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).

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

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.

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.

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.

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.

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 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.

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?

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.

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.

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’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.

Thank you Dan!!!!!!

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.