Friday, February 29, 2008

The Polls are Closed and here are the results...

Thank you to all who voted in the MathNotations survey over the past month. It's only appropriate that the voting ended on February 29th. Guess we will have to wait 4 more years before we ask this same kind of question again!

Those of you who cast your vote saw the results at the time you voted, however, you might not have seen the final results. Also, our feed subscribers may not have seen the survey or the ongoing stats. At any rate, I want to make sure we all understand that, as mathmom pointed out early on, this or any other web survey does not constitute a scientific poll, i.e., we cannot draw firm or possibly any conclusions. Even though we collected 165 responses, here are some specific reasons why we have to be cautious:

(1) The wording of the question and the choices may have been ambiguous or unintentionally biased.
(2) I originally had (E) None of the above, but then decided to remove it. This choice might have picked up 10% or more. It may also have deterred several from actually voting as was suggested in the comments early on.
(3) There is certainly no basis for asserting the randomness of the sample we received. In fact, it is more likely those who voted tended to be those who have visited the site before and therefore might share sentiments similar to those of the author. This would surely skew the results.

In spite of all these reasons not to take the results too seriously, the distribution was decidedly 'normal!' About 3 out of every 4 votes were cast for the balanced choices (B) and (C), with (C) representing the greatest balance between traditional and reformed approaches, IMO. I lean toward (C) myself. Yes, I do recognize that the balanced choices were in the middle so it sets up a bell-shaped curve!



Which best describes your preference for teaching multidigit multiplication?


15% (A) Teach only the traditional algorithm and expect mastery

36% (B) Teach the 'partial products' method to develop understanding of place value and the traditional algorithm; teach the traditional algorithm as a more efficient method and require it; expect mastery

38% (C) Teach the 'partial products' method to develop understanding of the traditional algorithm; teach the traditional algorithm as a more efficient method; give students a choice of methods; expect mastery of at least one method

9% (D) Model other methods (e.g., 'lattice method') and encourage students to invent their own method; do not require any particular method or mastery

Obviously there's a 2% error here (these numbers were copied from the screen)! I'll let our readers try to figure out the software/math issue that led to this!

Your thoughts....

Ratios, Ratios, Ratios...

More of the same...

In a certain group:

The ratio of males to females is 4:5.
The ratio of left-handed people to right-handed is 1:11 (assume no one is ambidextrous!).
64% of of the left-handed people are males.

(a) What % of the males are left-handed?
(b) What % of the females are left-handed?


  • If students or any of us see enough variations of these, will they become almost mechanical or does one have to decide which method/model/representation is needed for each problem?
  • How many models should students be shown for these? Usually students or our readers will find a method or model no one else imagined!
  • Is algebra the most powerful method? The most efficient? How many variables? Is it usually best to use one variable and let it represent the total number of people in the group?
  • Anyone ever use a matrix/spreadsheet/table/Punnett square model to represent the data for these kinds of relationship problems? Specifically, problems in which the entire group (the universe) is divided into either groups A and B or groups C and D. This will become less cryptic as the discussion unfolds.
  • Do you think most students today would feel more comfortable working in %, decimal or fraction form? What about rest of us out there?
  • Too challenging for middle schoolers or not? Math contest problem or just a challenge to develop facility with ratio thinking? How would most algebra students fare with this?

Wednesday, February 27, 2008

Kyosito Ito - our Mystery Mathematician - and our Winners!

Kyosito Ito

I chose Dr. Ito for many reasons. Here are a couple--

(1) He is the father of stochastic analysis/stochastic differential equations, which, for me meant that he found precise mathematical equations and formulas to describe apparently random fluctuations and motion like Brownian movement. I remembered taking a grad course in Ergodic theory which demonstrated to me how there can be order in chaos.

(2) This quote in his bio made an impression on me:

"Although today we see this paper as a fundamental one, it was not seen as such by mathematicians at the time it was published. Ito, who still did not have a doctorate at this time, would have to wait several years before the importance of his ideas would be fully appreciated and mathematicians would begin to contribute to developing the theory."
I believe it is not unusual that important theories and discoveries have been ignored or dismissed by 'experts' who don't want to take the time to consider that someone of lesser renown might have actually discovered something they could not! Years or generations later, the research often receives the recognition it deserves.

