Episode 403 - “Velocity” airdate: 10/12/07
This episode opens with the dramatic crash of a car slamming into a busy coffee shop. The apparent accident turns suspicious as the FBI team discovers evidence suggesting foul play. Charlie and our new character, Ray Galuski, take on the task of developing a computer simulation to reconstruct the crash scene. Charlie’s father finds it peculiar that Charlie and Ray’s simulation does not include a slew of equations plastered all over the garage. Enter the first “CharlieVision…”
| The Vision… | Charlie says… |
| An apple tree | “Newton observed a real world event…” |
| The apple falls | “…and created a mathematical model to represent the phenomenon. That model could be extrapolated to predict the orbits of the planets, even the general structure and movement of the galaxies.” |
| The apple freezes in mid-air and Newton’s equations of falling bodies and his Law of Universal Gravitation appear. |
The concept of finding mathematical models can be found throughout the high school mathematics curriculum. What immediately pops into my mind is the idea of fitting data with mathematical models and using these models as a way of quantifying a real world situation. Here are some activities (and there are many, many more) that were developed for earlier episodes that you might want to check out:
- Episode 109, “Sniper Zero”
Exponential Modeling looks at the rate of growth of cell phone subscriptions over time, with a particular focus on how the values of a and b impact the graph of y=ab2. - Episode 202, “Judgment Day”
How Tall is the Criminal explores the linear relationship between femur length and height. - Episode 224, “Hot Shot”
Parabolic Food Fight investigates finding a quadratic equation to model the path of a grape in flight.
Sometimes we perform simulations and gather experimental evidence to support a claim, and then use our conclusions to derive a theoretical result, much like Newton’s Law of Universal Gravitation. Simulations also allow us to “see” a result that may otherwise escape us.
Take for example this classic problem posed by the Ukrainian theoretical physicist George Gamow in 1947 that describes the location of a buried treasure:
From the palm tree, walk directly to the Falcon-shaped rock and count your paces to the rock. Turn a quarter circle to the right and go the same number of paces, and put a stick in the ground. Return to the palm tree and walk directly to the Eagle-shaped rock again counting your paces to the rock. Turn a quarter circle to the left and go the same number of paces, and put another stick in the ground. Connect the two sticks with a rope and dig in the middle.
Upon landing on the island, you discover the rocks, but the palm tree is no longer there. Where should you dig to find this treasure?
Using dynamic geometry software, students can reconstruct this situation – or essentially simulate the location of the buried treasure described in the paragraphs above, much like the way Charlie and Ray simulated the crash scene. Students will see very quickly that the location of the tree has nothing to do with the location of the buried treasure!
Not only does the use of dynamic geometry software help produce an answer, the construction itself is the first step towards proving that T is the midpoint of S1T and TS2.
There is a very comprehensive article about this very problem (titled “Dynamic Visualization and Proof: A New Approach to a Classic Problem”) in the September 2003 issue of Mathematics Teacher. Download this article for a small fee.
Later in this episode, Charlie is found analyzing the forces acting on the car as it turned. Enter Charlie’s second audience vision:
| The Vision… | Charlie says… |
| The wheels of a car making a turn | “When the car turns, it’s on a circular path. The further from the center of the circle, the greater the distance traveled and velocity…” |
| A carousel ride | “…like on a carousel. A horse on the outside travels farther than the one on the inside. So it must go faster. Same for the drive wheels on a car. The outside wheels travel further - to keep up with the inside tires, they have to spin faster.” |
This rather simple concept is often difficult for students to grasp. It also offers an opportunity to talk about two different velocities: Angular velocity and linear velocity. Provide students with a simple example where they must calculate the linear velocity of two points, A and B, that lie on concentric circles and rotate at a fixed angular velocity. You may also want have students consider expressing the angular velocity in revolutions per second, radians per second, and degrees per second.
The Sketchpad construction shown above gives a top view perspective of a car (yes, that’s a car) turning on a circle. By animating point A, the car will travel around the circle and students can measure the length of time it takes for the car to make one full revolution. Using this information, as well as knowing the lengths of OB and OA, students can see that the linear velocity of points A and B are different. As I’m writing this I’m thinking of an extension idea – What is the angular velocity of the wheels located points A and B? What additional information do you need to answer this question? Download The Geometer’s Sketchpad® file.
The issue of maintaining constant angular velocity versus constant linear velocity with regard to rotating objects presents an interesting challenge for the sound recording industry. For example, traditional vinyl records operated at a constant angular velocity (such as 331/3 rpm), whereas audio CDs operate at a constant linear velocity (which allows for a constant rate of data transfer). See a Wikipedia article that provides some additional information on this topic. I wonder if traditional cassette tapes, or reel to reel tapes for that matter, are also impacted by the angular versus linear velocity dilemma? Am I dating myself by bringing up these questions?
While I’m still on the topic of circles, check out Cycloid 1, an activity based on Season 3 episode “The Mole”. In the activity, students study cycloids by examining the movement of a point on the wheel of a bicycle as the bicycle travels in a straight line.
Here’s a TI-Nspire™ document and Microsoft Word® file that provides a dynamic representation of linear velocity, cycloids, and circles rolling on other circles (called Hypocycloids and Epicycloids), with supporting documentation and activity ideas to accompany the TI-Nspire file. Download a free 30-day trial of the TI-Nspire software here. Many thanks to Chuck Vonder Embse and Dennis St. John for their contributions to this blog!
