Saturday, August 29, 2009

Batteries Required! A Combinatorial Problem MS /HS Students Can Use...

Have you ever inserted batteries in a device only to find that it didn't work? You reverse the batteries and try again, but no luck. You can't find the polarity diagram to guide you and you're dealing with 3 or 4 batteries and all the possible combinations! Well, that just happened to me as I was inserting 3 'C' batteries into a new emergency lantern I just purchased. There was no guide that I could see. I knew there were 8 possibilities but it was late and my patience quickly ran out. I tried it again the following morning, shone my small LED light on it and saw the barely visible diagram.

After seeing the lantern finally operate, I realized I should have used a methodical approach -- practice what I preach!! Then I thought that this might be a natural application of the Multiplication Principle one could use in the classroom. Of course, it would work nicely if you happened to have the identical lantern but you might have some of these in the building or at home which take 2 or more batteries. IMO, there's something very real and exciting about solving a math problem and seeing the solution confirmed by having "the light go on!" I'll avoid commenting on the obvious symbolism of that quoted phrase...

Instructional/Pedagogical Considerations

(1) I would start with a small flashlight requiring only one battery to set up the problem. For this simplest case, students should be encouraged to describe the correct placement in their own words and on paper.

(2) Would you have several flashlights/lanterns available, one for each group of 2-4 students or would you demonstrate the problem with one device and call on students to suggest a placement of the batteries? Needless to say, if you allow students to work with their own flashlights, they will look for the polarity diagram so you will need to cover those somehow. That is problematic!

(3) Do you believe most middle school students (if the polarity diagram is not visible) will randomly dump in the batteries to get the light to go on and be the first to do so? Is it a good idea to let them do it their way before developing a methodical approach? Again, if a student or group solves the problem, it is important to have them write their solution before describing it to the class. If there is more than one battery compartment, students should realize realize the need to label the compartments such as A, B, C , ... Once they reach 3 or more batteries, they should recognize that a more structured methodical approach is needed so that one doesn't repeat the same battery placement or miss one. One would hope!

(4) Is it a drawback that the experiment will probably end (i.e., the light goes on) before exhausting all possible combinations? How would we motivate students to make an organized list or devise a methodical approach if the light goes on after the first or second placement of the batteries?

(5) I usually model these kinds of problems using the so-called "slot" method. Label the compartments A, B, ... for example and make a "slot" for each. For two compartments we have

_ _

Under each slot, I list the possibilities, e.g., (+) end UP or DOWN (depending on the device, other words may be more appropriate). Here I would only concern myself with labeling the (+) end, the one with the small round protruding nub. For this problem I would write the number (2) on each slot since there are only TWO ways for each battery to be placed. Note the use of (..). In general, above each slot I would write the number of possibilities. For two compartments (or two batteries), the students would therefore write (2) (2). They know the answer is 4 but some will think we are adding rather than multiplying. Ask the class which operation they believe will always work. How would you express your questions or explanation to move students toward the multiplication model? The precise language we use is of critical importance and we usually only learn this by experimentation. If one way of expressing it doesn't seem to click with some students, we try another until we refine it or see the need for several ways of phrasing it. This is the true challenge of teaching IMO. We can plan all of this carefully ahead of time, but we don't know what the effect is until we go "live" (or have experienced it many times!).

Perhaps you've already used a similar application in the classroom - please share with us how you implemented it. Circuit diagrams in electronics also lend themselves nicely to this approach. Typically, I've used 2, 3 or more different coins to demonstrate the principle but the batteries seem to be a more natural example, although I see advantages and disadvantages to both. At least with the batteries, students should not question the issue of whether "order counts!"

I could say much more about developing the Multiplication Principle in the classroom, but I would rather hear from my readers.
If you've used other models to demo this key principle, let us know...

MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates (Pls Read!!)
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
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Sue VanHattum said...

Fun stuff to think about.

But (in real life) some permutations wouldn't occur. Either the batteries will go +-+-etc or they'll all be in the same direction. So there are really only 4 possibilities in any battery-powered device I've seen.

Dave Marain said...

Agreed, Sue, but with 3 batteries it proved to be infuriating since it turned out that two were + up and one was + down!

Of course math only provides models which simulate the real world! Well I tried...

I'd love it if you or other readers would share the models you have or might use.

Sue VanHattum said...

Sorry if I wasn't clear. I didn't mean to criticize. I like this.

I was just thinking about how well it connects with the real world. I don't think our models ever do that completely. (See this.)

