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At the NCTM Institute last month, we broke into task groups to discuss reasoning and sensemaking (the conference themes) in content focus groups. I slipped into Geometry a little late and found a seat. The group was discussing approaches to this problem:

This was the session immediately following my keynote and the difference between the tasks I had described and the task they had just finished was stark. Someone asked, “How would we apply Dan Meyer’s approach to this problem?”

I ducked.

It isn’t fair. It’s apples and oranges. Paper is a great medium for a lot of math problems. Paper is a terrible medium for representing how people apply math to the world outside the math classroom. My techniques for one problem type have limited use for the other. My enthusiasm for one problem type shouldn’t be mistaken for a lack of enthusiasm for the other.

That said, I don’t find myself terribly enthusiastic when I think about assigning this problem to Geometry classes I have taught. As a challenge problem or extra credit, sure, but in its current form — with the abstract mathematical language and symbology smacking you right in the face — students are going to wonder, “Who comes up with these problems, seriously?”

If we make a better first act, though, we can engage, I dunno, 17.2% more students without any cost to the math. That’s empirical, friend.

Here’s the redesign:

  1. Show how this new, difficult problem arises from an old, easy problem.
  2. Make an appeal to student intuition.
  3. Introduce abstraction (labels, notation, etc.) only as a necessary part of solving a problem that interests us.

Act One

Some Really Obscure Geometry Problem from Dan Meyer on Vimeo.

  1. Start with a square.
  2. Draw the diagonals of the square.
  3. Ask students to tell you what percent each of those regions is of the whole. This is insultingly easy and that’s the point.
  4. Drag the endpoint of one diagonal halfway down the side of the square.
  5. Ask them, “How about now?”
  6. Ask them to guess the percents again.

Watch the video. Basically, we’re applying pressure to their confidence, which is how I try to approach pure math problems. Start from what they know. Then mess with it in some trivial way (eg. we just dragged the endpoint down a little) that requires math that is anything but trivial.

Act Two

You and your students will begin to find it very difficult to talk about all these different segments and regions without labels. So add them. A recurring point around here is that if you want to disengage a lot of students who might otherwise be engaged in the math, simply start the problem with as much abstraction as possible. If you want to engage those students, don’t introduce that abstraction until students know why they should care about it.

Act Three

You’ve been walking around and taking note of different solution strategies, right? Have students come up and explain those different strategies. Then show use this Geogebra applet to show the percentages changing, in case anyone still needs convincing.

Sequel

The sequels here are really, really great.

Suppose M cuts side CD so that MD = n • CM. What are the ratios of the areas of the four regions?

Send n to infinity and watch the fireworks.

Again, though: print-based media require you to keep everything on the same page — the sequels in the same visual space as the original problem. I realize that math teachers by nature don’t mind that. Do students?

Featured Comment

J Michael Shaughnessy, President of the National Council of Teachers of Mathematics and designer of the problem under discussion in this post:

Not all math problems have to be posed everytime in a a high tech environment. Sure, it’s ‘cooler’ that way, but i completely disagree with your comment on this one, about ‘how the problem was posed.’ It’s only boring in the beholder’s eyes, depends on how it’s pitched to a group.

2011 Aug 29: My response here.

54 Responses to “[3ACTS] Some Really Obscure Geometry Problem”

  1. on 18 Aug 2011 at 6:39 amBrian Frank

    Link to geogebra app has extra gunk in it, at least for me. ;)

  2. on 18 Aug 2011 at 6:56 amRaymond Johnson

    Let me guess – that problem came from Mike Shaughnessy’s session, right? I’ve seen him present that problem before and I agree – for the type of problem it is, it’s a pretty good problem. The challenge for a task designer is pushing the boundaries of what we think of as “type,” and incorporating the same mathematics into a task that better engages students.

  3. on 18 Aug 2011 at 7:19 amNathan Chow

    Really enjoyed this discussion on twitter last week, Dan. It was revelational for me to think of how easy it is to go from an intuitive result to a rich perplex problem… with little confidence drop off.

