[For a detailed conceptual discussion of several methods, read the excellent comments by Mike and novemberfive.]
It's 'probably' too early in the year for this, but...
This is one of those typical probability questions that all students struggle with until they have enough experience to feel comfortable. When away from these for awhile, most everyone needs to revisit this kind of question and work through the theory again. It's so elusive...
Here's a typical version of this question:
Three cards are drawn at random from an ordinary deck of 52. What is the probability that they will consist of a king, a queen and an ace?
Do you feel totally comfortable when seeing these? Imagine how most students feel. They either have a strong concept here or they get that queasy feeling and express, "I don't like these!"
So here's the issue:
Is there one method you have found for these that makes sense to most students or should multiple methods be demonstrated (and shown to be equivalent)?
To get you started:
Method One: For the denominator, select all 3 cards 'simultaneously' in C(52,3) ways. The numerator is more interesting, so I'll leave that to you.
Method Two: Analyze one card at a time without replacement. Thus, the probability that the first card will be a king is 4/52, etc. Of course, most of you know the 'catch' here, but the method can still work if...
Method Three: Yours!
Friday, August 31, 2007
Drawing 3 cards one at a time vs. Drawing 3 cards simultaneously - Those nasty probability questions!
[For a detailed conceptual discussion of several methods, read the excellent comments by Mike and novemberfive.]
Thursday, August 30, 2007
Here's a warm-up you can give to your Algebra 2 (and beyond) students to welcome them back to math class after a summer of brain drain. NO CALCULATOR ALLOWED! This oughta' set the tone...
A total of 9991 M&M's were eaten by a group of Freshmen. Here are the facts:
(1) Each freshman ate the same number of M&M's.
(2) There was more than one freshman.
(3) Each freshman ate more than one M&M.
(4) The number of freshman was less than the number of M&M's each freshman ate.
How many M&M's did each freshman eat?
Work in your groups of 3-4. You have 3 minutes. First team to arrive at the correct number AND explain their method, gets to eat ______________.
Let me know how many groups solved it or your thoughts about the appropriateness of this question.
Wednesday, August 29, 2007
You may want to read the comment I posted on John Armstrong's, The Unapologetic Mathematician. It's the 23rd psalm, oops, I meant comment, listed there and it does represent my strong feelings about where we are in all of this 'bifurcation' business. I'm hoping we can move on now.
The next edition is coming out next week over at Learning Computation and I will be hosting the following one on 9-21. I will have several suggestions for the format of submissions at that time, so I'm hoping you will follow this thread over the next few weeks. Actually, Kurt over at Learning Computation has anticipated most of what I will be requesting so please read his article on the Carnival thoroughly - it's very thorough and well-written. He also has some fascinating information on the mathbloggers.com site and Social Rank. Why should we be surprised that there are computer science whizzes trying to develop new tools for ranking blogs and websites. Why would they do it other than for commercial gain? Because they can, I suppose! Why did a teenager in Glen Rock NJ spend countless errors trying to unlock the secrets of the iPhone? Because he could! (earning instant fame and a new car in the process). The opportunities for technophiles to instantly change the world are now endless...
Tuesday, August 28, 2007
Those of you who found your way here via Carnival of Math Edition XVI, may also want to read some of the more recent posts:
A Preview of an Interview with Lynn Arthur Steen
Algebra 2 End of Course Exam - Latest Info
Singapore Math - Part III - Info from the 'Source'
Another Problem from Singapore Grade 6B Placement Test
If you're interested in more geometry investigations, look through the labels on the sidebar. There are many geometry activities and challenge problems for your students or for your pleasure...
[As promised, scroll down to the end of this post to see some links to the best references I have found for the golden triangle and related topics.]
The following detailed investigation is designed for geometry students. Questions 7 and 8 are designed for students who have had some basic trigonometry, which is sometimes included in a geometry course. You may want to save this for an extended activity or project for later in the year, but if you want to inspire your group earlier, show them what they will be able to solve by the time the course is over! Feel free to comment and make corrections and suggestions. There are many excellent online references for the triangle below. I will provide these links after you've had a chance to explore and to feel the 'Gold Rush'.
The isosceles triangle shown above doesn't seem very special but you may feel differently after the journey we're about to embark on.
Here are the given: ΔPQR is isosceles, PQ = PR. RS bisects ∠PRQ. ∠P = 36 degrees, sides PQ and PR have length x and base QR has length 1.
1. Draw a regular pentagon and all of its diagonals. How many triangles can you find that are similar to ΔPQR?
2. Determine the measures of all the angles in ΔPQR.
3. Explain why PS = SR = QR.
4. Here is the key step: Using similar triangles, show that x satisfies the equation:
x2 = x+1.
5. Solve to show that x = (1+√5)/2.
6. You've now shown that the ratio of the longer side of the isosceles triangle to its shorter side is (1+√5)/2. Note that this number is also the ratio of PS to SQ (why?). You've struck gold! This number is known as the Golden Ratio and is usually denoted by the Greek letter Φ (phi), pronounced 'fye'. Research this number and write five fascinating facts about it.
7. Think you're done? You've only scratched the surface. We will now relate all of this to a trigonometric value of 36°. You recall that cos 30° = √3/2, cos 45° = √2/2 and cos 60° = 1/2. Well, these are not the only angles whose cosine or sine can be expressed in exact form using whole numbers and square roots! Here's your challenge:
Show that cos 36° = Φ/2 using 2 methods:
(a) The Law of Cosines in ΔPSR
(b) Let M be the midpoint of segment PR. Now work with with ΔPSM. Show your results are equivalent.
8. Since we're having fun with trig, let's show that sin 18° = (Φ-1)/2 using 2 methods:
(a) Bisect ∠SRQ and use a triangle.
(b) No geometry - just use a half angle identity. Show that the results are equivalent!
9. If you feel there's much more to be mined here, you're right. Perhaps you could find the sin or cos of other angles that are multiples of 18°! Have fun!!
(a) cos 36
Click on 'One solution" to get more details. Prof. Wilson from UGA has an awesome problem-solving page that is accessible to high school students as well.
(b) Exact trig values
An exceptional site from UK that develops the theory of exact representations of certain trig values using radicals. Worth reading through it. I found this after the fact but I learned a great deal.
(c) Pentagram and the Golden Ratio
Another wonderful site with excellent diagrams and an historical account showing the development of the golden ratio. Once you start reading this, you will not want to stop - bookmark this one!
Saturday, August 25, 2007
I always enjoy reading quotes (excellent bathroom reading), particularly those that challenge and provoke (kind of like teaching!). I also admire the brilliance of those who can speak volumes in the fewest possible number of words.
Here are a few I found with my own interpretations as well as a couple that have been guiding principles in my career. Some are loosely paraphrased from memory, so if you know the original, please share...
Let's start with some humor, although you may not find it so.
(Bumper sticker) If you can read this, thank a teacher.
My addition : Never follow too closely. (there's a double meaning there!)
(Seen on a T Shirt): 3/4 of all students don't understand fractions; the other half don't like them.
(This is not the original which I can't quite recall - but the message is the same)
Now onto more serious thoughts...
An understanding heart is everything in a teacher....One looks back with appreciation to the brilliant teachers, but with gratitude to those who touched our human feelings. The curriculum is so much necessary raw material but warmth is the vital element for the growing plant and for the soul of the child. [Carl Jung]
My thoughts: What did a wise teacher once say to me -- Dave, just remember, you don't teach math, you teach children.
Good teaching is one-fourth preparation and three-fourths theater.
My thoughts: I almost want to make another joke about fractions but this truism should stand alone. I never thought of myself as a great performer but in reality I gave 5x5x40x35 performances, some of which were definitely better than others! Ever notice how some former teachers become stand up comics...
Those who are incapable of teaching young minds to reason, pretend that it is impossible. The truth is, they are fonder of making their pupils talk well than think well and much the greater number are better qualified to give praise to a ready memory than a sound judgment.
My thoughts: This should provoke some strong feelings. I've asked many teachers why they don't ask more probing questions in a lesson. I often get the following type of reaction, "I tried, but I wind up having to answer the questions myself, so what's the point." Perhaps, it's enough to ask the questions... the next quote says it better...
To know how to suggest is the great art of teaching. [Henry Adams]
When the National Science Foundation asked the "breakthrough" scientists what they felt was the most favorable factor in their education, the answer was almost uniformly, "intimate association with a great, inspiring teacher." [R. Buckminster Fuller]
My thoughts: None needed
The mediocre teacher tells. The good teacher explains. The superior teacher demonstrates. The great teacher inspires.
My thoughts: I guess I was mediocre, good and superior in my career. I'll leave it to others to judge how great I was.
More important than the curriculum is the question of the methods of teaching and the spirit in which the teaching is given. [Bertrand Russell]
My thoughts: I believe they work hand in hand. It would be hypocritical of me to downplay the importance of what is being taught.
And last but not least, the oft-quoted but still the most meaningful for me...
A teacher affects eternity; he can never tell where his influence stops. [Henry Adams]
I was hoping not to get melancholy about not starting a new school year with the rest of you but these quotes are having an effect on me. Have a wonderful year as you touch another group of lives. Please share your favorite quotes here with all of us.
Friday, August 24, 2007
Outrageously funny, bizarre at times, brilliantly clever...
There's no other way to describe the latest edition of the Carnival of Mathematics over at John Kemeny's website. From the wildly amusing rating system for classifying math posts to the over-the-top graphics, I'm amazed that John was able to get this published on time. Well worth waiting for and it leaves little doubt that the CoM is alive and well. And, oh, by the way, John absolutely did justice to each of the submissions. Enjoy it!
The only problem for me (and every other host) is that I have to follow John's act in a few weeks and this is one tough act to follow.
Congratulations, John and in the true spirit of your blog entitled, a mispelt bog, let me add that this edition of the Carnival was mind-blogging!
Some time ago, I posted a piece about math mnemonics. Buried near the bottom was my feeble attempt to make a table showing a well-known (?) fascinating pattern for sin and cos values for the common angles in Quadrant I. Over the years, some students have found this to be as useful as memorizing ordered pairs on the unit circle or deriving everything from 30-60-90 and 45-45-90 (which I still prefer personally). I've seen students make this table at the top of their trig unit exam - they figured it was worth the effort! I'm reprinting this today using an image created in LaTeX and the absolutely wonderful and easy to use Texify website. This has been a real boon for those using Blogger since LaTeX has not yet been supported. Many math bloggers have been using it for a while now and I'm sure they appreciate its power and simplicity as much as I do. Its author is Andrey Burkov and Ars Mathematica gives him proper credit here. Certainly if an old dog like me can learn new tricks like this, anyone can! By the way at the Texify site, there is an extremely well-written tutorial with many examples to follow. I suspect I will be using this a lot for my new posts and perhaps cleaning up my old. Let me know if the table below is as readable as it appears to me. and, of course, if you like the pattern, you can tell me that too!
The original post used the klutziest of notations and was barely readable. This should be a lot better! I omitted the row for the tan function which is just the quotient of rows 2 and 3:
Wednesday, August 22, 2007
[Wonderful discussion going on in the comments dealing with math terminology and the underlying concepts embedded here. As always, my post is just the appetizer. The main course is to be found in the comments section. If you enjoy this discussion you may also want to visit some of my recent posts touching on Singapore Math and a review of an important book by Alec Klein on teaching gifted children.]
The following is similar to a question that appeared on the PSAT a couple of years ago. The College Board has been testing recursively-defined sequences for some time. Students take the test, then it's 'out of sight, out of mind'. Educators may want to explore the ideas behind these problems in much greater depth. Recursive sequences are an important topic and are now included in many sets of standards. Should this be viewed as exclusively a high school topic?
Consider the sequence 11,6,5,... defined as follows:
Each term after the second is the nonnegative difference of the two preceding terms.
What numbered term has a value of zero?
Things to ponder beyond the answer...
(1) Is this question appropriate for middle schoolers?
(2) Would the terminology nonnegative difference be problematic for some students?
Note: This is the actual wording from the exam. Would you have reworded this using absolute values?
(3) If an educator wants students to develop these ideas in a more extended investigation, how much preliminary groundwork needs to be laid if any? It's important for our students to understand that mathematical researchers often start with specific examples like this without knowing the general relationship. They analyze many specific examples looking for patterns just like students can be trained to do.
(4) How many particular examples would students need before they can begin to formulate an hypothesis? Should the first two terms be restricted to positive integers (or even integers for that matter!)? Should the instructor suggest different starting values or allow the students to explore? As always, the issue is the role of the instructor while students are exploring. Teachers inexperienced with facilitating these kinds of investigations can benefit from observing those who have been doing this for awhile. It ain't obvious and it sure ain't easy!
(5) Would most students conjecture that the sequence will eventually become 'stable' at some point if the initial two terms are integers? Would middle schoolers be able to explain why this would happen?
(6) Do you think some students would be able to formulate a general theory for this type of sequence? Develop a formula for the number of terms required for the terms to reach zero? How many students in middle school or high school would suggest starting with decimals or even irrational numbers like pi? Is there ever an end to these investigations?
Well there is an end to this post! [Notice that the original question seems less significant now.]
Sunday, August 19, 2007
I just received a complimentary copy of a new book, A Class Apart, by Alec Klein, an award-winning reporter from the Washington Post. I accepted this with the understanding that I was under no obligation to review or promote the book on this blog. However, I did read a few advance reviews and a blurb I found online. And now I've read the book. Not quite Harry Potter but this is an honest and well-written view of one of the highest-rated high schools in the country, Stuyvesant HS, with its long tradition of excellence and famous alumnae. Stuyvesant is a selective (based on an entrance exam) school for the gifted, particularly in math and science.
Coincidentally, this week's Time Magazines' feature story is entitled, The Genius Problem, focusing on the lack of attention being given to our most talented youth. Both the book and the article strike the same chord: "But often overlooked are gifted and talented students."
What particularly hits home for me is that I live less than 10 miles from Stuyvesant and less than 3 miles from a similar school in Bergen County, NJ, the Bergen Academy, which ironically is linked to Stuyvesant in the book. My links to both schools are also ironic, but I won't go into that here.
Mr. Klein has written a powerful story about 'Prodigies, Pressure and Passion.' I related to it on so many levels, both as a student and as an educator who recently stepped out of the classroom. The book evoked a flood of bittersweet memories. Just as Mr. Klein became emotionally involved (yet somehow able to remain objective) with the four or five students ( a small but representative sample of the student body) whom he followed about for one school year, I recalled one particular student I had in my first or second year of teaching in an affluent Bergen County, NJ, high school. John was a misfit in the school. He arrived in his junior or senior year, living with his grandmother. He was a hulking 6'4" whose clothes were not only out of style with his classmates but had holes and were ill-fitting. John sat quietly in my calculus class and one day presented me with a sheet of yellow paper on which he sketched his alternate theory of limits, a central concept in calculus but one he could not accept. I shared some of this with the class. I tried to comprehend what John was doing but really only grasped a small part. He attempted to explain it to me after class but I was too dense and communication was not his strongest suit. John did acceptably in the course although he struggled to show his methods, since he did most of the problem-solving mentally and resisted or had difficulty in explaining his reasoning. My memory dims at this point. I don't recall if he made it into college. I never heard from him or about him again but I will never forget him. He taught me to open my mind and my eyes and to appreciate the extraordinary uniqueness of each of my students. Perhaps John would have flourished at a school like Stuyvesant where he would not have been viewed as so different, where students might have looked beneath the veneer...
If you haven't already gathered this, I strongly endorse Mr. Klein's book. It is inspiring and presents a strong objective argument for demanding excellence and having the highest expectations for all our children. More than that, it reminds us that our nation cannot afford to overlook our best and brightest minds, who require challenge to reach their potential, not the cavalier attitude that they will succeed no matter what school they're in. It will be available August 25th and it is published by Simon and Schuster. You can read the publisher's notes here.
Saturday, August 18, 2007
I've always believed we can learn so much from observing what and how children learn in other cultures. I've received permission from Jenny, a representative of Singapore Math, to post a few of the questions from the Grade 6B placement Test and discuss them. This post will focus on Question 13:
Last month David and Mary saved some money in a ratio of 3:5. This month they saved an additional $154 together, and David now has three times as much money as he had last month while Mary has two times as much money as she had last month. How much money did they save last month?
Jenny is the curriculum advisor for the US based distributor of Singapore Math, has an intimate knowledge of the materials and how they they are implemented. From looking at the 6A Placement Test, which included some algebra, I had assumed that children would solve this using a variable x as follows:
Originally 3x 5x
Additional ___ ___
Afterward 9x 10x
Therefore the additional savings would be 6x and 5x for Dave and Mary respectively.