Do you want to see things like the 'What Can You Do With This?' series? I posted one at my blog, using a scene from Holes. If that's what you'd like to see, google WCYDWT for lots more. Or just go to Dan Meyer's blog, dy/dan.

Dave Marain said...

No offense taken, Sue! I depend on my readers to make constructive suggestions and help me improve.

Actually I was looking for examples and instructional strategies used to help students develop understanding of and facility with the Multiplication Principle in particular. How do we enable students to grasp that counting by enumeration and grouping is equivalent to the Principle. Knowing when it is appropriare to use it and how to use it properly is not always transparent to Middle or HS students.

Thanks for the links - I will check them out.

Eric Jablow said...

I thought your question was going to be "Is there a procedure you can do to the batteries that when repeated will make them set correctly?" And that's a good introduction to group theory.

The actions you can do to the batteries (not exchanging them, just flipping them) form the group Z_2^n, the product of n copies of the group with two elements. Every element of this group other than the identity has order 2. It's pretty obvious, but that means that repeating any single procedure is unlikely to place the batteries the right way.

You might want to look up Brian Hayes' recent book, Group Theory in the Bedroom, and Other Mathematical Diversions, ISBN 0809052172. The title refers to the problem up flipping a mattress efficiently, and that deals with the dihedral groups in much the same way.

Dave Marain said...

Thank you for bringing the group theory back to my consciousness!

I now see another aspect of this problem I had missed. The different "states" of the batteries can also be modeled by the Binomial Theorem (not for Middle Schoolers):
Let U represent (+) up, D be (+) down for each battery. Then
(U+D)^3 = U^3 + 3(U^2)(D^1) + 3(U^1)(D^2) + D^3.
The sum of the coefficients of course is 2^N.

Now, what would be the most efficient algorithm for 3 batteries (if there is one)?

Eric Jablow said...

Thanks. I should have added that if you allow changing the order of the batteries as one of the operations, then the group of actions G becomes larger. Ignoring the orientation of the batteries provides a map from G to S_n, the symmetric group on n elements, and so one has an 'exact sequence'
0 →(Z_2)^n → G →S_n → 0. (I haven't used those in a while!) In fact, G is a semidirect product of S_n and (Z_2)^n.

For example, in the case of two batteries, the action "Swap the two batteries and then flip the one in the top position" works. It turns out to have order 4, and its orbit has all possible orientations of batteries.

I don't think abstract algebra is of that much interest to normal high school students, however. I wonder what area of mathematics would be most accessible to elementary school students, middle school students, or high school students.

I'm guessing that basic topology would work for the youngest; they can have fun with Möbius strips and the like.

Middle schools students able to do manipulations with fractions with confidence could do reasonably well with probability theory. This would be around the age when the students memorize baseball statistics, or get into role-playing games.

I'm not sure what would be most accessible to high school students. Perhaps number theory would be best.

Dave Marain said...

Thanks, Eric. This is why abstract algebra (along with number theory) was my favorite in grad school. It enables one to create general structures that model so many particular phenomena.

IMO, without extensive development,students of all ages can be introduced to the groups of symmetries using actions on regular polygons. For example, a middle schooler solves a 3x3 magic square and the instructor asks: "Can you find other solutions by flippling or turning? How many can you find?

I'd have to say that number theory is appropriate for students from grade 3 on (or once multiplication and division are set). I've actually done this with 4th graders as part of my job as a staff developer. We played the "Million To One" game:
Two competitors took turns dividing one million and its resulting factors by either 2 or 5. The child who reached 1 wins. It didn't take long for the children to realize that the student who goes 2nd will win, but getting them from there to the factors of one million was fun and challenging. The divisions were done on a calculator. One child noticed that it always took 12 divisions and I asked why that was. A young lady said that a million is just 10x10x10x10x10x10 so we have to "knock out" 10 six times! This was a very precocious group of 4th graders, since many middle schoolers would have difficulty making that observation!

So what mathematics is appropriate for different ages? I believe ALL of it is as long as children develop a strong foundation of arithmetic skills. Without that, there is little to build on. This is why we assume that younger children can do sophisticated spatial problems like tangrams and pentominoes but we avoid doing number theory and probability with them.

For example, the concepts of probability can be introduced to kindergarteners, long before they've seen a fraction. I've seen this done and I have done it myself by asking chances such as "Are the chances the same" or "You have one chance out of how many to win?" I'll cut a circle into two unequal parts, I'll take the bigger part and ask, in a coin toss game if that makes a "fair game" and the children scream NO!

See what you started!