  4. on 18 Aug 2011 at 7:48 amSue VanHattum

    I like the estimating. I estimated 40%, 30%, 20%, 20%, and could easily see that was not quite right, since the two smallest sections must add to 25%. I haven’t solved it yet. Gotta get ready for work.

  5. on 18 Aug 2011 at 10:33 amDavid Wees

    There are some interesting ways to estimate the percentages. For example, if you draw the shape onto a piece of cardboard, and then try and drop tokens (as randomly, or alternatively evenly distributed) onto the piece of cardboard, the relative percentages of the tokens in each area will be close to the actual percentages of the areas.

    I’ve used a similar experiment to show that the ration of a circle inside a square is about π and it worked really well, especially when we reduced the effect of the experimental error by finding the mean of our results.

  6. on 18 Aug 2011 at 11:02 amAlexander Bogomolny

    There are many components in making a good teacher. Charisma is certainly among the important ones. What works for a charismatic teacher may not work for another one. Many months ago, when you were still a teacher, I posted a warning that there is a danger that there were going to be attempts to extrapolate your success, to lift it to a pedestal and declare it universal truth. This happened to Sol Khan. This is happening to you, as your post shows.

    The problem is trivial actually. As Sue has observed, the two smallest parts add up to 25%. The key is to notice that P divides BM in the ration 2:1 = BP:PM. To see that draw BD. In triangle BCD, BM and CP are medians and P is its center.

  7. on 18 Aug 2011 at 11:05 amAndrei

    An excellent little problem – thanks for sharing. And I agree that your way would definitely go down better with a class.

    Sue, having solved it I can reassure you that you won’t need to resort to trigonometry or angle chasing to put this one to bed. Get the right insight and you can draw a diagram which makes the solution look completely natural. In fact there are several…

  8. on 18 Aug 2011 at 11:16 amNathan Chow

    @Alexander

    I agree that the problem is simple _for us_. I think that for students it’s a great exercise in using the available information and finding a result (depending on what skill level they are). And there are many pathways to the solution for the students to explore and discover.

    For instance, you don’t have to draw BD (and use properties of medians). Just recognize that AC has a slope of 1, so that perpendicular lines to P from BC and CM will be equal, hence the 2:1 ratio of BPC:CPM areas by (1/2)*b*h.

  9. on 18 Aug 2011 at 11:29 amR. Wright

    This is a charming problem when posed simply and innocently, not flayed alive by terminology, labels, and notation.

  10. on 18 Aug 2011 at 11:40 amBon Crowder

    I wonder what would happen if you gave the kiddos the problem like it is and said – make this more interesting to solve. Or better. Or snazzier.

    Might they come up with the backed off and “normalized” figure you started with? Or would they come up with something else?

    Could they find a freaky shape like this in nature that they could DanMeyer all over?

  11. on 18 Aug 2011 at 12:25 pmR. Wright

    “Ew, somebody DanMeyered all over our old tile patio!”

  12. on 18 Aug 2011 at 1:58 pmMarc

    “This is insultingly easy and that’s the point.”

    I’ve heard of the adage “write what you know” and this made me think a corresponding teaching adage might be “teach what THEY know” – or at least start out with that. Regardless of the context or the problem (I’m not a teacher and my math days are 15+ years past) most of what I’ve read Dan say seems to start with that idea – although this is the first time I’ve made that particular connection.

    (Long time listener, first time caller, by the way. Thanks for helping me keep my brain in tune with your ideas.)
    marc

  13. on 18 Aug 2011 at 2:30 pmGilbert Bernstein

    I like your observation about the difference in media here Dan. I hadn’t thought about the ramifications of being in control of the reader’s attention afforded by video/new media compared to traditional text. Something I hadn’t thought of in that way before…

  14. on 18 Aug 2011 at 3:33 pmAlexander Bogomolny

    @Nathan Chow

    This is in fact an excellent problem. It may be brought up several times at a geometry class prompting students for a different solution. The problems serves an example where solution is not unique.

  15. on 18 Aug 2011 at 7:56 pmCarlo Amato

    What program do you use to construct this video?

  16. on 19 Aug 2011 at 12:56 amSteve Phelps

    Taking a static figure, turning it into a dynamic figure, and examining the functional relationships between the different parts of the figure is what, I think, a good geometry teacher would do anyway. It is what should happen in geometry, but I know it doesn’t.