However, Ito didn't respond with bitterness:

"When I first set forth stochastic differential equations, however, my paper did not attract attention. It was over ten years after my paper that other mathematicians began reading my "musical scores" and playing my "music" with their "instruments."
And now for our winners...
Those who correctly identified Professor Ito (and selected comments/anecdotes):

"Kiyosi Ito --
I have an interest in the history of mathematics and the contribution of great mathematicians. Even when I do not completely understand their contributions (which happens all too frequently,alas!), following that history puts mathematics into a beautiful, cohesive whole. Euler remains my favorite mathematician for this reason because he did just that - he put so many prior contributions into an understandable whole that we still use today. Of course, he made some pretty outstanding contributions in his own right. -- Looking at mathematics though an historical perspective from its earliest roots of simple counting - i.e cave drawings of the successful hunt - to the establishment of number systems and their expansions to meet the problem solving needs of society - is a wonderful guide line for what we should be teaching and when."
Thank you, Hypatia for your continued interest in the recognition of some of the world's greatest mathematicians and this contest. You are one of our most consistent winners!


"Was it the same guy who found a way

to divide time into infinitely small
chunks so that rocket trajectories
could finaly be calculated in an
uninterrupted continuum?
If so it must be: Kiyosi Ito"

I'd also like to personally thank Florian for excellent suggestions to improve our contest, some of which I've begun to implement!

Totally Clueless

"The mystery mathematician is Ito san of Ito calculus fame. I remember

struggling through stochastic differential equations in grad school. I
think his work might have had an influence on the work of Black and
Sholes on the pricing of options on the stock market."


Kiyosi Ito (last o should have circumflex).

I only know of him because he won the Gauss Prize in 2006; his work is outside my expertise.

The March 2007 volume of The Japanese Journal of Mathematics had a number of articles devoted to Ito's life & mathematics (including a photo). There are a few interesting anecdotes: a) Ito, about 90 years old at the time, expresses regret that his mentor had recently died at 100, and so did not hear of the award and could not enjoy the success of his student. Mathematics contributes to longevity perhaps ? b) Ito observed the good sense of there being no age limit on awarding the Gauss Prize (as for the Fields medal) since it often takes a long time for applied mathematics to be recognized. c) It is recalled that in the late 1930s, while working for a Japanese statistical bureau and before obtaining a doctoral degree, he wrote two fundamental papers. Since no copy machines were available, he needed to copy by hand in the library a text of P. Levy that included ideas he developed, added to and made rigorous.

Kevin added a fascinating extra piece which could easily be a discussion on its own:

As an interesting aside, in the same journal issue, there are some articles about Teiji Takagi. An interesting quote from one of them:

There is an interesting historical document. A copy of a Japanese
journal, The Journal of Tokyo Suugaku Kaisha (Tokyo Mathematical Company),
No.44, 1882, has an appendix1(pp. 24–26) which reports the fifteenth meeting
on Japanese translation of the word ‘arithmetic’. The point of the argument was
whether they should choose a word meaning ‘art’ or ‘science’ for ‘arithmetic’.
Kikuchi strongly appealed for ‘art’ and finally won 9 votes among 14 committee
members excluding the chairman. Since then the word ‘san jutsu’ in Chinese
characters, which was originally found in old Chinese books of mathematics, is
still used to mean ‘arithmetic’ in Japanese.

Congratulations to our winners and a deep expression of gratitude to our readers for their continued support of my efforts.

Finally, only a few hours left to vote in our survey! I will try to post the final results and a brief discussion on Friday, Feb 29th as we leap into the future!

Monday, February 25, 2008

Mystery Mathematician Week of 2-18-08, etc...

As the number of MathNotations feed subscribers has dramatically increased, it seems unfair to leave the Mystery Mathematician just in the sidebar. One of those who solved this week's contest suggested I post it in a separate entry -- great suggestion. Thus far we have three winners and I'm hoping others will recognize the image to the left or use their research skills to identify him. I will announce all of the winners on Wed 2-27 so you still have time. I'm tempted to give a hint, but I'll hold off..

Don't forget to email me (dmarain at gee-mail dot com) with the information - PLS DO NOT POST THE NAME IN A COMMENT!

Also, try to include some anecdote about the mathematician and your source if possible. If you wish, you might also include some personal information so that I can get a feel of who is responding to this contest.

The responses may not be random nor the poll scientific, but the results are nevertheless interesting! You need to visit the site to vote in the sidebar.

Sunday, February 24, 2008

Beyond Mixed Nuts --A More Challenging Ratio Problem

Readers are strongly encouraged to read the extensive discussion of methods in the comments.

In Virtual HS, the ratio of the number of juniors to seniors is 7:5.

The ratio of (the number of) junior males to junior females is 3:2.
The ratio of senior males to senior females is 4:3.
What is the ratio of junior males to senior females?

