Friday, June 14, 2013

The Famous Jinx Puzzle



If I was forced to put my money where my mouth is about alternative paths to Algebra, I would start with a very simple, yet significant puzzle. I call it The Famous Jinx puzzle. It was inspired by Harold Jacob’s introductory lesson “A Number Trick.” (See pages 1-4 in his Algebra book (1).) Here is a description of how I do it with kids.

I start off by saying, “Today, I’m going to jinx everyone. Take out a piece of paper and do the following steps.” (I read the instructions one at a time and wait an appropriate amount of time for them to complete each step.)

1. Choose a number from 1 to 10
2. Add 11
3. Multiply by 6
4. Subtract 3
5. Divide by 3
6. Add 5
7. Divide by 2
8. Subtract the original number that you chose.
9. Your answer is. Write it down. Don’t yell it out.

When I see that most of them are done, I ask one of those students what they got. If the student says 13, I ask another student the same question. I continue until at least 3 students said 13.

Note: For the time being I acknowledge other answers but don’t make a fuss about it. I just move on and ask other students the same question. The students who get a different answer have made a mistake somewhere in their calculations. So for the time being I ignore the problem though I return to it with those students on an individual basis because, at this point, I don’t want to change the focus of the lesson to correcting computational errors. This is also a good assessment whether the kids can do this kind of computation. What I do with that information is different lesson.

Then I say playing "dumb" (2): "Thirteen? So that must mean you all started with the same number?" Negative head shakes. So what numbers did you start with?" I list 4 or 5 of their numbers on the board. “Do you think all numbers from 1 to 10 will end up jinxed?” I ask. Not being sure, the students now try it with the missing numbers on the board.

Note: While the students are working on the remaining numbers, I check in with those students that didn't get 13 and help them spot their mistakes. Obviously, if most of the kids were having trouble with the computation, then this lesson could have easily bogged down since we can't go on without them knowing how to compute. (But this was not the case here, so I continued and left the few computation concerns for another time.)

“How about other numbers? Bigger ones? What do you think? Or what about fractions? Say ½?” I continue. The students respond with mixed results. Some kids plow ahead and try it using paper and pencil. Others grab a calculator. If it's a calculator with fractional entry capability then they proceed without much difficulty. But others with scientific calculators (who are in the know) will enter .5 and continue from there.

Note: It's very interesting to watch from a teacher’s standpoint their facility with numbers. Each time I teach it I'm surprised by what happens. One student once told me you can't enter fractions into a scientific calculator. You can only use decimal numbers. This started a wonderful conversation about whether fractions are decimals or not. We came up with the fact that fractions are masters of disguise and can take on an infinite number of forms including decimals!

Since it's time-consuming to do this puzzle by paper and pencil or with a calculator tedium starts to set in. Sooner or later (timing is important here) I stop and say: "Would it help if we had a calculating machine that would do all the calculations for us? I usually get a resounding “yes”. That's when I turn on the projection device that’s connected to my laptop and show this file (figure 1) on the screen and introduce them to the Jinx calculator. (Actually, it’s really a spreadsheet file in disguise that does the calculations for them so they can try more ambitious numbers. An Excel version is available here.)


Figure 1

I start by showing them only the left side (column A) in Figure 1. Then I drag the lower right hand corner of the window to reveal the numbers in column B. I then replace the number B1 with another whole number. They see the results of the calculations change, but not the 13 in cell B8.

Figure 2 shows what happens when I enter .25 in B1. Decimals  appear everywhere except in B8 where 13 holds steady.


Figure 2

Note: This Excel file has all the instructions encoded in cells B2 through B8.  Figure 3 shows what they look like. After a few tries I ask Do you think it matters what number you started with? (The “correct” answer of course is no, but we will need the Algebra to prove it.)


