Math Circle – Billiards

A new year of the Mercer County Math Circle will be gearing up soon. I’m also teaching a class once a week this year at PLC called “Math to Love”. It’s meant both for folks who like math and want more, as well as for those who don’t like math and so need something to love all the more. Hopefully I can attract some people to attend.

For our first Math to Love class, I made a lesson that I’m also hoping to share at MC^2 soon. It’s about billiards.

circle-billiards trapped pentagon

There is sooooooo much cool math that involves billiards.

The reason I picked billiards to feature at this particular moment is because two of this year’s Fields Medalists study billiards: Maryam Mirzakhani and Artur Avila. To find out more about these amazing mathematicians, see our recent Math Munch post.

Here are some ideas for sharing billiards with kids that I used, or will use, or that someone (you!) could use.

A couple of cool facts to mention, illustrate, or expand on:

  • the short videos about Maryam and Artur, linked to above.
  • the reflection property of the ellipse (billiards video), and potentially the other conics
  • the fact that it’s unknown whether every triangle has a closed billiards loop. (Richard Evans Schwarz has made substantial progress on the problem.)
  • Using Cinderella software, you could share this applet that can show a variety of billiard phenomena.

A couple of tasks:

  •  this investigation of billiards in rectangles that I made. It’s something I first learned from Avery Pickford years ago. There are all kinds of great questions that arise through this investigation about topics like scaling, common factors, odds and evens, division with remainders, and more. (Similarly, at Illuminations.)
  • some version of the “bouncing icon” WCYDWT that Dan Meyer shares in these posts.
Which corner pocket will the billiards ball land in?

Which corner pocket will the billiards ball land in?

One more way to bring this year’s Field Medalists into your classroom? Print out a copy of this poster that I created.

I had a lot of fun sharing these billiards ideas with my “Math to Love” class and I’m looking forward to doing it again at the math circle. Hope you found something here to enjoy and to use!

fields medalist poster final with blurb

smOOC!

Last summer I ran a smOOC (small, open, online course) designed for teachers called Math is Personal. I’m running it again this month. If you think you might be interested in unpacking your relationship with math—and also building it up—you can find all the details on this page. The course starts on August 10 and runs through August 31. Registration opens on Monday at noon. Please share this opportunity with folks who might be interested!

Math Circle – Halving Fun

I recently led a recreational session at (MC)^2—that’s the Mercer County Math Circle. I called the session “Halving Fun”. I don’t remember exactly how I hit upon halving—maybe because of a partition puzzle I’ve been contemplating—but I knew I was excited to share how much great mathematics can come from such a seemingly simple concept. Here’s the session description:

Join us as we “halve” some fun while creating some art and solving some puzzles that involve breaking shapes in half. What’s the most beautiful way you could cut a square into two equal pieces? Is it always possible to cut a funky shape in half through a specified point? Whether on grids or with free-form shapes, halving tasks provide many fruitful mathematical opportunities. Come explore and these questions and more!

I kicked off the session by showing off some examples of halving art, and giving a little spiel about each.

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Here’s the slide deck if it’s useful to you. I also shared the Yoshimoto Friends video that Paul and I made.

Then I set everyone loose to either make some art or try out some of the problem sheets I’d brought. I also had some physical “halving” puzzles and other models for attendees to play with.

Here are the problem sheets:

  • It’s Half – I’ve gotten a lot of mileage out of geometric fraction problems like these. On this sheet, you already know that each picture represents a half. The goal is to make clear just why this is the case.
  • Half Frames – These are two problems that anyone can work on, but that have rich connections to algebra and geometry.
  • Halving – I always share this sheet with my Geometry classes, ever since I found it in Girls’ Angle Bulletin a number of years ago. It’s also a great jumping-off point for students to make problems of their own!
  • Half Tetra – This is a template for a 3D puzzle to cut out and fold up. Two identical pieces combine to make a regular tetrahedron. Cardstock recommended. Via Futility Closet.
Can you say why the hexagon is half shaded?

Can you say why the hexagon is half shaded?

Attendees made some lovely Truchet-inspired drawings, and one attendee especially dug into the half frames problem—coming up with an interesting conjecture in the process. :) A middle school math teacher who attended even planned to share some of the problems with her students. Maybe you’ll find fit to do so, too. Let me know if you do!

