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

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.

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 Ï€ and it worked really well, especially when we reduced the effect of the experimental error by finding the mean of our results.

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…

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

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

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…

What program do you use to construct this video?

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

Carl

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!

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.

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!

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?

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.

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.

@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.

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

@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.

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:

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.

“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.

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 

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

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).

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.