  17. on 19 Aug 2011 at 4:34 amCarlMalartre

    Is paper enough for the student’s explanation or do they need a better medium to explain their strategy?

    Carl

  18. on 19 Aug 2011 at 5:27 amShari

    I attended an NCTM Institute last week. It was about teaching PreAlgebra in grades 3-8. I would love to compare notes. I wasn’t impressed with the Institute at all, and neither were most teachers in attendance, though we had different reasons.

    I was surprised that the “deep thinking tasks” they presented were very much like this task. They were not engaging in any way. (If you can’t engage math teachers, how do you hope to engage students?) They didn’t even try to connect the tasks to something in the real world, which is ok sometimes. They were all number crunching activities that could be “algebrafied”.

  19. on 19 Aug 2011 at 6:04 amDvora

    I love your video for this question. It is true about the limitations of a printed page versus other media for furthering a problem like this.

    On the tech side of things… how did you create the video? What programs did you use?

    Thanks for inspiring!

  20. on 19 Aug 2011 at 8:35 amSteve Phelps

    @CarlMalartre

    I will often have my students record themselves as they investigate a sketch using screencast-o-matic.

  21. on 19 Aug 2011 at 8:43 amDebbie

    I think you’ve hooked me in. In the past I’ve been tempted to think, “well that’s all very well if all you have teach is algebraic graphs”.

    By starting off with a very familiar problem-style and seeing you apply your approach to it I think I’m finally convinced that this isn’t a one-trick pony but something that can work with all sorts of maths.

  22. on 19 Aug 2011 at 9:21 amPatricia

    I attended the same session and I was trying to solve the same problem you show here. I also asked myself how to convert this problem using Dan´s approach. Thanks for posting your conclusion! It is much more clear now for me what you are trying to say about math teaching.

    I teach mathematics in Buenos Aires, Argentina (7/8 and 9th grade) and next Tuesday I will have the opportunity to talk about my experience at NTCM Reasoning and sense making sessions. I will talk about you and show some of your videos. Do I need to ask for your permission…?

    Thanks a lot, it was very inspiring to know you in person

    Patricia

  23. on 19 Aug 2011 at 2:03 pmAmbika

    I was actually at the session that you were in and when someone asked that question I looked at you to see if you were going to respond to it, I’m glad you finally did :)
    I can see how it’s hard in the moment to revamp task questions of this sort to something more engaging and worthwhile for our students, but you do an excellent job of it. Thanks for sharing Dan!

  24. on 19 Aug 2011 at 2:41 pmClimeguy

    @Shari: NCTM always tries to show how dynamic and innovative they are and sometimes they pull off a piece of it. For example, inviting Dan. But the NCTM “insiders” tend to always want push the beauty of the conventional math that they have been teaching forever. And they don’t understand why its not engaging for many teachers like you. I’m surprised you think that most of the attendees were not impressed. Usually (and unfortunately for those of us who want more creative Dan-like approaches in general) most teachers are OK with what gets presented at these NCTM gatherings. Maybe what you report is an optimistic sign for the future so we’ll see more changes. Especially at the middle school level where the “algebrafying” is way over the top.
    -Ihor

  25. on 19 Aug 2011 at 3:32 pmluke hodge

    What an interesting problem. I initially thought of putting it on a coordinate plane – took me quite awhile to get that thought out of my head and see it in terms of geometry.

    I think Dan’s suggestion of modifying a very easy problem sets the scene very nicely and efficiently. I don’t think I like video for this particular problem. A timer or speedometer or ruler or something of that nature is technology we see in the real world, and helps to connect the math to the real world. Math world technology that purports to show the percentages for each region isn’t that compelling and doesn’t add insight to this problem in my view. Depending on the class, it might be a good idea at some point to draw the diagram on grid paper and do some measuring. I am not knocking technology like sketchpad or video, but don’t see the benefit in this instance.

    As far as paper-medium issues go, why would you want a handout for this great question? It is an easy diagram to draw and the question doesn’t require explanation. There are so many additional questions and issues that might come up, why commit on paper early and comprise your ability to adjust the lesson?

