If you have never rolled a cup across a flat surface and marveled at how precisely it returns to the same place you rolled it from, it’s possible you’re the wrong audience for this post.
There is math here, certainly, but I have made it a goal this year to stall the math for as long as possible, focusing on a student’s intuition before her calculation, applying her internal framework for processing the world before applying the textbook’s framework for processing mathematics.
Bad First Question
This one sucks the air right out of the room. We’re into the math immediately, having bypassed several easy opportunities to pull in our students who hate math
Jason’s First Question
Jason Dyer suggests handing out plastic cups, letting students roll them around, then asking “why do they do that?” I have no problem with this approach. I would like to start from a position of stronger student investment, though.
My First Question
Have them roll some plastic cups around. Then toss up this slide and ask them a question that has a correct answer, yes, but which attaches little stigma to the wrong answers. It’s an educated guess and different students will make persuasive cases for all three of these. Ask them to write their guesses down, to put them on the record
A Lesson Sketch
The conversation can then proceed along some interesting lines where you ask the student to:
- justify her guess.
- draw the kind of cup that will roll the largest circle using a fixed amount of plastic. This is fun. Many will draw a really tall cup, which isn’t the best use of limited material. A two-inch-tall cup can roll a circle that’s a mile wide.
- make their ideal cup from a page of card stock. The fixed size of the card stock will normalize the results.
- draw a complete picture of their cup including the auxiliary lines. Can they find the invisible center of the circle it will roll? What’s the method?
We do all of this before we start separating triangles, before we write up a proof, before we generalize a formula. We ask for all this risk-free student investment before we lower the mathematical framework down onto the problem.
A cool feature of this formula is how well it handles degenerate cases. For example these two:
- A cone’s roll-radius is the same as its slant height so letting d = 0 should (and does) eliminate D from the formula.
- A cylinder will roll forever so letting D = d should (and does) return an undefined answer.
From there you can pull out of your cupboard (digital or otherwise) any random set of cups and the students should be able to predict the roll-radius within a small margin of error.
And the framework grows stronger.
A Parting Swipe At Textbooks
I didn’t dig this out of a textbook
MathewJune 10, 2009 - 11:03 am -
Will we ever math textbooks that start lessons with experimentation and conceptual understanding before bringing in the mathematics?
Or by nature of being in a textbook are all lessons doomed?
RichJune 10, 2009 - 11:08 am -
I follow — and like — the progression of what the first question should be….
But how about starting off with “What shape or path will be made if we roll this cup?” In other words, why give up the fact that the cup will roll in a circle at all? Or is that too soft of an opening? Would some/most students’ intuition tell them that it’ll roll in a straight line instead of a circular path?
And I really appreciate the wealth of the directions that we can go in after getting our feet wet.
JazoJune 10, 2009 - 2:25 pm -
This is why I love math so much – it comes up in the most random places. I didn’t realize the triangle made by the cup until you drew it out. I’m liking the bridge from intuition to math.
BTW, I got a 100 on that Projectile Motion project. Thanks!
MichaelJune 10, 2009 - 2:26 pm -
Perhaps this is overreaching but what about giving them a wine glass, a champagne glass, or a martini glass at some point in the lesson, if not tangible make them virtual. None of these glasses initially seem to fit the “format” of the other glasses but do end up following the same principles as the other glasses and leave the students to come up with the question of “why.”
Additionally, does this principle only apply to glasses or can it be true for trash cans without handles, barrels, and other non-glass objects?
Dan, I am amazed at the simplicity and the complexity of this lesson and yes it does make sense to open a lesson up with such a great attention getter. I think this is what my credential program was trying to teach me, but failed to elaborate so effectively as you have done with this series of WCYDWT. Thank you for sparking a wonderful collaborative conversation!
BonnieJune 10, 2009 - 5:44 pm -
This is an awesome lesson. Steal-worthy!
Jason DyerJune 10, 2009 - 8:28 pm -
Side question: what are the graphs you have in the background tacked to the wall?
Maria DroujkovaJune 11, 2009 - 6:43 am -
A discussion of this post at the Natural Math Google Group: http://groups.google.com/group/naturalmath/browse_thread/thread/a87065bc712bb7af
Dan MeyerJune 11, 2009 - 9:46 am -
Not that my approval’s necessary here but I dig both Rich‘s updated opener (“what path will this make when it’s rolled?”) and Michael’s closing extension (put up a margarita glass with it’s bizarre undulations and ask the same question). I like the former because there is a distinct possibility that a few students will self-assuredly write down “a straight line” and it is almost always a good thing when self-assured students get their expectations checked a little. I like the latter because it forces you to look a little harder for the large diameter.