Thus, 11x = 154 and x = 14. Originally, Dave and Mary saved 8x or (8)(14) = $112.
Well, the children in Singapore are taught a visualization for complicated ratio problems like this which, in my opinion, powerfully lays the foundation for algebra. Instead of the variable x that I used, children are shown how to represent the given ratio using unit bars and solving for the value of a unit:
David and Mary saved money in a ratio of 3 : 5
After this month, David now has three times as much, and Mary now has two times as much.
The total additional amount is $154.
From the diagrams, you can see that there is an additional 11 units. Therefore:
11 units = $154
1 unit = $14
Last month they saved 8 units together.
8 units = $14 x 8 = $112
I had to adjust to this when first reading it. I had mistakenly assumed that the children were introduced to traditional variables earlier on and would be encouraged to use them. But then I began to realize what was different here. Many of our children (including secondary students) struggle with complicated ratio problems (even uncomplicated ones!). I needed to imagine seeing the unit bar construct through the eyes of a young child. The idea of using a visualization of a unit bar for a 3:5 ratio doesn't seem to be that significant at first blush, but now I think it is. Instead of representing the original quantities as 3x and 5x, children can see these quantities in a tangible way. More significantly, they can draw the effect of multiplying Dave's savings by three and Mary's by two. I needed to step back here to appreciate this. Jenny explained that children do not use unit bars for all ratio problems, just for the most complicated ones. So what are we saying here? You mean, it's not enough to just buy Singapore Math materials and give it to kids in our classrooms? Teachers need to be trained in how to implement them successfully? You mean there's no easy short-cut here? You think!
I want to personally thank Jenny for her graciousness in replying so thoroughly to my naive questions. I will have more to say about this but I await your thoughts...
Wednesday, August 15, 2007
John Kemeny is hosting the 15th Edition of the CoM on 8-24 and you can submit your articles here. However, there are no more Carnivals scheduled at this time.
In the most recent edition of the Carnival of Mathematics, Vlorbik reported that there has been a call for change in the structure of the carnival. This idea was first raised by yours truly in the Tenth Edition of the Math Carnival and recently again by Michi over at his blog. Look here for the ongoing dialogue I've been having with John Armstrong and Michi (read the comments). John feels strongly that the Carnival has been heavily weighted toward math education of late and I agree. Since that is my main area of interest, I'm not overly upset about this but I do agree that the research-advanced mathematics side has taken a hit. I made a proposal that we split into a Carnival of Math Ed (K-14 oriented) and a Carnival of Research Mathematics (John's and Michi's idea). Although previous comments have suggested many are satisfied with the status quo, John feels that we already have a 'de facto' separation. I suggested we take the current temperature of the water by inviting comments. I volunteered to be the home base for the Carnival of Math Ed site but I would welcome support from others, particularly the most frequent contributors to the math ed blogosphere who have the time to take on this project. John volunteered to be the moderator of the Research group. He also suggested that we run the Carnival monthly and I agree. This would enable more to contribute and make the day of the each Carnival more of a significant event. If you're wondering why we seem to be ignoring Alon, the original architect of the Carnival of Mathematics, that is not the case. I am deeply appreciative of the monumental efforts made by Alon but he recently has decided to drop from the blogosphere and, presumably, the Carnival as well. I know he would want someone to pick up the reins and keep this moving. Alon, if you read this, I hope you will share your thoughts. There will be some technical problems to overcome involving the submission of posts and registering the 2 Carnivals of Math with the main Carnival site, but I'm sure we can work this out.
If I receive very few replies to this proposal, I may move forward on my own or with John , since we are willing to accept this challenge. Waiting to hear from you...
Tuesday, August 14, 2007
[Note: As promised, I have now posted a lengthy comment to this post, which only scratches the surface. I welcome your comments.]
Although Singapore Math is no secret to many homeschooling parents and districts around the country, I thought that the sample placement test for Grade 6B, available for free downloading, might lead to some fruitful discussion. Pls try all 14 questions before commenting. I did and I will have much to say later. There are many other placement tests at the same site so have fun! Go to the Singapore Math home page for background if needed.
Here are two standardized type questions that students sometimes struggle with. Those who do well on these kinds of questions know the key is to understand the basics of arithmetic sequences. The 2nd question is a bit more sophisticated. What changes did I make to complicate the picture?
