Make a mathematical mess in Challenge 08: The Confetti Problem. Or, be the first to take a stab at a number that's nearly 6000 digits long in Challenge 09: The Awe of Large Numbers.
We'll be back again in April with Challenge 10, so stay tuned!
We're taking a break this month, but there are still problems to be solved! Check out our recent challenges:
Make a mathematical mess in Challenge 08: The Confetti Problem. Or, be the first to take a stab at a number that's nearly 6000 digits long in Challenge 09: The Awe of Large Numbers. We'll be back again in April with Challenge 10, so stay tuned!
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Challenge 08 has received its first video responses, courtesy of Andrew. Check them out on the Challenge 08 archive page and prepare to be astonished by the size of some of the numbers he encounters!
Speaking of the Awe of Large Numbers, Challenge 09 Part 1 is now available. Have a look and share your thoughts in a response video! And while I have your attention, please take a moment to nominate the Collaborative Mathematics YouTube channel for inclusion in YouTube EDU. You'll need two links to do so:
Thanks for your support! Jason We're keeping up with our experiment in a new rhythm: a video every two weeks!
Now available: the second half of Challenge 08, which continues the paper rippingtheme, and provides teachers and students with yet another pedagogically sound reason to make a mess. Challenge 08, Part 1 explored a simple procedure for ripping up a piece of paper. Challenge 08, Part 2 presents a new procedure, with a small twist that leads to a whole bunch of new problems to investigate! Enjoy! Math can be messy sometimes. Grab some scratch paper and get a rippin' with Challenge 08: The Confetti Problem (Part 1).
Also, I'm going to try out a slight adjustment to the usual format. I've typically published one video per month, and that video has contained both a main challenge and some extension or elaboration. This spring I'm going to try separating those two aspects. There will be two videos per month, connected to the same theme. The first video will contain the main challenge, and the second will have questions that extend or augment the ideas from the first. I'll be interested to hear your thoughts on the new pacing! Plus, I look forward to seeing some response videos about Part 1. Stay tuned for Part 2 in about two weeks! We've been inspired by the internationalization (or is it internationalisation?) efforts of other video producers, like Minute Physics and Numberphile to launch the Collaborative Mathematics Translation Project.
In an effort to make our videos available to as many people as possible, we'd like to recruit help creating subtitles in languages other than English. Here's how you can help out:
Mange takk på forhånd for deres hjelp! ;) We're taking a break for the holidays and we'll be back in January 2014 to celebrate our one year anniversary with new video challenges!
In the meantime, we would love to: (a) try and spread the word to more teachers, and (b) get some feedback about how to make the resources here of more value to students and teachers. Please share the site with anyone you think might find it interesting or helpful. Please also consider taking a moment to complete this survey. Of course, you are also welcome to share your comments, ideas, or suggestions via Facebook, Twitter, or Google+. And, as always, check out the Challenge Archive for access to the challenges from this year. Consider joining participants from Belgium, Norway, Peru, and the United States by adding your data for Challenge 07: String Theory! Thanks again for your support! See you in the new year! This is part 2 of a discussion about Maile's solution to Challenge 03: Finger Counting. If you haven't explored that challenge or thought about her solution yet, please start here. In her solution, Maile observes that the numeral 1000 can represent different numbers. It might represent one thousand. In fact, I think it's fair to say it "usually" represents one thousand because we usually interpret numerals in base 10. But, the symbol 1000 could just as easily represent eight if we interpret it in base 2. (In fact, it could just as easily represent a lot of numbers: sixtyfour, four thousand ninetysix, negative twentyseven... but let's not get ahead of ourselves.) Maile answers the challenge by claiming that we can stop finger counting at eight rather than counting all the way to one thousand, since 1000 represents the number eight in base 2. She's right in that we do in fact land on the same finger. "But," I wondered, "Does that always work?" (As a quick aside, I have to appreciate the lucky randomness of the fact that I chose 1000 as the target number. That is to say, if I had asked what finger we'd be on when counting to 500 or to 2000, the connection to binary numbers would never, I suspect, have come up.) To answer the question "does it always work?", it's helpful to watch Paul's solution to the challenge, in which he makes the connection between this kind of finger counting and modular arithmetic. If a number is congruent to either 2 or 0 modulo 8 it will end up on your index finger. Since \(1000 \equiv 0 \mod{8}\), we end up on the index finger when we count to 1000. Said the other way around: if two numbers are congruent modulo 8, then they will end up on the same finger when finger counting in this way. So here was the question I had: Given a numeral made up only of the digits 1 and 0, are the base 2 value of this numeral and the base 10 value of this numeral always congruent modulo 8? This is a cool problem to solve, and you might want to kick this problem around before reading on. My gut feeling was that the answer was yes, but I needed to crack open a number theory book to prove it.