Mathematically yours,
Steve Ouellette, Westwood High School, Westwood, MA
October 20th, 2007 at 1:08 am
I don’t want to sound pessimistic or ingrateful, but are there going to be any changes to the format for the activities? The activities lag behind the episodes. They refer to previous episodes that the students may or may not have seen. It seems like there has been no response to all of the math teachers (like myself) who have written only to say, “Keep the blogs, but give us back the PDFs.” I can make my own connections to previous activities. Plus, I (& other teachers, I’d think) don’t have computers for students to access on a regular basis.
Part of what made WAUMED a success was it’s ease for teachers & it’s real life examples for students. I’ve read the blogs so far, & (unfortunately) many have been negative to the new format. Is there a better way to express our concerns? Maybe, but there seems to be no direct links to the bloggers. Instead, we write responses hoping someone will listen.
If I wanted to feel like I wasn’t being listened to, I’d hang out w/ some of my sophomore classes. But if those of us who (used to) routinely check this website don’t feel like we’re being heard & can’t use the activities, then we will stop visiting altogether.
October 21st, 2007 at 6:50 pm
I must agree with Jkonow, I don’t care for this new method of getting my lessons. The old way was much easier and faster. I didn’t have to go through all these blogs and blah, blah, blah. I aplologize, but I don’t have that kind of time each week to read all of this, just get me the information so I am ready to go when the kids are. Sorry to be so blunt. I hope we can get a better forum for getting information. I am not a blogger, and don’t like to waste time doing, or reading it. Some of us out in the real world actually teach all day, and take Grad classes at night. Not to mention our families. Just make it easy to get to the info so we can use it in the classroom. Thanks
October 25th, 2007 at 7:15 pm
Your passion for the activities is appreciated. There are many reasons for the change from having weekly activites created before the show to blogging after the show, so I apologize for the longish post. The last two years the activity writing teams recieved a synopsis of each show’s math approx three weeks before it aired. That synopsis — which was well-written and detailed, focused only on the math and didn’t include any creative direction as to which math is the “hero” math that saves the day, and which math is “other.” This resulted in lots of time and effort being put toward creating activities around math the ended up being cut from the final show. Not a good use of anyone’s time and frustrating when such good work is put forth. Also, for anyone who’s watched the series for a couple of seasons, many math concepts are revisited. And while WAUMED had some of the most creative and dynamic math teacher activity writers available, it was a challenge even to them to create the Nth compelling activity for the same concept. (Particulary when they had already created other well-done activities that could work as well.) And do it in the 3 week timespan (more frequently, shorter.) And speaking of that timespan…the planned length of time from synopsis to print-ready was 3 weeks. Sometimes (often) it was 2.5. At times there was great, great stress. Not the environment anyone wants to promote. Finally, none of the feedback indicated teachers were using the activities prior to the show airing on Friday nights. We talked to people who came to the WAUMED booth at the NCTM regionals and national conference…to those who stood line for autographs when Dylan was at NCSM and surveyed the thousands who received the weekly emails (36 responded). People knew of the show, and some had heard of the WAUMED program, but most who went to the website said they printed out activities to use when it was appropriate for their class. This, of course, made sense — the math used in each week’s episode wasn’t correlated with any math curriculum. So having the activities before the show aired didn’t appear as important as previously thought.
So given all of this we detemined that it was important to continue providing support for those who were fans of using NUMB3RS as a teaching tool but try something different that could address the problems noted above. We thought a blog would offer a good option — hopefully providing a more conversational way to talk about each show and how it can be incorporated into the classroom AND use more of the 200 or so activities already created for previous seasons. Before, the writers could only imagine how the math was used in the show…but after they see the show and hear the script it’s much easier to know the context. And finally, since there didn’t seem to be a strong need for having anything appear before the show aired, this could help reduce the stress of the 3-often-less- week turnaround. And we wanted to provide a venue for educators who were fans of the show and WAUMED to share with one another how each one of you uses this to promote math.
Does having the background above help here? What has the WAUMED team not thought of that could help in the future? In the meantime, thank you for trying out this new format…
October 29th, 2007 at 7:33 am
I appreciate the difficulty of having activities posted before the episodes. I don’t really care if they are not posted until several weeks after the episode as I tape the episodes and use the clips when it fits into my curriculum. I would still like to see you go back to the style of posting activities that was used in seasons 1-3.
October 29th, 2007 at 11:05 am
tjelinek, pls help me understand. What do you mean by “the style of posting activititivies that were used in seasons 1-3?”
If you click on the linked activities (highlighted) in each blog, it takes you to the same format/activities from those seasons. Is it the format on the blog or the format of the activities themselves?
Also, all the activities from season 1-3 are still on the site, listed under the “activities tab.”
Let me know what your envisioning…
many thanks!
November 7th, 2007 at 10:17 am
I’m not sure what tjelinek means about the old format but I’d like to be able to click on season 4 from the activities menu and then see what activities are available for the episodes from season 4.
November 9th, 2007 at 7:12 pm
tomlinson, who are you? I’m sorry but I cannot find your name or bio on this site, and when I searched the site for “tomlinson” the response was “no posts found.”
November 27th, 2007 at 3:37 pm
It’s been a while since I’ve posted, since (for some reason) I’ve been locked out. But come on! Everyone who has commented a preference has said PDFs are preferable. Please read what others have to say.
April 29th, 2008 at 12:49 am
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