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Archive for the ‘Learning Through Problem Solving’ Category

In my quest for a good Three Act Problem for elementary level students, I’ve come up with two more ideas. It’s actually the same idea presented in two different ways. It’s going to hit division at the grade 3 or 4 level, I hope.

Hay Bales – Act I

Cars – Act I

These videos are rough. The car one could be fantastic if a car lot would let one of us come by with a video camera and film them loading a carrier. The hay bales one would be great if we could get a farmer to let us film him loading a truck. In the past week, I have driven to Calgary and back, Stettler and back, and Red Deer and back. I’ve seen a ton of those bales in fields. I haven’t, unfortunately, come across any of them being loaded up or hauled along the highway.

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I get asked frequently if anyone is compiling 3-Act Math stories in the style of Dan Meyer or learning through problem solving activities specifically for elementary school. I’m not aware of anyone cataloguing them at the elementary level, probably because most elementary school teachers have to teach everything, not just math. It’s daunting, and probably hard to focus so much on one subject.

Here’s the thing that occurred to me, though. Because Act I is typically completely visual, and we ask students what THEY wonder about, won’t they automatically wonder about things at their own level? I think that many of the ones presented on this blog would work in an elementary classroom with minor tweaks to Act II (and maybe Act III).

I’ll use the ticket roll video I posted yesterday to explain.

Act I – Same Video Used for High School Students

I assume elementary school students will also wonder how far Sarah will get. The high school kids will get dimensions of a ticket and work with area and volume to determine how many tickets are on the roll. What if we just change the Act II information we provide for elementary school students?

Act II – Modified for Elementary School

Tell the students that there are 2000 tickets on the roll, and show them the video below.

Act III – Same Video Used for High School Students

Play the same answer video from my previous post. The answer is the same. The payoff is the same. The math is more elementary.

The answer the will come up with (by dividing) is actually a little too close. I was hoping it would be messier in the end so the students could discuss why it might not have been so accurate.

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The first time Dan Meyer came to Edmonton, he had us work on this problem. It was engaging, and a room full of teachers dug right in. The second time he came to Edmonton, he explained his 3-Act Math format. During that session, which was almost two years ago now, I came up with an idea for a video Act I and Act III for the ticket roll problem. My idea was to film a roll of tickets unravelling along a marked football field. The math remains the same.

I didn’t film it for a number of reasons. Marked football fields that don’t have Eskimos playing on them are scarce in Edmonton. It would require several cameras, and I had only one. It would require editing above my level of ability. This week, it all came together for me. We are doing more and more video editing in our work at AAC. We recently hired a NAIT student to do some of that work for us, and bought a shiny new computer and the Adobe suite. One of the student’s jobs was to teach us how to use some of that fancy equipment.

The way I learn technology is to immerse myself in a project, and seek support when I need it. Today, I did just that. I put together my vision of Acts I and III for this problem, with the NAIT student nearby for support. I learned a lot about video editing. It was a great day of learning for me. The videos are not perfect, but they’re better than I could have done last week at this time.

Enough Preamble. Here’s what I did. I give you Ticket Roll Reworked. Presented in 3 Acts, of course.

Ticket Roll Act I

Ticket Roll Act II

As near as I can tell, Canadian ticket rolls are identical to the American one Dan photographed. His Act II information should work out perfectly here. I have a different Act II filmed and not posted yet. I’m on it. I’m hoping it will take this problem to an elementary level.

Ticket Roll Act III

Sequels

Dan has sequels listed here.

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You may not know it, but the Amazing Race is big in Canada. It is so big, in fact, that they are planning on running a Canadian version. I’ve already started training and I am currently accepting applications from people who would like to join me on the winning team. But I digress…

Last week’s season finale (season 21, I think) included a challenge that asked contestants to put banners containing the words “hello” and “goodbye” beneath country flags, in that order, and in the language from that country. The contestants struggled and the challenge took over 2 hours, but one contestant tackled it systematically by trying all possible combinations. It was made for a math classroom. In the WNCP, this fits Permutations and Combinations from Pre-Calculus 12 (Math 30-1) in Alberta. It also fits Math 30-2 in Alberta. Here it is, in 3 Acts.

