Kicking off the year in my modeling physics course means practicing working with and interpreting linear data. Some students quickly pick up the modeling method of describing slope and intercept, while other students just need more practice for the data to speak to them in the same way.

I use the spaghetti bridge lab to introduce graphing linear relationships and have a pretty good handout^{1} with a few more linear data sets as practice.

However, my experience has taught me two things:

There will be students that just need a bit more practice to really nail down the skill.

It's deceptively hard coming up with linear data sets.

So, I sat down earlier and surfed the internet, found some real data sets, cleaned them up a bit, and imported them into Excel & Google Sheets (from whence they are easily copyable and paste-able). I might as well share them, because I know you'd prefer to avoid converting .txt data files to .xlsx. In fact, I'll make the Google Sheets version editable, so you can add your own awesome data sets).

Download the data sets

The Excel version includes graphs with the equation for best fit lines. Google Sheets doesn't do best-fit lines yet, so those have the graphs (as interpreted by the Google Sheets converter), but no equations.

Google Sheets (editable- so you can add more data sets)

Enjoy!

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from the AMTA Modeling Physics curricular materials, which is why I'm not sharing them here. But, seriously, you should just join the AMTA so you can access the huge wealth of resources they provide. Here's the link. Do it. (back)

As I mentioned earlier, I'm teaching Physics at a new-ish high school this year. I've been spending a large chunk of time designing the curriculum and materials for this class. So far, the year has been a bit hectic (thus the lack of posts here), but the school community is really amazing, supportive, and progressive. A few things that are making what can be a difficult first year much better than average:

Experts at my fingertips & time to develop curriculum. The curriculum people at my new school were very proactive in trying to connect me to experienced physics teachers. I was (and continue to be) impressed with the level of support they're providing for teachers developing new curricula. Unfortunately, none of the teachers had used Modeling Instruction. Fortunately, I've curated a twitter feed that includes 15-20 active modelers and I've found countless helpful resources from those very helpful people. We've also had dedicated time to work on curriculum development. Besides a (paid) week in June, we've also been given time during our professional development time to simply work on building the curriculum. As someone new to the school developing the curriculum for a class that has never before been offered at this school, this has been invaluable.

The willingness to help a n00b. Here's a Venn Diagram showing Teachers using Twitter/Blogs and Teachers willing to help out a poor Modeling Instruction rookie who wasn't able to make it to a Modeling Workshop this summer due to his crazy schedule:
Perhaps this shouldn't be surprising- I mean, if someone is actively spending time writing a blog or sharing via twitter they're more than likely into the whole "sharing" thing. I'm sure I've asked (and will continue to ask) more than my share of dumb questions. Amazingly, despite my frequent questions that surely induce heavy eye-rolling on the other side of the Internet, I've continued to receive an amazing amount of help with zero snark (and zero snark when we're talking about The Tweeter is nothing to shake a stick at!).

The huge resource of online materials. I chose Modeling Instruction as the curriculum for my Physics classes because I believe in the process it supports- not because it's the easiest to design and implement. To be honest, it's a bit scary (especially because I couldn't get to a Modeling Workshop prior to implementation). However, there is no shortage of materials to be found online- and not just general "modeling-instruction-is-great-and-here's-why" materials (there's a lot of that too, though). There are detailed descriptions of labs and their results, handouts, tips for whiteboarding, worksheets, etc., etc., etc.

Here's a partial list:

The American Modeling Teacher's Association. Yes, you need to be a member to access the resources, but the resources are huge. I shelled out the $250 for a lifetime membership. The materials and support I gained access to for that money is easily worth the $250 by itself.

Kelly O'Shea's Model Building Posts & Unit Packets. Kelly's an expert modeler. Her posts really helped me first visualize what a modeling classroom looks like. Her materials are also excellent.

Mark Schober's Modeling Physics. Contains materials and resources for every modeling unit, along with calendars- which was nice as someone new to modeling to get a rough timeline for each unit.

Paying it forward. It's my plan to make the materials I develop and implement for my Physics classes readily available online in some format, at some point. I've gained so much value from the resources others have posted that it is (perhaps with some hubris) my hope that others in the future might gain something from my experience. Obviously I'm no expert- but my hope is that through sharing both the materials and my reflections on how they were implemented will, if nothing else, help me to become a more purposeful and reflective educator.

Perhaps I'm odd, but I really enjoy designing new curricula- which is a lucky break since I'm responsible for designing the Physics curriculum from the ground up. So far it's been a challenge given the specifics of my particular situation (which will undoubtedly be a topic for future post), but as I come to know my students better and gain more experience implementing modeling instruction, I've found the process more and more enjoyable.

Well...to be precise, it's titled "Implementation of a technology-rich self-directed learning environment in a ninth grade Integrated Science classroom." Catchy, I know.

To be honest, this is a bit old. I thought I had posted this a long time ago, but recently realized I never had despite always meaning to do so. I implemented this project in the spring of 2010 and officially submitted my project in June of the same year. It won me a "Scholar of Excellence" award, so it must be at least somewhat decent. 😉

The Goods

Though the full paper may not be of interest to you, let me recommend the Lit Review. I went through many, many papers on constructivist environments and instructional technology's impact on student learning. It'd make me very happy if anybody found this even remotely useful.

