Dorothy Revisited -- Another View...

Mathmom contributed some insightful thoughts about how most middle school students might feel about the probability investigation from the other day. I agree with her that some would be able to compute the results or even devise a general formula but "proving" it in the general case might be too ambitious. In my reply, I suggested there might be another way of deriving the formula 1/N for the probability of losing the game. Here's what I came up with. It still requires some careful development to show that the outcomes are equally likely but I will indicate how it could be done in the particular case where N = 10.

Brief Explanation of Method:
There are N equally likely (to be shown) ways for the game to end (i.e., when the red card is selected). Of these, only one will result in a loss -- when the red is the last card chosen. Therefore, the probability of losing is 1/N, hence the probability of winning is 1 - 1/N or (N-1)/N.

Demonstrating "Equally Likely" for N = 10:
P(game ending after one card) = 1/10
P(game ending after 2 cards) = P(black selected followed by red) = (9/10)(1/9) = 1/10
P(ending after 3 cards) = P(black,black,red) = (9/10)(8/9)(1/8) = 1/10
etc...

The general case is similar using N in place of 10. I do think that students with some understanding of algebra could follow it but deriving it on their own is another story!

I also indicated that I might provide a program for the TI-83 or -84 which could be used to simulate the game. The programming skills needed are not that advanced and some high schoolers or even middle schoolers can pick up on the code and begin writing their own programs - I've seen it happen! Here it is...











T represents the number of times the game is played with 3 cards. I entered 100 for the number of trials. K stores the number of times Dorothy won when playing 100 times. Can you make sense of the rest of the code?

The experimental probability of 0.68 is reasonably close to the theoretical probability of 2/3. I often feel more confident of my reasoning in difficult probability problems when my simulation approximates my answer. This doesn't prove anything but it does have value IMO. There is also the opportunity to demonstrate some important stat concepts by running the program several times and having students plot the experimental probabilities and observing their distribution.

Please Help Dorothy Go Home - A Probability Fantasy for Middle School and Beyond

Summer vacation is an appropriate time for fantasy. Enjoy the hiatus!



The following investigation is not intended to be a math contest challenge. It reviews fundamental principles of probability and you might want to bookmark it for the fall. We can also simulate the first problem using the programming capabilities of a graphing calculator. I may post a simple program for this later on.


The wizard will let Dorothy go home if she can pass three challenges.

He shows Dorothy 3 playing cards, 2 of which are black and one is red. He shuffles them and turns them face down. "Dorothy, here's your first challenge."

"You will pick a card. If it's red the game ends, you win the game. If it's black, I will remove the card and you will pick a card from the remaining two. If it's red you still win! Ah, but if it's black again you and Toto and your weird friends will remain here for at least one more month."

Well, Dorothy won the game and said, "Now, I want to go home!" But the crafty wizard said, "You weren't listening carefully, Dorothy. I never said you can go home if you won the game. You've only passed the first challenge. You must still pass two more." "That's not fair!" Dorothy protested but the wizard makes his own rules in Oz.

"Alright, Dorothy, you won the game but you knew the odds were in your favor since you had two chances to win. Here's your next challenge:

"What was the probability of your winning and you must give me two correct but different methods?"

Dorothy asked, "These are the remaining challenges, so if I get them right, I can go home, yes??"
"I will not lie to you, Dorothy. This is your 2nd challenge. There will still be one more."

Dorothy was upset but knew she had no choice but to trust him. She thought about the problem for a minute and replied, "The probability of my winning was 2/3. I know I'm right!"
"Very good, Dorothy, but you must explain that answer two different ways." Fortunately, Dorothy was a very responsible middle school student back in Kansas and had learned the methods of compound probabilities and the idea of complementary events (this is a fantasy after all!).

Dorothy was able to provide two correct methods. Can you?

"Very good, Dorothy! You only have one more challenge to conquer and you can go home.
This time there are N cards, one of which is red while the remaining cards are black. N is a positive integer greater than 1. Same rules as before. The cards are shuffled and laid out face down. You pick a card. If it's red the game is over and you win. If it's black, the card is removed and you try again. The game continues until you pick the red card. The only way to lose the game is if you pick all the black cards and the last card remaining is red."

"In terms of N, what is the probability that you will win? Oh, yes, you again have to show two different methods in detail on this magic board over here."

This time, Dorothy needs your help. She can guess the formula but she needs our help to show two ways to derive it. Please help Dorothy go home!