Figure 3

After a few more tries I ask the class if they think that this puzzle works for all numbers. Most of my students answered "Yes". The others weren’t sure. One student said “Don’t we have to test all the numbers?”  “Yes,” I responded emphatically. “But can do we do that?” It was pretty obvious that that was not possible. "Instead of testing all the numbers," I said, "let’s see if we can find just one number that will "break" the Jinx puzzle, then we can say it doesn’t always work.” With the convenience of the spreadsheet Students try larger numbers, fractions, negative numbers, etc. but come away frustrated giving up declaring that “it always works” . But strange things did start to happen when the numbers got very large. One student tried 30 billion (3 followed by 10 zeros.) Notice a couple of strange results (Figure 4.). The “Multiply by 6” command results in an approximation of the answer in Excel’s version of scientific notation. Same thing for Subtract 3. But the computing recovered in the next step and finished with the expected result.


Figure 4

Another student noted that 3 followed by 15 zeros produced the results in figure 5.
Figure 5

All the calculations are approximations in Excel notation except for the final 13. Another student added a 16th zero. Surprise! The result now is 0. (Figure 6.) Now does that mean we’ve broken the Jinx puzzle? Or is it that our Jinx calculator has some flaws? (It’s the latter.)


Figure 6

What happened here? Was it something mathematical that broke down here? No, it was the spreadsheet that misfired. It's not that the puzzle doesn't work for a number three followed by 16 zeros, it's just that this version of the spreadsheet doesn't handle such big numbers. So it appears that the spreadsheet is not good enough to test all numbers. But even if we could does that mean we can guarantee that it would work for all numbers? Unfortunately, math is very fussy sometimes. To prove that it works for all numbers Algebra needs to come to the rescue. We have to test all the numbers but how do you do that?

Another number that might come up and challenging the jinx calculator is Pi. Here the best you can do is enter an approximation like 3.14 since Pi has an infinite string of digits after the decimal point. We need a better way to show that it works for all numbers. This is where using a bag which symbolically can contain any number we choose including Pi and any other pesky irrational numbers. I use a bag metaphor (inspired by W. W. Sawyer (3)) for younger students because it's more intuitive, since a bag usually contains something while using X is more abstract. To add to the confusion the variable X is the name of the “container” that holds an unknown quantity usually a number. You can see why algebra can become messy, very quickly for young minds. Here is a suggested approach to the doing the puzzle. 

Step 1: Pick a number.
 Instead of picking a specific number, let’s choose a bag to represent any number.


Step 2: Add 11

. To show 11 we will use 11 small circles or “marbles”. Now we have a bag and 11 “marbles.”




Step 3: Multiply by 6. 


We now have 6 bags and 66 marbles.



Step 4: Subtract 3. 


Now we have 6 bags and 63 marbles.



Step 5. Divide by 3. 


What's left? 2 bags and 21 marbles.



Step 6. Add 5. 


There are 2 bags and 26 marbles.





Step 7. Divide by 2 


and we have 1 bag and 13 marbles.






Step 8: Subtract the number you picked in step 1. 



But that’s the bag! You subtract the bag. So you are left with only 13 marbles or just plain 13.

Can you explain why this puzzle always gives you 13? Watch a Flash demo of the proof. X aficionados can watch the more typical Flash version

In my teaching I used bags, boxes, and triangles to represent variables. I wanted to reinforce the notion that there is a distinction between the picture or name of the variable and the contents that it holds so I used boxes instead of X even after Harold Jacobs stopped using them in his book. I eventually switched to using Xs after my students told me to stop with the "baby boxes" because they "get it." But even then on occasion I had to remind them that X means the "value of X" quite often.

References
1. Harold Jacobs. Elementary Algebra. Freeman and Co. (1979)
2. Robert Berkman defines Kimp in his February, 1994 article "Teacher as Kimp" in the Arithmetic Teacher as someone who is so stupid that they are unaware of how stupid they are. Here I'm pretending to not know the answers to the questions I'm asking.
3. W. W. Sawyer. Vision in Elementary Mathematics. London: Penguin Books. (1964). Chap. 4. P. 65.
4. Online lesson the Famous Jinx Puzzle

Wednesday, June 5, 2013

Focusing on the Right Problem...Craps to the Rescue


How can I create the conditions within which people will motivate themselves? - Edward Deci & Richard Flaster 
In a recent guest post "Why Khan Academy is not the answer"  Daniel Kitts writes: "Popular efforts to improve education are focusing on the wrong problem. Millions of dollars and hours of innovation are being spent on improving how we deliver content in an era when content matters less and how we interact with it matters more. [...] Future education is better served by investing in and developing tools that support discussion and interaction, not improving content delivery."