Challenge 07 from Collaborative Mathematics is here! Watch the video, try out the probability experiment, and send in your data. Or, tackle it theoretically!
Plus: we've received a great response to Challenge 06 from Dani, and a fantastic new approach to Challenge 03 from Paul. There are several very cool blogs out there  among them Math Mistakes and Bridging the Gap  that highlight the value of thinking deeply about student responses, the mistakes they make, and the explanations they give for doing things the way they do.
One of the ideas behind Collaborative Mathematics has always been that participants could learn something from exploring other people's solutions, in addition to exploring the challenges. Some of the response videos in the archive contain mistakes, and I think that's fantastic. Rather than being an answer key of tidy solutions, they represent a library of arguments and ideas that require additional critical thinking. One aspect of this is that I avoid posting my own answers to the challenges. I don't want my way of solving a problem to be interpreted as "the right way" just because I'm the one who posed the challenge in the first place. Over the next few posts, though, I would like to share my thoughts on a particular solution as a way to talk about my own mathematical curiosity, and what I've learned from a really creative approach to one of the video challenges. But before I start going on and on about what I think, I want to give you a chance to explore for yourselves. Here's what I suggest:
Next time I'll share how I've come to think about the mathematics of Maile's approach. This is my response to the first mission from Exploring the MathTwitterBlogosphere. My whole thing lately has been about trying to bring more openended problem solving into my teaching, so I'll write a bit about a trio of connected problems that I really like, and which I hope others might find interesting.           A note from the future: I've got to say that as I'm writing these problems, I'm both tempted and reluctant to talk about the solutions. I'm tempted to talk solutions because I find myself wanting to be specific about the kinds of conversations I like to have with students about these problems. I'm reluctant because I, personally, would (almost) always rather explore a problem on my own, at least for a few minutes, before reading someone else's ideas about it... and maybe some of you are like me in that way. I've chosen to avoid solutions, though I think it might make this post sound a bit vague and handwavy. I welcome your thoughts in the comments.           You may already know the first one. It's a classic: The Handshake Problem. If everyone in our class shook hands with everyone else, how many handshakes would occur? I like this problem because it lends itself to a variety of approaches. Students might try acting it out or drawing a picture. They might jump right in and try to answer the question for a large number of participants. Or, they might try solving a simpler problem: starting with smaller numbers of participants, organizing their data in some way, and looking for a pattern. Then, once we find the pattern (and perhaps even an equation for the number of handshakes given the number of people), I like that we can explain that pattern in a number of different ways. We could explain it purely using the context, or using arithmetic in a clever way, or using algebra, or by making a geometric argument... And, even better, we can connect all of these different explanations. I like to followup the handshake problem with a seemingly unrelated problem: The Polygon Diagonals Problem. A convex polygon has 170 diagonals. How many sides does it have? I like this problem because it forces students to employ an often helpful (and often necessary) problem solving strategy right from the start: solve a simpler, related problem. Students realize immediately that 170 diagonals is ungrokable; we have to start smaller. But then students usually start by drawing polygons and counting their diagonals, without realizing that they have inverted the problem. That is to say, the problem gives you the number of diagonals and asks you for the number of sides. Students usually start by fixing the number of sides (for example, by drawing a pentagon) and counting up the diagonals. I like that the strategy of solving a simpler, related problem emerges so naturally. Plus, the polygon diagonals problem has all the great features of the handshake problem when it comes to explaining the pattern and the equation that are eventually revealed. Then, I like to conclude with a problem that seems both (a) unrelated and (b) totally tedious: The Multiplication Table Problem. What is the sum of all the entries in the standard 9by9 multiplication table? After their initial groan, I'm quick to remind students that we've learned a lot that might help us to solve this problem without having to slog through adding up all of the numbers. This becomes an exercise in creative application, as well as a chance to explore other interesting patterns that emerge.
In fact, if you haven't ever explored the multiplication table problem, I encourage you to kick it around for a few minutes. I was surprised at the number of cool connections hiding in there. I look forward to hearing from people in the comments! 
CollaboMath:
