## Linear data sets (for your enjoyment)

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 handout1 with a few more linear data sets as practice.

However, my experience has taught me two things:

1. There will be students that just need a bit more practice to really nail down the skill.
2. 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).

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.

Enjoy!

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1. 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)

## My first weeks as a Modeling Physics teacher

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:

1. 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.
2. 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!).
3. 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.
• Todd K's DHS Physics Site. Even more modeling materials and calendars.
4. 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.

## The spring constant of Winston Churchill's belly

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 upon1. 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:

$mg\Delta y = \frac{1}{2}kx^2 \\ \\ k = \dfrac{2mg\Delta y}{x^2}$

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):

$k = \dfrac{2mg\Delta y}{x^2} \\ \\ \\ k = \dfrac{2(79.4\text{ kg})(9.8\text{ m/s}^2)(0.65\text{ m})}{(0.28\text{ m})^2} \\ \\ \\ k = 12,900\text{ N/m}$

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 meter2. 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:

$m = \dfrac{kx^2}{2g\Delta y}$

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:
$m = \dfrac{(12900\text{ N/m})(0.16\text{ m})^2}{2(9.8\text{ m/s}^2)(0.77\text{ m})} \\ \\ \\ m = 21.9\text{ kg}$

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:

Watch animaniacs - what are we? in Animation  |  View More Free Videos Online at Veoh.com

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1. 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)
2. I'm not sure if Churchill actually has a meter of stomach to depress, but you get the idea. (back)

## Worksheet labs aren't that great: Hooke's Law

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:

#### Analysis

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:

$T = 2 \pi \sqrt{ \dfrac{m}{k}}$

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

1. The period of a spring's oscillation depends on the mass attached to the spring.
2. 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:

1. The nature of the relationship between the force applied to a spring and the spring's elongation.
2. The slope of a Force-Elongation plot is the "spring constant."
3. 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 $F \propto x$. 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: $F=kx$. 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 $T \propto \sqrt{m}$.

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 $T = 2 \pi \sqrt{ \frac{m}{k}}$. 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. 🙂

## What is the purpose of Physics class?

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

1. 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.
2. 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.

## Pipe Insulation Roller Coaster Assessment

Welcome back. If you haven't joined us for the last two posts, let me recommend that you first read about determining rolling friction on the coaster and the project overview.

On to the assessment...

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.

1. 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 footnotes1. 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.

2. 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.
3. 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?

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1. Here are posts I've written about my experience implementing standards-based assessment. I'm not an expert, so let me also direct you my bookmarks related to standards-based grading, and some resources written by a couple people who are more expert: Shawn Cornally and Frank Noschese (who offers blog posts, a shared google doc foler, and a collection of bookmarked links). There are certainly other great resources out there, but these are a great starting point. (back)

## Pipe Insulation Roller Coasters

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.