A good example of content that will matter "less" in the future is Algebra. That's not to say that Algebra will not be important, it will just be approached differently or so I hope. For those of you that think this is a bad idea, don't worry there will be plenty of students that will continue on the geek track devouring Algebra in its current form. What I'm getting at is that the traditional, step by step approach - as outlined in the table of contents of almost any algebra textbook - will become an option, rather than requirement. Math courses that will be a subsitute for traditional algebra will focus more on real world problem solving, answering questions that are more meaningful to students than learning to solve quadratic equations in isolation.

The heart of effective problem based learning is this: the teacher sets the stage with a problem, puzzle of game containing an interesting context where math isn't mentioned, but when all is said and done the students learn some powerful mathematics.

I learned this lesson in the early 1980s when microcomputers were just beginning to enter the schools. I had a 10th grade high school student - I'll call him Don - who challenged me to respond to his complaint that this Algebra 2 stuff he was doing in my class was a complete waste of time. I sympathized with him. I knew the way I was teaching Algebra made no sense to him. What he did for me was to bring out into the open the conundrum that I faced every day as a teacher: how to motivate a subject that most students see as irrelevant. For them Algebra was something to be avoided unless absolutely necessary (e.g. passing my class) and where a C grade was fine with them.  Even when I offered them a blueprint for getting an A students like Don never tried even though it was there for the taking. I offered it up guaranteed if they worked for it. For me an A was short for Achievement (reaching a goal) not an indication of how clever or smart they were. Though Don was taking Algebra it was clear to me he had no understanding or appreciation for anything mathematical that was different from the everyday math that he could handle. (Mostly in relationship to money which he knew a lot about.) If I wanted to have any success with Don I had to dig deeper into his story to find a way for him to feel successful and appreciate Algebra. I didn't need to look at his school history. All I needed to know was that he loved to play Craps the 2 dice betting game. The basic game is played as follows.
The player throws two dice. If the sum is 7 or 11, then the player wins.  If the sum is 2, 3 or 12, then he loses.  If the sum is anything else, then he continues throwing until he either throws that number again (in which case he wins) or he throws a 7 (in which case he loses).  (Source: http://mathforum.org/library/drmath/view/70263.html)
Don knew these rules cold. But would he have ever thought to ask: is Craps a fair game? Or what are the chances of winning? Are they 50-50? Did he understand that playing the game over and over again could help him determine whether it was fair or not?

All I knew was that he thought Craps was a fun game. That's all.

So I asked him, "Do you think it's a fair game?" I didn't have to explain what I meant. He understood. "Sure it's fair," he said. "Sometimes I win and sometimes I lose." "But what if I told you that the dice were stacked against you," I replied. "In other words, in the long run you will lose more times than you win."

"How do you know?" he asked.

"Do you think gambling houses play games that are completely fair meaning that both player and the house have an equal chance of winning?" I queried. No response. He was confused. I went on.

"Does a gambling house make money?" I asked. Don replied, "Yes."
"Then they must have an advantage, right?" I continued.

At this point I suggested that we play the game in class to find out if that really is true.

But before we started rolling dice, I wanted my class to understand how probability works so I played the coin flip challenge with them. I split the class into two groups creating two teams. I would play as the third team. Now I tell them that I'm going to flip 2 coins. Team A will score a point if both coins are heads, Team B will score a point if they both are tails. And I will get a point if it comes up a head and a tail. We played a few rounds. It wasn't long before the class realized that I had an unfair advantage since after 10 tosses I was well ahead with 5 points. (A had 3 and B has 2 points.) Why is that? Well, there are actually 4 possibilities not 3 (as was assumed by my students.)