1. The median of a set of 20 consecutive integers is 14.5. What is the mean of the first 10 of these?
2. The mean of 98 consecutive odd integers is 44. What is the greatest of these numbers?
BIG IDEAS for ARITHMETIC SEQUENCES:
(a) MEAN = MEDIAN!
(b) MEAN = MEDIAN = (LAST + FIRST)/2
(c) N = (L-F)/D + 1
Can you guess what these variables represent?
Hint: This is a variation on a well-known formula for arithmetic sequences.
Saturday, August 11, 2007
BACKGROUND and OVERVIEW of ACTIVITY
Middle schoolers are often introduced to the famous sieve mentioned in the title to find which numbers, say from 1 to 100, are prime. This is a common activity in which all the multiples of 2 are first crossed out, then multiples of 3 and so on. The following is a combinatorial (counting) activity that may help them (and more advanced learners as well) appreciate just how beautiful this method is and how it can be generalized to demonstrate the endlessness of primes. In the process, middle schoolers will review the concepts of multiples, common multiples as well as composite vs. prime numbers. The 2nd activity below is for upper-level students although middle schoolers can certainly try it.
Consider the first 3 primes; 2, 3, and 5. Children know what the next one is and that there are many more after that up to 30, but, for this activity, tell them that they will find, by elimination, the remaining primes up to 30. Specifically, using the famous sieve algorithm, they will determine there are SEVEN more primes up to 30 by eliminating all the numbers that are divisible by 2, 3 or 5! Sounds like you've seen this many times? Wait...
Note: DO NOT have students use colored markers or pens to cross out numbers. It tends to obliterate the marks underneath that are needed for analysis.
Middle School Activity (standard sieve approach):
1. List the positive integers up to and including 30.
2. Cross out the multiples of 2 in your list using a slanted /. Explain how you could have determined that 15 numbers were crossed out without using your list.
3. Now cross out all the multiples of 3 from the original list using the \ mark. How many numbers did you cross out this time? How could you have determined this without your list? Count how many numbers have been crossed out twice. How could you have determined this without your list?
Note: In some applications of the Sieve method, students cross out only from the remaining numbers, not from the original list each time. Since our objectives here involve developing the idea of common multiples and also combinatorial methods, students are instructed to cross out some numbers more than once. This is not unusual in many texts or workbooks.
4. Now cross out all the multiples of 5 from the original list. Use the --- mark for this. How many numbers were crossed out? How could you have determined this without your list?
5. Count how many numbers were crossed out exactly once. Describe these numbers.
Note: Students may have difficult expressing this and some discussion is needed. For example, they might at first say "Numbers divisible by only two." This is a fine response but how can the instructor build on this?
6. Count how many numbers were crossed out exactly twice. Describe these numbers.
7. Explain why there was only one number crossed out exactly 3 times.
8. There should now be EIGHT numbers remaining which have not been crossed out. Are these numbers all prime?
9. Ask more questions!
Notes: We know that students (of all ages!) have difficulty with the issue of the number 1 not being regarded as prime. The accepted definition of prime requires that the number have exactly two distinct factors. Seems arbitrary, huh? Besides 1, most students will assume that the remaining seven numbers necessarily have to be prime, however this method does not guarantee that! If we used the above sieve up to 50, then 49 would be left over as well! Subtle...
OVERVIEW OF ADVANCED ACTIVITY
How could older students have attacked this without making a list, using more sophisticated combinatorial methods? The idea behind the above activity was to first identify the numbers that were divisible by 2, 3, OR 5. After eliminating these and the number 1, students were to consider the remaining numbers, which all happen to be prime. The remaining part of this activity deals with combinatorial methods needed to COUNT how many numbers are divisible by 2, 3 or 5 without first making a list. Some will still need to make the list!
MORE ADVANCED ACTIVITY
Consider the list of the positive integers up to and including 30. The following set of questions is designed to get an accurate count of the numbers that are multiples of 2, 3 or 5 and then to consider the remaining numbers.
1. Explain why there are 15 multiples of 2 in this list without actually counting or listing them!
2. Explain why there are 10 multiples of 3 in the original list without...
3. Explain why there are 6 multiples of 5 in the original list without ...
4. Thus far we appear to have accounted for 15+10+6 = 31 numbers in this list which are multiples of 2, 3 or 5. Since there are only 30 numbers to start with, what went wrong! Explain carefully.
