I worry that there isn’t enough processing time in between the first two examples that you show. If I were doing this with a class, I’d leave up those images for a lot longer.

Pretty slickety animation!

Questions I had about the video while watching:
How does the final rectangle communicate any rectangle? Why not an (n,m) vertex to echo Schoenfeld and your first examples? Or a variety of rectangles as the final image?

Why 7×10 for the first non trivial? Are you consciously suggesting classes or scaffolding approaches?

After 7×10 I could imagine a quick montage or 3-7 other rectangles.

When you want to ask a specific question, why not ask the question?

I like the video too, but I guess I’m puzzled by what it adds to experiencing the problem statement that is not provided by my three successive questions (that include an invitation to explore any number of rectangles – and the implication that the learner will need to decide what different cases are interesting) + the ggb to play with to take the imprecision out of hand-drawing? It would be nice if the intersected squares could light up on the ggb. Hm.

Is it just that the video provides a way to beg the interesting question instead of explicitly asking it?

And is that really such a pedagogical advantage, that it would be worth my time to learn how to produce such a video?

I swear I’m not being defensive; I’m thrilled that I’m famous on dy/dan today! Woo! I’m sincerely curious about these questions.

Using Schoenfeld’s or Kate’s treatment of the question dumps the kid into the open ended abyss that can be very overwhelming. Using two simple examples where counting is quite easy and then opening it up to a much larger example where counting is still possible puts the kid in a position where she says, “yeah I can count, but dang, it’s gonna take a long time. I wonder if there’s a shortcut.”

I kinda favor your treatment for that reason. I’m still struggling to reconcile the message of your previous post of “just ask the question” with this one. That is, unless your intention is to start with a closed question (ie. How many squares does the diagonal pass through in a 67×35 rectangle?) and eventually open it up to an open ended create-a-rule type question.

I favor Dan’s video, if students are not given the chance to explore their own virtually created rectangles in Geogebra or GSP. Kate’s prompt seems to be one of directions to a hands-on experience, which I think wins best learning experience nearly every time.
But I would ask the question, and in this way: “IS there a pattern to the number of squares the diagonal passes through in a rectangle?” after showing them Dan’s video. I’m not sure a typical high schooler would be convinced there was a pattern. Then follow it up with, “So what’s the pattern?”

I’d use Kate’s method to get them creating rectangles of their own and have them log their findings.

Schoenfeld’s version is dizzying it moves so quickly from specifics to generalization. There has to be an invitation to join the square counting party, which it lacks.

@Michael nothing special about perfect squares, but it does matter if the sides are co-prime, I think …

Keep in mind, Kate, that I’m looking at this through the lens of a middle school teacher who has 2/3 of his kids less than proficient. As soon as I say, “develop a rule…” they’re gone. Dan’s version let’s the student decide whether or not to create the rule.

And yeah, what you’re talking about is at least half our job.

Better or worse is a difficult question. I think it depends on the audience. I think David is right that Nowak’s (you’re published now, so last name it is) or Schoenfeld’s approach wouldn’t work for lower level students as well as Dan’s video. But for upper level students, I think it is important that they are able to translate the problem on their own without the frosting of a video. I think they’ll hook onto the problem just as well (or even better) using Nowak’s approach.

I like the visual progression of larger and larger rectangles in the video. To me it suggests that if I managed to count the squares touching the diagonal then there would be an even larger rectangle next. I’m not sure if that’s just because I know the general question is coming though.

Also, having taught many kids with EfL recently, I definitely agree with your point about language demand.

To me, the benefit of Nowak’s approach is that the kids will spend a little more time thinking about the specific cases and will be able to use them for reference when they get to the general case. Possibly that would be an example of making decisions for them on how to solve the problem, however(?)

I like the fact that Dan & Alan quickly get to the point of the puzzle, though Alan’s would require an audience that is comfortable with that level of abstraction. The shading on Dan’s video could make for some interesting patterns on bigger rectangles. I also like Kate’s approach, but would rather use verbal/visual instructions — maybe just project the first diagram, count the squares together, then ask them to make other rectangles & try to develop a rule. I don’t see video adding or subtracting value in this case. Depending on the class, and the amount of time you wanted to spend on this, I could see any of the three being the best approach.

Bowen: I think the choice of rectangles to use in the intro depends on the class. If you want a slam dunk start, 2×1 & 5×5 are nice choices. Kate also made a visually simple first case, though not as simple as 2×1 & 5×5. In some ways 3×2 is springing the meat of the problem on them right away.