HH
HT
TH
TT

HT and TH are separate events so my probability of winning is 2 out of 4, not 1 out of 3 as I led the class to suspect. I believed that playing this coin game would help my students to understand the dice probabilities better later on.

Now it was time to play Craps.

"I'll be the house and you and the rest of the class will be the shooter," I said, speaking to Don.

A big smile appeared on Don's face. Finally we were doing something in Algebra class that was fun and as far as he knew had nothing to do with math. Or at least that's what he thought.

We had time to play a few games. Fortunately, I had a supply of dice in my desk drawer. We finished playing 5 games when the bell rang. Don and the class won 4 out of the 5 games.  Feeling smug Don left the room with a smile on his face. The following day I asked the class if they thought this game was fair? The class had different responses: yes, no, maybe and I don’t know. "How can we find out?" I asked. Don said for us to keep playing. "I bet ya if we play enough games we will win more often than lose." "Ahh, what an excellent idea," I thought. "What does he know about the law of large numbers or Monte Carlo methods?" I wanted to find out.

"But how can we keep playing a lot of games quickly?" I asked rhetorically. "What if I could build a machine that would play the game over and over." "You mean like a computer program?" asked Don. "Exactly," I said. With the help of several of my students we eventually had these instructions written on the board.
Table 1. To see actual code written in Scratch click here.
After programming the game in BASIC, Don and I ran the simulation 1000 times. The "shooter" won 497 times and lost 503. I was pleased with the results and told Don that this was very close to the actual probability of winning which is 49.29%

Don wasn’t impressed. He still thought it should be 50-50. How to convince him that knowing that its closer to 49% than 50% makes a big difference in the real world of gambling? Our next conversation would be about that.

My conversation with Don went something like this:

Ihor: How many possible outcomes can you get when you roll 2 dice?
Don: 2 through 12. Let's see.

Then with his fingers he counted. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.

Don: There’s 11.
Ihor: That's right. Those are all the possible outcomes. But how many different rolls can you have? Snake eyes [two ones] is one roll, Boxcars [two sixes] is another. How many are there all together?

To help them I made a tansparency template with the 6 dice roll possibilities along the top and the left side of the grid and put it on the overhead projector.
Figure 3: A grid overlay for possible outcomes when rolling 2 dice.

We filled in the numbers together. Each number inside the grid is the sum of the column and row headings (photos of the actual dice faces.). For example there are 3 ways to make a 4 (3+1, 2+2, and 1+3.)
Figure 4: All possible outcomes when rolling 2 dice.
"What are you chances of rolling a 7 or 11 to win?" I asked. No response. I followed up with "Can you see that there are 6 ways to make a 7 and 2 ways to make 11?" Positive response. Don now relishing his spokesman for the class role said, "So there are 8 ways you can win on your first roll and since there are 36 possible ways [events] the probability of winning is 8/36 or 2/9."

The class got it.

Ihor continues, "2, 3 or 12 on the first roll are losers and since there are 4 ways you can lose the chances of you losing is 4/36 or 1/9. So what that means is that a one-third of the time you will win or lose on the first roll." (2/9+1/9=3/9=1/3)

"But what if you roll a 4?" I asked.

Don immediately responded, "You don’t win or lose. Four becomes your “point” and you roll again. If you then roll a 4 you win. If you roll a 7 you lose. If you don’t get either, you roll again until you get either a 4 or a 7.

"What do you think your chances are of making your point?" I asked.

Don said, "Since there are 3 ways to make a 4 and 6 ways to make a 7, your chances of winning are 3 out of 9 or 1/3."

