Sunday, December 14, 2008

Teaching Probabilities and Strategies Via Games!

NCTM Teaching Standards:

  • Develop and evaluate inferences and predictions that are based on data
  • Understand and apply basic concepts of probability

  • Target Audience: Grades 7-12

    Tools Needed:
    Graphing calculator with a random integer generator or an online random number generator (look here for example)

    Classroom Organization
    (After demo mode): Students working in groups of 3 (two opponents, the 3rd calls out the numbers and keeps score; roles are rotated)

    Sample Classroom Scenario
    Who thinks they can beat me at a game of chance? I will demo the game, then I will play against an opponent. If you beat me two out of three, you are the new champ and you can pick your opponent. After 10 minutes, you will be playing in small groups and recording the results.

    The Play
    Using a random integer generator we will generate random digits, one at a time, from 1 through 9, inclusive (no zeros). The object is to build a 5-digit integer which is greater than your opponent's by placing each 'called' digit into one of the five place-value positions. Once you place a digit you cannot change it!

    Let's try it... Ok, Marissa, turn on the random integer generator, press Enter and call out the first integer. FOUR!
    Ok, I'll place it here: ___ ___ 4 ___ ____
    Call out the next integer: SIX!
    I'll place it here: ___ 6 4 ___ ___
    Next: TWO!
    ___ 6 4 ___ 2
    Next: FOUR!
    ___ 6 4 4 2
    Last digit! FIVE!
    5 6 4 4 2
    How did I do? Could I have used a better strategy? Do you think you could have beaten me?
    Who wants to play! To win, you have to beat me two out of three. Ok, Dimitri, I will work on my paper and you work on yours. Remember, you cannot change a digit's position once you place it...

    Brief Discussion of Strategy Based on Probability Arguments:
    Suppose the first two digits called are 3 and 6 in that order. Would it be better to place the 6 in the thousands' place or the ten thousands' (leftmost) position? If you place the digits here:
    ___ 6 ___ 3 ___, what is the probability that at least one of the next three digits chosen will be 6,7,8, or 9. (Otherwise, your strategy would have backfired). To compute this, we look at the complementary condition, i.e., we determine the probability that the next 3 digits chosen will all be in the range 1 through 5. The probability of this is (5/9)(5/9)(5/9) or approximately 17%, so the probability that our strategy works is about 83%, odds that seem worth playing! Experienced game players often compute these probabilities mentally or have seen these situations so many times they know these probabilities by heart!

    (1) Students may not know there is a Random Integer generator built into many graphing calculators. For example on the TI-84, press MATH, then PRB, then 5:randInt(.
    From the home screen, Enter randInt(1,9), ENTER.
    Each time you press ENTER another "random" digit will be displayed. The person calling these out must be instructed to announce only ONE digit at a time!
    (2) Why 5-digit numbers? This seems to make the game fairly interesting and moving at a good pace. Expect ties of course!

    Perhaps this is a good activity before the holidays. Have fun and let me know how it goes!


    Pat Ballew said...


    I think this could be a great game, and in high school I would want them to finish up with a strategy summary,... Where would you put the first number if it was a 2, 3, etc... explain why... Conversly you could ask them to explain Which position and why they would put the different digits if they occured on the first draw.

    Dave Marain said...

    Thanks, Pat and Happy Holidays!
    I also believe it has that potential. I was hoping my readers would make these kinds of suggestions rather than my fleshing it out in too much detail.

    I've also played this game with younger kids (as early as 4th or 5th grade when I was a Staff Developer). I know you would not be surprised by how much intuition kids bring to this problem. I can recall losing more than once to some 10- or 11-year olds!

    Certainly, experiences like this as well as dice games (sorry, we have to say number cubes these days) and other games of chance are engaging and highly instructive in developing a conceptual basis for the later laws of probability and expected values.

    I particularly like your idea of having them express their strategy. Thus is so important to their mathematical growth.

    Jim McClain said...

    Couldn't you just use a 10-sided die as the random number generator, ignoring a result of zero?

    Dave Marain said...

    Absolutely, Jim.

    However, I have found that pressing the Enter key on the TI-84 is quick and easy. I also wanted my students to be aware of this capability of the graphing calculator. Actually I haven't seen a 10-sided die recently! Not to mention that, knowing some middle schoolers, we might find ourselves searching for the die on the floor!

    Aside from the logistics, what do you think of the game itself and could you suggest some variations?

    Jim McClain said...