Note: The method of 'overcounting' is a critical one for many set-theoretic and counting problems.
5. By now you realize that we need to compensate for the numbers that were counted more than once. Show that 10 numbers were counted more than once.
6. To compensate for these duplications, we can adjust the count: 31 - 10 = 21. Thus it appears that there are 21 numbers in our list that are divisible by 2, 3 or 5. In fact, there are 22! What went wrong! [If you don't believe this, make a list and count!!].
Note: This is subtle. The number 30 still needs to be counted.
7. Ok, we have hopefully established that there are 22 numbers that need to be eliminated from our original list of 30. Thus, there are 30 - 22 = 8 numbers remaining. One of these 8 is not prime. Which one?
8. If you haven't already done so, make a list of the 30 numbers and actually work through each of the steps above to verify your results.
9. What do you think? Is this a good method for counting how many primes there are in a particular list? Would it be practical for counting how many primes there are up to, say, 210 given that 2, 3, 5 and 7 are prime? Where did 210 come from? Why might this method fail to produce only primes?
Although this post seemed to be about a sieve for primes, you've probably figured out that it really became an activity to solve the problem of counting how many of the 30 numbers in the list were not divisible by 2, 3 or 5. I'm sure you are thinking that there are many others ways we could have counted the multiples of 2, 3 or 5. Your students may object to the above method and suggest a 'better' one. For example, count all the even numbers up to 30 first. Then count the odd multiples of 3, then the powers of 5 . No duplications, short and sweet, right? However, the set-theoretic method of counting with duplications, then compensating is actually more powerful and can be generalized to longer lists of primes. Embedded in this activity is the subtle notion that there cannot be a finite number of primes. Do you think students would recognize that on their own?
Thursday, August 9, 2007
[A special thanks to Mike for correcting the error I made in question 3 below. It has now been updated!]
Students in geometry often see problems involving inscribed and circumscribed circles, squares, triangles and other polygons. Questions involving such diagrams often appear on standardized tests and on math contests as well. Although this particular post focuses only on ratios of areas involving squares, equilateral triangles and circles, I am planning a series of investigations which delves far more deeply, requiring students to discover and verify more general relationships for polygons of n sides. Further, as the number of sides of the polygons increase, students will be asked to analyze the ratio of the areas of the circumscribed to the inscribed polygons and consider if they approach a limiting value. Thus this series of activities prepares students for the calculus as well. Students are also introduced to the duality principle, useful for later study in advanced geometry. A strong background in geometry is needed and, at some point, a knowledge of trigonometric ratios is required. This could be a culminating project for a marking period or the year. I hope you enjoy this and will save it for the school year or ...
Note: Although it would appear to be more logical to have students determine ratios for the triangles first (n = 3), the diagram shows the squares (n=4) on top because the analysis is somewhat easier.
Note: Although this is couched as an investigation for the classroom, my readers are invited to attempt the questions below and suggest various approaches to finding the ratios. Those experienced in this topic will find each question fairly straightforward, however, consider the bigger picture here! Those whose geometry is rusty will need to review some basic properties of circles and polygons.
FOR THE STUDENT:
1. The diagram at the upper left depicts a circle circumscribed about a square and a second circle inscribed in the square. Verify that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:2.
2. The diagram at the upper right may be thought of as the dual to the first diagram, in that each circle has been replaced by a square and the square by a circle. Show that the ratio of the area of the inscribed figure to the area of the circumscribed figure is invariant, that is, it remains 1:2.
3. The diagram at the lower left depicts a circle circumscribed about an equilateral triangle and a second circle inscribed in the triangle. Show that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:4.
4. Similar to #2 except that the circles and the triangle have now been interchanged. Again, show that the ratio is invariant.
Wednesday, August 8, 2007
The 14th edition of the Carnival of Math is out there for your pleasure a couple of days ahead of schedule over at Vlorbik on Math Ed. This is an excellent edition with links to a wide variety of worthwhile math blogs at all levels from K-8 through advanced. Similar to a suggestion I had offered in the 10th edition of the Carnival (which I hosted), there is now a call to consider a Carnival of Research Mathematics. Although the consensus back then was to keep one math carnival intact, the idea of splitting it appears to be gaining momentum, particularly since the last couple of carnivals seemed to be weighted more heavily toward math ed. As long as each has enough of a following, I believe this can benefit both ends of the spectrum. I could add more here, but go enjoy the latest Carnival!