Math Circle – Celtic Knots

On Saturday at (MC)^2, Harini and I led a session on Celtic knots. Many of the ideas for it were taken from these great articles on the NRICH website. Here’s the description we sent out ahead of time:

It’s St. Patrick’s Day this week! We’ll look at some ornate Celtic knots, observe and explain some patterns, and make some knot drawings of our own. We’ll also discuss the difference between knots and links, ways to identify each, and some other basic topological ideas. Come shamrock some great math with us!

Here is the warm-up sheet we gave to people on Saturday as everyone arrived and got settled.

How many bands are each of these designs made up of?

How many bands are each of these designs made up of?

We started with some introductory remarks and showing a few examples of Celtic knots:

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Here are all of the examples we showed.

Then we played Vi Hart’s video “Snakes + Graphs”:

examples curveAfterwards we focused in on the first doodle game Vi shares. We drew and talked through a simple example on the board, then game folks time to try some of their own or to construct the over and unders on this example curve.

Next we shared a second, much more restrictive method of drawing knots. It produces images like the A and B weavings pictured above. We showed Alison’s video on this NRICH page twice and let folks follow along with this sheet (or work on whatever they liked).

Then I shared a third method for drawing knots that I learned through these lecture notes. Here was my first try, which I made in preparation for the session:

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Here’s the handout of dot paper that we passed out for people to try on. You’ll need to tilt the page at a 45° angle.

We showed this photo of a project Harini did at a Bridges math art workshop a couple of summers ago, as inspiration for them to take next steps on their own at home.

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Much of the 45 minutes of the session was giving attendees time to try out these methods on their own and giving them support and answering their questions as they arose. As a result, it was a much quieter math circle than what I’ve been used to, but still a lot of fun. People seemed pretty happy, and lots of great knot drawings got made.

Dan’s Circle-Square Challenge

Dan Meyer has posed some questions in a recent post about the merits and potential of the following problem:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

Dan asks:

What can you do with this? How can you improve the task?

Here are the thoughts I had (comment-turned-post) and an attempt at reworking the task.

Experiences with problems often become more powerful when they’re not just one-offs—when they’re surrounded by some kind of context. One kind of context that I find fruitful is the notion of a problem space. Most of the interesting aspects of the circle-square problem that Dan highlighted—like “assigning a variable” and “constructing a model” and “using operations on it”—aren’t specific to the geometric figures of circle and square. They equally well apply to similar questions (some easier and some harder) involving many different kinds of shapes. The circle-square combo does have a nice “classic” feel to it, but if we care most about the mathematical practices and methods involved in solving it, then we might do well to also explore variations on it.

In addition, the circle-square task as it stands puts the bar at one specific level of difficulty and requires certain algebraic and geometric skills to be firmly in place for a successful solution—and really even to make any significant headway. Whenever possibly I prefer to lower the threshold for success in a task while also raising the ceiling—an easy way to let students self-differentiate a task.

Here’s a first shot at a warm-up worksheet to introduce the problem space:

stringchop

I might have string and scissors on hand for students who were interested in using them, but my sense is that the problems as I’ve stated it—and the solution methods students are likely to take—is sufficiently concrete to make manipulatives perhaps useful and fun, but not necessary.

What would happen after student had a chance to work on this sheet on their own for a bit? Well, they’d have time to share what they came up with in pairs or small groups. Then we’d do some sharing as a whole class and discuss solutions and conjectures. That’s another thing—there are so many equally and perhaps even more interesting mathematical questions that can arise in this particular problem-space than the circle-square one that we’re (likely) headed to and that prompted this whole adventure. I’m sure the directions that students would take this in and the questions and problems they would pose would be fascinating and worth pursuing. They’d also likely provide richer and more powerful mathematical experiences than the circle-square problem on its own would, and solving this problem would carry a new significance in light of the other problems that had been posed and solved prior to it.

I can imagine using this warm-up worksheet with middle schoolers, but I also think it would be an appropriate onramp to the circle-square problem for sophisticated high school students. For both audiences it will clarify the circle-square problem and build context and experience for the students. If the circle-square problem is the main course, it deserves both to have appetizers to whet everyone’s appetites and to make sure everyone has a seat at the table before the meal is over. Because let’s face it: mathematical meals in school are all too often hurried affairs.

Those are my thoughts so far. What do you make of them?

I Choose Fractions

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On Monday I was walking with one my colleagues (Paul Scutt) as we tromped around Cranbury, NJ, on an excursion to the local historical museum and the store fronts of the town’s main street. We’d just had a bit to eat at Teddy’s, a local diner. Paul was wondering about a possible discrepancy in the bill and was calculating what the price of one drink must be if the total we’d been charged for three drinks was $5.25.