(1) Isn't it a shame that the Jn:Sn ratio isn't 5:7!
(2) Algebraic methods seem to be the most reasonable here, but would some students attempt straight numerical methods using common multiples? Can you find such a way?
(3) Does anyone ever use or teach tree models for ratio problems? We know they are useful for probability problems, but isn't there a similarity here?
(4) Would a Singapore bar model approach work here? I'm counting on those who are proficient with this method! This is not my area of expertise...
(5) Is this question too difficult for the SATs? More appropriate for a math contest question?

Friday, February 22, 2008

MIXED NUTS - A Middle School Activity to Promote 'Ratio Sense'

[Note: Readers are strongly encouraged to read the extensive comments to this post, which clarify the ideas and provide more detail for introducing ratios to younger children.]

R A C T I O N....
a:b a ÷ b

An expression showing how two quantities compare?
A fraction? A percent? A probability?
Other definitions?

Developing ratio concepts is generally considered to be a crucial part of a middle schooler's math development. The following activity is designed to help students develop an understanding of the equivalence of ratios, fractions and percents. Just as importantly, it provides an application of part:part and part:whole relationships.

On a table in front of the room the instructor has placed a small empty cup, a large container filled with cashews, a large container filled with peanuts and four large empty containers. In this activity, students will make different mixtures of nuts that are respectively 50%, 20%, 10% and 40% cashews by volume. Students will need to describe the methods to accomplish this and provide mathematical details to support these methods.

Let's get started, boys and girls. Notebooks open, dated ________ and title today's activity MIXED NUTS. Before we are all tempted to enjoy these delicious nuts, does anyone know which nuts are generally more expensive: cashews or peanuts? Other than for flavor, can anyone suggest why cashews are sometimes mixed with other nuts? Ok, suppose a company or grocer wants to sell a mixture that is 50% cashews by volume. Think about
(1) What you think this means
(2) How
you would do this

Now write your ideas and methods in your notebook.
Be very specific - like a recipe.

To be continued...

Wednesday, February 20, 2008

Thank You...

As you may have noticed recently in the sidebar, MathNotations has risen dramatically in Social Rank according to Math Bloggers. In fact, at one point in the past week, MathNotations reached #1 among all math blogs and is currently ranked #4 as of this posting.

Regardless of how one might feel about the reliability of the index/algorithm used, one thing is clear: Your support for this blog has been overwhelming and is deeply appreciated.

My goal has not changed: To provide rich engaging content for K-14 educators, as well as for students, parents and anyone out there who is as fascinated by mathematics or is as passionate about the math education of our children as I am.

Again -- Thank You...

Important Reminder
Don't forget to check out the 27th Edition edition of the Carnival of Mathematics over at jd2718 coming 2-22-08! Submit your own posts or recommend some of your recent favorites from fellow math bloggers!

Odds and Evens - Week of 2-18-08

  • New Mystery Mathematician has been posted in the sidebar! Please DO NOT submit your answer as a comment to this post! simply email me at "dmarain AT GeeMail dot com'
  • Only a few days left to vote in the MathNotations survey in the sidebar regarding your preference for how multidigit multiplication should be taught. The results thus far are interesting, but I'll say more about this when the polls close. No early exit poll results at this time!
  • As previewed several weeks ago, I've just received the Mimio equipment for making Math Casts or mini video math lessons. I'm learning how to use it - my first few attempts are fairly crude - and I hope to post one of these in a few days. I may also upload it to You Tube. Of course it won't get as many views as "An Inconvenient Truth!" Here is a link to the company to give you some background before I post a more detailed discussion of this technology.
  • The following is the body of an email I sent to a close personal friend in the math blogosphere in response to an article this week in the Washington Post (Parents Rise Up Against A New Approach to Math). This article is getting lots of attention among other math bloggers as well, particularly over at KTM. I feel like I am reiterating the same position I have stated on this blog dozens of times, but the message still is not being heard by those making decisions about our children. I will never give up, no matter how many times I am ignored or criticized:

"There is a grassroots revolution among parents going on right now, but the issue is far from black and white. If most teachers felt comfortable blending teaching procedurally with teaching for comprehension and concept, we'd be moving in the right direction in my opinion. BUT curricula generally take an 'all-or-nothing' approach.

Centrists such as myself are looked on as hypocrites or 'waffling'. Teachers need a text that combines the best of both approaches, balancing understanding with practice and repetition. WHY IS THIS SO DIFFICULT FOR EVERYONE TO GRASP?"