Posted by Dave Marain at 10:35 PM
Tuesday, August 7, 2007
I can't believe I'm doing this but one of my students asked me if I heard the other half of the joke. Here's the original:
"What did pi say to i?"
Here's the other half:
"What did i say to pi?"
[Ok, at least it's not about Math Wars and Learning Styles!]
At that moment I became possessed and asked:
"What did one-half say to one?"
"You're so full of yourself!"
I'm afraid of where this will lead so I'll sign out now. So, do math teachers really have the corniest sense of humor and do they really think they're funny?
Monday, August 6, 2007
ABCD is a rectangle in both figures. Figures are not drawn to scale.
1. In Figure 1, E is the midpoint of side BC. If the area of AECD is 1, what is the area of ABCD?
2. In Figure 2, E is a point on side AB such that AE:EB = 3:1. F is a point on side BC such that
BF:FC = 2:1. If the area of DEFC = 1, what is the area of ABCD?
(a) Although the first question is very similar to an actual SAT problem, both questions admit multiple solution paths and strategies that can help students develop both their spatial reasoning and analytical skills.
(b) The first question is similar to some textbook questions and would be rated as above-average difficulty on the SAT's, although most visiting this site would not consider it difficult. There are many variations on these kinds of problems. The most famous one is to consider the figure formed by connecting the midpoints of the sides of a rectangle and showing its area is half of the whole.
(c) The 2nd question is more discriminating and requires more than intuition.
(d) How many of today's students have a well-developed sense of fractional parts of the whole?
(e) Try to imagine how a variety of learners, say 9th or 10th graders, would approach these. Do you think the majority would attempt a visual approach - cutting up each irregular shape into common figures or perhaps dissecting the entire rectangle into equal parts? Would any students consider plugging in arbitrary lengths for the sides of the original rectangles even though specific values are given (one could then use similarity to finish it)? Is it a good strategy to encourage students to assume each rectangle is in fact a square? How many students would attempt an algebraic setup?
(f) What is the point of spending so much time discussing various approaches to a problem? Is it really worth the time expended when there is so much more material that one must complete? You all know my "less is more" mentality and that there are no shortcuts to developing problem-solving facility.
(g) Which is more important for learning to take place: Allowing the student to struggle but arrive at a solution with some strategic guidance from the instructor OR Allowing dialogue to occur enabling students to see how their peers are approaching it? What kinds of questions might the instructor ask in facilitating the activity? What exactly is our role when students are engaged? Is there a course one can take or a book from which educators can learn such pedagogy or does one learn from experience and/or from their peers?
(h) Is this more about solving geometry problems or developing problem-solving skills? Will I ever stop asking such inane questions...
Thanks to Mathmom!
She reminded all of us that there is an RSS feed available for you to receive automatic updates for comments on MathNotations from your favorite reader. I use Google reader personally but there are other options available. I am so lucky to have knowledgeable caring readers who are not an afraid to teach an "old dog (me!) new tricks!" Thanks also to Jackie for bringing this to my attention in the first place.
Posted by Dave Marain at 3:07 PM
Saturday, August 4, 2007
Very nice logic problem in the sidebar over at Text Savvy.
Because it involves only a few possibilities, it is a great way to open the school year without causing undue frustration. Students will enjoy the feeling of accomplishment when they solve it!
Also, I posted a comment on a fascinating piece of research over at Joanne Jacobs blog. The post is entitled No evidence for 'learning styles'.
While I agree with the researcher and the other commenters that a school district's approach that tries to match individual instruction to each child's modality is unreasonable, I take exception to research that attempts to invalidate theories by only looking at extreme cases of its application. Certainly, competent caring teachers try to create a classroom environment that is accepting of different strengths and weaknesses and attempt to provide explanations of concepts in a variety of ways. I referred to the Rule of Four in math pedagogy, a critical part of current thinking about maximizing learning in the math classroom. 'Learning styles' has become an inflammatory term that raises the hackles of most bloggers and educators in general. I expect that my comment will not be well-received and most likely will be ignored!