Archive for the ‘Permutations and Combinations’ Category

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.


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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Edit – March 13, 2014 – The original YouTube video I reference below has been removed. It’s better to section this one into pieces anyhow, and I have done that for you in higher resolution at the links in the February 25, 2012 update. Those links should be working. There’s also a less cluttered version of this entire task here: Basketball Free Throws

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.

Part I

Part II

Part III

Part IV

 All 4 parts for download. Zip file at 11.57 MB.

Original Post

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.

  1. 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. 
  2. Ask the students what they wonder about. I suspect they will wonder who won.
  3. 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).
  4. 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.
  5. Let students work on the math. Who wins? What are the probabilities of each of the outcomes?
  6. 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.
  7. 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%


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I got this idea from a teacher I marked diploma exams with a couple of years ago.  I’d love to credit him, but I can’t remember who he is.  This is a WCYTWT submission, or as I like to call it, WWDDWT (What Would Dan Do With This).  I used it with a couple of classes last year, and they really enjoyed it, but I used it poorly.  I did it at the end of a unit on Permutations and Combinations, so the students knew exactly what to do with it.  It wasn’t truly a problem, because they knew exactly how to solve it when I gave it to them.  I should have used it on day one of Permutations and Combinations, and let them invent the fundamental counting principle on their own.

Mozart’s Dice Game

Mozart is credited with creating a dice game, whereby you roll a pair of dice 16 times to select 16 measures to insert into a minuet section, and then a single die 16 times to select 16 measures to insert into a trio section.  For example, if you roll a 6 for the first measure, you consult a chart to see what measure number to insert into the first measure of your minuet, and so on.  The idea is that no matter what you roll, you always produce a minuet that fits whatever rules go along with a minuet.  This site explains it in a little better detail.  Mozart’s Dice Game.


You give the students the history, and then you throw this site up on the SMART board.   Play Mozart’s Dice Game has a chart that looks like this:

The drop down menus let the students enter numbers that they roll on dice, so give the kids some dice, and let them enter their rolls.

Notice the link below the chart that says “Make Some Music!” Once the students have all entered their rolls, you click here and the newly created minuet will play.  There’s even a “Generate Score” link below the media player that lets you generate the score for the minuet they created.  My students loved this, and printed theirs off to try to play it on the piano themselves.

After the minuet plays, you tell the kids that even though this was written more than 200 years ago, you are pretty sure that nobody has heard the particular minuet the class just created ever before.  Ask them to discuss this statement.  They’ll say things like, “Why, did they lose the score until recently?” and dance around it until one kid finally asks, “How many minuets could be made in this game?”  Then you’ve got them and you let them play around with it.  They will invent the fundamental counting principal, determine that there are an incredibly large number of possible minuets, and even create their own interesting extensions.

One student even came in the next day with an iPhone app that generates minuets using Mozart’s Dice Game.  The only  problem with the app is that it only randomly generates minuets, and doesn’t allow you to enter your own rolls.

Update: March 11, 2016

My greatest fear was realized. The site I talk about above is no longer functional. Something about Midi Files, Flash players, and the like. I couldn’t get it to work in any browser.

I found a functional site. https://www.mozart-game.cz/

It randomly selects the measures to play, as highlighted below. You can play your own measures by clicking on the un-selected ones, so you could still have your students roll dice and create their own minuet, but you’d have to click your way across the chart. If you can keep time, it’s pretty easy. The vertical columns represent one roll of a pair of dice. So if the first roll was a 2, you’d play measure 96. In the screen shot below, the first roll was 3, the second roll was 6, and so on.



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For your viewing pleasure, here are what I believe to be all the entries in the binomial expansion contest.  They are presented without bias, and in no particular order other than putting the best one at the end.


James Tanton’s Part 1

James Tanton’s Part 2

Jason Dyer’s


Edit (May 30):  Jason Dyer’s was judged the winner in the competition.  See here for the rationale for the decision.  Congratulations, Jason.