Tuesday, February 19, 2008

The temperature changed from -5 to 5 degrees. The percent increase was (A) 10% (B) 50% (C) 100% (D) 200% (E) 300%

Many readers of this post objected to the scientific fallacy in asking for per cent change in temperature even if the arithmetic is correct. I completely concur with these criticisms and have a posted a definitive explanation below quoted from a knowledgeable meteorologist...


Just something for you or your students to consider as we are still in a deep freeze in some parts of the country.

Some thoughts...

  • Why multiple choice? Would the question be better asked in an open-ended way? Could this appear on a standardized test in this form?
  • Does the negative sign affect the outcome?
  • Is % change an important topic?
  • When should students be expected to know how to do this?
  • What are some effective instructional strategies and/or methods for this?
  • (F) None of the above?

"Your arithmetic is correct but your result is arbitrary. When expressing a temperature change as a percentage, one must use a temperature scale whose zero point is the temperature of absolute zero, and then use that selected scale consistently. The Kelvin scale is such a temperature scale. All other temperature systems, like Fahrenheit and Celsius, have arbitrary "zero points," and calculations of percent temperature change using those scales will give arbitrary results.

In the example you provided, on the Fahrenheit scale the temperature rise is 100 percent (from 50(degrees) to 100(degrees) F.), but that same change on the Celsius scale is 280 percent (from 10 to 38 C.). Using the Kelvin scale, whose zero point is absolute zero (-460(degrees) F., -273(degrees) C.), the rise is actually 10 percent, from 283(degrees) to 311(degrees) K."


Tom Skilling is chief meteorologist at WGN-TV.

Friday, February 15, 2008

Top Ten Lists of Common (Student) Math Errors!

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

If you enjoy this post, you may want to read a newer post on developing Ratio Sense for Middle Schoolers.]

In truth, we could drop the 'student' from the title and simply enumerate common math errors or, even better, fallacies. The latter term is more the flavor of this post, rather than simply careless student errors. This is because fallacies imply that there is a plausibility to some of these errors, i.e., they are somewhat natural errors to make if one doesn't fully grasp the ideas. Experienced math educators anticipate these errors and caution students about them during the lesson, preferable to commenting on these errors on their tests! In fact, it is possible that students can learn considerable mathematics by being asked to comment on these and explain each error.

IMO, helping students understand the underlying concept in each of these common mistakes is an essential part of teaching and learning mathematics. This is my approach in this post, rather than a "Can you believe I saw a student do this once!" attitude.

There are many such lists easily found by searching the web although most refer to common college errors (in reality, they include many precollegiate errors). Here is one of the best from a wonderful professor at Vanderbilt University. It is thorough, contains excellent discussion and categorizes the errors. Rather than copy from these sources, I wrote a few off the top of my head. I will also include Eric's which he posted in a recent comment to the 16/64 = 1/4 post.

Eric's Excellent List (and I know he has a few hundred more):

1. √(a²+b²) = a+b

2. (-x)ⁿ = - xⁿ

3. (fg)ʹ = fʹgʹ

These are wonderful. #1 and #3 could be classified as 'everything distributes' errors, although in verbal form, it could be interpreted as an 'everything commutes' error:
"The derivative of a product is the product of the derivatives" error.
Similarly for #1:
"The radical of a sum is the sum of the radicals" error.
#2 is an order of operations type of error and I included this in my list in a particular form.

Here's my initial offering for Grades 7-12. Feel free to bring your own list to the table for other grade levels. Common errors in calculus and beyond are fair game as well. I will try to classify some of these...

Radical Errors
√49 = ±7 type; similarly 161/2 = ±4
√(n2) = n

Operation Errors (Exponents)
-42 = 16
an = bn ↔ a = b
x2/3 = 16 → x = 163/2 or 64
[One possible correct method: x2/3 = 16 → x2 = 163 → x = ±√(163) = ±64]
Fraction Errors - Algebraic or Arithmetic (limited to the top 10000 errors please!)
(12x+7)/(4x+9) = (3+7)/9
[What name would you give this one? 'Cancelling error', 'Cancelling terms not factors error'?]

Rather than continue my list, I welcome offerings from our readers. A fairly thorough compilation of these could become a book! Perhaps, in a serious vein, a small monograph that could be helpful to both students and math educators...

Thursday, February 14, 2008

16/64 = 1/4...How to Reduce Fractions the 'Easy Way'!

Totally Clueless sent me an email reminding me of some famous 'fractured fraction' examples like the one in the title. Can you think of a couple of other two-digit examples of the same type that 'reduce' this way? Note that 10/30 = 1/3 doesn't qualify (the digits have to 'cancel' diagonally!).