@Mathew, your question is unclear.
@Jason, that’s some preliminary Feltron work right there.
@Maria, that discussion seems to neatly illustrate the difference between the objectives of public schooling and homeschooling while, as a bonus, highlighting the hazards of criticizing one model without really understanding its constraints.
If I’m missing an obvious way to achieve an instructional objective on a timetable without any sort of coercion, it would be awesome if someone would throw me a line here. I don’t have the luxury of telling kids they can just check out if they aren’t into it. This has made me all the better a teacher, incidentally, because I’m forced daily to determine interesting on-ramps to these topics and instructional strategies (a lá WCYDWT?) that will maximize student investment.
MichaelJune 11, 2009 - 10:25 am -
Here is a question that this lesson does not consider.
Do manufacturing companies, glassware designers, and artists consider what size circle the glass they are making will roll in when creating new glasses? Do such people consider volume and stability as constraints for the diameter of the top and bottom of the glass? There seems to be other math here other than radii and circles.
Also, how would students handle a triangular beaker where the mouth is smaller than the base? Would the results surprise them? How would that example help them in their generalization?
Maria DroujkovaJune 11, 2009 - 10:28 am -
I wish there were a way to join mailing list and blog conversations together, like wiki “includes.” I know how to do it for myself through subscriptions, but not for two groups.
“Meaning” is a theme to consider here. What makes learning meaningful? Social considerations, such as graduating school, can provide a powerful source of meaning, if extrinsic to subjects.
The extreme (or ideal?) case is, “Study math like your life depends on it.” We see this type of learning in fairy tales, like Harry Potter. I don’t think urgency is the only way to make learning meaningful, though.
Beauty is a strong source of meaning, but I find it finicky when it comes to long-term motivation in math.
After I visited his webinar on Futures of Education, I’ve been following Michael Wesch cultural anthropology work with some interest. What amazes me is his ability to create and pose tasks students find deeply meaningful. Forty out of forty students gave him positive ratings on “Rate My Professor.” Their classroom videos are watched by millions of people on YouTube. Here is the webinar page with further links: http://www.futureofeducation.com/forum/topics/michael-wesch-a-cultural
I want my students to have “veto rights” and freedom to check out. At the same time, I want to convey the message that the tasks I offer them are deeply meaningful and very important, for the world and for them personally. The message, “You can either do it or not, nobody cares” begs the question, “Why then do you offer me these tasks in the first place?”
The balance between meaning and personal choice is a challenging theme for me.
Maria DroujkovaJune 11, 2009 - 10:37 am -
Sorry about the meta-discussion, Dan. On the subject of glasses, I would roll them on non-flat surfaces. Maybe in those museum coin trap thingies? On giant balls? I don’t quite know how to accomplish this in our physical world, for real glasses would surely slide rather than roll. Metal glass-like shapes covered with something rough like sandpaper, rolling on sandpaper cones, may do the trick. Ruby Goldberg for the win!
We did a good somewhat related activity in our Math Club Jr. last week, which simply started with, “What shapes can Spiderman make in the air as he hangs on his thread?” Those five year old boys are all Spiderman fans.
Matthew BardoeJune 11, 2009 - 6:07 pm -
First, I did an experiment with my exercise ball. It has nice “lines of longitude”, and the glass nicely hugged one of those lines of longitude. So I think it is somewhat safe to say that it will make a circle on the sphere. I recommend the exercise ball. The rubber was perfect for keeping the glass in place. Which isn’t too surprising…
My question, and I haven’t thought about this at all is how does the shape of the glass effect the number times it must rotate to create one rotation around the carpet. I guess that there is no limit to the number of times that it will rotate around. I realize that this is blatant math question, but can a formula be found?
Great question, great activity.
MichaelJune 11, 2009 - 9:05 pm -
Sure, there are many formulas.
A straight Algebraic approach would take the radius of the large circle (x+y) and find the circumference of that circle 2*Pi*(x+y). Then find the circumference of “mouth” of the glass which is Pi*D or 2*Pi*R (Where D is the diameter of the “mouth” of the glass and R is the radius of the “mouth.”)
Next, divide the circumference of the large circle, created by rolling the glass and traced by the “mouth” of the glass, by the circumference of the “mouth” of the glass. This is the number of completed rolls made by the “mouth” of the glass.
[2*Pi*(x+y)] / [2*Pi*R] = (x+y) / R
We can likewise find the number of times the bottom of the glass completely rolls around the smaller circle. If we let d be the diameter of the bottom and r be the radius of the bottom of the glass and y be the radius of the smaller circle, then we get
[2*Pi*y] / [2*Pi*r] = y / r
But, since the “mouth” and the bottom of the glass are attached, they should both roll the same number of times. So that means that
(x+y) / R = y / r
But, does this makes sense if R = r which would give a cylinder which should roll in a straight line?