  26. on 19 Aug 2011 at 6:17 pmBowen Kerins

    It’s a good redesign, and good recognition that, if possible, symbology and labels should be part of the work students do on problems. Definitely supportive of good mathematical practice.

    I’d love if students learned to visualize the movement and change as shown in the video. To me this is a major goal of a geometry course: visualization of change and non-change. For example, picture a vertex of a triangle moving on a line parallel to the opposite side. The more experience students have with dynamic geometry (Sketchpad, Cabri, Nspire, Geogebra) or with this sort of video, the more likely they can succeed at these thought experiments, and the more they’ll know about continuity and invariance. (It’ll serve them well in later courses, too.)

    I also want to point to some language used in the discussion here. The initial problem is “insultingly easy”, while the later problem is “trivial” (Alexander’s comment). This is in the eye of the giver of the problem, not in the eye of the recipient.

    Saying that something is “easy” means one of two things. If someone succeeds at a task and it’s declared “easy”, there’s no excitement or celebration, because sheesh, it’s easy. But someone who has trouble with a task that’s declared “easy” shuts down completely, because if they can’t handle the easy stuff, hoo boy.

    “Easy” really means “easy for me”, and I think we would do well to avoid the use of the word with students and even colleagues. Explanations can be simple; sometimes there are great, simple, elegant solutions to very challenging problems. “Easy” is loaded, except in the Staples sense of the word.

  27. on 20 Aug 2011 at 2:42 pmluke hodge

    Bowen: I agree that that sketchpad and the like are valuable in demonstrating dynamics. Questions about changes resulting from dropping the diagonal might be a nice way to wade into the problem and give students practice visualizing (pencils down): describe some changes to the diagram if you (describe the plan), which of the 4 regions will be larger/smaller/same size, what would happen if you slide the diagonal down even further, do any of the 4 regions get bigger at first and then smaller, etc. Maybe show the video if they seem to be having trouble visualizing the changes.

    I can’t quite put my finger on what bothers me about the video. It seems a little to passive of an introduction. Maybe a little too close to being a sparkly item used to grab attention – especially with the percent boxes moving around. A lot of what I like about many of Dan’s videos is the lack of distractions you so often see that might be superficially engaging but really just take attention away from the math. This one just seems like it was forced on the question to me.

  28. on 20 Aug 2011 at 3:53 pmAlexander Bogomolny

    Bowen, hi:

    there is no need to pontify.

    When I call a problem trivial, as in my previous post, I mean it should be trivial for you teachers of geometry, not for students. Note also that I give a solution that I can’t imagine you find difficult. There is another one in one of the comments. So you can pass a judgement.

    In my opinion, a teacher who is about to teach geometry and who is not able to solve this problem should immediately disqualify self from that class.

    Dan has my deep appreciation for his mastery of video technology. Even though, it must be admitted that his approach does not (and is not capable to) bring students one inch closer to solving the problem.

    Percentage-fying a beautiful geometric problem is distorting the essence of geometry. Mathematics has many applications in real world. Much of mathematics came about by abstracting from real world phenomena. However, mathematics has its own subject and its own methods that could not be understood by an artificial linkage to an “artificial” real world.

    I must confess I could not believe some of the responses here were written by math teachers. This is just deeply distressing.

  29. on 20 Aug 2011 at 4:55 pmlouise

    I’d also like to see the ratios approached in fractional form, not percents.
    As far as trivial goes, start off with a square divided into 4 equal squares – each 1/4. Then go diagonally. How can the triangle be the same area as the small square? You just lost 50% of my class.
    The rest is of the problem is algebra and arithmetic.
    I am not sure that any of it is reasoning. Not that I’m complaining about teaching math, but if students struggle, you don’t know if they’re struggling because they don’t know the algebra, they don’t know how to multiply, they don’t know how to approximate fractions into percents, or they don’t know how to find the area of a triangle.
    None of these are about reasoning.

  30. on 20 Aug 2011 at 7:05 pmR. Wright

    …it must be admitted that his approach does not (and is not capable to) bring students one inch closer to solving the problem.

