Stimulating Interest In Science By Simulating Labs

Boris DirnbachHSCAPA

Three Simulations

My RET lab experience has consisted of collaborating with Profs. Youngmoo Kim & Gail Rosen, Electrical Engineering, on several HS simulation labs they plan to offer as after school enrichment next year to students at CAPA:

Three of the labs are sketched out below– 1) What’s Up with Will Smith?– 2) The Sound of Music.– 3) Let’s Take a Random Walk.

(1) Fighting Corruption, or What’s Up With Will Smith?

Smith’s picture at left has been corrupted by adding “noise.”

How do we restore Will back to his old self?The students will answer this question using the “Matlab” program to analyze Will’s image pixel by pixel using two mathematical procedures to change each pixel’s value. Student’s will analyze the resulting images and evaluate the effectiveness of the two methods.

Standards Addressed

The lab addresses a number of PA Science and Technology State Standards, for example for Standard 3.1 (Unifying Themes) students use the concept of a model and “apply mathematical models to science and technology.” (Grade 10, p. 8) Additionally, in Standard 3.6 (Technology Education) students “apply knowledge of information technologies of encoding, transmitting, storing and retrieving and decoding, etc. (Grade 10, p. 23)

How will the restore Will?

They will learn that using two simple statistical concepts, taking the mean and finding the median, can help correct defective pixels. The center pixel’s value can be altered by either replacing it with the mean or the median of the set of nine numbers.

(2) The Sound of Music

Say a tuning fork, a saxophone, and trumpet are all playing the note A (440 Hz or 440 cycles/sec) at the same loudness (and this will be played). The sounds are different because of timbre. By varying up to seven sine curves and their amplitudes and superposing them, students can experiment with producing sounds more complex than that of a simple tuning fork.

An example of how sounds can combine (superpose).

A fundamental note and selected harmonics can produce a special “square” wave.

Standards Addressed

The lab addresses some PA Academic Standards for Science and Technology, for example, for Standard 3.4 (Physical Science, Chemistry and Physics); students "describe sound effects (e.g. Doppler Effect, amplitude, frequency, reflection, refraction, absorption, sonar, seismic)." (Grade 10, p. 16) For the Academic Standards for Mathematics, the lab addresses Standard 2.8, Algebra and Functions, because students will "use patterns, sequences, and series to solve routine and non-routine problems." (2.8.11C - p. 13) and they will "represent functional relationships in tables, charts and graphs." (2.8.11Q - p. 15)

(3) Let’s Take a Random Walk

What happens over time if you flip a coin over and over, walking one foot to the left when heads comes up, and one foot to the right on tails? On average, where will you be? And how is this related to the perpetual random motion of a molecule in either a liquid or gas? The (Matlab) enrichment lab to be conducted after school will follow labs that I propose for my students as part of the regular curriculum.

One dimensional walk: (up is right, down is left -->)

Two dimensional Walk:

Examples of Random Walks

Standards Addressed

The PA Academic Standards for Science and Technology briefly mentions the objective in Standard 3.4.10A: [Students will acquire the knowledge and skills needed to] . . ."Describe phases of matter according to the Kinetic Molecular Theory." (p. 15) Additionally, The National Council Of Teachers of Mathematics (NCTM) recommends that students intending to go to college "become familiar with the normal distribution …" (see Standards for School Mathematics, grade 9-12, Standard 10, p. 169) These math goals are seconded by the PA Academic Standards for Mathematics which has an eighth grade standard, “E. Analyze and display data in stem-and-leaf and box-and-whisker plots.” (2.6.8E) and an eleventh grade standard, “ … Describe the data as an example of a distribution using statistical measures of center and spread. Organize and represent the results with graphs…” (2.6.11A)

A Normal (Gaussian) Distribution

The kinetic energies and speeds of molecules in liquids, and their displacements after collisions,

takes the pattern of a normal distribution.

Students Will Flip a Coin & Walk

First students will take a one-dimension random walk in the classroom. Heads they go one floor tile to the right, tails they go left, for (say) 100 trials - for as many times as they can manage in one period. The students will summarize the class results with a stem-and-leaf plot and graph the results using a histogram. Ditto for a two- dimension walk.

What Can we Expect?

Surprisingly, the results cluster around the prediction that, on average over many trials, the distance after N steps = √(N) * length of the step. For 100 steps at a length of 1 foot, the class results should cluster around a distance of 10, and we can see whether we get the beginnings of a normal distribution around a mean of 10.

Einstein (in 1905) was the first to analyze Brownian Motion mathematically showing that root-mean-square distance (for us equivalent to the average distance) varies as √(Time). Feynman (Lectures on Physics, v. I, ch. 41) argues from a diagram like the one at right. You can seethat vector R2 = R3 + L, R3 = R2 + L, andin general Rn = R(n-1) + L. Taking“Dot Products” of R1 with itself, and R2with itself, N times, yields N*L2 plusmany terms that look like R4*L*cos(angle).Summing up all the cosine terms ultimatelygoes to zero because the angles will varyfrom 0 -> 360, and the cosine terms will randomlybe positive and negative, canceling out when added.Given that the number of steps varies with time, then in D = √N * Lmeans that D varies as √(Time), Einstein’s result from 1905.

What Justifies the Distance Formula?

We Will Simulate These Walks in Class with the TI-83

Sample TI-83 Program (Simulating 100 steps takes about 14 seconds): Program:WALK :ClrHome :Disp “RND WALK IN 2D” :INPUT “NO STEPS=”,S :0->X:0->Y :For (n,1,S) :X+randNorm(1,.333)*cos(360rand)->X :Y+randNorm(1,.333)*sin(360rand)->Y :END:Fix3 :Disp “X=”,X :Disp “Y=”,Y :PAUSE :Disp “Dist=”, √(X2 + Y2) :Disp “Prediction=”,√(S)

The Drexel Connection

The final phase of this series of lessons will involve the Matlab simulation program as an enrichment lab coordinated by a Drexel University graduate student.

As an example, students can simulate a walk of 1000 steps, and run this 1000 times and collect the distance data for all these runs. Does the data show a distribution of distances around the √(N)*L prediction? Students can vary the step length L to see whether Feynman’s argument holds for other situations.

Assessment

One assignment will be an embedded assessment; the students will simulate a random walk at home using graph paper. Once all student data is collected from all classes, then students will create a stem-and-leaf plot, a histogram, assess whether there is a normal distribution, and check on the prediction of the average distance.

Additional Assessments

Can students analyze data which does not show randomness, but exhibits a bias instead? In general, the average distance traveled in such a situation will be different than the Feynman prediction. I would like to assess students on data that does not show a Normal Distribution to see how they evaluate the situation, but this still needs to be worked out.

Don’t have Matlab or aTI-83 handy? Try theSimulation below.

A 2-D simulation from Boston University’s Center for Polymer Studies:

http://polymer.bu.edu/java/java/2drw/RandWalk2D.html

Brownian Motion under a microscope.

http://video.google.com/videoplay?docid=-485060005033050010&q=brownian+motion&total=22&start=0&num=10&so=0&type=search&plindex=5

That’s all for now!