Comparative Paper: Change and Permanence

[HA]

Defending: Writing Communication, Self-Regulation

Introduction

-what is it & why i decided to do this

this is a milestone paper I wrote in my English class at the end of this semester. it is a paper that compares three novels The Good EarthThings Fall Apart, and Of Mice and Men that we have read during the semester. these novels describe the life and story in three different places: China, Nigeria, and America. 

-how did i do

to write this comparative paper, first, I have to finish the three books and got some general thoughts on the key information that the author wanted to present in each book. also, it is even better than I can have some personal feelings toward each novel. based on this information, I was going to find one point that involves all the books whether it is by similarity or inversion, and choose it as my paper topic. 

secondary, I have to comes up with a thesis statement within my topic by analyzing the tendency of the point I chose as my topic. a thesis statement is a point that I want to argue throughout the paper. 

thirdly, I was going to collect evidence (quotes) in the three books that support my thesis and organized them into three key pieces of evidence and a paper outline. 

now, it is time to write the first draft of my paper. after one peer review, I revised my paper and give it to another peer. 

after I received the second round of peer review and also the feedback from Joel I did my last round of revise. 

lastly, the comparative paper Change and Permanence was been completed. 

-the knowledge i applied

in this comparative paper, reading and writing skill is throughoutly applied. I have to know how to read novels efficiently and recognized the key information that I want to use in my paper. also, I have to know how to write a good paper with an interesting introduction, convincing evidence, and a conclusion that can sum up my ideas. 

the other skill that applied in my paper is self-regulation. I have to ask others for feedback and consider their ideas seriously when revising work. I also exhibit a strong determination to finished the paper.

Self-reflection

-what have i learned

about writing skills:
while writing this comparative writing, I learned how to write writing that involves the information in three books. I also learned how to compare one element in three different reading and to find the similarity between them. 

about self-regulation:
while editing my paper after the peer review, I learned how to improve my work by asking for feedback from my teacher and my classmates. I got more proficient on revise my draft and produce the best final version. 

-skills i have used and improved

I have used almost all the skills in writing communication and this helping me to improve a lot in writing. 

I have also used and improved some of the skills in self-regulation such as regulation and stress management. peer review and revise my work help me improve my regulation while a milestone that worth 1000 points developed better my stress management. I have to control well my stress in order to finish the paper and other works at the same time. 

-skills i still have to improve

I still have to improve thoroughly evaluates points of view, purposes, or other context information to assess the credibility of sources.

Reflection Essay on Modeling the London Eye

Laura Liu 2020年4月10日

不知道你是否乘坐过伦敦的千禧轮,也称为伦敦眼?或者乘坐过任何在空中旋 转的摩天轮?不知道你在由地面而上,逐渐看到远方的地平线,又在黄昏的光线里下 降的时候有没有思考过每一刻流动的时间,还有每个时间里你距离地面的高度?这篇 文章记录了一次数学的探究,如何运用周期现象来建模伦敦眼的乘坐时间和所在高度 之间的关系。你将在这里看周期函数在生活中存在的一种形式,以及如何寻找被定义 的实际问题的解决方法和整个解决、验证过程。最后,揭晓可以表示乘坐时间和所在 高度的方程。文章的前半部分记述了我和我的合作伙伴kolento的整个模拟伦敦眼去解 决问题的过程,包括从最初的收集信息到最后找到公式并在desmos上绘出图形。后半 部分是我个人对于此次数学探究的反思,主要包括我学习到的新东西和未来可以继续 使用的技能。文章的最后一段是一个小总结。

过程

收集信息&定义问题 

3月12号,在数学课上我们开始了一个新的任务。这次任务的第一页很特别,一打开是一张蓝色天空背景的照片,照片里竖立着一个摩天轮,旁边的文字里解释道: 伦敦眼,或称千禧年之轮,是位于伦敦泰晤士河南岸的一个悬臂式观景轮。

我一下子兴奋了起来,很期待这次任务,也随之看到了我们的要解决的问题:在任何 时候我们离地面有多高? 任务的第二页上,每个人要根据这一页的信息计划在数学的 角度我们希望发现的东西,于是我写到:“每秒钟从你所在位置到地面高度的精确值/ 时间和高度的关系。” 随后,数学课上两个人组队形成了这次的探究小组,我和 kolento一组,我们又在选择任务难度的时候选了较难的、根据基本原则去解决这个问 题的方法。后面就进入了小组自己的工作时间,我和kolento收集好yif老师给的已知信 息,开始讨论。已知信息包括:直径120、旋转一周需要30分钟、整个高度是135。

解决方法 

根据手里的已知信息,我和kolento互相分享了一下自己对于此问题的头绪。kolento想到了用物理公式计算了伦敦眼需要多大的力才能抬升多高的高度,算出它的 机械效率然后反推出时间。而我第一时间想到的是几天前circular functions任务里的一 个动画图。在这个图里,有一个圆以及一条穿过圆心的线段,这线段就像时钟的指针 一样在圆上旋转,同时这条线形成的角度的sin和cos值也显示在动画里。