Paul was working out the quotient aloud in the style of long division. “3 into 5 is one with 2 left over…” Going through the steps was tricky—there was some retracing steps and checking involved—but things carried through and the answer came out as $1.75. This squared with the prices on the menu. All was well.

Through some of this I was only half listening. My thoughts were distracted by something or other; perhaps I had already begun my alternate calculation. Soon enough I was sharing my thoughts with Paul: “$5.25 is 5 and 1/4 or 21/4. 21/4 divided by 3 is 7/4, and that’s $1.75.”

It was an occasion to recall that over the course of my years of teaching math—and middle school math, in particular—I’ve developed a preference for fractions over decimals. Decimals, I’ve been inclined to argue, are just a special case of fractions—fractions with powers of ten in the denominator that we can suppress with a bit of ingenious and pragmatic notation. And decimals are a very slick trick—calculating with them is practically verbatim from whole-number arithmetic.

But then again, seen in a certain light, so is calculating with fractions—whole numbers that we calculate with while carrying them around in separate buckets. And since calculations with fractions have their own shape to them—their own kind of mental juggling—their meaning isn’t sedimented and murked up with their relation to ordinary place-value arithmetic. What is gained in ease is often purchased with opacity. This is true throughout math, and I’ve certainly found it to occur with decimals in particular. From these experiences, I’ll claim that it’s easier to help kids to think about fractions than about decimals, and that a teacher with a particular bent of mind can safely leave decimals as an afterthought. (Note: not “should” be.)

The ability to think flexibly about both decimals and fractions and to move between them comfortably is certainly the ideal. But how do we help students get there? What are the paths that we carve?

This post was prompted by an exchange on Twitter:

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What thoughts do you have to add? How do you think about the relationship between fractions and decimals? How do you talk about this with your students, or how might it affect your classroom silently? What is your own relationship with computations involving fractions and decimals—any preferences, for any reason? And what are good examples of their relative strengths and weaknesses?

Screen Shot 2014-02-19 at 9.45.18 PMScreen Shot 2014-02-19 at 9.45.24 PMWhat lines do you draw?

Mission #8: Sharing is Caring in the MTBoS

Justin Lanier:

We’re wrapping up “Exploring the MathTwitterBlogosphere” this week, which makes it a great time to share your favorite corners of the MTBoS with colleagues, students, and others!
Thanks, everyone who participated in Explore MTBoS! :D

Originally posted on Exploring the MathTwitterBlogosphere:

It’s amazing. You’re amazing. You joined in the Explore the MathTwitterBlogosphere set of missions, and you’ve made it to the eighth week. It’s Sam Shah here, and whether you only did one or two missions, or you were able to carve out the time and energy to do all seven so far, I am proud of you.

I’ve seen so many of you find things you didn’t know were out there, and you tried them out. Not all of them worked for you. Maybe the twitter chats fell flat, or maybe the whole twitter thing wasn’t your thang. But I think I can be pretty confident in saying that you very likely found at least one thing that you found useful, interesting, and usable.

With that in mind, we have our last mission, and it is (in my opinion) the best mission. Why? Because you get to do something…

View original 501 more words

A Day in the Life at PLC

ditlifeI haven’t blogged about my day-to-day at PLC at all, so I’m glad to have #DITLife as a kick in the pants to do some writing and reflecting.

We’re on a weekly schedule at PLC. Classes happen once a week. Every day of the week is different, but each week has the same rhythm to it. That said, every day is different, because a lot of what happens at PLC doesn’t happen in classes.

Note that we call the teens who attend PLC “members”, rather than students.

8:30 am – Arrive at PLC. Say hi to folks who are already there in the common room. Find a spot on the couch to organize for the day—email, browsing some videos about Thomas Was Alone for my video games class, and putting together some feedback on an essay one of my mentees wrote to apply to Simon’s Rock College. Chit-chat with members and staff. Lay plans with a member to make a giant pompom—more on that later.

9:15 am – Help to greet a potential new member who is visiting for the day–A, an eighth-grader from a local public middle school. She and her mom had come by on Wednesday for an hour or so, for a tour and an initial conversation. (The conversation was exciting, because it was the first time I did one of those initial meetings on my own.) They also attended our Outside the Box event last night. I showed A the day’s schedule of classes, then passed her off to a couple of members who were nearby and went back to my couch and computer.