I've decided to release it under a Creative Commons Attribution license, so have at it. Here's the full paper in variety of formats for any of your consumption needs:

Implementation of a technology-rich self-directed learning environment in a ninth grade Integrated Science classroom

Simply put, students worked in teams of four to five and shared a team blog. Students investigated any topic that interested them around the general theme of climate change. Students were tasked with researching the topic and sharing their learning and questions on their blog. There were no due dates (other than the end of the school year), though students were all required to write a certain number of posts and comments on their classmates' posts (for more details, check out the Project Design section of the paper). For a bit on the rationale, here's an excerpt from the Introduction and Rationale:

The purpose of the educational system in the United States has been described in many different ways depending on the viewpoint of the individual doing the describing. Creating individuals able to become positive members of society, providing skills for the future workforce, or preparing individuals for an uncertain future have all been cited by various people and organizations as the purpose of schooling- each relying on their own value set and particular social and political biases. While there is no doubt that these various beliefs about the purpose of the American educational system have been true, and may continue to be true in various times and places, it is this author's belief that one of the more important goals of the educational system is to create life-long learners who will be able to actively and knowledgeably engage in whatever ideas and issues may cross their paths. As specific information and skill-sets are quickly changing due to the rapid increases in knowledge and improvements in technology the importance of teaching students specific content information decreases while the importance of teaching students how to locate, evaluate, and interact with knowledge increases. As what it means to be productive members of society or effective members of the workforce changes, the ability for individuals to understand how to learn new knowledge when they need it is more valuable than simply falling back on information learned through formal schooling.

If schools are to become a place where students learn how to interact with, challenge, and develop new knowledge, then the traditional classroom structure- that of the teacher as the primary source of knowledge and assessment- needs to change as well. Students should be given a chance to work out the solutions to problems that do not have predefined answers. In doing so, students lose their status as passive recipients of information and instead become active creators of knowledge. A method of implementing this might be built on the problem-based learning (PBL) model that has been used for many years in many content areas with various age levels. The incarnation of PBL envisioned here provides students with real-world problems to solve that do not already have easy or "neat" answers, gives students the freedom to explore down side canyons as part of the problem solving process, allows time for students to share their ideas and work with others, and provides support and time for students to document and reflect on their learning and problem solving process.

Let me know what you think or if you found anything useful for your own purposes.

This will be the most scientific and precise post regarding Winston Churchill's belly you'll read today. Maybe all week.

Today, we'll be analyzing the following video:

After randomly embedding the preceding video while thinking about Hooke's Law and the spring constant in my last post, what I, and I'm sure you as well, immediately wonder is, of course, "I wonder what type of spring constant Winston Churchill's belly had?" This seems like something worthy of my time.

Here we go!

If we're going to figure this out, we need some data. First, we need some sense of scale. Since I have no idea the how tall the Animaniacs are, let's focus on the historical figures. I'm going to go with Winston Churchill's height to give the video some scale since he's pretty stretched out whilst his belly is being jumped upon^{1}. It's surprisingly hard to find Churchill's height online with any sort of citation. I found what seems like a pretty solid source (via Wikipedia) for the height of Harry S. Truman (1.75 m). Using that information along with the following picture, I can figure out Churchill's height after throwing the image into Tracker:

Churchill and Truman were nearly the same height. I got 1.76 m (5 ft, 9 in) for Churchill. That seems pretty close to most of the unsourced figures for his height I found online.

I think the best way to go about finding the spring constant for Winston Churchill's belly is to use gravitational potential energy and elastic potential energy. If we can find the gravitational potential energy Stalin has at the top of his bounce and the maximum compression of Churchill's belly, we should be able to do the following:

Where m is Stalin's mass, Δy is Stalin's maximum height above Churchill's belly, and x is the maximum compression of Churchill's belly.

I can fairly easily find Δy and x using Tracker to analyze the video.
I used 1.70 m for Churchill's height in the video instead of the 1.76 m figure above since his knees are bent slightly. Using that information to scale the video, Stalin's maximum height (Δy) is 0.65 meters and the maximum compression of Churchill's belly (x) is 0.28 m.

Finding Stalin's mass will require another long and probably fruitless internet search. Instead, I'm going to assume from the above picture Stalin is approximately the same height as Harry S. Truman and then assume Stalin's BMI is slightly above average (he was a dictator- which means he has access to lots of food). I'm going to say Stalin's BMI is 26. According to this BMI calculator, that would give Stalin a weight of 175 lbs, or 79.4 kg.

Now we've precisely (ha.) figured out all our variables, so we can go ahead and solve the equation for the spring constant (k):

OK, so what's that mean? It means that if you could compress Winston Churchill's belly by a full meter it would require 12,900 Newtons of force. On the surface of the Earth, that would take a mass of 1,315 kg (2,900 lbs) sitting on his belly to compress it by a full meter^{2}. WolframAlpha helpfully notes that this is approximately a mass equivalent to approximately 2 "typical dairy cows."

We can also learn something about the Animaniacs' collective mass now that we know the spring constant. If we rearrange the previous equation to solve for the mass, we get:

It looks like the maximum height the Animaniacs attain is 0.77 m with a maximum belly compression of 0.16 m. Now solving for the mass we find:

Collectively the three Animaniacs have a mass of 21.9 kg (48.3 lbs). Wow. They're lighter than I anticipated. If you divide that figure evenly by three, the average Animaniac weight is 16.1 lbs. Clearly Dot and Wakko are smaller than Yakko. This may, in fact, prove Dot's hypothesis that in addition to being cute, she's a cat:

Also, I came across a few places that speculated that Stalin may have use elevator shoes to make himself seem taller, so it might be harder to get an accurate figure for him. However, this isn't exactly going to be a super-accuracy fest anyway, so maybe I shouldn't let that bother me. (back)

I'm not sure if Churchill actually has a meter of stomach to depress, but you get the idea. (back)

In a recent post, I strongly suggested that a physics class should be a place where students are actively involved in the exploration of the relationships that exist between different variables (force and mass, for example)- not a place where students are simply given a list of equations they are told explain how the world works. Let's continue down this line with an example.

Example: Simple Harmonic Motion and Hooke's Law

This is a lab from a college class I took last semester:

This lab isn't terrible. I mean, who doesn't like bouncing springs?

In the first part, we were required to find the spring constant by examining the relationship between the force applied to the spring and the spring's elongation using a graph. That's not too shabby, right? Well...no...but...

What the lab doesn't require is any thinking about the relationship between force and elongation. You make a nice graph, but are told right in the instructions that the slope of the graph is this thing called the "spring constant." We aren't expected to know anything more about how the relationship between force and elongation and the spring constant works.

In part two, we varied the mass on the spring and measured the period of the spring's oscillation, which we then compared to the expected period based upon our calculations and a formula we were given ahead of time:

I didn't need to know much to write up the lab report:

The period of a spring's oscillation depends on the mass attached to the spring.

The formula we were given to find the period of a spring's oscillation works.

That's it. If I was an astute student I might've realized that the slope of a Force-Elongation graph will give you the spring constant- but we were walked through that step in such a way that it would have been easy to miss that tidbit. Never mind understanding what having a larger or smaller spring constant would mean in real life.

Rethinking the lab

So now you're thinking that I'm just a cranky-pants who likes pointing out the failings of other people's labs. Let me try to improve your perception of myself by explaining how I'd like to run a lab covering the same content.

First, I think it's important to identify what I want students to understand as a result of completing this activity. I'd like them to understand:

The nature of the relationship between the force applied to a spring and the spring's elongation.

The slope of a Force-Elongation plot is the "spring constant."

The nature of the relationship between the mass hanging on a spring and the spring's oscillation period.

Second, I want the students to play be the primary investigators. I'm not going to give them a sheet explaining step by step exactly what they have to do. I want the students to handle that part. Maybe I give each group of students a few springs and a set of masses and simply set them free to play around and make observations for 10 minutes or so- after which we discuss as a class observations they have made and decide upon a path for further investigation. Maybe I give some guidance right away and tell them to investigate the relationship between the mass on the spring and the elongation of the spring.

Third, we draw some Force-Elongation graphs. We discuss the relationship between force and spring elongation (it should be pretty obvious it's a direct linear proportionality- i.e., if you double the force on the spring, you double its elongation). So now we know that . Next, we look at the difference in the graphs for each spring. Why are some lines steeper than others? What is the difference between a spring with a steep slope and a spring with a more gradual slope? Then explain the slope on a Force-Elongation graph is called the "spring constant." So now we've figured out that if we know the force acting on a spring and that spring's spring constant, we can figure out how much the spring will stretch: . Hey...that looks an awful lot like Hooke's Law...

Fourth, I'd play this video clip:

Fifth, I'd tell students to investigate the relationship between the amount of mass on a spring and the period of the spring's oscillation. We'd collect data, make some graphs, and hopefully come to the conclusion that .

If we stop here, we've already done a lot. We've discovered Hooke's Law. We understand a stiffer spring has a bigger spring constant. We know how doubling the mass on a spring will affect the spring's oscillation. At this point I could introduce the equation . Maybe we could then do the second part of the lab posted above and see how close the observed periods of the springs match the values calculated with that forumla. We'd probably notice all of our observed periods were off by a little bit. This opens up a discussion of why we all have this systematic error. Why are we all off? What could be off? Looking at the formula, there are really only two places we could have error: the spring constant or the mass. Maybe we draw a free-body diagram for the mass on the spring. At this point a student will probably suggest we need to draw a free-body diagram for the spring as well. Hmm...you know...this spring has mass too...could the mass of the spring itself be affecting the spring's period? Now we've independently figured out we need to consider the spring's mass as well. From there we could figure out a test to determine how much of the spring's mass we need to include.

Overcoming the traditional lab format

If you randomly visited physics courses in high schools and colleges across the nation, you'd most likely see a lot of labs similar to the first lab. Traditionally physics labs have been designed so you're given a formula and are asked to make observations that fit with the formula. This is despite the fact that the student-led investigation requires deeper thinking, encourages greater engagement and thinking about the concepts, a better understanding of how the world works, and an understanding of what an equation actually means.