How does the last equation support the real life observation of a cylinder?
Since this seems to have a periodic pattern, how could trig functions be worked in to model this mathematically?
How many other ways can this simple thing be modeled?
AlexJune 12, 2009 - 8:45 am -
(x+y)/R = y/r only makes sense for R = r if R = infinity then you get 0 = 0. A straight line can be thought of as a circle with infinite radius so the result does hold for a cylinder.
Looking at limiting cases is a great way to check if a result makes sense and any time you can get students to do this I like it.
Dale BaslerJune 12, 2009 - 5:46 pm -
What natural selection has done with this:
Here’s a biology tie-in.
Eggs laid by cliff dwelling birds have evolved to be far more lopsided so they roll in circles rather than off the cliff.
SueJune 13, 2009 - 6:11 am -
Wow! I love that this can be tied back to the natural world!
Maria DroujkovaJune 13, 2009 - 2:01 pm -
Dan, this is an incredibly rich and excellent topic! Searching for biological connections, I found a neat word:
Apparently, it’s a technical term widely used in manufacture, biology, and warfare (shell stress) research… Check out these neat phrases fished out from Google Scholar:
Conicity index and waist-hip ratio and their relationship with total cholesterol and blood pressure …
High speed stability for rail vehicles considering varying conicity and creep coefficients
A comparative study of root canal preparation with HERO 642 and Quantec SC rotary Ni-Ti instruments… Coronal preparation Instrument no. 1 (6% conicity, size 25, working length: 14 mm); … 2 to no. 4 (2% conicity, sizes 15-25)
Dan MeyerJune 13, 2009 - 2:32 pm -
“Conicity” and those resilient bird eggs are the reasons why we keep the comments turned on around here. Keep ’em coming.
ElissaJune 13, 2009 - 3:22 pm -
I love this idea but I am not smart enough to teach this lesson. Maybe I chose the wrong content area.
Jackie BallariniJune 13, 2009 - 8:22 pm -
Matthew Key Curriculum’s Discovering Series does a decent job of incorporating discovery and conceptual understanding.
ElissaJune 14, 2009 - 4:46 pm -
But what does the D/(l + x) = d/x mean?
Where did the cup radius become that?
What does the final answer mean?
I guess I don’t get the entire problem.
Dan MeyerJune 14, 2009 - 9:40 pm -
D = the larger diameter of the cup.
d = the smaller diameter of the cup.
l = the slant length of the cup.
The goal is to determine l + x, the roll-radius of the cup (not the radius of the cup) where x is the distance between the bottom of the cup and the invisible center of the roll-circle.
Because D and d are parallel, we have two similar triangles and that’s where that proportion comes from.
From there we solve for x and then add it to l in order to get the whole roll radius. If that doesn’t make sense, say something. If I can’t work this out with another math teacher, I’ll be sunk with a geometry student. Do you see the similar triangles?
ElissaJune 15, 2009 - 6:18 am -
I do see the similar triangles and I understand the D, d, and l. I get it up until we take what x equals and add it to l and call it the cup radius. How is it that the radius? I get the math and so then the final answer equals the radius of the cup, right? But how?
Dan MeyerJune 16, 2009 - 12:50 pm -
We aren’t calling L+X the cup radius. We’re calling it the roll-radius, the radius of the circle the cup makes when we push it. See the first photo, perhaps. That radius is the sum of the slant-length of the cup and the invisible distance x between the bottom of the cup and the center of the roll-circle.
Maria DroujkovaJune 16, 2009 - 5:07 pm -
Elissa’s questions led me to this task, probably not a good start, but a good middle:
– What are things you can measure/change about the situation?
I expect a lot of math fun in sorting measures, for example:
funny or not: conicity is funny, and so is glass purpose as far as cocktail mixing
relevant to the rotation path shape or not
affecting other properties: glass weight may affect the speed but not the shape of the path – stuff the glass full of modeling clay to prove it experimentally
I am trying to think of a good way to help kids quickly make a bunch of cup shapes for experimentation with measurements. Probably just heavy paper plates, easily made into cones and then bottomless “cup” models.
anonJune 18, 2009 - 6:40 pm -
How did you determine (mathematically) that a cup rolls in a circle with a radius equal to the slant height of the cup’s cone?
sam shahJune 29, 2009 - 5:56 am -
I want to throw out that this is a good question for calculus also… because after doing the straight edged glass case, you can move onto funny shaped wine glasses like @michael suggests… because to find the “triangle” you have to find where the wine-holding part of the glass hits the ground…
which is found using…
dum dum DUM!
the tangent line from the base disk of the glass to to curvy part of the glass. And how do you find lines tangent to curves? derivatives! But it has a harder element to it… it’s a bit trickier because the problem reduces to: “Find the point on the curve whose tangent line contains a point (which is the point on the base disk of the glass).”