    I was thinking of creating a target number, and allowing students to use addition, subtraction, multiplication, and division on the randomly generated numbers to see who can get closest to the target number. This would reinforce not only computation, but the order of operations as well. I suppose you could even throw in exponents and grouping symbols after they get used to the four basic functions.

    At Gen Con, a big gaming convention, this summer I picked up a lot of polyhedral dice with operation symbols, but haven't formalized what I want to do with them yet. But I think with a few tweaks, your game could be even more useful to my purposes. I teach 8th grade math, by the way, which is why I like using dice. Kids love to roll dice!

    Dave Marain said...

    Nice extensions, Jim. The target game reminds me of one of my favorite order of ops games, Krypto (google that if you want more info), although it's not so much a game of chance. BTW, polyhedral dice are good to have around in the classroom for other purposes as well, counting faces, edges, and vertices for example. And, I agree, kids always like to roll dice!

    Here's a variation I just thought of for two of your polyhedral dice:
    The players roll the dice and earn:
    (1) point if the dice form a 2-digit prime in some order, e.g., 1 and 4 can make "41" which is prime.
    (2)points if they form a 2-digit primes both ways, e.g., "79" and "97" or "11"!
    The player's turn ends when neither order forms a prime.

    Kate said...

    Any particular reason to exclude zero? It's taken quite a while for the 9th graders to remember that there are ten digits, not nine, I think I'd want to include zero. :-)

    I might try this tomorrow, was going to do a day on geometric probability (throw a dart at a weird shape, etc), but I'm not married to the idea.

    Dave Marain said...

    Zero is fine, Kate.
    If the student has the misfortune of choosing 0 as his/her ten-thousands' digit, that would result in a 4-digit number of course, but it might be fun to watch the student's reaction! I've played this game many times with and without zero and the latter seems to work better overall but it's fun and instructive either way. Let me know how it goes if you use it. Now that I have automatic updates of your blog on my sidebar, it will be easier to see this!

    Happy Holidays to you and your family...

    Kate said...

    Did it today - this was 7th and 8the period the day before break, so a bit of a challenge to get any discussion of strategy going. Dave do you have any suggestions for compelling questions to ask? I found myself stopping the game periodically to ask things like "what's the probability Nick gets two 9's in a row?" "what's the probability Andrea won't get any 8's or 9's in the next three turns?", but it was pretty much me doing the figuring and it really interrupted the flow of gameplay.

    They were playing for the chance to get their picture taken with my wrestling belt. Competition was fierce. We had a bracket and everything. :-)

    Dave Marain said...

    Sounds like the group had fun - nothing like competition!
    I agree that you don't want to interrupt the flow of the game. There can be some initial questions in the beginning when you're doing a demo with another student. Students can make some conjectures perhaps and record those for later discussion.

    Some youngsters will be able to formulate strategies after a few plays but they surely don't want to stop playing to share those. Afterwards there can be a discussion of strategies and probability concepts. Even if the group has spent some time with probability problems there needs to be review of some basics here. One can do some of this review prior to the game but I think it's effective afterwards as you bring the group together to reflect on the mathematics of chance. I do think a key question involves the issue of pulling a 6,7,8 or 9 at least once with 3 digits left or 2 digits left, etc. They should make a table of these probabilities and compare these against the conjectures and strategies they were formulating on the fly. Most of us are capable of making intuitive decisions without even realizing we are thinking 'mathematically' and it's instructive for our students to see that our intuition is often quite good. However, mathematics may have some surprises for us and inform us when phenomena are counterintuitive! You also will have some students who don't asee the point of probability since the theory never guarantees what happens -- they're not into the Law of Large Numbers or the "Law of Averages." As they mature, they will be!

    Wish your group a Happy Holiday from their friend in NJ and tell them to invent their own game of chance and share it with me!

    I really appreciate you're trying this, Kate. Did the kids seem to enjoy the game itself or was it more the competing to take a picture! BTW, are you an Olympic wrestler when you're not in the classroom!

    Kate said...

    Thanks for the suggestions - I'll need to sit down before trying this again and make a more structured plan. Today was mostly about getting them to do anything math-related. :-)

    A few did suggest some additional rules, like "if you roll the same number twice in a row, you can keep it or choose to roll again".

    I'm no athlete - another department teacher has a friend who works for WWE wrestling magazine and sent him a pretty hefty, official-looking belt. We use "get your picture taken with the wrestling belt" as a prize when we don't have anything else available. :-)

    Happy holidays to you, too! Relax and enjoy!