Why should this be so? I believe it's because traditional labs are easy. Print out a sheet with a step by step procedure. Hand out the supplies. Make some measurements. Maybe make a graph. Answer a couple quick questions. Done. The student-led investigation is tricker to share and explain. The entire process I described in the student-led investigation could be preformed without any worksheets whatsoever. It's harder for teachers looking for a new lab to stumble on a description of this type of lab. It's really easy to hit up The Google and find a lab handout, save it, print it, and pass it out. Student-led investigations also lead to potential student errors. Students may struggle. It may take more class time. Sometimes you'll get data that doesn't turn out as well as you'd like. This can be scary and frustrating for teachers. And yet...

Struggling with what this or that graph is telling us, or being forced to think about where errors came from, or having to defend your results and process requires a lot of thinking. Critical thinking. And helping students learn to think critically is worth the extra time and effort. As a bonus, they'll also actually understand the physics better, which is also a good thing in a physics class. 🙂

These aren't brand new items, as they're things I came across awhile ago and am just getting around to posting now. In addition, I realized that the anniversary of this blog just passed. My first post was published January 12, 2008. As I look back at my first posts, it's clear that I've come a long way (hopefully for the better)- in my location, in my career, and in my thinking. So, in celebration of the 4th anniversary of this blog, let me present you with the following interesting tidbits:

Thanks to the ActiveGrade blog for bringing this to my attention. I don't know how many times I've had discussions with other teachers on the topic of what constitutes fair and effective grading. Often the most heated topic (where I never made any headway) involved the giving out of zeroes for either missing or poorly done classwork. Rick Wormelli gives a great explanation of why grading scales matter- and specifically why zeroes are no good. It's long for YouTube at 8+ minutes, but it's worth it:

The post is ostensibly a take down of Judith Curry's claim's that recent studies and reports on the topic of climate change are "hiding the decline^{1}." However, the real appeal of this post (for me) is how it so effectively describes how science and scientists work. He goes through the data, the uncertainties in measurement, and explains how exactly it is that scientists determine that some effect is real and not just a statistical fluke.

Somewhat related, the Skeptical Science blog (one of the best places to find science-based information about climate science) released The Debunking Handbook a while ago and just recently updated it. The Handbook provides guidelines for communicating about misinformation and gives tips to avoid falling into common pitfalls. In their own words, "The Handbook explores the surprising fact that debunking myths can sometimes reinforce the myth in peoples' minds. Communicators need to be aware of the various backfire effects and how to avoid them..." The handbook is a free PDF download available at their website.

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"Hiding the decline" is the (totally false) idea that climate scientists are tweaking their graphs to make it seem like the Earth is getting warmer, when it really has been cooling the last decade (which it hasn't). Read the full article for more details. (back)

I took three physics classes through a local community college last semester. From how the content was presented in each class, it would be fair to say Physics is primarily concerned with learning a set of equations and then figuring out which equation you need to use in order to find the right answer.

This is not a very useful skill. People wiser than I have pointed out similar things. So why, in high school and introductory college physics classes, do they lean so heavily on "learning the formulas?" Here are the two arguments I've heard the most often:

They'll need it in college/their careers

It could be argued, perhaps, that it is good preparation for students who will be pursuing engineering or scientific careers- after all, they'll be taking college classes and graduate classes and probably use a couple equations during their careers. However, there's a big problem with this line of thinking. Are all the students in a high school physics class there because they're planning on becoming scientists and engineers? A few, maybe. Most of them will not- and that's OK, but this realization should cause us to rethink how we present the material.

The equations explain the relationship between variables

I'm sympathetic towards this line of thinking (more on this later)- but not enough to think it's valid. Whenever I hear this argument the first question that comes to mind is "Is this the best way to explore those relationships?" In my experience, students who have struggled understanding physics often did so because they couldn't make sense of what the equations actually describe. Given an equation and all the variables but one and they'd be able to work though a problem, but they weren't understanding why that answer makes sense and any further obfuscation of the problem quickly threw them off track. I agree that the relationship between variables is an important bit. I don't believe that equations clarify that relationship for the vast majority of students.

How I'd like to teach physics

Understanding the relationship between variables, in my mind, is the key to a useful understanding of physics. If I push twice as hard on this shopping cart, what happens to the cart's acceleration? That's a tangible situation that is easier to understand than simply throwing out F = ma and hoping students figure out that relationship on their own. Further, students should discover these relationships. Give students some equipment and tools and have them measure what happens to an object's acceleration as they apply more or less force on the object (some tracking software would be really handy for this). Then have them apply the same force but change up the mass. Chances are pretty good they'll be able to discover F = ma on their own. Chances are they'll have a much better conceptual understanding of what F = ma means at this point than if you simply gave them the equation and had them do some problems. Or if you simply had them prove the formula is correct in a lab.