9:40 am – K arrives to PLC and have a brief mentoring meeting—checking in on how his application to Simon’s Rock is going, chatting about the fun we had ice skating on Monday, and finding out how his week’s been.

10am – Introduction to Programming. Three members, me, and A  worked on Scratch projects in a classroom. One member is working on a platforming game with some unusual physics, another is just starting a new game after his successful “Underwatermelon”, and the third searched the web for sound effects for her Dragon Ball Z fighting game. A was new to programming, so I set her up with the tutorial on scratch.mit.edu, and before long she has a couple of sprites dancing around with sound effects. Alongside of checking in on their progress and chit-chatting with them, I started trying to make a Scratch “Pong” game, but then headed in the direction of making a “Pong” musical instrument instead.

11am – Last week we had a wine-and-cheese outreach event for local mental health professionals. It was held at K’s family’s house. As people were arriving and mingling, K had the idea of showing me his pompom making supplies. (Like this.) And then, of course, we made one. :D So this morning K came up to me and said we ought to try making a huge pompom with a hoop-thing we’d make ourselves out of cardboard. And so we dove into that during this hour.

I also shepherded A off to go with a staff members and a couple of members to a construction site nearby. Bob Hillier, a local architect, was going to give them a tour of the site. A had at first thought she’d stay at PLC to see me work with another member for our calculus tutorial—but it’s especially good that she went on the trip, because my calculus student had been busy with other things this past week and so we decided not to meet. And I made a giant pompom with K  instead.

12:15 pm – Recess! This is a class I run on Thursdays during the noontime lunch period. We go outside (typically) and play some sort of athletic game. Today we played Spud. I think we got up to eleven participants at our max. A simple game, not as exerting as some of the other games we’ve played, but a lot of fun. We adapted the rules so that we “ran away” in different styles, like “baby steps” or “spy crab”.

1 pm – Briefly checked in with a member who’s one of our photographers. We’ve been trying to put together a system for getting photos they take onto PLC’s website, Facebook page, and elsewhere. B laid out a sensible plan that we’ll run with.

1:10 I usually have a Geometry tutorial this hour, but I checked in with N and he decided to hang out in the common room instead for the day. He’s really self-motivated and has been working through a Geometry book on his own and with the support of his grandma. Our meeting times are really just him working on his own, with him asking me an occasional question or me offering some additional resources or ideas.

So for a chunk of this hour today, I went back to doing some of my own odds and ends of work. I did, however, meet with a member M who doesn’t do classes very much but who has been talking off and on about getting back into doing some math. He had asked me about my geometry tutorial earlier in the day and I said I’d print him out a couple of angle chasing worksheets. I saw him going into the workshop during this 1:00 hour and poked my head in. He’s in there a lot, working on various construction projects. So I invited him over and he and I sat down and did some geometry for part of the hour.

2 pm – Video Games class. I had sent around a link to Thomas Was Alone in the interval since our last class. Some kids had played it some, others weren’t interested in it. We chatted some and played the video games we wanted to some. Pretty low key.

3 pm – With the day wrapping up, I found A to chat about how her day had gone. For a while it was just me and her and later another staff member joined us. A talked about details she noticed at the construction site, about how her long math session with a staff member had given her the chance to ask many of the questions that she hadn’t felt comfortable asking in class, and about how at PLC there are so many opportunities to interact with people and how everyone is so friendly.

So it seemed like she had had a good day. :)

We then talked about different timelines for joining PLC that are possible—right away, after Thanksgiving, at semester—and how she might approach explaining her choice to the people in her school life—if in fact she decides to come to PLC.

3:30 pm – Chatted some with A and her mom and two staff members out in the common room as they (and we) wrapped up our day. Hopefully they’ll have a good chat tonight, and maybe we’ll see them again soon!

Come Explore the MTBoS!

“The time has come,” the Walrus said,
“To talk of many things:
Of shoes–and ships–and sealing-wax–
Of cabbages–and kings–
And why the sea is boiling hot–
And whether pigs have wings.”

Yes, the time has come. But not for talk of cabbages or flying pigs or any of that. No, the time has come for…

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Are you…

  • just becoming aware that math teachers learning from each other online is a thing that happens, and your interest has been piqued?
  • new to Twitter and blogging, with your feet wet, but looking for a chance to really take a deep dive—to  get more involved?
  • someone who’s blogged and tweeted for a while, but wanting to try out some new things, meet some new people, and engage in a little MTBoS fiesta?
  • reading this post?