Act I

Play the video by clicking the photograph of one of the contestants working on the challenge.

Flag Challenge

With any kind of luck, the students will wonder how many combinations of the “hello” and “goodbye” banners are possible. They will require more information.

Act II

This video is longer than the Act I video, and by watching it closely, they should be able to determine that they are working with 9 country flags, and 20 banners with words on them. There are 2 extra banners.

Act III

I don’t have a video with the answer. It is fun playing with this problem, though. Initially, there are 1,216,451,004,088,320,000 combinations (20 x 19 x 18 x…x 3). By getting France and Spain correct immediately, the contestants reduced that number by a factor of 116 280, and now only have 10,461,394,944,000 possible combinations to try. If they had truly had to guess them all, they’d still be at it.

Enjoy. Fix my math.

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In my school days, I remember (and not fondly) problems like:

A train leaves Toronto for Montreal at the same time as another train leaves Montreal for Toronto. The cities are 500km apart. The trains pass each other 2h later. The train from Montreal is traveling 50km/h faster than the one from Toronto. At what distance away from Toronto do the trains pass each other?

I’m no whiz with a video camera or script writing. I’m not much of an actor (which you’ll see if you bother to watch the video below). But I think this is a more compelling way of presenting the same problem.

Act I

Act II

http://vimeo.com/51113026

Act III

Answer

Sequels

I need some help here. Any ideas? Is it worth bothering?

  • John was driving slower because he thought Darlene would drive farther to the meeting point. His plan is to drive 110 km/h all the way back, thinking that this would save him time overall. Would he have been better off driving 110 km/h the whole way?
  • What if Darlene left an hour later?

Production Notes: My wife says that there’s no way I’d be that calm if I had to drive her purse back towards home.

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At a recent session in Wainwright, one of the participants, Mary Frank, showed me this demo. It’s similar to Dan Meyer’s Popcorn Picker, but I really like the payoff in Act III. Presented in Dan’s three act format, here’s the materials.

Act I – Video

Act II – Information

Have the students make predictions about whether the wider cylinder will overflow, fill right up, or have space left in it. If they want to run calculations, the tubes are simply 8.5 x 11 pieces of overhead paper. One is rolled vertically, and the other is rolled horizontally.

Act III – The Answer

Sequels

  • How tall would the skinnier cylinder have to be to completely fill the wider one?
  • By what factor are the volumes different? Why?

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I’m starting to see this stuff more places. This one could be fun. This one is easier mathematically than the Penny Pyramid that Dan Meyer describes here. In Dan’s 3 Act Format, here it is.

Act I

Video

Ask the kids what they wonder about. They could go lots of interesting directions with this one.

Act II

Video

This video will give them the information they need, assuming their perplexing questions require them to know how many watermelons are in the pyramid.

Act III

The Answer

The answer is in the form of a Word document with a photo and some calculations. I’d love a video. If anyone has the budget for 385 watermelons, the patience to stack them, and the video editing skills to insert a counter to the filmed stacking, I’d gladly take it.

Sequels

What is the mass of the pyramid?

What is the cost of the pyramid?

How many watermelons could we stack in this room?

If the truck is 20 m from the pyramid, how long would it take to build?

Any other sequels? If so, throw them into the comments.

Download

This zip file (29.7 MB) contains both videos.

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I’ve been getting a kick out of the first season of this show.

Act I

Just to pique their interest, play this clip.  Ask them what they wonder about. Hopefully they talk about the number of possible codes.

Act II

Scene 1

Play this clip. Let them work.

Scene 2

Play this clip. Let them work.

Act III

I have no video that reveals an answer here. Let them share their solutions with each other. Then let the watch the clip below so they can at least find out if Fusco manages to get the file.

I may have learned a new trick. It’s possible that this link will take you to a zip file (10.2 MB) that will allow you to download all 4 videos. It’s possible that it won’t. Let me know, either way.

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Last week, I had the pleasure of attending a workshop with Peter Liljedahl from Simon Fraser University. The session was on assessment, but he opened with his take on problem solving. I’m going to discuss the assessment stuff in a later post, but I really wanted to talk about his problem solving process while it was fresh in my mind.

Peter grouped us randomly by having us select cards, and then gave us the following problem.