Here is TC's version:
Note that the product 16 x 4 can be obtained by deleting the '1' and the 'x'!


(a) Find the other two instances of this 'weird' multiplication. The two factors have to be of the same type as in the example, i.e., a 2-digit number by a 1-digit number and the tens' digit of the 2-digit number must be 1.

(b) Most would find the other instances by guess-test. Here's a more significant challenge. Verify algebraically that there are exactly three such solutions.

(c) Is this problem equivalent to the 'easy way' to reduce fractions mentioned in the title of this post? Why or why not?

02-14-08 The 'Binary' Power of Love --- Happy Valentine's Day!

Holidays mean different things to different people and some may not have such warm fuzzy feelings today. Some may even associate it with computer viruses. The average adult will spend about $100 for candy and other gifts today. Over a billion dollars will be spent, 75% of which will be on chocolate. How about all those people who are allergic to chocolate, like me? Just some random number thoughts...

Wednesday, February 13, 2008

The [I.N.]Comparable I.N. Herstein - Our Math Icon of the Week Revealed

[Only a few more days to vote in the poll in the sidebar! To see the results thus far, you have to cast your vote.]

Well, the secret has been out of course if you read the comments to Doors and Windows Left Unopened, but it's time to officially recognize one of the great teachers and writers of modern mathematics. Israel (Yitzchak or, affectionately, Yitz) Nathan Herstein wrote, IMO, the clearest exposition of Abstract Algebra I have ever read. His Topics in Algebra was a major influence on my development as an undergraduate math major. Whatever ability I have to write a logical structured proof probably came from reading and re-reading his classic. To this day, I can picture his presentation of theorems, particularly how he developed the set of numbers that can be represented as the sum of two squares. Yitz, you can still run rings around anyone!

Our winners...

kevincp wrote:

Mystery mathematician: I N Herstein. I used his unsurpassed text, "Topics in Algebra" as an undergrad in the 60's. A curiosity I came across when browsing his name today was his acute use of logic to demonstrate the superiority of the latke over the hamantash as quoted in R.F Cernea's "The Great Latke-Hamantash Debate". The final line of the "proof" is: "Has anyone here ever seen anyone eating a hamantash with sour cream? Q.E.D."

and our other winner is...

Hypatia who wrote

...your mathematician is Yitzchak Herstein.

Congratulations to our winners!

I will leave Yitz's image up there a few more days - he deserves that recognition.
I will end my tribute to him by quoting one of his 30 research students:

"He was someone of great warmth who took an intense personal interest in his students and had a knack of getting them to believe in themselves."

Tuesday, February 12, 2008

[1,2]-3-[4,5]-6-[7,8]...21 Helping Children Devise and Understand Winning Strategies

Do you remember playing those fun counting games in elementary school? No, well, play along as if you do! The teacher or a friend would go first and always seem to win or you would go first and always lose. You knew there was a trick and if you figured it out it was exhilarating - like understanding the key to a magic trick.

Like most parlor games, there's genuine mathematics underlying these counting games. In this post we will describe a few of these and an investigation to help students not only devise a winning strategy (or algorithm) but to come to an understanding how division and remainders play a significant role.

Variation #1: The Game of '21'

Age Group:
Certainly appropriate for children even as early as 1st grade (however, devising winning strategies and explaining why they work might be a bit ambitious!)

# of players:
2 is best

To win, make your opponent say some target number like 21

First player starts counting from 1 and says either '1' or '1-2'; Other player then says the next number or the next two numbers; play continues in this way until someone is forced to say the number '21'. Verbal or written directions here are far more confusing than just demonstrating actual play.

Sample Play: See title of post for a partial play

Winning Strategy (partial): If you go first, say '1,2'. If you don't, your opponent can beat you if she/he knows the strategy.

Further Discussion: For the younger children, let them play against each other in pairs for a few minutes to allow them to feel comfortable with the game. Then you can ask if anyone wants to 'challenge the master' - you, that is! Tell them because you are older, you deserve the courtesy of going first (that will last for about 30 seconds or less!). After playing against students for a while, they will figure out that part of the winning strategy is to go first and say '1,2' but most will not pick up on the rest of the method. To mystify them even more, you can let them go first. You most likely will still win because you know the strategy and they will most likely not catch on for some time! There's always one sharp youngster even in the primary grades whose eyes will start glowing and will say, "Let me go first. I can beat you." At that point, you may want to say, "Game over!"