    I disagree. Aside from issues of visual style and the weird use of percentages instead of fractions, I think Dan’s approach is a much better way to pose this problem. In fact, if I had encountered the problem as originally posed, I would probably have no interest in solving it. Fortunately I saw something like Dan’s approach first and felt compelled. The original problem is completely overburdened with terminology and notation, most of which isn’t necessary (and even if it all were, it should be of the students’ devising).

    But I do agree with the sentiment of you and others that this problem has no business being related to the real world.

  31. on 20 Aug 2011 at 7:37 pmAlexander Bogomolny

    @R. Wright

    Could you please post your solution.

    I am prepared to admit that Dan’s video might be a catchier way of presenting a problem. I am curious if it had any bearing on the solution.

    Do please post yours here. Thank you.

  32. on 20 Aug 2011 at 7:59 pmR. Wright

    My point is that I probably wouldn’t have bothered to come up with a solution to the problem as originally posed. It is in that sense that an approach similar to Dan’s can “bring students closer to solving the problem.” To begin, they must be at least a little interested in solving it.

    The details of my solution are immaterial, but I’m pretty sure that if I could have been coaxed somehow into solving the problem as originally stated, I would have used essentially the same reasoning, though I would probably have expressed it using the tedious notation given in the problem. (As it happens, in my solution, I merely referred to the four regions as “A,” “B,” “C,” and “D.”)

  33. on 20 Aug 2011 at 9:35 pmBowen Kerins

    Hi Alex, sorry for pontification. I just feel it’s a good thing to avoid words like “trivial” and “easy” since they are never objective. I don’t know if you were deeply distressed by my comment or others’ — who and what was distressing?

    I am not currently teaching geometry, so there may be some comfort there. I do not feel this problem is trivial for all teachers; maybe it actually is.

    Your solution included an additional drawn segment and used the property that a triangle’s medians create a 2:1 ratio on either side of the centroid. The thinking to generate such a proof is what matters: what made you want to draw in BD then look for medians? This is at the heart of good geometric reasoning and what I feel should be at the heart of a geometry course. Coming up with a proof is more important than presenting it.

    The dynamic representation in the video led me to an observation I might not have made otherwise: the heights of the right and bottom triangles are always equal. This led to a proof I now realize is the same as Nathan’s comment #8:

    – The left and right triangles are similar by AA, and the side ratio is 2:1, therefore the area ratio is 4:1.
    – The bottom and right triangles have the same height (since the vertex is on the diagonal). Their area ratio is the ratio of the base lengths, 2:1.
    – The left and bottom triangles therefore have area ratio 4:2. They also combine to form half the square, so the left triangle is 4/6 of 50% (or 4/12 of the total), the bottom is 2/12, and the right is 1/12.
    – The other shape is what’s left: 5/12.

    I still wouldn’t call that easy, but it has the benefit of being general (it can work for a 3:1 ratio instead of only working for 2:1) and relies only on facts readily available to all high school geometry students. (I do feel all students should learn the property of medians you use, but not everyone learns that.)

    A dynamic representation helped me come up with a proof I would not have otherwise seen, so I do feel it had some bearing on the solution. Aren’t there lots of problems in geometry like this? The shape formed when you connect the midpoints of a quadrilateral, for example: dynamic geometry makes it much more tractable to find the property, and can offer insight on how to solve it, including the ability for students to investigate unusual or limiting cases. I have seen students discover a lot of mathematics — properties AND proofs — more efficiently through use of dynamic geometry.

    Anyway, thanks for your terrific website and best wishes in your
    work.

    – Bowen

  34. on 21 Aug 2011 at 4:44 amAlexander Bogomolny

    @R. Wright and Bowen

    OK, I do not insist. If it takes an animation to get one to interested in a problem, so be it. My warning to Dan (it was at the post of several cars and vans and making transportation arrangements) stemmed from the belief that there is no one right approach; so that Dan’s could not be that. I would be inconsistent to expect that my opinion is universally correct.

    One thing I still have to say. To be successful a teacher should know more that is going to be expected from students. It’s absurd I think to talk of problem solving with practicing solving problems. Thus I would expect from a teacher to be an easier target for a problem bait than a student.