我在思考,如果我们可以画出时钟的指针,我们就能求出它所在角度的sin值,通过用 sin值加上75(摩天轮底座15加半径长60),我们就能求出任何时候从时钟指针到地面 的垂直高度。(见下图)


计算

我和kolento决定用我的思路来解决问题。kolento搜集到了可以用于解决此问题的公式,也就是一般形式的周期公式 y=Asin(ωx+φ)+b,并开始研究怎么运用它,而我 则开始寻找那条时钟的指针。我没什么思路,于是找到yif老师,他给了几个启发,可 是我能总结出来的点只有1.它是一条直线,可以用y=kx+b模拟,2.它的k值在一直变 动,从0开始增加到无穷大,再到负的无穷大,逐渐增加回到0。可惜这些信息不足以让我找到时针线段的k值,而我人在美国,那时候已经是晚上了,所以第一天的工作到 这里我就去休息了。

得出答案 

第二天早上,微信里发来了kolento的完成作品,我看见了一个紫色的,会旋转的摩天轮! (如下)

根据kolento发过来的截屏,我开始摸索,尝试画出来自己的摩天轮。我首先找到了在 kolento的图里模拟时钟的表针的公式,并且利用此摸索出了所有摩天轮的零件,绘制 出了自己的摩天轮。也得出了可以表达时间和高度关系的公式:y=60sin(x1/5*pi/ 180+3/2pi)+75,其中y代表单位为米高度,x代表单位为秒的时间。

到这里我们就完成了问题的求解,找到了可以模拟伦敦眼时间和高度关系的周 期函数,我现在来解释一下为什么函数是这个样子。根据一般形式的周期函数 y=Asin(ωx+φ)+b,在函数y=60sin(x1/5*pi/180+3/2pi)+75里,60是此函数的A,也就是 函数的振幅,根据已知信息伦敦眼的半径是60m,所以A=60;1/5x代表的是从时间转 换到角度的计算,一圈360°=30分钟/1800秒,所以每1°就是5秒钟,1/5°=1秒;乘以pi/ 180是因为在desmos数学画图的网站上我们要以弧度进行输入,根据180°=π rad,我们 将角度值x1/5乘以pi/180;括号里面加的3/2pi是因为sin值在原点的值是从0开始,而我 们需要让它从最低点开始,因为摩天轮都是从最低处开始乘坐的,这里要加上3/4圈或 减去1/4圈,让函数走完这圈或者退回去一部分;最后加75是摩天轮的底座15m加上半 径60m的值。

去总结的一下的话,这个方程表达了y高度值,等于摩天轮半径60m与 一单位时 间内的旋转角度加上3/2pi的乘积,再加上75m(半径和底座)的和。

验证

得到答案后我们就进入了验证结果的环节。借用desmos,我们使用最直观和便捷的验证方法。首先,进入desmos页面,输入我们找的方程。随后开启时间t值的滑动 条,这样就可以调动t值,查看不同时间下y值,也就是所处高度了。根据已知条件, 摩天轮的起始高度应该为15m高,而因为转一圈耗费30分钟,则时间为30分钟的时候 应该再次回到15m的高度。当时间为旋转一圈的1/2时,其高度应当在摩天轮的最高 处,也就是直径加上底座的距离,等于120+15=135m。

有了这些信息,就可以进行验证了。首先我们把时间调到0,可以看到y值的结 果是15,信息正确。

下面时间为900秒,也就是15分钟,其y值为135m,信息正确。 最后当我们把时间调到180秒、30分钟的时候,y值又回到了15m。

现在,对于公式的测试全部符合给出的已知条件,我们验证了产出的公式,完成整个 数学探究任务。

反思 

关于策略

在此次数学探究问题里,我和kolento使用了“from first principles”的策略去解答 问题,也就是根据第一原理去找到能模拟此情境下的方程。我们两个人主要从两个角 度出发,去寻找这个“第一原理”。我是先观察到之前出现过的类似情况,然后分析它 的原理,就像我会先找到那个表针,从中去找到也可以模拟摩天轮的方程。而kolento 则会直接去调研,找到此类方程的一般式y=Asin(ωx+φ)+b,然后去研究这个方程的不 同部分的特质,再把给出的已知信息对应套入,最后找到答案。

通过这次数学探究的策略,我开始意识到了有的时候可以去做一些调研,并且 仔细阅读那些看上去像“天书”的内容,理解它的每个重要的部分,再回到眼前的问题 之下,找到每个已知信息所对应的部分,带入、解决问题。这样不但能让自己对问题 的理解有很大层面上的提升,对于问题的理解也会更加有清晰和结构化。希望在以后 的数学探究任务里,我可以更多去尝试和使用这样的策略。