They’re all great.

1. I’m drawn to Schonfeld’s because of the shaded squares. I know, I know: be less helpful, not more. What can I tell you? It got me where I needed to go, faster. Plus I don’t know, I like that NxM conversation.

2. Nowak’s gets the job done, like usual. Right mix of exploration and explicit instruction. But my favorite thing she’s ever done is this. Her original materials average a 99.3 out of 100. That absolute value revelation is a 648 on the same scale.

3. Meyer’s is the sleekest (shocker), the most visually compelling (durrr), and the clear winner for ELL. Just a monster effort. If it’s too fast, pause it. If you want to run something back, rewind it.

Audience-wise, I think Schonfeld works for a particular (possibly idiosyncratic) kid, Nowak’s works beautifully if you set the stage right, Dan’s gives the teacher more room for error.

I liked Schoenfeld’s best (other than the use of the word “comprise” which seemed a little too formal). I could see the advantage of giving several examples (both with relatively prime n and m and ones with common factors), but the video seemed like a waste of time, both to produce and to watch. Doing the same thing on the board (without erasing) would give the students several examples to look at and not take any longer.

Or another idea for inspiring a productive investigation:

1. Have each student draw a 2×1 on a sheet of graph paper and ask them to draw a diagonal. How many squares does it go through? They all certainly agree that it is 2.

THEN, have them all draw large rectangle (let’s say 20×35 or whatever) and have them draw the diagonal. How many squares does it go through? I’m betting money they don’t all get the same answer. So, which is correct? How can we be certain? Why are we getting different answers? I think most students are now ready to figure out if theirs was right.

Dan, I’m generally not pulling your leg. I think that you are sometimes so enamored of your video skills that you miss simpler and more effective low-tech solutions.

Having the students able to simultaneously see both rectangles with relatively prime edges and ones with common factors is far more important than having them watch an animation of a rectangle being drawn and counted, then expecting them to remember what they have seen.

Some of your videos have been good starting points for math problems (like the basketball parabola, which would be difficult to do on a whiteboard). But this one gains little or nothing from the video. Having a computer draw larger examples may be useful, and the Geogebra applet may be a good way to do that, if every kid gets to play with Geogebra at the same time. Otherwise, it is better just to have the kids sit down with squared paper and draw and look at a dozen or so examples.

I have watched the video a few more times. Do you have the boxes turning grey as a hint or to prevent a hint? At first I thought it was a hint. As a look at the problem more, I think seeing the diagonal cross a horizontal line is actually more helpful, and the grey boxes prevent the viewer from seeing this.

The language involved in presenting this could be minimal without using video. I am not sure I see the type of student you believe will benefit more from the video than pictures side by side. I don’t buy the “story telling” argument some have offered. The story is boring, you want to zip through it quickly to get to the interesting question.

For some classes the video might be great, for others it might actually be de-motivating and mildly insulting.

Has anyone done this on graph paper or on sketchpad? Graph paper could be a nightmore for students that struggle with motor skills, but I think having the diagonal drawn instantly makes it tougher to see some of the key ideas.

Why shouldn’t the teacher simply pose the question as follows:

“Consider a rectangle with sides of length M and N, divided into squares of equal size arranged in M rows and N columns. How many of these squares will a diagonal of the rectangle pass through?”

Wouldn’t the students be able to understand this?

Granted, the students will probably need to look at pictures of some examples in order to find the solution, but couldn’t they select and sketch the examples themselves?

Does the teacher really need to be more helpful?

@Zeno: To answer your questions (at least with regard to my students): no and yes.

Tony (and others above) have mentioned that Dan’s approach is better because students get immediate access to the question without needing to go through some technical language. One can argue (as some some have in comments above) about how much of a barrier this really is, but certainly both were written in the “math dialect” that exams, assignments, etc. are typically written in. Avoiding technical language and the math dialect can make it easier for some students to understand the problem.

But there is another result of using technical language to pose a problem. It sets the stage for the discussion that will follow. If a student is not very comfortable with this math dialect, it takes real guts to join the conversation. Not so with Dan’s approach. He mentions the questions:

“How many squares do you think it’ll cut through? Write down your best guess. While you’re at it, write down a number you think is too high, and too low.”

It’s easy to jump into that conversation, and that’s where I think Dan’s approach shines brightest.