    Bowen, thank you for the kind words.

  35. on 21 Aug 2011 at 6:10 amShari

    @Climeguy
    The teachers at the NCTM Institute were unimpressed for different reason than mine. I saw a problem with engagement and making the activities “work” in a classroom of students with a broad range of abilities. I also was disappointed in the complete disconnection to the real world.

    Most teachers, on the other hand, didn’t have any problem with the lack of connection to the real world. They felt that real world problems, as they have experienced them in textbooks, are fake making them more confusing than helpful. It sounded like they had given up on real world problems because they can’t find a good source. Even a teacher of “enrichment” classes felt that he might be able to write a good, real world problem once a month. And that only HE could write problems appropriate for his students because only HE knew their abilities and their interests.

  36. on 21 Aug 2011 at 8:10 amGarth

    As I see it the problem can be “easy” and “trivial” or it can get pretty mathematical. Being a trivial kind of guy I prefer that type of solution. My approach would be to draw the square and segments on paper, weigh the paper, cut out the shapes and weigh them. Poof, all done. Not quite as fancy as all the ratio and slope business but it sure makes a lot more sense. Years ago the Montana State governor wanted to know the geographical center of the State for some strange reason. The story goes that he calls up the math department at the University of Montana to get some mathematicians on the project. Of course being mathematicians they start on all these complicated geometric solutions. Then some brilliant soul (supposedly a kid) says “Why don’t we draw the outline of the State on a piece of cardboard and balance it on the tip of a pencil. Won’t that give a good enough solution?” So much for complicated solutions to easy problems. Math teachers sometimes need to step out of the mathematical box and just solve the problem.

  37. on 21 Aug 2011 at 2:53 pmNathan Chow

    On a completely different aspect of this video problem…

    I thought the original video posed, without Dan’s narration, was more powerful engagement for the student. Instead of some narrator directly asking them the question, the question seemingly arises naturally from the student’s own curiosity (of course, the video has been carefully constructed that way via #anyqs). The question is in their _own_ voice. It’s a more authentic problem from them to tackle, as it’s their question.

    (The original video tweeted is here: http://vimeo.com/27451133)

  38. on 22 Aug 2011 at 7:02 amMathcurmudgeon

    I, too, would rather see the question posed as “How big is each piece?” without the % marks because it fairly screams fractions.

    A possible refinement: a sheet cake could be cut and one person claims the smallest piece as being “a good serving size for one.” How many servings are the other pieces?

    My solution had to do with the two lower pieces adding to 1/4 and then noticing that each small triangle had the same height (P – MC = P – BC) but that one had twice the base of the other. Therefore one was twice the size of the other, making 1/12 and 2/12. Subtraction from 1/2 gets us the other two areas of 4/12 and 5/12.

  39. on 22 Aug 2011 at 7:04 amDenis Roarty

    This thread was fascinating, as occasionally happens on Dan’s blog because of the dialog between those defending “pure math” or traditional math and those with a pragmatic preference for simplifying things…

    Perhaps it is the engineer in me that values both approaches… but I couldn’t have been a successful engineer if I hadn’t been taught the symbolical language. More complex problems require that you be able to communicate this way with others. That said, I think I got to the “pure” approach, as all of us do, via more pragmatic, real world, and scaffolded examples which I take to be the point of this blog.

    So this problem, as originally presented would certainly be a killer for any student who hadn’t already been scaffolded to that point. I would want my students to all be able to do this problem as a test of their reading of the symbols and application of the concepts. But that would come after I was sure they had been taught to be literate of the symbols involved and after they had experience with the concepts involved. I believe both are important. Equally important is the scaffolding that gets them there… whether it is Dan’s video or a set of simpler problems that lead up to this one.

    I can’t imagine anyone presenting such a problem to students out of the blue, as may have been done at the conference. So maybe presenters of cool math problems need to talk about how they scaffolded the thing and why the fun little problem has value.

    It would also be nice if cool little problems did actually have a real world application. But I am OK with some of them not having this. This one appealed to me for its simplicity and its versatility.

  40. on 22 Aug 2011 at 2:53 pmTom I.

