What Can You Do With This: Annuli

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?



Toilet Paper

Dental Floss

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.


  1. Reply

    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).

  2. Reply

    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.

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    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

  4. Reply

    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.)


    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


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    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.

    Z. Shiner 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.

    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?

    Sam:How long is the TP? How many tickets? How much time can you record on the CD? How long is the dental floss?

    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.

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    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.

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    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.

  8. Daniel Schaben

    June 14, 2010 - 5:04 pm -

    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.

  9. Reply

    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.

  10. Reply

    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.

  11. Reply

    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.

  12. Reply

    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. :)

  13. Reply

    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.

  14. Reply

    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.

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    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.

  16. Reply

    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.

  17. Reply

    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.

  18. Reply

    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?

  19. Reply

    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.

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    Chris: Why is it that you can see mathematics in one situation but extending it to others never enters the mind…

    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.

  21. Reply

    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.

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    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:

    A toilet paper roll has inner diameter 5 cm and outer diameter 10 cm. You unroll 200 squares off the roll and now the outer diameter is 8 cm. How many squares are on the roll in total?

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    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?

  24. Reply

    I’m a little late to the party here, but I sure had fun with this! I’m a former high school math and physics teacher who may return to teaching sometime in the next few years. Apologies in advance for some awkward equation representation in this comment.

    What came to mind for me was, “What percent of the toilet paper is left when the roll is down to half the original thickness?” The first fun thing with that question is that there are two interpretations for kids to find/investigate: Do we mean “half the original radius” or do we mean “half the annular thickness”? The latter is more meaningful and more realistic, but the former is also interesting. And neither case results in “half” being the correct answer, which will be obvious to some kids and really perplexing to others right off the bat. What fun! And if you use variables instead of actual measurements, you can extend the problem further:

    Consider the case of the roll used to the point that it is down to “half the annular thickness.” Using r for radius of the toilet paper tube and R for the outer radius of a full roll, algebra gives me “proportion of toilet paper remaining”=(R+3r)/4(R+r). Well, what if we manufacture toilet paper rolls with lots of different R’s and r’s. (Or, what if we can vary the size of the tube?) What happens when we use smaller and smaller r’s? Gee, now we can start introducing limits, and we see that really tiny r’s lead to “proportion of toilet paper remaining” approaching 1/4! (Which just happens to be exactly what happens to the area of a circle if you cut the radius in half, which is a toilet paper roll with no tube! Not so unexpected, but great to discuss, and gives you a chance to reinforce proportional reasoning: formula for area is pi*r^2, so what happens if r doubles? triples? halves?) Ok, now what would it take to get HALF the toilet paper to be left when we’re at half thickness? Fun! Turns out it’s not even possible, but you come close if you have just have a really thin layer of toilet paper wrapped around your tube, as confirmed by the fact that if r approaches R, then “proportion of toilet paper remaining” (at half annular thickness) approaches 4R/8R=1/2.

    Now consider the case of the roll used to the point that it is down to “half the original radius”. Again using r for the tube diameter and R for the original full radius of the toilet paper roll, “proportion of toilet paper remaining”=(R^2/4-r^2)/(R^2-r^2). So, this time what happens if r gets really small? Yep, you get 1/4 again, as expected. But what happens if r gets big? HOW big is r allowed to get this time? If you let r approach R, you can’t even do the problem anymore! So r can only approach R/2. And then the proportion remaining becomes 0! With a little thought or the right picture, kids should be able to explain this result to each other pretty quickly. But to catch the kids that are still fuzzy, and solidify everybody else’s understanding, I’d want to ask, “In this case, the proportion left gets smaller with increasing r. But in the first case, the proportion left gets BIGGER with increasing r! Why?” I think I’d like kids to output their answer as a journal entry or video, to force them to clarify their thinking, streamline their logic, and communicate their logic clearly, with good use of both language and images.

  25. Reply

    I tried the “How Many Tickets?” last week, and it was a great moment in Geometry for my class. It was funny to see students going up to the board and trying to look under the black rectangle.

    I had almost total buy-in for this problem, probably 24 out 30 students working in organically formed groups. The students who were not engaged were either confused by the problem to the point of quitting, or they were just chatting and unfocused.

    One of my freshman was the first to come up with the right process. He told me how he thought the problem needed to be done, and I gave him a nod that he was taking in the right direction… and then 5 students ran over to him and they began with the calculations

    I definitely recommend asking the class to first write down a guess that is definitely too big, and one that is definitely too small.

    It was a good day in Geometry.

  26. Reply

    Another type of annulus problem is the amount of water that drains from a sink. Some sinks use 1.25″ drain pipe, while others use 1.5″ drain pipe. Is there much difference between the two choices, if the drain runs 15 feet to the main sewer drain of the house?

    The pictures and discussion have been enjoyable to read.

  27. Randy Blackwood

    August 17, 2012 - 8:18 pm -

    Today was our first day of school, so as an intro activity in my PreAP Algebra 2 class, I took an old roll of tickets that was being thrown away last summer and as the students came in the door, I would greet them, write their seat number on the ticket and hand it to them. After the some typical 1st day logistics. I held up the roll of tickets I had and asked them what they would want to know about the roll. After some thought the typical question was “Hhow many tickets were on the row?” I wish I could have recorded the discussion in my 2nd hr. I had a student who proposed measuring the thickness of the ticket, count how many tickets were on the outer layer, measure the width of the roll to determine how many layers there were. After some discussion another student piped up, but the inner diameter doesn’t have as many tickets. So we decided to try the first method anyhow, as I said well at least we will know the maximum number of tickets there would be. I think our calculations were around 1300. Then another student proposed averaging the number on the inner layer with the outer layer for a better estimate. I believe the number was around 910. I used a set of micrometers to measure the thickness of the ticket, and a set of calipers to measure the length of the ticket, as well as diameters, etc.
    It was amazing how accurate the correct calculation was. Prior to class the roll had 1000 tickets. When I did the measurements I calculated 999.5 After the second hour class I think the number of tickets left was around 940. I don’t remember the calculated number but is was within 8 of the actual number. And after my last class of students they calculated 914 tickets, when the actual number should have been about 910. Pretty awesome stuff.

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