Winning Strategy: Those of you who are familiar with these kinds of counting games, know that they are all variations on the same basic theme and are simpler versions of the classic game, NIM. In this version of '21', some children will quickly see that, whoever gets to 2o has to win. It will take them a little longer to work backwards from there to see that to get to 20 you have to get to reach 17, which is 3 less than 20. To get to 17, you have to reach 14, which is 3 less than 17. Thus, working backwards, the winning positions, or 'magic numbers' if you will, are 20-17-14-11-8-5-2. Reversing this provides you with a guaranteed win but of course you need to go first and say '1,2'! But learning and using this strategy does not imply that the child understands WHY it works!

Using questions to help children begin to grasp the underlying idea: Children will immediately see why '20' is a winning position but ask them to explain why 17 also is (Possible student response: "Because if you say '17, then the other person can only get to 19 and you will be able to get to 20"). Continue to subtract 3 to obtain other 'magic numbers.' Ask the children why subtracting 3 is critical. Why 3? Children, even older ones, will soon see what is going on. Some may ask if one has to memorize all of these numbers. Don't answer that! Just smile and let them figure it out for themselves. Allow the children to practice the winning strategy on each other until they feel comfortable. They will surely want to try this out on other friends, teachers or family members!

Underlying Concept: At what grade level are children expected to grasp the essential idea that repeated subtraction is equivalent to division? Thus, in our problem, working backwards, starting from the winning position of 20 and continually subtracting 3, is equivalent to dividing 20 by 3:
20 ÷ 3 = 6 with a remainder of 2.
This can be interpreted to mean that after performing six subtractions by 3, the number 2 will remain! Of course, the repeated subtractions reveal all of the winning positions so children may not be appreciate the benefit of division. Help them to see that the remainder does reveal that one needs to go first and say '1,2' to guarantee a win.

A Million Variations
Well, maybe not that many in this post, but I'm sure you can see the possibilities are endless. You may want to ask children to devise their own version and a winning strategy as an outside project or assignment. They may invent something really cool no one has thought of! You might want to first ask the group how they could modify the game: "If you were going to invent your own game, what might you change about the game of 21?"
Some suggested variations:
(1) Whoever says '21' wins
(2) '21' loses but this time students can say the next number or the next two numbers or the next three numbers. Thus, if you go first and say '1,2,3', I would say '4'; then you might say '5,6' and I would say '7,8'. Am I guaranteed to win if I play correctly?
(3) Start from some number like 50 and allow children to subtract any number from the set {1,2,3,4,5,6}. Then you subtract one of these 6 numbers from the result and repeat play until one player reaches the number '1'. That player wins. This 'Game of 50' is also famous and will mystify adolescents as much as younger children! I'll let our readers explain the strategy and why it works! By the way, don't underestimate how much reinforcement of basic subtraction skill this game provides!

Saturday, February 9, 2008

Find all combinations of 3 distinct primes whose average is 13

[Have you voted yet in the survey in the sidebar? Time is running out...]

Just an isolated middle school mini-challenge to get the day started? Perhaps...

Those of you who are familiar with this blog know that MathNotations is dedicated to providing activities/investigations for middle and high school teachers to use or modify (provided proper attribution is given of course). In this post, I will demonstrate how one can build an extended or richer activity from a math contest or standardized test problem.

It is important to remind our readers here that these kinds of activities and problems do not constitute a curriculum. Students need to first develop proficiency with skills and procedures. These explorations are only intended to extend and enrich student learning. They can be used in part or in whole, as a long-term project outside of class, a team activity in the classroom or a myriad of different ways. All of this is at the discretion of the educator.

First of all, the problem in the title, in its present format, would not be an SAT or a standardized test question, unless the standardized test included free-response or open-ended questions.

In SAT format, the question might be changed to:

Which of the following can be expressed as the sum of three distinct primes?
(A) 6 (B) 9 (C) 12 (D) 15 (E) 17

Not a particularly challenging problem, but some students would struggle with comprehending the wording or paying attention to details ('distinct') or because of lack of knowledge about primes. This type of question is fairly common.

Let's return to the original question:

Find all combinations of 3 distinct primes whose average is 13.

I've administered this type of question to students and observed their methods. Sadly, some do not immediately recognize that the problem is equivalent to:

Find all combinations of 3 distinct primes whose sum is 39.

Most students do see this at once, but there are a few in middle and high school who have not developed sufficient conceptual understanding of averages or have simply not been exposed to enough problems.

As far as methods and approaches go, I'm always surprised that many middle and high school students use fairly random listing methods rather than a systematic approach. After all the years now of instruction in problem-solving techniques, one should expect that students would make an organized list as follows:

2,2,35 Discard this for two reasons! Would most students recognize the logic behind concluding that 2 cannot be one of the three primes?

