dl, 1, 0)会判断第行的三个 数值是否满足条件,即leftseg”、“middleseg”和“rightseg” 下拉填充这一列的数据, 直到第100个单元格。如下图。0.1555310 5123870-321970.0940730.0721930 0051S90.2301930 72?5850.342444 0.35SW

23、leffoeg=min 吸管问题n jffiH i I I I020406080trial0.45 -0.30-0.15-0.00：-0.15.1003 2000 3000 4000totaltiak3 1 |3.2 |B3破営i可翹p03-2 on.C2b 气汶 6b 80 0300.150.00-0.45吸管问题qoCL 二 m-0.15.0 WOO 2000 . 3000 4000 totaltrialcExplain why the two scatterplots show such different distributions.解禅为什么两组散点图的分布会有区别。Given th

24、e evidence produced by this Monte Carlo Probability Simulation, what is the probability that a triangle is formed when the soda straw is bent randomly in two places along its length?通过蒙特卡洛概率模拟实验，请说明、”l把吸管上的任意两个点弯曲形成三角形的概率是什么？10Bending Soda Straws电 Texas InstrumentsKoen StulensGeometric Modeling of t

25、he Soda Straw Problem管问题的几何模型Open a Geometry page (hidden scale) and draw a triangle AABC. 新建一个几何页面并画一个三角形。Construct a point P inside AABC to act as the driver point for this investigation and construct lines through the point P parallel to side AB and BC of AABC.在AABC中构建一个n丁动点p.并过P点作与AABC的边AB和BC的平行

26、线。As you drag the point P around inside AABC, the two parallel lines intersect side AC. 在AABC内拖动P点，使两条平行线与AC边相交。For the Soda Straw Problem, let side AC of AABC represents the straw The places where the lines constructed through point P intersect side AC will be the points where the straw is bent dur

27、ing the experiment Even though P can move anywhere on the screen, the model only makes sense when P is inside zXABC 对于这个折吸管的问题，假定AABC AC的边为吸管。穿过P点构造的两条平行线和AC边相交 的点，我们可以看作是折吸管实验中的两个点。即使P点可以在屏幕内的任意移动，这个模醴 只有在P点在AABC内部时才有意义。Construct circles centered at the intersection points with radius points at the

28、 vertices A and C ofAABC as below Construct the intersection points of the two circles Consider the point of尊 Tkas Instruments卜创隔szwsKoen StulensBending Soda Straws电 Texas InstrumentsKoen Stulensintersection of these two circles outside AABC as the third vertex of a triangle with its other two verti

29、ces at the points of intersections of the parallel lines and side AC.我们分别以过P点的两条平行线与边AC的交点为圆心，这两个交点到点A和点C的距离为半 能作I员I。构造出这两个圆的交点。这两个鬪的交点可以分别看作是一个三角形的顶点。We know that all the radii of a circle are equal in length and the intersection point of the two circles represents the radius point of each of the c

30、ircles. This means that the triangle formed by these three intersections points has perimeter equal in length to side AC of 4ABC.我们知道两个【员I半径的氏和两个圆圆心之间的K度等于AC边的长度。Use the Attributes, Hide/Show and Color tools to change the layout of the figure as below.Drag point P around inside AABC and observe the

31、behavior of the shaded triangle for different positions of point P.使用属性菜单中的“颜色”和“显示/隐藏”工具，改变下面的图形的相关属性。 在AABC内拖动P点，观察P点在不同位置借况下阴影三角形的变化。12Bending Soda Straws电 Texas InstrumentsKoen StulensAccording to this model, what is the theoretical probability that a triangle will be formed if a soda straw is r

32、andomly bent in two places? Explain!根据这个模型，一根吸管的任意两点被弯曲形成一个三角形的概率是什么？请解释！IntermezzoIf we make the same construction on a Graphs page (hidden Axes) we can determine the coordinates of P and store them in variables xp and yp.如果我们在“图形”页而(隐藏坐标系)内构造同样的图形，我们可以定义P点并使用XP和 YP存储P点的纵横坐标值。If we （manual） data ca

33、pture xp and yp in separate lists in Lists & Spreadsheet we can plot a scatter plot on top the figure to indicate the area where a triangle will be formed if a soda straw is randomly bent in two placesCTRL + . performs a manual data capture.如果我们使用“列表与电了表格”程序（于动的）捕捉XP和YP的值，我们就可以绘制 个关 于P点的散点图，这些散点会在一个

34、像三角形的区域形成，这就是一根吸管（边AC）上的任 意两点被弯曲形成的三角形区域。请按【CTRL】+ .进行于动数据捕获。13电 Texas Instruments#电 Texas InstrumentsBending Soda StrawsKoen Stulens#电 Texas InstrumentsAn Algebraic & Graphical Model of the Soda Straw Problem代数和图形模型解决吸管问题In the figure below, the left and middle segments of the bent soda straw are r

35、epresented by the variables x and y respectively. The right segment is represented by l-(x + y) since the soda straw is assumed to have a length of 1 unit.在下面的图形中，变最X和Y分别代表吸管左边和中间的线段。我们假定吸管的长度为1,则 右边线段为i-(* + y)onuddte = y电 Texas Instruments电 Texas InstrumentsAccording to the logic of the problem, i

36、n order to have three segments to compare, it must be true 三条线段若要组成一个三角形，必须满足下列基本条件：fx0that: yx + y v 1By the Triangle Inequality Theorem, the following inequalities must all be satisfied for a triangle to 根据三角形不等式原理，下列不等式组必须满足：be formed:x+yl-(x + y) x + l-(x + y) y y + 1 (x + y) xWe manipulate 一 in

37、 the Calculator - the inequalities (if possible) into a uy form. 用Tl图形计算器的“计算器”程序，输入不等式，如下图。Bending Soda StrawsKoen Stulens电 Texas Instruments51 巨.2 | 6/1(X41-(X4)-X (“2QH4)ty 27L-2x2iSQy-2x-+l2yl-2x亠(2*1)2靛吸管问题7QE3Z 2旷2Axl-(y 卜 yx4 1-(x*y)7(l-yyly127L2y1Y22,| 5.1 | 5.2 | 6 12yl-(x*y)xyl-(x4)xl-xx(l

38、-rr)4-xl2x2214ZX):靛吸管冋啟vWe converted the constraints on x and y into:把不等式化简:x 0y 0&y11xv-2x 0 y 0 tells us already that (x)is part ofx +y 1the first quadrant and x 1 and y v 1 这个不等式组告诉我们点（x,y）在第一彖限的部分区域，xliylTo determine the area which fulfill the inequalities we enter them as a Text Box and drag and drop them on the axes For each step (inequality) we will draw a polygon corresponding to the condition. 为了定义一个满足不等式条件的区域，我们将在文本框中输入不等式，并放到坐标轴上。把每 一次绘制出的不等式区域用有颜色的多边形进行填充。yl-xResults into the yellow triangleEl田EH 吸弐何商bTB|yX1x0 5IX301l 2在黄色