Why it matters

I believe the focus on relationships promotes a better conceptual understanding of physics- the students can more effectively internalize the way the world around them works. A populace with a healthy baseline of physics knowledge could prevent silly and potential harmful pseudoscience such as magnet therapy from becoming an issue.

There's been a focus on increasing interest in STEM careers- and a special focus on recruiting women and minorities into STEM fields (see this White House press release). An equation-focused physics curriculum can seem intimidating to students. A collaborative, constructivist approach can be perceived to be less intimidating and more welcoming (I'd recommend giving Episode 32 of the Shifted Learning podcast for some interesting bits on gender issues in STEM).

Modeling Instruction

I don't have much experience with Modeling Instruction, but it seems from my reading that the instruction I've been describing is essentially what it is. As a bonus, it's well developed, well researched, and well used instructional method to improve students' ability to construct a better understanding of the physical world around them.

If you're interested, I've found both Kelly O'Shea's series on model building and Frank Noschese's primer to modeling instruction to be great resources. Check their blogrolls for even more good stuff from teachers using modeling.

As I look forward to potentially teaching physics next year I want students who take my classes to come out with a lasting understanding of the topic. I don't want them to half-heartedly memorize equations that they'll forget two weeks after we finish a unit. I'd like to teach for all the students, not just the future scientists and engineers.

Assessment is extremely important. It explicitly informs students what things we value (and thus the things we value). If we assess the wrong things, students will focus on the wrong things. This can turn an otherwise excellent project into a mediocre project. For this post, I'll share two methods of assessment: First, the "old" method I used when I last taught physics (in 2008). Second, my updated assessment scheme that I'd use if I did this project again.

The old assessment strategy

Embedded below is the document I gave to students at the beginning of the pipe insulation roller coaster project. Most noticeably it includes a description of the assessment scheme I used way back in January of 2008.

As you can see, I split the assessment of this project into two equal parts:

An assessment of the finished roller coaster

I wanted students to think carefully about the design, construction, and "marketing" of their coasters. I wanted them to design coasters that not only met the requirements, but coasters that were beautiful and interesting. Individual items being assessed under this rubric were weighted differently. For example, "Appropriate name of the coaster" was only worth 5%, while "Creativity, originality, and aesthetics" was worth 20%. Here's a link to the sheet I used when assessing this aspect of the coaster project.

An assessment of the physics concepts

In the embedded document above, you can see the breakdown of what items were being assessed. In my last post on pipe insulation roller coasters, you can see how students labeled their coasters with information on the marble's energy, velocity, and such along the track. Groups were required to turn in a sheet with the calculations they performed to arrive at these numbers. These sheets were the primary basis for determining whether students understood the physics concepts.

Problems

There are a lot of problems with the assessment scheme as described above. I'm not going to try to address them all, so here are a couple of the biggest issues:

Assessing coaster design

I'm a fan of elegant design. For this project I'm a fan of finished coasters that look well designed and exciting. That's why I included the first part of the assessment. I wanted to incentivize students to think about the design and construction of their coasters. In retrospect this is probably unnecessary. Students generally came into this project with plenty of intrinsic motivation to make their coaster the best in the history of the class. While I'd still stress the importance of quality design in the future, I'd completely cut this half of the assessment. Students already cared about the design of their coaster. If anything, awarding points for coaster design had an net negative effect. Especially because it doesn't assess anything related to the understanding of physics.

Assessing student understanding of physics concepts

As a normal part of working in a group while attempting to complete a large project in a limited time, students split up the work. Students are generally pretty smart about this in their own way. While I stressed that everyone in the group should contribute equally towards the calculations. Most groups would have the student who had the best understanding of the physics do most of the calculations. Why? Because it was faster. They needed to finish their coaster and just having the fastest person do the calculations meant more time for construction. While I generally knew when students in a group were adding very little to the calculations (and would assess them accordingly), on the whole this method didn't give me a good picture of each individual students' level of understanding. There were certainly students who skated through the project while minimally demonstrating their understanding of the energy and friction concepts involved.

The new assessment strategy

You've probably already picked up on a few of the improvements I'd make for this project.

Use standards-based assessment. Standards-based assessment is an integral part of the classroom throughout the year- not just for projects. If you're unfamiliar with what this "standards-based" business is all about click the little number at the end of this sentence for plenty of links in the footnotes^{1}. Here are a list of standards that would be assessed through this project:

Content standards assessed

Energy

Understand and apply the law of conservation of energy.

Explain and calculate the kinetic energy and potential energy of an object.

Explain and calculate the amount of work done on and by an object.

Solve basic conservation of energy problems involving kinetic energy and potential energy.

Solve conservation of energy problems involving work and thermal energy.

Circular Motion

Solve basic circular motion problems using formulas.

Habits of Mind

Collaborate and communicate with others to meet specific goals.

Handle and overcome hurdles creatively and productively.

The specific standards used can vary based on your specific implementation.

No points for coaster requirements. As I mentioned earlier, it proved unnecessary to incentivize their coaster designs and meeting the basic requirements of the project. This decision also comes out of standards-based grading, which focuses assessment around, "Do you know physics?" instead of "Can you jump through the right hoops?" That isn't to say we don't talk about what makes a coaster "exciting" or "aesthetically pleasing" or whatever. It just means a student needs to demonstrate their understanding of the physics to earn their grade.