3,11,25 (discard)
(I'll let the reader finish the list!)

If I were to assess the value of this single question, I might give it a 7 on a scale of 10. I'm sure some would rate it as 1 or 2 since some perceive these kinds of questions as useless. However, my feeling is that the question does develop mathematical thinking and there's something to be said for attention to detail and a systematic approach.

But this is not the end. Suppose the educator finds this problem in a book or math contest or online. How can one extend it to a richer experience for all students, not just the accelerated, honors or gifted child? Although it may appear at first that the primary intent of the question is to encourage a systematic approach (making an organized list) or reviewing ideas about averages or primes, the content of the question is essentially about writing a number as a sum of 3 primes, distinct, in fact. Is this an important question that has occupied the minds of our greatest mathematicians for years? Uh, actually, yes! Look here!

Students need to be encouraged to ask more questions after the problem is solved. The instructor guides this exploration by modeling some of the questions students need to ask: Is there anything special about 13? Can every prime be written as a sum of three distinct primes? Every odd? Three primes, not necessarily distinct? Does the original number 13 have to be prime or even odd for that matter? Why are we using three primes in the sum? Why not two? Your turn, boys and girls!

You get the idea. This isolated problem becomes a springboard for deeper mathematical research. Here is one possible assignment:

Write your own challenge problem of this type? Make sure you can solve it and be prepared to present it to the class!

What would you expect your students to come up with? You can't be sure until you try it of course, but can you anticipate some of the responses?

By the way, I have already heard most of the arguments for why this type of research is impractical in a math classroom:

"My students don't even know their basic facts and you want them to become mathematicians!" "This is for the math team geniuses."
"I don't have time for this - I have a real curriculum to cover and if this not going to be tested..."
"Teach children the basics, not this 'fuzzy' math!"

Oh well, enjoy it anyway!

Thursday, February 7, 2008

Fascination with Pyramids again...

Do you recall the post about pyramids from last April? This is a continuation of that investigation and is a problem that has appeared on standardized tests. There are several approaches and the learning objectives for geometry students are many:

(1) Develop spatial reasoning
(2) Review terminology of space figures, pyramids in particular
(3) Make connections to the real-world problem of finding the height of an Egyptian pyramid
(4) Apply the Pythagorean Theorem or special right triangles
(5) Justify (prove) one's methods

The figure attempts to depict a special regular square pyramid.
The 4 lateral edges and the 4 base edges all have the same length x.
Show that the height PT of the pyramid has length x(√2/2).

(a) It may be instructive to encourage students to approach this by more than one method. One could ask students to find 30-60-90 as well as 45-45-90 triangles in the pyramid (lines may need to be constructed of course).
(b) Many students may assume ΔPTS is 45-45-90. Challenging them to prove it is an important objective here (there's more than one way).
(c) One could begin with a specific value of x, such as x = 10 (see the original pyramid post).
(d) I strongly urge you to have students research the Great Pyramid of Giza. Is it approximately a regular square pyramid? Are its faces equilateral triangles as in this post? There are many classic math problems associated with this Wonder of the World and it's not all about geometry!

Tuesday, February 5, 2008

Doors and windows left unopened...

Many many loose ends...

1. The poll in the sidebar is still ongoing. Have you voted yet? It is Super Tuesday after Super Sunday after all! Jonathan suggested I submit the link to this poll to the Carnival of Education. I will consider that as well as the upcoming Carnival of Mathematics on 2-8-08 over at 360.

2. The poll has generated some interesting comments on this blog and elsewhere. Some readers and highly knowledgeable individuals are offended by the implication that the poll is somehow suggesting that teachers must teach a particular way and students must learn a particular way. I do feel there is a fairly wide range of options from extreme traditional to extreme reform, but others may see it as more biased. Similarly, some are bristling at the implication of Standards. I won't get into an exposition of my views at this point but I will share my wife's insights. She cuts through all the b******* -- that's her style. When I told her that some are having a problem with the poll, because it seems to suggest restricting what or how children should learn, she looked at me incredulously and stated:
"First of all, you're surprised someone is offended by a position you're taking! I'm not an educator, but, why do all of you make things so complicated. Who decided that the way we learned was broken and needs to be fixed? I see it like this -- Children should learn the traditional method, then when they can do that, they can be shown other ways. If they like another way better, they can decide for themselves but only after they know one method well."