I was impressed. But I wasn't sure any one else in the room understood. So I said, "Let me see if I can sum so far. The chance of rolling a 4 on the first roll is 3/36 or 1/12. The chance of you rolling 4 again before you roll a 7 is 3/9." Then I took it one step further and said, "Since the two outcomes are independent of each other the probability of you winning when you roll a 4 is 3/36 x 3/9 = 9/324=1/36 which is 2.78%. The same argument applies to the other 5 numbers (5, 6, 8, 9, and 10) that escape the immediate win or lose test.

I showed table 2 to the class.
Table 2: Probability of winning at Craps (1)

Showing this table at this point was a mistake. I got a lot of eye rolling. Don didn’t get my explanation about why I multiplied and asked, "What do you mean they are independent?"

I replied, "You rolled the dice twice. Were the probabilities of each roll effected in any way?" "No," said Don unsureadly. "If dice rolls are independent you can multiply their probabilities to determine the probability of the outcome." Now Don (and probably everyone else in the room) was really confused. He did with me what he always did in math class  he capitulated to my authority by saying "if you say so." The chart claims that the shooter has a little better than a 49% chance of winning, but why?" Table 2 had to be right because Mr. Charischak said so, but the reasoning was just hocus pocus to Don as I'm sure it was to the rest of the class.

"Can we verify this claim about the probability in another way?" I asked trying to relieve them of their math anxiety. "No, its not going to be on the test," I said answering what I'm sure what was on their mind.

"What do you think will happen if we run the experiment 1,000 times?" I asked.  "We're going to beat the computer," was the reply meaning that the shooter would win. I estimated it would take about 2 hours since the output was going to the printer so we waited until the following day to check the results. Don was pleased to see that “the shooter” won 511 times. "Does that make the math [in the chart] wrong?" I asked. The responses from the class were split between yes and no. "What do you think will happen if we run the simulation 10,000 times?" "I don't know," said Don. "But let's try it and see." I reprogrammed the simulation to do the additional rolls and I also had the computer print the results to the screen instead of the printer. I don't remember the exact results but the shooter was the loser. We then talked about the law of large numbers and how the experimental results came closer to the actual probability because of the large number of rolls.

For fun I ran the experiment last night using a program I wrote in Scratch and it approximated the actual probability (49.29%) well. 4,961 wins in 10,000 games. This time I smiled because it confirmed the mathematical analysis.

Some afterthoughts
After this experience I vowed that Craps would became a permanent part of my Algebra “curriculum” without going into the details of determining the theoretical probability. Just highlighting how the computer program uses variables to produce its results was a dynamic example of using variables in Algebra. Unfortunely, I never got to write it since my days at BFS were numbered and I was going to move on to other adventures. Looking back I realized that the the seed of what I now call the Wannado curriculum were planted firmly, but making it happen was another story.

Don and I had a productive learning experience. Don learned how to use variables in programing the game which we wrote together. He understood the "law of large numbers" after we ran the simulation a 1000 times. The light bulb went on when he realized how the casino had the financial edge so playing Craps in the long run was a losing proposition. The key to the success was that I was a student, too. I was in the beginning stages of learning to program myself. When he realized he was my partner in programming the game I saw the sparkle of engaged learning in his eyes.

Years later I learned from Elliot Soloway (2) that he did research in how programming supported students ability to problem solve. For example, he gave this problem to college students to model algebraically: "There are six times as many students as professors at this university. Use S for the number of students and P for the number of professors." The most common answer was 6S = 1P with the students thinking that this equations means there is a large number of students associated with each professor.  Then he asked the group to write a simple program that would take as input the number of professors (P) and output the number of students (S.)

They realized that the number of students S was equal to 6P not the other way around as previously thought. So programming a computer helped the students realize the correct use of the variables involved.

References
1. The original source for this chart is long since forgotten. You can s a similar chart here: Smith, S. (2009) Craps ... A Casino Game of Pure Chance.
2. Soloway, E., Lochhead, J. and JohClement. Does Computer Programming Enhance Problem Solving Ability? Some Positive Evidence on Algebra Word Problems. In Computer Literacy, edited by Robert J. Seidel, Ronald E. Anderson, and Beverly Hunter, 171-185. New York: Academic Press, 1982.