"On The Road Again" With 'TC' -- A Real World Application of Geometry

As my devoted readers know, Totally Clueless, affectionately known as TC, has contributed many insightful comments and profound ideas for us to think about. His sobriquet belies a brilliant creative mind of course. He recently sent me a geometry problem which was motivated by his own experiences driving to work. The problem itself is accessible to advanced middle and secondary students but the result is interesting in its own right and should generate rich discussion in class. I recommend giving this as a group activity, allowing about 15 minutes for students to work on, then another 15 minutes to discuss it. Save it for an end-of-year activity or bookmark it for the future. Beyond the problem, there are important pedagogical issues here:

• How to introduce this
• Asking questions to provoke deeper thought
• Drawing conclusions and further generalizations
• Connecting this problem to other circle or geometry problems
  
• Maximizing student involvement
  

I told TC I would need some time to rework the original problem for the younger students so here goes...





Diagram for Parts I and II







Part I (middle and secondary students)
In my city, there are two circular roads "around the center" of the city, of radii 6 and 4. There are a number of radial roads that connect the two loops. Points A and B in the diagram above are at opposite ends of a diameter of the outer loop and the dashed segment is a diameter of the inner loop.

If I have to go from point A to point B on
the outer loop, I have two options:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially (blue) to the inner loop, drive along the inner loop (red), and then drive radially out (blue). (Assume that there are radial roads that end at point A and point B).

Show that Option 2 is shorter than Option 1.

Part II (middle and secondary students)
Same diagram but now the radii are R and r with R > r.
Show algebraically that Option 2 is shorter.


Part III (secondary students)












To generalize even further, points A and B are distinct arbitrary points on the circle, central angle AOB has radian measure θ where θ ≤ π. OC and OD are radii of the inner loop; OA and OB are radii of the outer loop. Again the radii of the two circles are R and r, where R > r.

As before, there are two options in going from A to B:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially from A to C (blue), then along the inner loop from C to D (red), then radially outward from D to B (blue).

Show that Option 2 will be shorter provided π ≥ θ > 2.

Click Read More for further discussion...


Further Comments
(1) TC's original problem was Part III. I decided to add Parts I and II to provide 'scaffolding' for students. Was this really necessary in your opinion?
(2) The results of these questions are independent of the actual radii. TC felt this was an interesting aspect of this problem and I agree. Do you think students will be surprised by this? Do we need to point this out to them? Are there other circle problems you can recall which have a similar feature?

Thanks TC for providing us with another stimulating challenge!

...Read more

An Equation Which May Be More 'Complex' Than It Appears!

Maybe I should rename this blog to Saturday 'Morning' Post. After all, no one reads that either anymore!

As the school year comes to a close (and I'm assuming it's already over for some), here's an innocent-looking equation which might be worth discussing with your advanced algebra/precalculus students now or next year. I might have considered saving this for our next online math contest but it's complex nature makes it more suitable for discussion in the classroom than on a test. Have you seen exercises like this in your Algebra or Precalculus texts? Do students often delve beneath the surface of these? It's kind of like a black box. We often feel we simply cannot reveal too much of the mystery here or we will not finish required content. Well, you know my philosophy of 'less is more' and I don't even live in Westport, CT. (Ok, that's a post for another day!).

SOLVE (by at least two different methods):

2a^-3/2 - a^-1/2 - a^1/2 = 0

Preliminary Comments/Questions/Issues

• Is the term solve ambiguous here, i.e., should we always specify the domain to be over the reals or over the complex numbers or is that understood in the context of the problems? I'm guessing that most advanced algebra students learn that the domain of the variable or solve instructions may impact on the result, but, that is precisely one of the objectives of this problem.
  
• Should students immediately change all fractional exponents to radical form? OR use the gcf approach (which requires strong skill)?
  
• It's not hard to guess that 1 is a solution but is it the only solution? Can we make a case for -2 being the other solution? The graph doesn't reveal this and surely, -2 doesn't make sense or does it....
  
• Is there ambiguity in raising a negative real number to a fractional exponent (never mind raising i to the i)? Why? Isn't there a principal value for such an expression? How is it defined? This problem raises fundamental and sophisticated issues about numbers which can be taken as far as one chooses to go Just how complex can complex numbers get?
• What is the role of the graphing calculator here? Mathematica? Wolfram Alpha? In addition to verifying solutions or determining answers, can these tools also be useful in clarifying ideas or raising new questions?
  
• Students (and the rest of us) are now capable of quickly filling in the gaps in their knowledge base by visiting Wolfram's MathWorld or Wikipedia for more background. Should this impact on how we present material? Typically, in the pre-web days teachers would avoid opening up a can of worms like complex solutions here, but, with your more capable groups, the sky's the limit now IMO...
  

Two Geometry Problems To Sharpen The Mind - Never Too Late In the Year For That!

Well, the June SATs have arrived today so these problems come too late for that, but these kinds of questions can be used to review basic ideas while strengthening thinking skills. Both questions below are appropriate for both middle and secondary students, although the second requires knowledge of a fundamental geometry principle regarding the sides of triangles.

There are other important principles embedded in these problems as well. In the end, I believe that students need to be exposed to many of these "contest-type" challenges to improve reading skill, learn how to pay attention to detail and think clearly. As a separate issue, performing well under testing conditions requires extensive training. You may not feel this is an important objective for math teaching in the classroom, but testing is a reality for the student...

These questions may appear fairly straightforward at first but be careful! I believe the second is more challenging than the first. These are not so different from the "gotcha" problem on our latest online contest.

1) The dimensions of a rectangle are odd integers and its perimeter is 100. How many different values are possible for its area?

2) The perimeter of an isosceles triangle is 96 and the lengths of its sides are even integers. How many noncongruent triangles satisfy these conditions?

For my "unofficial" answers, click on Read more...


Unofficial Answers (no solutions):
1) 13
2) 11
Feel free to challenge these answers or express agreement!

Comments

Which of the following do you believe would cause the most difficulty for students?

• The wording/terminology (e.g., noncongruent); general reading comprehension issues
  
• The sheer number of details (e.g., odd vs. even, perimeter vs. area, integer values)
• A precise counting/listing strategy vs. an abstract or commonsense approach
• The "square is also a rectangle", "equilateral is also isosceles" traps
  
• The issue of different areas for #1
  
• The triangle inequality for #2
• Other concerns?
  

...Read more

RESULTS OF 2nd MATHNOTATIONS CONTEST!!

It took a couple of weeks but the results are in -- finally!

FIRST PLACE

PINK PANDA TEAM
Canadian Academy
Kobe, Japan
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SECOND PLACE (TIE)

WALLINGTON HS (SENIOR TEAM)
Wallington, NJ

THE BLACK SWAN TEAM
Canadian Academy
Kobe, Japan
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THIRD PLACE (TIE)

WALLINGTON HS (JUNIOR TEAM)
Wallington, NJ

DECATUR AREA HOMESCHOOLERS
Decatur, Il
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FOURTH PLACE (TIE)

LAKE STEVENS HS TEAM I
LAKE STEVENS HS TEAM II
Lake Stevens, WA
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Comments

• Winning score was 9 pts out of a possible 14
• Median score was 4
• Participation was down for this contest but opening it up to more than one team from a school proved very successful. Some schools which had planned on participating found the timing at the end of May to be very problematic and had to drop out after registering. I'll remember that for next year!
  
• Hardest problems involved trigonometric derivations and a probability question requiring an infinite series.
  
• This contest definitely proved harder than the first and several questions were designed for the upper level secondary student.
• Future contests may be split into a 9th-10th grade version and an 11th-12th grade version similar to AMC-10 and -12.
• Students indicated the contest was challenging but expressed interest in participating again.
  
• The open-ended questions required considerable effort on students' parts and mine in grading them!
• The winning team sent a highly detailed and original solution to one of the trig questions. They wrote it by hand and scanned it. This technology works very well for this kind of contest.
• I will probably publish 1 or 2 of the questions with answers and partial solutions on this blog in the near future.
• Any schools interested in participating in the fall should send me an email now ("dmarain at gmail dot com") to get on my mailing list. I've already received several emails. At this time I plan on keeping the contests "free to a good home"!
• I am very excited about the international flavor of these contests -- this seems highly appropriate given the culture being established by our current administration. The world really is becoming "one out of many!"
  
• As mentioned previously, one of my goals is to publish 10-12 of these contests with detailed solutions as a book or pdf document which can be downloaded for a nominal fee.
  

CONGRATULATIONS TO ALL OF OUR PARTICIPATING TEAMS!