What Can You Do With This: Annuli
June 14th, 2010 by Dan Meyer
I don't foresee any slack to these features when I'm in grad school. Math is just too fascinating; problem solving is just too fun.
What fascinating math can you find in these scans? What fun problems could we solve here? Are these multimedia in any way superior to the annulus problems in your Geometry textbook? Are they just shinier?
Tickets
CDs
Toilet Paper
Dental Floss
CDs are cheap and plentiful enough (lots of freebies in shops and in junk mail). Students could be given an actual CD so that they are taking their own measurements for questions on area and perimeter. (Not just of the CD but the packaging and storage requirements). This would be somewhat better than a labelled diagram in a textbook.
Same with the toilet paper, but I dread to think where it might end up with my students.
(My first comment on a blog ever btw – discovered your blog recently following a link on http://www.ncetm.maths.org.uk).
The thing I really like about the CD image is that I am sure there is a very noticeable relationship between burnt area on the CD and length of music when the CD is played.
This is a great problem to solve! I like the idea of using tickets, because the content is the most tangible. My son came up with the idea of calculating the _area_ which the mass of tickets consume (2 pi R squared minus 2 pi r squared).
This divided by the side area of a single ticket, which is easily measured, will give you the total number of tickets.
And of course it applies to the contents in the CD, toilet paper and dental floss forms. Figuring out the density of the media when given the total content is an obvious progression.
Good thinking, Dan
How long is the TP? How many tickets? How much time can you record on the CD? How long is the dental floss?
(Other examples: Masking tape. Streamers. Records. Cassette Tapes.)
HELLO SUMMATION!
For CDs, I wonder if something can be done relating linear to angular velocity? I just looked it up (http://www.pcguide.com/ref/cd/constSpindle-c.html) and apparently CDs don’t spin at a constant rate (meaning their angular velocity is not constant). The linear velocity is constant. A question might be what is the fastest RPM that the CD is spinning? The slowest? What would the CD sound like if it played at a constant rate? Or something like: if you have a record player (which plays at a constant angular velocity), how do you have to design the grooves so the music doesn’t sound crazy?
Sort of related, but not quite, is this proof without words. Which I now think would be an interesting visual in class if there is something you could cut that would do this (instead of being one long piece of something wound around, it would be a lot of concentric circles… can’t think of something that’s like that at the moment): http://www.youtube.com/watch?v=i1Nfp2Ow-C4
Sam
Debbie, welcome to the social! Nice to have your input. Good suggestion with packing problems. Jason Dyer had a fun one recently, if you’re just jumping into the math edublogosphere.
Goosebumps. That was the question that originally brought this whole thing on. I’m going there in the follow-up post. Consider, if you would, how you’d structure a long-ish thirty-minute activity around that question. What materials and supplements would you need? What sequences of questions would you ask to help students make their way through the narrative?
All of those questions have a huge instructional advantage over the textbook stand-in, “What is the area of the annulus?” I’ll make it explicit in the follow-up post, but I’m curious if any commenters besides Sam know what’s so great about those questions.
I’m not seeing a straightforward way to translate area into number of tickets or sheets of toilet paper. Wouldn’t thickness of one layer and length of each individual thing play a part. Now I want to do some math and I have so much other crap to do tonight GRRR MEYER.
It would be interesting to have a full roll of tickets where you know the number and then show a picture of the roll after a sporting event and then have them come up with how many are left.
Also if you had a roll you could have them come up with a relationship between the size of the roll and the number of tickets in the roll, if one exists.
The same process could be done with a variety of things (toilet paper, paper towel, etc.) which would be easier to get large amounts of.
The greatness of these examples for me is the facts that students can touch the math and not just stare at it.
This would be the curveball but what about the bar code you have pictured in the ticket roll? What do those bars mean? How do we decode that? just thinking MOD. Not sure why. It is just something sticking out in the picture. And a different topic to consider. The amount discussions-way more grounded with the students I teach. But just as well milk it for all you can.
Also this is my first Blog post. My school filters all blogs so this is the first time I have had a chance to peruse your site. I hope your Doctorial does not stop you from contributing to my education and your blog:) I may start my own after viewing yours great stuff!! I showed it to other teachers today.
Caution… I am a student-teacher… but I might try something irreverent here… tell the class “today we’re talkin’ toilet paper”, throw up the slide and ask them to find the area… let them estimate any information they might need, by looking at the picture or perhaps an actual roll… check answers, and then put up the grid lines for actual measurement… and let them see how close they got. Then say something to the effect of “toilet paper is much too important to stop here”, and then give them a challenge problem (given the thickness of a sheet of toilet paper, come up with a way to measure the area of the roll without using a measuring device… do a think-pair-share and maybe have students try out their methods — scary I know)
Oh man, this is kind of crazy, but you could ask something like, “what would take more toilet paper?… toilet papering every inch of a 10 X 10 dance floor, or using the toilet paper as a replace tire for a car that needs 10 inch tires, or skate board tire might be better” (que the photoshop’d image… or a fast forwarded video of the unrolling of a 10″ roll of toilet paper (where to get one of those???))
As long we don’t get stuck trying to present it as a real-life application, but rather as a way to throw around geometric concepts in an amusing manner, we could get students to engage in some good problem solving. I believe there is a place for this type of lesson.