    To me this seems like a classic, Martin Gardner-style math problem : enticing in its abstract simplicity. I don’t see what’s obscure about it, and I wouldn’t worry about how to make it hip or “meaningful.” I bet 9 out of 10 readers of this blog thought it was a fun problem and felt an itch to solve it. Why wouldn’t students feel that way?
    I like this blog, but it seems like Dan is always recommending that we (more or less) apologize to our students for the abstractness of math. The abstractness makes it hard, but must we assume that it makes math pointless and uninteresting for our students?

  41. on 22 Aug 2011 at 5:14 pmHeber

    “enticing in its abstract simplicity”

    I am not suggesting that anyone should being apologizing for their personal enjoyment of abstract math. To assume, though, that everyone must share that same enjoyment is perhaps a bit naive (like when my brother thinks I should like rap music just because he does). For many (if not most) people, abstract math doesn’t seem to have any significant value (and I think this is sad). Math is a tool for understanding, defining and solving problems, and for most, that means real world problems, or at the bare minimum, a problem whose result has some kind of value or meaning to them. I will admit, though, that appreciation of abstract math is definitely something that can be gained (and have seen some of my students gain it). Using purely abstract models for math with success may be possible (or even preferred) after reasonable time has been spent to teach appreciation of abstract math.

    The original format of this problem is very abstract. Most students will see no reason to think about it, much less solve it. The video presents a model that most students will find easy to solve (1:1:1:1 or 1/4 or 25% or .25 or however the students want to understand/represent the area relationships). At this point, a slight change is made to the model, and students are asked again to try to represent the area relationships. They see the change as it is being made. Their intellect is being directly challenged. The original model convinces them that the problem is solvable, the video suggests some possible clues to how to solve the problem. This, I believe, makes the abstract more enticing and engaging.

  42. on 22 Aug 2011 at 8:09 pmAdam P.

    Is anyone else having trouble downloading the GeoGebra Applet?

  43. on 22 Aug 2011 at 10:27 pmJ Michael Shaughnessy

    Been reading your sound-off and blog about the geometry problem from the institute–after your talk. 

    In this case, it was ‘me’ who came up with that problem–borrowed from an MT article a number of years ago. You can check the way it appeared back a while in my PTP column in my President’s message a few months ago. Indeed we did pose the general case–after ‘what about the 1/3 mark’ the ‘1/4′ mark? It is a really cool generalization. And I have gotten a LOT of mileage out of this problem with middle school kids, high school kids, perspective teachers, and, they actually don’t have to have a dynamic geometry tool to make a lot of cool headway with the problem. Actually, elementary teachers I’ve given it to do a great job attacking it using paper folding strategies. Not all math problems have to be posed everytime in a a high tech environment. Sure, it’s ‘cooler’ that way, but i completely disagree with your comment on this one, about ‘how the problem was posed.’ It’s only boring in the beholder’s eyes, depends on how it’s pitched to a group. 

    J Michael Shaughnessy
    President of the National Council of Teachers of Mathematics 

  44. on 25 Aug 2011 at 6:58 pmYost

    I also can’t download the GeoGebra applet :(

  45. on 26 Aug 2011 at 8:57 amClimeguy

    Reply to Shari and Mickey:

    @Shari As long as the curriculum continues to be revisions of Traditional Math (1.0) teachers will continue to try to make their teaching more interesting by writing their own problems. But alas that does not scale much as you have pointed out. We need bolder, more creative “curriculum” that engages more students and is not so labor intensive for the teachers. Sadly, these our not coming out in our mainstream curriculum pipelines any time soon as far as I can tell.

    With all due respect to our current NCTM’s president Mickey Schaugnessy’s comment:

    Sure, it’s ‘cooler’ that way, but i completely disagree with your comment on this one, about ‘how the problem was posed.’ It’s only boring in the beholder’s eyes, depends on how it’s pitched to a group. 

    I say, you are right. It’s always how its pitched, but why choose problems that are difficult to motivate for most students? I’m not talking about the honors level students or the kids who like math, but the average ones that we fail especially in the urban school districts? And that attitude is what keeps us stuck in the old paradigm where the vast majority of students don’t get much out of math and just go through the motions to satisfy course requirements.