She then added the following: "Perhaps those who are upset about children and educators being 'boxed in', fail to recognize that leaving everything open-ended is just another box." My wife has always seen things differently - I guess she thinks outside the box! She also added that if her position offends anyone and she gets attacked, she can handle that. After all, it's not her blog!

3. Anyone notice there is a new Mystery Mathematician in the sidebar. Daniel Gorenstein has now been replaced by another contemporary mathematician. Sorry, no hints at this point, other than to say that he has had a profound influence on myself and many students of undergraduate mathematics.

4. Ok, so anyone can prove that they made some prediction about the Super Bowl that sorta kinda turned out to be true. I suggested a couple of days ago that perhaps the number 8 would play a role in the game. Since I'm a 'digit-man', I'll let you scoff at the fact that the number 17 played a key role in the final score and Plaxico's uniform. Well, what do the digits of 17 add up to? Yes, we all know how fortune tellers make their predictions! Here's a little more coincidence:

Last night, as the Giants were disembarking from their plane at Newark's Liberty International Airport and boarding their team bus, my son was disembarking from his flight at the same terminal at approximately the same time and getting on his team bus going back to his high school after a competition. So what, you ask? Uh, well, my son's name is the same as some MVP quarterback. Alright, we all know how common that name is...

5. Many other loose ends from 'boring a hole in a sphere' to clock problems, but I'll stop here. Good morning and g'day!

Monday, February 4, 2008

My dad was born 2-4-08. He would have been a centenarian today...

Happy 100th dad. I will never forget you. (My dad passed away in '88). You will always be the greatest teacher I ever knew...

Sunday, February 3, 2008

Saturday, February 2, 2008

'Left-Overs' before the Super Bowl: Crazy Eights, Squares, Remainders and Algebra

Ok, so most normal people are not thinking about the significance of the digit '8' in 2008 the day before the Super Bowl. Sorry, but in this post there will be no predictions about the score, no 'over-unders', no boxes, no betting at all. You do have to admit that this is a great time for lovers of mathematics. People are actually interested in mathematical odds and chances of all kinds of weird number combinations occurring in the score on Sunday night. However, this post will focus instead on the number 8, the units' digit in 2008. The Super Bowl comments above will no doubt soon become outdated but the mathematics below will live on! Who knows, maybe the number 8 will turn out to have special significance on Feb 3, 2008? Remember, I said that here before the game!!

2008 is a special number for so many reasons, being divisible by 4 of course: Leap Year, Prez Election year, Summer Olympics and much more. In fact, 2008 is divisible not only by 4 but also by 8 itself. In the good ol' days, some students were even taught the divisibility rules for 2, 4 and 8:
Divisible by 2: If the 'last' digit is divisible by 2 (of course!)
Divisible by 4: If the number formed by the last TWO digits is divisible by 4
Divisible by 8: If the number formed by the last three digits is divisible by 8.

Let's demonstrate this for 2008:
2008 us divisible by 2 because 8 is divisible by 2
2008 is divisible by 4 because '08' is divisible by 4
2008 is divisible by 8 because '008' is divisible by 8

A little weird with those zeros and not particularly interesting, right? Anyone care to guess a rule for divisibility by 16? Interesting, but none of this is the issue for today....

Today, we are are interested in the squares of numbers and their remainders when divided by 8. Notice that 42 is divisible by 8 but 62 is not. So we cannot say that the square of any even number is divisible by 8. What about the squares of odd numbers when divided by 8?
12 leaves a remainder of 1 when divided by 8
32 leaves a remainder of 1 when divided by 8
52 leaves a remainder of 1 when divided by 8
72 leaves a remainder of 1 when divided by 8

What is going on here? That's for your crack investigative team to decipher.

TARGET AUDIENCE: Our readers of course; Middle schoolers through algebra

1. Discover, state and prove a general rule for the remainder when the square of an even number is divided by 8.
2. Discover, state and prove a general rule for the remainder when the square of an odd number is divided by 8.

(1) These are well-known relationships and not very difficult questions. Just something to extend thinking about divisibility, remainders and the use of algebra to deduce and prove generalizations. Prealgebra students may be able to explain their findings without algebra!
(2) 'Discovering' or stating the rule for question (2) is transparent from the examples above. Instructors may prefer 'data-gathering' and making a table first. That is, have students develop a table for the squares of the first 10 positive integers and their remainders when divided by 8. Proving the result for the squares of odd integers is more challenging, even algebraically. Most will see the remainder when dividing by 4, but 8 is slightly trickier.
(3) Those who are more comfortable with congruences and modular arithmetic can approach these questions another way.