Have a bunch of rolls of toilet paper with varying amounts of tissue left. Ask them to find a function that models the number of squares of tissue left as a function of the radius of the roll.
Then stand back and see what they do.
Measure the active area i.e. donut less hole with one measurement. Show that length of line tangent to inner circle inside the outer circle is “average” diameter.
Thinking about Sean’s idea to look at the tickets before and after the game…
How about an image of the roll of toilet paper in each stall before a high school football game and then an after shot. Which stall used the most? I think I heard/read once that the stall closest to the door is used the least and the one farthest is used the most. Would this confirm that theory. Each group could have their own stall to get data on and then post class results to compare.
Came across this blog about 3 weeks ago. Had no idea about edublogs. Darn if you all post blog rolls. I’m never going to get through all of them before summer ends. It’s just so hard not to want to read each and every post when google reader says I haven’t read it yet.
Good motivation for introducing polar coordinates.
Also, “Why do we need calculus?!” answered. Because, lol@ trying to find the total of tickets, tape, songs, paper, floss with algebra.
Interesting questions you have brought up. I guess I would say all these items though different in size are all around the same price.
I wonder….
I wonder how far the roll of tickets would extend if you unrolled all of it and measured from the first ticket to the last ticket.
I wonder if we gave students some short strips of tickets and individual tickets, and the ability to measure any aspect of the large roll that they wanted (radius, circumference, inner hole, ?weight?, etc.), if they could try and form a reasonable argument as to how long the roll would be…. of course allowing them to make predictions before hand and writing those predictions on the board
I wonder if most math teachers could form a reasonable argument given this task…
I wonder if students could share their ideas with one another and discuss which ideas seem to have the most validity and why…. and then it would be interesting to actually go outside and unroll the tickets to see whose answer was the closest.
Its always good to wonder, thanks Dan.
My colleagues and I did a problem with toilet paper a few years ago at a before school “algebra week”. The question was similar to Jackie Ballarini’s about finding a function to model how much toilet paper is left. We were asked to find some way of telling how much paper was left on the roll. I think it was phrased in the manner of inventing a device–which could have been a function–that told you how much TP was on the roll at any particular time.
Each group got one full roll of toilet paper to work with however they saw fit. Hilarity ensued. As well as a lot of good mathematics.
My first thought was how many tickets are on the roll. I hate buying tickets for school functions and not having a clue how many rolls I’ll need.
Give students varying numbers of tickets (10,20,30, etc.). Have them create a table showing number of tickets and radius of the roll. Perhaps graph the point. Model the relationship using a function and have them extrapolate to a roll of any given radius.
@Brian-
In addition to ‘talkin’ toilet paper’, and I’m trying to think a little like Dan here, there’s a Seinfeld episode where George talks about toilet paper, something along the lines of how it has remained unchanged over the years, you could throw that in there, too.
[...] there any advantage to these images over the analogous problem in a [...]
Hell, get’ em into the school bathrooms and take field notes. Assuming the rolls aren’t replaced during the day, what are the differences in area between 9 AM and 3 PM?
@ Daniel Schaben I’ve run into the same problem with some blogs. Especially blogspot… Try using google reader…unless that’s blocked too!
[...] Jun Dan Meyer, in one of his recent WCYDWT, posted a picture of a roll of tickets (among other [...]
Why is it that you can see mathematics in one situation but extending it to others never enters the mind…
I have tried something similar (I used a roll of tape). The question (given the roll, how much tape is left) occupied a class of 30 for a full minutes or more. In the end, they found a reasonably justified process to answer the question. If you haven’t tried this, it is an awesome problem for any Geometry class.
I figure it’s something like the question, “why can I benchpress 120 lbs. but not 130 lbs.?” I think there are certain habits (exercises, if you will) that strengthen the “noticing” muscle. Wish I knew ‘em all.
[...] well with some highly combustible mathematics. Per Brian Lawler's suggestion, we began with the ticket roll and the water tank (which, for the record, I had never attempted to solve until last Tuesday with [...]
Ooooh! I wish I hadn’t missed this one when it came out! I’ve done the TP roll and it has some great math behind it. Especially suitable for more experienced students and teachers.
One strategy (like Dave L) is to measure the diameter, then unroll a few squares, then remeasure the diameter, repeat for a while. Then graph diameter versus squares and try to decide when the diameter will reach the cardboard core. This makes sense; and
* the squishiness of TP creates yummy measurement problems (so I have used adding-machine tape if it looks like too much) and
* the data look linear enough for a while to seduce impressionable folks to fitting a line. Which will give a seriously wrong answer; a good advertisement for figuring out the geometry.
Ref: Erickson, 2001. Data in Depth, p 119. Key Curriculum Press. Note the vague “Sonata” assignment form; I’m hoping to use more of these in the coming year.
Squares v. diameter. That’s great. That’s the kind of fun that results when you don’t prescribe the method of solution too heavily, a la:
Wow funny. I was just thinking about this the other day. I was wondering how how to calculate dr/dL (L) for toilet paper, where L is length remaining in the roll, and r is radius. It did not seem immediately obvious to me, due to the discrete nature of the layers. Have you solved this?
I didn’t. But Sam Shah can get you halfway there.