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A few weeks ago, Kate Nowak posted on her blog that she was frustrated at the lack of online resources to support the teaching of the binomial expansion.  She threw out a challenge asking people to create a better way to teach the binomial expansion.

Here’s her challenge:

“Objective : Present the binomial expansion in a way that makes sense. Bonus points if students are able to as a result completely expand a power of a binomial and find a specific term in an expansion.”

Our revised curriculum in Alberta includes a pedagogical shift that asks teachers to create opportunities for students to establish their own meaning through exploration, problem solving, investigation and developing personal strategies.  One of the common concerns I hear from teachers is that we can’t teach higher level mathematics this way, because the material is so hard that students won’t be able to construct their own meaning.

Kate’s challenge gave me an opportunity to try to teach in this manner in one of our academic courses.  In Alberta, binomial expansion shows up in a pre-calculus course called Math 30 Pure.  Students go from that course into Math 31, which is a calculus course covering limits, derivatives, application of derivatives, and basic integration.  Binomial expansion, and the binomial theorem are two pretty dry topics.  I took the challenge and attempted to create a lesson that was both engaging, and allowed students to construct their own meaning through looking at patterns.  A colleague was nice enough to let me try the lesson in his class.

My lesson went over fairly well with the students, and I think I met all of Kate’s objectives.  The introduction/hook was more compelling than anything I had done in the past on that topic.  Students were able develop the patterns in the binomial expansion on their own, and were able to apply those patterns to expand more complicated binomials.  What surprised me was that they were able to come up with the formula to find any term in an expansion all on their own.

I’d love to hear some feedback on the lesson.

An 11 minute compressed version of the lesson can be watched here:  Binomial Expansion Lesson

If you’d prefer to just read the lesson plan, here it is.

Lesson Plan

Introduction:  I showed them a mathemagician video from TED.

Hook:  I claimed that I was a mathemagician, too. I wrote (a + b)^2 on the board, and asked them to expand it. Then I wrote (a + b)^3 on the board and had them expand it. Then I wrote (a + b)^4 and (a + b)^5 and challenged them to a race. They could use friends, pencil, paper, calculators or whatever they wanted, and I would just use my brain. I pretended to struggle, and then wrote down the answers as quickly as I could. I had to hurry because one kid was darn quick, and was getting at (a + b)^4 by multiplying the answer to (a + b)^3 by (a + b) rather than expanding the whole thing as I had expected them to do.  If you watch the video, we appear to tie on that one, but he already started while I was blabbering.

Students look for patterns:  I explained that I am not really that smart, and that I was cheating using a pattern. I wrote “In the expansion of (a + b)^n, ” on the board, and asked them to spend a few minutes together coming up with ways to complete that statement. They got that each term had the same degree as the exponent on the binomial, and that there were n+1 terms in the expansion. It took a little longer and some direction from me for them to notice that the a’s started at a^n and decreased to a^0, while the b’s did the same thing in the opposite direction. One girl put Pascal’s triangle on the board when she noticed the pattern of coefficients.

Expand a binomial:   I asked them to use all that they had learned to expand (x + 2y)^5. They worked together and there was much discussion about how to handle the 2y part. They managed to figure it out.

Develop a formula for a general term:  I gave them one like (x + 2y)^12 and asked them if they could figure out the 8th term without writing out the first 7. I gave them three blanks to fill in: coefficient, “a”, and “b”. They got the a and b part just fine, but struggled with whether the coefficient should be 12 choose 7 or 12 choose 8. They figured it out by counting.   I told them that they had just figured out the formula. I wrote the formula tk+1 = nCk x^(n-k) y^k on the board.

(At this point I fell back into my old bad lecture style, and did two examples with them using the formula, rather than making them do it themselves.)

Example: Find the term containing y^7 in the expansion of (x – 3y)^7.

Example: Find the constant term in the expansion of (3x – 2/x^5)^12.

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