Then Exploring the MathTwitterBlogosphere could be right up your alley!

ExploreMTBoS is low-stakes but high-reward. It’s eight weekly “missions” to scope out different ways of engaging with people in the MTBoS, plus an opportunity to write about your experiences. That is, eight weeks of “here’s a thing you could do” and “here are some writing prompts about it or something related”. You’ll have the chance to read other participants’ reflections, chat with them on Twitter, and…flash mob?

So head on over and sign up. You just have to introduce yourself in a comment. Then be on the lookout—through the blog, Twitter, Facebook, or email subscription—for your first “mission” on October 6!

***

On a personal note, one part of why I’m excited to be helping out with Explore MTBoS and participating in the missions is that it’ll help me to stay connected with a community I care about and to make new connections. I’m in a different headspace these day and have different kinds of tasks and challenges on my plate—like coordinating volunteers, putting together field trips and events, and helping with open houses for potential PLC members. And while I’m working with a few kids one-on-one with math and I’m running a Math Munch class once a week, much of my teaching is happening in other subjects—physics and programming and video games and poetry and PE.

Anyway, this is just to say that I’m having a blast with new things, but I’m grateful to have this way to stay engaged with the MTBoS. I hope you’ll join up for Exploring the MathTwitterBlogosphere, too, and encourage others to come share in the fun of the MTBoS. There’s something for everyone!

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Daily Desmos: Phase Two and You

If you haven’t scrolled through the goodies over at Daily Desmos before, scurry over there now! (If the link brings you nowhere, know that Dan Anderson is doing a bang-up job of migrating the site over to a snazzy new WordPress venue, and that it’ll be back up soon.)

One of my favorite Daily Desmos challenges that I've created so far.

One of my favorite Daily Desmos challenges that I’ve created so far.

What’s Daily Desmos? Well, I’ll link you to our How to Play page, but basically it’s a collection of graphing challenges created by a motley band of math teachers. Each week, Monday through Friday, someone presents two graphing challenges—one basic, one advanced. The most straightforward challenges are just a graph and the challenge is to create an equation that will create that graph. Straightforward, but necessarily all that easy. And other kinds of challenges have cropped up along the way—dots to be hit or avoided, animations, partial pictures, and more. At this point we’ve created hundreds of challenges, and there’s no sign of us slowing down. Definitely try some out for yourself!

So what’s Phase Two for Daily Desmos, you ask? Well, we thought that we’d try to take all of the groovy graphing juices we’ve got flowing and focus them through some new constraints and with some new goals in mind. For the next several weeks we’re going to be operating with a theme of linear functions. We hope that this will help us to create challenges that might more easily fold into classroom use. Also, it’ll be interesting to see what new perspectives on the good ol’ straight line we’ll produce, given the milieu and habits we’ve worked to establish over the last months. We’ll tackle other function families in the future.

And there’s another component to Phase Two. These weeks will be a testing ground for interesting linear graphing challenges—generating ideas, throwing them up against the wall, and seeing what sticks. In the background we’ll be working to craft a sequence of linear graphing challenges that could help a student who’s new to linear functions to ramp up their understanding and fluency. The end product will be posted as a stand-alone problem sequence on the Daily Desmos site. The effort will be spearheaded by the inimitable and prolific Michael Fenton. I can promise that whatever we put together won’t be a mere bottomless pit of procedurally generated “graph this linear function” exercises.

And where do You come in, dear reader? Well, as Dan already blogged, we’re looking for some new Daily Desmos crewmembers. What does that entail, you ask?

  • Every couple of weeks, you’d create two graphing challenges. Whatever floats your boat. According to a theme, if one is operative. And you can even tap your colleagues and students as collaborators!
  • Then you’d post your challenges on the site. This is pretty simple—if you write a blog, you basically already know how. And if you don’t, we can have you up and running in a few minutes.
  • You can sign up for a fixed term—a couple of months, say—or have it be open-ended and then bow out and take a break when you feel like it.

That’s it!

Creating challenges for Daily Desmos has been a ton of fun for me. I’ve learned some graphing tricks in the process, it’s been a creative outlet, and I’ve gotten to work with some really fabulous and passionate tweeps. I can heartily recommend signing up. If being a part of Daily Desmos sounds intriguing to you, just tweet at me or another crewmember, or shoot one of us an email.

Tallyho!