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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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?
I was in a meeting this morning, and we were discussing how to connect literacy across the curricular areas. I flashed back to high school, and a great short story we read. I started wondering whether I could use Shirley Jackson’s “The Lottery” in a math class. Then I began to wonder if a 3773 short story would fit with Dan Meyer’s 3 Act Mathematical Story Telling. Here’s what I would try with this story.
Act I
Have students read The Lottery, by Shirley Jackson. Ask them what they wonder about. They will probably wonder about lots of things non-mathematical. Eventually they might wonder (Spoiler Alert!) what Tessie Hutchinson’s chances of winning the lottery were.
Act II
Ask the students what information they require to be able to answer the question. If they wonder how many families were in the first draw, you can have them look back through the story and count, or tell them that there were 16. They will also need to know that there are five members in the Hutchinson family in the second draw.
Act III
Students work it out. I still need to come up with a better way to reveal the answer, which is that Tessie had a 1 in 80 chance of winning the lottery.
Sequel
If this lottery has been going on all of Old Man Warner’s life, what is the probability that he survived to age 77?
Edit
Kendall reminded me that I started with connections to English class, and I meant to close with connections to English class. I would totally do this in collaboration with my school’s English teacher.
I support mostly high school math teachers. I work with colleagues who support K-9 teachers. Last week, I eavesdropped on two of them as they tried to come up with a 3 Act Math Story in style of Dan Meyer that would apply to division 1 students. This week’s Parks and Recreation may have provided us with one. You be the judge.
Act One
Click on Andy to play the movie.
Act Two
Find out what the students wonder about and what information they will need to answer their questions. I suspect they will wonder whether it will really be a billion nickels. Depending on how young a group you give this to, they may need to know that nickels are worth $0.05 or that there are 20 of them in a dollar. Canadian kids may need to be told that those wacky Americans use paper for $1 instead of coins.
Act Three
The good folks over at Parks and Recreation didn’t film the right answer for us. If anybody wants to withdraw 20 000 nickels, stack them up in some way, film it or photograph it, and send it my way, I would appreciate it. Otherwise, this is the best I can do. Give them a photo and some information.
$1000 = 20 000 Nickels
Sequels
Could Andy hold 20 000 nickels? How much would they weigh? What size container would he need? Would they fit in his trunk? If he piled them all in a giant stack, how high would they reach? What about a billion nickels? How much would they weigh? How high would they reach if all stacked up?
This idea is another I picked up at ISTE2011 in Philadelphia in a session on technology in math with Frank Sobierajski. Frank mentioned using the 2005 Conference USA championship between Memphis and Louisville in a math class. He didn’t flesh it out, but I immediately knew how I would use it. I remember the game vividly. As a basketball fan and a math fan, this one really speaks to me.
Before we get to it, consider the following question. One like it could be found in any of our Math 30 Pure textbooks.
A basketball player is successful on 72% of his free throw attempts. What is the probability that he makes at least two of his next three free throws?
The question is fine. We’ve all given similar practice questions to kids. But in this format, it is nothing more than practice on binomial probabilities. It evokes none of the emotion of sport. Is the player under pressure to make these shots, or is he just messing around in the gym? Does it matter?
This same question could be given before the students know anything about the binomial probability distribution, making it a rich learning through problem solving experience. In the textbook format above, though, students would not be remotely compelled to struggle with it long enough to get a solution. But if you present it in the format I describe below, I contend that students will be hooked. They will want to know if he can make at least 2 of 3 shots, and will invent binomial probability on their own. Try it.
In the frame of the 7 steps of learning through problem solving, here is how I would use it before students have been taught anything about binomial probabilities.
Play the first 0:17 of this video. Consider downloading it first, and changing the title, because the title kind of gives away what is going to happen. Pause, and make sure the students understand the context and the magnitude of the situation. This is a conference championship game. The winner will go into the NCAA tournament of 64, and the loser’s season will end. There are 6 seconds left. Memphis, in blue, has the ball and are trailing by 2. Once students understand the context, play the video up to the point where Darius Washington steps up to the line for his first free throw, and the announcer says, “For the season – 72%”. Make sure to pause it before he shoots the first shot. Make sure students understand what is going on here. Some don’t know basketball rules. Make sure they understand that he will get three shots, and each shot is worth one point.
Ask the students what they wonder about. I suspect they will wonder who won.
Have students make a prediction. Let the discussion go on and I’m confident they will talk about the three possible outcomes (If he makes 3, Memphis wins. If he makes 2, they go to overtime. If he makes 0 or 1, Louisville wins).
Ask them if they require further information. I would make sure they heard the relevant parts of the video, which are the facts that Darius Washington is a 72% free throw shooter on the year, and that he is 2/3 in the game.
Let students work on the math. Who wins? What are the probabilities of each of the outcomes?
Play the video to 1:40, so students can see him make the first shot. Repeat steps 3-5. Does Memphis have a better chance of winning now? Play the video to 2:00, so students can see him miss shot #2. Repeat steps 3-5. Play the rest of the video.
Share student solutions. How did the predictions (and math) change after each shot? Why did the math not match the outcome?
My answers (please correct if wrong).
Based on Season 72%
Before first shot: Memphis win = 37.3%, Tie = 43.5%, Louisville win = 19.2%
After first shot: Memphis win = 51.8%, Tie = 40.3%, Louisville win = 7.9%
After second shot: Memphis win = 0%, Tie = 72%, Louisville win = 28%
Based on Game 2/3
Before first shot: Memphis win = 29.6%, Tie = 44.4%, Louisville win = 26.0%
After first shot: Memphis win = 44.4%, Tie = 44.4%, Louisville win = 11.2%
After second shot: Memphis win = 0%, Tie = 66.6%, Louisville win = 33.4%
February 25, 2012 Update
Based on session feedback, I have broken the video into four parts, so you don’t have to keep hitting “Pause” and risk showing too much too soon. There was also the slight problem of the title of the YouTube video kind of giving away what was about to happen to poor Darius.
While in the Pennsylvania Convention Center for ISTE 2011, I came across this intriguing piece of art wedged under an escalator.
I have since learned that the piece of art is by Mei-Ling Hom, and is called “China Wedge”. It is composed of many Chinese cups, bowls and spoons wedged into a space under an escalator. It appears that the space is not entirely filled with cups, bowls and spoons, but they go about 2.5 deep all around. The sculpture, which was commissioned for the Center’s Arch Street Concourse, pays homage to Philadelphia Chinatown, which is adjacent to the Convention Center. (Source (Spoiler Alert): http://philadelphia.about.com/cs/artmuseums/a/paconvention.htm)
I took some photos, because I wanted to turn this into a math story. I didn’t get all the shots I needed, so I have to give a big thank-you to Max Ray, who went back and took care of the Act II photos with his girlfriend, who happens to be a photographer and got great shots. The Act II photos containing referents and shots of Max are all the work of Kaytee.
Here is my attempt at a three act math story as described by Dan Meyer here.
Word document containing all the information I could find about the China Wedge.
Sequels
I struggle with this one. Any feedback would be greatly appreciated. I’ve only one idea so far.
If the entire area under the escalator was filled with cups, bowls and spoons, how many more would have been needed?
In his last one of these, Dan asked whether this broad outline is enough for teachers to go by. I know it is enough for lots of us to run with. If you need more details about how to make this work in a classroom, contact me and I will spell it out a bit more. I would present it in much the same way I discuss in my learning through problem solving explanation.