1001 pennies are lined up in a row. Every second penny is replaced with a nickel. Then every third coin (might be a penny or a nickel) is replaced with a dime. Then every fourth coin is replaced with a quarter. What is the total value of the coins in the row?

He gave each group a whiteboard marker, and a window or small whiteboard to work on. We were to solve the problem on the space we were given. Peter circulated and encouraged us to look at what was written on the other windows if we were stuck. In the end, most of us were satisfied that we were right. Peter never told us the answer or revealed it in any way.

Here’s what interested me about Peter’s take on problem solving, which is very similar to what I have advocated as Learning Through Problem Solving.

  1. The problem wasn’t written down any where. He just told us the story, clarified, and put us to work. This isn’t a textbook problem.
  2. He grouped us randomly, rather than letting us sit with the people we came with. Throughout the day, he kept on re-grouping us. His feeling is that this frequent changing of the groups leads to more productive classes, and helps include all students. Kids are more willing to work with someone they don’t like when they know that it won’t be for long. This process helps include even the shy and “outcast” students in the classes. These students are included because the people working with them know that the groups will change again very soon, and so they can be patient for that long. Peter also has a way to assess group processes, which I will discuss in the subsequent post on his assessment material.
  3. Writing on non-permanent surfaces is critical to Peter’s process. He has observed that when kids work on paper, they take a long time to start. When they work on something like a window or a whiteboard, they start right away, because they know they can erase any mistakes easily. Groups that are stuck can step back and see what the other groups are doing as a way to help them get started. When finished, each group’s work  is displayed prominently around the room so that students can share solutions with each other in a non-threatening way. Students could do a gallery walk and discuss the different solutions, or sit down and write down one that works for them individually.
  4. This is the third time I have had the pleasure of working with Peter. He has never once told us the right answer. This drives teachers nuts. I’ve tried it in classrooms, and it drives kids nuts. The message, though, is that they have to figure out the answer. The teacher isn’t going to let them off the hook. An even more important message is that the teacher is less concerned about the answer than she is about the process. These are two pretty nice messages.

Peter’s problem solving process fits really nicely with the Learning Through Problem Solving discussed frequently on this blog. There’s always a “yeah, but” when I share some of this with teachers. They have one whiteboard, no windows, and nothing else to write on in their classrooms. Let me solve that problem for you.

Superstore has $20 whiteboards that are just the right size. Staples has tons of different sizes available for less than $30 each. I’m confident you could get enough whiteboard space in your classroom for 15 groups to work for less than $300. Your principal will find that kind of money in a hurry, especially when you explain that you are going to use it to enhance mathematics instruction by engaging students in group processes that will lead to better problem solving skills. I’ll write you a letter if you need one.

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The good folks at Vat19.com just keep making the Gummy Bear problem better for me. Thanks to John Burk for noticing this one and throwing it out on Twitter.

My favorite Learning Through Problem Solving activity right now is the Giant Gummy Bear problem. You can read my post on it here. I have used this problem with students and teachers, and it is always a favorite. Teachers who have used it have emailed me to tell me that they received Giant Gummy Bears from their classes as gifts after doing this problem. It really does go over well in class. Students tend to wonder how many small gummy bears make up the 5 lb gummy bear. They wonder about the dimensions of the 5 lb gummy bear. It’s fun, and it leads to good math. I’ve had trouble coming up with extensions for the problem until now.

Then the nice people over at Vat19 made me one. And I didn’t even ask for it. Check this out.

The beauty of this one is that I don’t even have to edit it. It works exactly how it is. It provides just enough information, but still leaves lots of math questions students could explore.

It’s an extension to the original because the math involved is going to be different than the math involved in the 5 lb problem. The  34 fluid ounce tummy throws a nice 3-dimensional wrench into the calculations. Students will have to compensate for this hole in the belly of the 26 pounder.

Questions I see them having include:

  • How many regular gummy bears make up the 26 pound one?
  • How tall is that 26 pound gummy bear?
  • How many small gummy bears would fit in the 26 pounder’s belly?
  • Is the cost of the 26 pounder proportional to the cost of the small ones and to the 5 pound one?

Question I have:

  • Can somebody with an extra $200 send on of those my way?

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