    This problem is fine for a traditional geometry course. if I was teaching such a course I would use it. But this is nothing new; it’s business as usual. And it’s definitely not one I would share at a showcase conference intended for teachers of grades 3-8.

    -Ihor

  46. on 26 Aug 2011 at 7:52 pmJohn Scammell

    Dan, I’m late to the party here, but this is some of your finest work. As posed initially, many students would be left wondering what was being asked of them, and wouldn’t really know where to start. By creating the simple animation, you have brought it to life, and made it compelling. Every student now knows what is being asked. They can make a guess. They have an entry.

    I say it’s some of your finest work, because of its simplicity. You haven’t taken a flashy or already compelling video and turned it into a math question. Instead, you have taken an existing question and made it compelling. No teacher can look at this and say, “Yeah, but how does it fit into my curriculum?” It’s straight out of their curriculum.

  47. [...] amigos. I'm back from Spain, back in the game after sidelining myself for a helluva comment thread. It turns out that NCTM President Michael Shaughnessy designed the task that I critiqued in a [...]

  48. on 17 Sep 2011 at 7:16 amnerdypoo

    how interesting. i am a math teacher who came to math teaching by way of an alternative certification program, and therefore have no real background in math. i teach only lower grades and have never taught a strictly geometry course, but the curriculum i currently teach in has a spiraling nature, so there is a lot of geometry covered.

    i called the biggest triangle region I, the triangle on the bottom next to it region II, the smallest triangle region III, and the non-triangle region IV.

    here were my thoughts:
    I + II = 50%
    II + III = 25%
    4*III = I

    i then began a process of guess-and-check, which was a great thinking tool for determining the reasonableness of my initial guess (which is something i tell my students to always think about). i began each time with guessing a value for III and seeing if everything would work out. i had a few a-ha moments and ultimately settled on I = 33 1/3, II = 16 2/3, III = 8 1/3, and IV = 41 2/3.

    what i think is fantastic about this puzzle is that it can be approached in so many ways. since i do not have a huge background in geometry, i found the problem to be a great puzzle, and i certainly did NOT think about medians, centroids, heights, or anything like that, as many of you did. i relied only on my knowledge of similar triangles & how to calculate the area of a triangle. i made a bunch of sketches and fooled around with ideas. this is why i think it could be good for kids of various levels.

    i teach in the IB and for me, this would be a great investigation where kids had to explain their thinking, what steps they took, what initial misconceptions they had (like perhaps that since the similar triangles were in the ratio of 2:1, their area was also in the ratio of 2:1).

  49. on 25 Oct 2011 at 10:18 amAdam P.

    I’ve talked to a few colleagues who also couldn’t open the Geogebra answer file. Any chance of you trying to make a new link here in the comments we could try? :)

  50. on 06 Nov 2011 at 3:11 pmNate L.

    This problem is similar to a problem Michael Serra spoke about at CMC South Palm Springs in Nov ’11.

    He uses patty paper Geometry and the Book “Origamics” (can’t remember author) to pose the problem.

    The solution is very elegant when done folding as well!!

    Patty Paper doesn’t = Video, but anoter medium and cool to see the versatility of this problem.

  51. [...] This is a more accurate representation of how I solved the problem. (I had to decide that a table and a graph would be helpful. I had to decide that a linear equation would be the best model. No one gave me any of that.) It's also more perplexing to see a problem as it exists in the wild, "posed simply and innocently, not flayed alive by terminology, labels, and notation." [...]

  52. on 03 May 2012 at 6:35 amYves Combe

    Thank you for this interesting question.

    Sequel:
    – if the shape is not a square? Is the result the same in a rectangle, trapezium, parallelogram, rhombus… what shape give the same result?
    – if the middle point is not at 1/2, but at p/q? at x (0<x<1) ?
    The smallest triangle is 1/12 of the square. Where put the point to get 1/144 ?

  53. [...] response one year ago to a commenter who said I was always recommending that math teachers apologize for the abstractness of math: [...]

  54. [...] Math teachers grossly undervalue these tasks. [...]