关于心得

这次数学探究是我练习过的最有意思的任务之一了,能解决一个和生活无比贴 近的问题,还能在desmos上绘画出数学解答很让人陶醉。通过后期的小组介绍成果和 提问环节,我反思、琢磨,也更加清晰我和kolento完成任务的整个过程了。我知道了 在求解时钟的表针时我的卡点的解答,也意识到当时我没能运用好之前学习的知识。 具体是这样的,在求解直线方程y=mx+b的时候,k值可以通过两个点的(y1-y2)/(x1-x2) 来获得,而表针线段的xy1和xy2分别可以为线段的两端,x2y2就在圆心上,都为0,所 以(y1-y2)/(x1-x2)=y1/x1,x1就是cos值,y1就是sin值。如果我能灵活运用此方法,当 时便能计算出k就等于sin值除以cos值了。经过这次的数学探究我不经意间就学习到了 很多东西,比如说我最初不知道什么是周期函数,也更不会使用它,在第一次看到其 一般形式 y=Asin(ωx+φ)+b的时候是完全懵的,但是通过这次任务我不但知道了公式中 每一个符号对于函数图的改变,也知道了周期函数是如何出现在我们的生活中。我之 前不知道角度和弧度的区别,在这次练习里也学会了如何转换它们。这次的探究让我第一次在desmos数学绘图网页上画出了一个真实的小零件🎡 ,颇为开心,当时甚至想直接在后面画几栋楼房,做成我的数学项目作业,但是看在项目已经有了一定的进展 最后还是放弃了。

我觉得这个探究的小组合作也进行的特别顺利。我和kolento有时差,所以是24 小时的工作,我睡觉了他在写,他睡觉了我又起来写了。我也发现和思维不太一样的 人合作很有益处,可以互补来完成产出,就像这次任务里按照我的思路,他找到了公 式,最后我们都得出了答案,也被yif老师称为“抄作业”行为。这些好的点我会希望在 未来的其他探究任务里我可以带着它们继续去探索,我也希望在未来如果自己再碰到 寻找不到表针的线段函数的情况下,可以沉下来在不同的角度去思考和尝试,而不是 选择直接撞上问题和怀疑自己的方法。

总结

这次探究绝对给我留下了深刻的印象。过程很快乐,产出的结果更是让人雀 跃。记得在任务完成的那一周末,我还在自己的网页上简略的记录了探究的过程。希 望有一天,我可以去到伦敦泰晤士河南岸,在伦敦眼上静静地数着每一秒钟,记忆里 desmos上那个红色的小摩天轮,也在转动着,显示着一串串变换的数字…

伦敦眼——Modeling Periodic Phenomena

Modeling Periodic Phenomena

3/16/2020
these days, we are learning circular functions in math class. one of our tasks is to model The London Eye, or the Millennium Wheel, which is a cantilevered observation wheel on the South Bank of the River Thames in London using the periodic phenomena.

we have paired up into groups. Kolento and I were groupmates and we were trying to discover the relationship of any time after a person goes up onto the observation wheel and the height of that position.

the information we have about The London Eye was only that it has a diameter of 120 meters, it cost 30 minutes to make one rotation, and the total height of the wheel is 135 meters.

when the task first started, I was thinking about the animation in the beginning task of circular functions. in that task, there is a graph of a circle and a line like the hand of a clock that rotates on the circle, and the sin and cos values of that line are also shown in the graph like the picture above.

I wondered if we know the hand of a clock line we would be able to find its sin value and by this, we can find the vertical height of the from the hand of the clock to the ground at any time. (see below

but how can we get the hand of the clock?

I asked our math teacher about it. he gave me some inspiration, but I could not get what is it…

Kolento’s Strategy

I was stuck. however, let us jump to Kolento’s work, see how he was doing and how did I steel elements from him in order to accomplish my work!

Kolento was at first thinking to do the task in this way, but I could not understand it haha.

Kolento suggests that we will do the task as my way of thinking. he found the periodic formula in general form for calculating the relationship between any time after you get up onto the observation wheel and the height of that specific time.
after many times of deducing the situation, questioning our teacher, and trying a guessed value, he figured out the equation while I was still in my dream(since we have time difference because he was in China and I was in the US)!

正弦函数在直角坐标系中,给定单位圆,对任意角α,使角α的顶点与原点重合,始边与x轴非负半轴重合,终边与单位圆交于点P(u,v),那么点P的纵坐标v叫做角α的正弦函数,记作v=sinα

——baidu
Kolento’s astonishing work🙀

after I finished appreciating his work, I began to try to understand how he graphed the lines from his half-baked screenshot. I figured out the equation for the hand of the clock! then, I realized that the function that shows the relationships between time and height must be y=the sin value of the hand of a clock + the base height, which is 70 meters.

the equation came up the same in our team: y=60sin(t/5*pi/180+3/2pi)+75

大a功ong告ao成eng

my final work

note: there are many edict versions later on, and more discussion, here is only a task introduction in my point of view. overall, I think it is really cool!