(To do this, you’d have to give students an equation for the curve of the wineglass, or have them figure it out by taking data measurements and doing some sort of regression, or just creating an hypothetica wineglass with a particular equation.)
Love it! If I can get the plan to run smoothly, I might just try this out next year!
(The concept question to end the class with would be if there are MORE than one tangent lines that exist that go through that point on the base disk, what would the radius be? How do you decide which tangent line to use?)
Hm. I’m not sure I’m being clear about this. I hope it makes sense.
Dan MeyerJune 29, 2009 - 1:30 pm -
Heh. Calculus really does require a certain kind of enthusiasm for the most ridiculous contrivances, for long, protracted solutions to problems which could be solved within a margin of error of 5% by simply eyeballing the thing.
Anyway, I read you clearly, Sam, but I reject out of hand a contrived solution to a contrived, fabricated example. If we’re going to run through all this work it would be nice if there were some sort of physical glass to roll around and verify a student’s answer. Which, of course, makes a perfectly cubic wine glass some kind of holy grail here. Anyone have a relative who works in glassware? Someone has to get this, size it up, and upload it for the rest of us.
anonJuly 1, 2009 - 7:49 am -
Let me rephrase my question.
Your demonstration seems to be missing a crucial step. How did you connect the slant height of the cup’s cone (which you ably calculated using similar triangles) with the cone’s roll radius?
Dan MeyerJuly 1, 2009 - 12:55 pm -
Yeah, that’s a good question.
Michael SerraJuly 2, 2009 - 10:58 am -
Last week I was reading some of your assessment ideas. BRAVO! Now this. Your name and ideas just keep popping up. Isn’t the classroom just a wonderful place for experimenting with ideas. Keep up the great work.
anonJuly 3, 2009 - 7:35 am -
Did any of your students ask you to justify your assumption that the roll radius is equal to the slant height?
TylerAugust 9, 2009 - 3:27 pm -
anon: Your question is a good one. I thought I had an answer until I started trying to explain it. What I have is an intuition, not a proof. Here it is anyway:
The cup is rolling, or to be more precise, rolling without slipping. Imagine that the rim of the cup is covered in paint so as it rolls, it paints an arc onto the floor. We only roll it a few inches for this particular thought experiment (less than the circumference of the rim).
Now imagine we roll it the same distance, but this time there is no paint on the cup, but there is paint on the floor. If you measure the arc on the floor in the first experiment, and the arc on the cup in the second, they should be the same length. If the cup were sliding as it went, one arc or the other would be longer.
Now, the bottom of the cup paints out a similar arc. Because the cup is solid, the bottom of the cup must rotate at the same speed as the top. And yet, by the time the cup has made a full rotation (around its own axis, not around the entire circle) the upper arc is equal to the upper circumference, whereas the lower arc is equal to the lower circumference. How do you have two circles rolling at the same rate, and yet one goes further in the same amount of time? Well they have the same *rotational* rate, but different linear rates. I think that basically proves it won’t roll straight. But why a circle? Why not an ellipse or a hyperbola? I guess because the problem doesn’t change once you’ve rolled the cup a few inches. The same thing applies, the top is still rolling faster than the bottom by the same factor, and so the curvature is constant.
And look! We have a new definition for a circle: It’s the shape whose curvature is the same everywhere. Of course that begs the question of how exactly we define curvature, potentially leading us to some circular logic. Sorry for the pun.
Another question that could come out of this lesson: Why don’t the wheels on a car slip when you make a turn? The answer is a clever little device ( http://en.wikipedia.org/wiki/Differential_(mechanical_device) ) your students may not have heard of, which might lead them to wonder how many other things are going on in their car that they simply never thought about before.
DavidAugust 17, 2009 - 9:23 am -
Wow! We can all spot in Tyler’s question – how do we define curvature – another pre-calculus investigation.
In other words, without any formal algebra, we have a locus that turns a bit for every step forward. You could model that with a velocity vector and an acceleration vector, if you like (or several other ways all relating to tangents as limits of slopes.)
I think that Alan Kay and Kim Rose’s car game is similar. I thought a rolling glass was a bit boring, but it is a surprisingly rich vein for investigation.