A focus on informal assessment. Rather than heavily relying on a sheet of calculations turned in at the end of the project (and probably done lopsidedly by one or two group members) to determine if the group understands the physics, I'd assess their understanding as I walked around the classroom discussing the coasters and their designs with the students as they work on them. Starting with questions like, "Why did you make that loop smaller?," or "Where are you having trouble staying within the requirements?" can be used to probe into student thinking and understanding. The final calculations would still be a part of the assessment, but no longer the single key piece of information in the assessment.

On the whole I was very happy with this project as I used it in the past. As I've learned and grown as a teacher I've found several ways I can tweak the old project to keep up with the type of student learning I want to support in my classroom. If you have other suggestions for improvement, I'd be happy to hear them.

As a bonus, here's a student produced video of the roller coaster project made for the daily announcements. The video was made by a student who wasn't in the physics class, so there's a little more emphasis on the destruction of the roller coasters at the end of the project than I'd like. Kids. What can ya do?

I like projects. I really liked this project. The pipe insulation roller coaster project is one of the most enjoyable projects I've ever used in class.

History

It was my second year teaching physics. During the unit on energy, the book we were using frequently used roller coasters in their problems. We even had a little "roller coaster" to use with photo gates. I thought we could do better.

My original idea was to get some flexible Hot Wheels tracks and make some loop-de-loops and hills. Turns out a class set of Hot Wheels track is pretty expensive. On an unrelated yet serendipitous visit to my local big box hardware store, I ran across the perfect (and cheap!) substitute: Pipe Insulation!. For $1.30 or so you can get six feet of pipe insulation- which doubles nicely as a marble track^{1} when you split the pipe insulation into two equal halves. It's really easy to cut pipe insulation with a sharp pair of scissors. Just be sure you don't buy the "self-sealing" pipe insulation, which has glue pre-applied- it's more expensive and it'd turn into a sticky mess.

At first I planning to simply design a one-period long investigation using the pipe insulation (my original ideas morphed into the pre-activity for this project). As I started to think through the project more and more, I realized we could go way bigger. And thus, the pipe insulation roller coaster project was born.

Building the Coasters

In groups of three, students were given 24 feet of pipe insulation (4 pieces), a roll of duct tape^{2}, and access to a large pile of cardboard boxes^{3}. All groups had to adhere to a few standard requirements:

Construction requirements

The entire roller coaster must fit within a 1.0m x 2.0m rectangle^{4}.

There must be at least two inversions (loops, corkscrews, etc.).

All 24 feet of pipe insulation must be used.

The track must end 50 cm above the ground.

Physics requirements
In addition to meeting the above requirements, students were required to utilize their understanding of the work-energy theorem, circular motion, and friction to do the following:

Determine the average rolling friction, kinetic energy, and potential energy at 8 locations on their roller coaster.

Determine the minimum velocities required for the marble to stay on the track at the top of all the inversions

Determine the g-forces the marble experiences through the inversions and at least five additional corners, hills, or valleys.

The g-forces must be kept at "safe" levels^{5}.

Calculations

Rolling friction, kinetic energy, and potential energy

The potential energy () is easy enough to find after measuring the height of the track and finding the mass of the marble. The kinetic energy is trickier and can be done by filming the marble and doing some analysis with Tracker, but since the speed of the marble is likely to be a little too fast for most cameras to pick up clearly, it's probably easier (and much faster) to simply measure the time it takes the marble travel a certain length of track. I describe how this can be done in a previous post, so check that out for more info. That post also includes how to calculated the coefficient of friction by finding how much work was done on the marble due to friction- so I'll keep things shorter here by not re-explaining that process.

Pro-tip: Have students mark every 10 cm or so on their track before they start putting together their coasters (note the tape marks in this pic). Since d in in this case is the length of track the marble has rolled so far, it makes finding the value for d much easier than trying to measure a twisting, looping roller coaster track.

Minimum marble velocities through the inversions.

This is also called the critical velocity. That's fitting. If you're riding a roller coaster it's pretty critical that you make it around each loop. Also, you might be in critical condition if you don't. While falling to our death would be exciting, it also limits the ability to ride roller coasters in the future (and I like roller coasters). Since we're primarily concerned with what is happening to the marble at the top of the loop, here's a diagram of the vertical forces on the marble at the very top of the loop:

So just normal force (the track pushing on the marble) and gravitational force (the earth pulling on the marble). Since these forces are both acting towards the center of the loop together they're equal to the radial force:

When the marble is just barely making it around the loop (at the critical velocity), the normal force goes to zero. That is, the track stops pushing on the marble for just an instant at the top of the loop. If the normal force stays zero for any longer than that it means the marble is in free fall, and that's just not safe. So:

Then when you substitute in masses and accelerations for the forces and do some rearranging:

There you go. All you need to know is the radius of the loop, and that's easy enough to measure. Of course, you'd want a little cushion above the critical velocity, especially because we're ignoring the friction that is constantly slowing down the marble as it makes its way down the track.