@Tony: Why wouldn’t your students be able to understand the question? Do they not understand what a rectangle is, what a diagonal is, what rows and columns are? This is all very basic. If students don’t understand these things, wouldn’t it make sense to teach them before introducing the diagonal square counting problem?

@StevePhelps: For this problem, “passes through” means “intersects the interior”. Schoenfeld makes this explicit. Including squares which the diagonal only touches at a vertex would be a different problem.

@aaronthill: Avoiding mathematical language may make it easier for some students to understand this problem. Will it help them learn to understand mathematical language? Is being able to understand mathematical language important?

@Zeno Yes, it is important to understand the “math dialect” that we typically use when writing mathematics. But that doesn’t mean it must be a part of everything we do (especially a classroom discussion that is primarily verbal).

I think the crux of this conversation is a question that hasn’t been posed: How much literacy should we teach in the mathematics classroom? In the past, Dan’s videos have been about storytelling – an approach that motivates students to want to find the answer to a math problem. In this case, I think all three of the approaches are still very abstract. The only difference in my mind is the level of language that is being used from nearly none in Dan’s case to very academic language in the original Schoenfeld problem.

While I want my (minority, inner-city) high school students to succeed in math, what I want more is for them to succeed in life. For them to do that, the first step is success on the state exam needed to graduate, and then maybe the SAT. While I appreciate the videos that engage their math curiosity, I don’t think videos that simply circumnavigate language are helpful. More important would be teaching the tools to understand the problems as they are worded on (unfortunately necessary) standardized tests. In other words, mastery of math is GREAT, but if they can’t prove to society that they’ve mastered it, they’re never going to get a leg up.

Hey Dan… Like the video. This same idea was used by the Maths300 people here in Australia. The software they have does a great job of inviting students in with the questions we WANT them to ask (or at least think!). It animates the diagonal and gives a count of how many squares as it goes. You input the dimensions of the rectangle (up to 1000×1000) so you can go for relatively prime numbers first, then when they think they have it, introduce like a 6 by 4 rectangle. The students are always keen to see the software verify their guesses.
They also couch the problem in a story of an electrician laying cabling diagonally across a tiled floor.

I hadn’t done this stuff in a while, so rather than think of this as a presentation issue, I tried to figure out how to solve the problem. The way _I_ ended up figuring it out required two insights that can be illustrated with two rectangles.

First, if you take a rectangle whose side lengths are relatively prime, you might see that the number of squares crossed is really the number of lines making up the squares crossed (except for that pesky place at the end where you cross both the horizontal and vertical line at the same time).

With that, a second rectangle, which had, say, some common factor to the side lengths could help one to see that the pesky place at the end isn’t really a special case any more than the other places where you cross at a vertical and horizontal line at the same time.

Anyway, to me, providing these two steps could allow one to see what was going on without getting embroiled in really big rectangles with lots to count. The point, I think, is to help students to look at the fundamental issues, rather than get into a fruitless counting exercise. And, while brute force can provide some information, one needs to always brought back to generalization and simplification – that’s what mathematics is about.

I like the paper (over something online), because I can draw on it as I’m thinking about the problem. Dan’s treatment has the rectangles get bigger fast, but at least for me, getting bigger didn’t provide any insight, and really made things harder to solve. It was the nature of the rectangles, not their sizes, that was important for me.

“I once had a teaching assistant who taught in a completely different style than I do. I thought — this is never going to work — but actually it worked out well.”

Linda, what a breath of fresh air it is to hear you say that.

I think a worksheet with a few examples followed by an invitation to find the answer for some larger (printed) rectangles would suffice, ending with the open-ended question. The back page of the worksheet could be covered in grid paper so students can draw some rectangles of their own.

The video would be a nice introduction to the worksheet. I feel that it’s not necessary and the effort required to create it is not worth it, but *given* it exists, I would use it enthusiastically. If it didn’t exist, I think the worksheet would still do the job quite nicely.

What can I say? I like a nicely conceived and executed worksheet.

I spent some time last night solving the general case, and having access to that Geogebra applet was great for my motivation and exploration. I’d happily give students access to that. Is the file available for download? I want to learn how it’s constructed.

Final comment on the video: I am gobsmacked at its production quality, offer my congratulations at your skills, Dan, and would love to know how it’s done. What software, and how long does it take when you’re reasonably experienced at using that software.

Many mathematical concepts are crying out to be visualised. Geogebra and friends are great — we are so lucky to have them — but sometimes a well-produced video would be fantastic.