Finding g-forces

An exciting roller coaster will make you weightless and in the next instant squish you into your seat. A really bad roller coaster squishes you until you pass out. This is awesomely known as G-LOC (G-force Induced Loss of Consciousness). With the proper training and gear, fighter pilots can make it to about 9g's before G-LOC. Mere mortals like myself usually experience G-LOC between 4 and 6g's.

As I mentioned, I set the limit for pipe insulation roller coasters at 30g's simply because it allowed more creative and exciting coaster designs. While this would kill most humans, it turns out marbles have a very high tolerance before reaching G-LOC.

To find the g-forces being pulled on corners, loops, or hills you just need to find the radial acceleration (keeping in mind that 1g = 9.8 m/s^2):

Raise the stakes

Students become fiercely proud of their roller coasters. They'll name them. Brag about them. Drag their friends in during lunch to show them off. Seeing this, I had students show off their creations to any teachers, parents, or administrators that I was able to cajole into stopping by for the official testing of the coasters. I even made up a fun little rubric (.doc file) for any observers to fill out for each coaster. This introduces some level of competition into the project, which gives me pause- though from day one students generally start some friendly smack talk about how their coaster is akin to the Millenium Force while all other coasters are more like the Woodstock Express. The students love to show off their coasters, and it seems the people being shown enjoy the experience as well.

Assessment

Assessment is massively important. However, this post is already long. The exciting conclusion of this post will feature the assessment piece in: Part 2: Pipe Insulation Roller Coaster Assessment.

The first day we played with pipe insulation in class I had students use some marble-sized steel balls. Unfortunately because the steel balls are so much heavier and the pipe insulation is spongy and flexible, there was just too much friction. When we switched to marbles the next day everything worked like a charm. (back)

Most groups typically use more than one roll of duct tape. My first couple years I bought the colored duct tape and gave each group a different color. That was a nice touch, but also a bit more expensive than using the standard silver. Whatever you decide, I highly recommend avoiding the cut-rate duct tape. The cheap stuff just didn't stick as well which caused students to waste a lot of time fixing places where the duct tape fell and in the end used a lot more duct tape. (back)

I had an arrangement with our school's kitchen manager to set broken down boxes aside for me for a few weeks before we started the project. If that's not an option, I've also found if you talk to a manager of a local grocery store they're usually more than willing to donate boxes. (back)

I made it a requirement for groups to start by building a cardboard rectangle with the maximum dimensions. This served two functions: (1) It made it easy for the groups to see what space they had to work with, and (2) it allows the roller coasters to be moved around a little by sliding them across the floor. (back)

Originally I wanted students to keep g-forces below 10. Very quickly it became apparent that under 10g's was overly restrictive and I upped it to 30g's. That's not really safe for living creatures, but it would certainly make it more "exciting." (back)

[Update: See the bottom of the post for a quick update thanks to some issues pointed out by Nicolas Marmet in the comments.]

[Update 2 (1/26/2014): I added graphs for the cost per hour instead of the cost per day assuming 3 hours of use per day. I s'pose it would've made sense to just start with these graphs. Oh well.]

In the 1960s Walter Mischel performed studies involving preschoolers and marshmallows. The "Marshmallow Experiment" involved sitting kids in a bare room and setting a marshmallow^{1} in front of them, then telling the preschoolers they could either eat the marshmallow now or wait 15 minutes. If they successfully waited 15 minutes then they'd get a second marshmallow to enjoy in addition to the first.

In addition to telling us something about deferred gratification, it's also immensely fun to watch preschoolers in agony attempting to defer their gratification:

This past week I needed to pick up two new lightbulbs for our oven hood. I noticed our local big box home supply store had a fancy new LED bulb that would fit into the outlet. On the downside the LED cost $32.98 compared to $7.98 for the same type of halogen bulb that just blew out. Wow. $32.98 feels really expensive for a lightbulb. I decided to get one of each then do a little cost analysis when I got home.

Here are the vitals for each bulb (according the packaging):

Halogen Bulb

LED Bulb

Power

50 Watts

7 Watts

Lifetime

2,500 hours

25,000 hours

Let me assure you our old halogen bulbs didn't last nearly as long as 2,500 hours. I'd roughly estimate those bulbs are on an average of 3 hours a day. The halogen bulbs should have lasted about 2.25 years at that usage. This is the second time I've had to replace these bulbs and we've only been in our house 3 years. Does this mean the LED bulb will have a shortened lifespan as well? I don't know. I'll cut the bulb makers some slack this post and assume their numbers for the lifetime of the bulbs is accurate.

How long before the LED pays for itself?

The LED bulb uses about one-seventh less power and has 10 times the lifespan of the halogen bulb. It seems pretty clear that at some point it'll eventually pay for itself. But how long will that be? Days? Years? Decades?

I hunted down a bill from the electric company and added up all the government surcharges, distribution rates, and so on. Hmm...it seems like they should print the total rate you actually pay instead of only listing the seven different surcharges individually. It'd certainly make it quicker to see what you're paying. Anyway, I pay $0.15373 per kilowatt-hour. What does that mean? It means that if I leave a 1000 watt lightbulb on for 1 hour, it'd cost me $0.15373. Knowing that, I can figure out how much it costs me to keep the LED and Halogen lights on for 1 hour:

That's nice to know, but the time has come to make a chart:

You might be wondering why the data points jump every so often on the graph. Let me explain: The halogen bulb line jumps $7.98 after every 2,500 hours of use. Why? That's the bulb burning out and me running out to the store buying a new bulb for $7.98. You'll notice the LED cost jumps $32.98 after 25,000 hours of use. Same deal.

I added in best-fit^{2} lines and had Excel whip up the equations for those lines. The trendlines' slope is the cost per day (assuming the bulbs are on 3 hours a day). I'm really interested in the intersection of the two trendlines. It's at that point where the LED bulb is actually saving me money. We can see they cross just before 1000 days. We can do better than that. If I subtract the two equations from each other, I should get an equation that gives the difference in cost between the two bulbs:

What does this tell us? Well, it says the difference in cost between the two bulbs in $25.00, and for each day of use (x), the LED is $0.0277 cheaper to operate than the halogen bulb. So how many days until the difference in cost between the LED an halogen is $0.00?

902.5 days
or 2 years, 5 months, and 20 days

.

Deferred gratification

LEDs save money. They're more efficient. They last longer. But...paying $32.98 for one lightbulb and then waiting nearly two and a half years before it pays itself off can be nearly as painful as a 4 year old waiting 15 minutes for a second marshmallow. In both situations the end result is desirable- but it involves subduing that part of your brain that says, "Mmmm...marshmallow...so tasty...must. eat. it. now." or "HOLY #$*&@! $33.00 for ONE lightbulb!" You just have to keep telling yourself that second marshmallow's on its way and that after 25,000 hours that LED bulb will have saved you over $200.

A related student activity

It seems that many sustainable technologies (LED bulbs, electric cars, photovoltaic cells) require more money up front. Over time, just like the LED bulb, they generally pay for themselves. Have students investigate whether these extra costs pan out by having them pick a sustainable technology (like buying an electric car) and by doing some research into, for example, their family's driving habits. After a few calculations they can determine how long it would take for the electric car to pay for itself compared to an equivalent gas-powered car. Open up the topic to students- let them pick topics that interest them. Have them do additional research into the costs and benefits beyond just dollars and cents. Encourage them to interview people who have already implemented these changes. Most of all, let them take the time to puzzle over how to figure this stuff out. Be there to help and guide them, but please, please, don't just give them worksheet that has them plug in some numbers to get some type of answer.

UPDATE

Nicolas Marmet (in the comments) pointed out a few things that I think are worth addressing, primarily the issue with not continuing the data in the chart above for several lifetimes of the LED bulb. I didn't think there would be much effect, but just because I was curious, I made a new chart (below) that goes through 11 lifetimes of the LED bulb:
As you can see, the slope of the best-fit line jumps from 0.0042 dollars/day to 0.0070 dollars/day. While that's still significantly cheaper than the Halogen bulb (at 0.0319 dollars/day), it's different enough that some new calculations are in order. After updating the Savings equation (from above), the new equation looks like this:

This tells us that it'll take an additional 102 days than the calculation above until the LED is the cheaper choice. It'll take 1004 days, or 2 years and 9 months until the break even point. Not too much longer, but still longer.

Also, a quick run through Nicolas Marmet's other points:

As far as the difference in lumens between the two bulbs, I no longer have the boxes pictured above, and can't make out any lumens listed on the boxes in the picture above. Qualitatively looking at the lights side-by-side in my house, I don't notice much difference in brightness between the two, and I'd pick the LED as the brighter bulb- though perhaps that's a trick due to the whiter quality of its light.

I can't say much about the premature failure of LED-bulbs other than to point out that my LED bulb hasn't failed yet- though as of this update it's only been 327 days since being installed, which assuming 3 hours per day of use, is only 3.9% of the bulb's 25,000 hour lifespan, and I'm still 2 years, 10 months, and 8 days away from the break-even point. I'll update this post again once either bulb has failed.

LED buzzing: The LED I installed doesn't buzz at all. It has made no noises at all that I've been able to perceive. My cats don't act weird (or I guess I should say, weirder than normal) when it's on, so that's not an issue.

While there's certainly a lot of time left where I could be proven wrong, it does still seem over the long-term the LED bulb is a smarter investment, though you're probably not going to be able to retire early from the savings gained from using LEDs.

UPDATE 2 (1/26/2014)

Here are two views of the same graphs using Cost per Hour instead of Cost per Day of 3/hr use:

Below I zoomed in to only include the costs up to 3200 hours so the break even point becomes more clear.

According to the Wikipedia, Mischel let kids choose whether they wanted a marshmallow, Oreo cookie, or pretzel stick. I guess the "Marshmallow, Oreo, or Pretzel Experiment" doesn't quite roll of the tongue as nicely. (back)

I set the y-intercept for the halogen and LED bulbs at 7.98 and 32.98, respectively. Since you're paying that money up front, it follows that at time zero (when you've purchased the bulbs but haven't used them yet) you're already out the cost of the bulb. (back)