A Digit Problem from Florian for 'Constructivists!'

First a humorous aside from one of my friends on another message board. A friend emailed it to him so it's probably making the rounds of the web. In case you haven't seen it, here it is...

A recent study found that the average American walks about 900 miles a year.

Another study found that Americans drink, on average, 22 gallons of alcohol a year.

That means, on average, Americans get about 41 miles to the gallon.

Kind of makes you proud!



One of our new and devoted readers, Florian, contributed the following unusual digits by algorithmic construction problem. This is a wonderful example of a different type of solution, since a standard algebraic approach should prove fruitless. Florian is our resident computer scientist. That should help you understand how he devised this question.

Suppose a₁a₂a₃...a_n-16 represents an n-digit positive integer whose units' digit is 6. Find the least such positive integer satisfying the property that when the number is multiplied by 2, the result is 6a₁a₂a₃...a_n-1 , the n-digit number whose digits are the same as the original number except that each digit is shifted one position to the right and the rightmost digit '6' rotates to the leftmost position.

Have fun looking for this 18-digit number! Would a calculator be useful here?

Variations and Extensions:

Here is how one could modify this for middle schoolers:
(i) Give them the 18-digit number to start with (sorry, I'm not giving this away yet), have them multiply it by 2 using paper and pencil and see how long it takes for various students to see the surprising result. (Yes, Steve, they actually are expected to multiply with accuracy!)
I guarantee they will express surprise!
(ii) Now ask them to figure out how they could construct the digits of the mystery number, one digit at a time. Some will catch on quickly, others will need guidance.
(iii) What questions should occur to students as they are building this number? You may need to ask them if they believe this process eventually has to terminate.

Extension for the Very Highly Motivated (or for people like me who need to get a life!):

Construct the 42-digit number a₁a₂a₃...a₄₁5 (ending in the digit '5'), which when multiplied by 5 is of the form: 5a₁a₂a₃...a₄₁, in which the result has the same digits as the original number with each digit shifted one position to the right and the rightmost digit rotated to the leftmost position.

Note: Check my accuracy on this!

Experimental Mathematics in the Classroom? Look at Investigations on MathNotations...

In the course of debating SteveH over the last few days, I had my own Aha! moment. I'm beginning to realize that the investigations I've been writing for several decades for my students were rudimentary examples of a somewhat new approach to finding mathematical 'truth' known as Experimental Mathematics. The Wikipedia article gives an excellent account of this from which I will excerpt the following:

Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns.^[1] It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit."^[2]

[edit] History

In one sense, mathematicians have always practised experimental mathematics. Existing records of early mathematics, such as Babylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.

Experimental mathematics as a separate area of study re-emerged in the twentieth century, when the invention of the electronic computer vastly increased the range of feasible calculations, with a speed and precision far greater than anything available to previous generations of mathematicians. A significant milestone and achievement of experimental mathematics was the discovery in 1995 of the Bailey-Borwein-Plouffe formula for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous proof found.^[3]

[edit] Objectives and uses

The objectives of experimental mathematics are "to generate understanding and insight; to generate and confirm or confront conjectures; and generally to make mathematics more tangible, lively and fun for both the professional researcher and the novice".^[4]

The uses of experimental mathematics have been defined as follows:^[5]

1. Gaining insight and intuition.
2. Discovering new patterns and relationships.
3. Using graphical displays to suggest underlying mathematical principles.
4. Testing and especially falsifying conjectures.
5. Exploring a possible result to see if it is worth formal proof.
6. Suggesting approaches for formal proof.
7. Replacing lengthy hand derivations with computer-based derivations.
8. Confirming analytically derived results.

High-speed computers allow for a different approach to 'proof', an idea that I used to find inconsistent with the formal structure of mathematical logic and truth. However, I'm beginning to open my mind to the possibilities afforded by technology and numerical analysis. In fact, I often wondered why, as a student, I was only shown the finished product, the formal abstract theorem, not the slightest hint of the trial and error and conjectures that were an integral part of the derivation. When I became a teacher, I continually reminded my students of this important facet of mathematical truth - the research aspect. Even though part of me continues to resist the idea of a computer-assisted proof by consideration of a large but finite number of cases, I cannot ignore the most remarkable result of this approach:

"The discovery in 1995 of the Bailey-Borwein-Plouffe formula for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous proof found."

More to follow...

A Very Big Number Question...

What is the sum of the digits of (googol + 1)(googol - 1) when expanded?

Comments:
(1) Google 'googol' if you need some background!
(2) Does the strategy of 'make it simpler' work well here?
(3) Can you invent a similar problem or, better, have your students devise their own!
(4) Oh, BTW, NO CALCULATORS!!
(5) I felt I needed a change of pace from the heavy math ed stuff from the past few days. You too?

Edspresso Reply from Barry Garelick and MathNotations Latest Response

A few days ago I posted on MathNotations a comment I had left on Edspresso, a reply to Barry Garelick's essay, Living in a Post-National Math Panel World.

My post was entitled, Balancing the Equation.

From edspresso:

Barry Garelick is an analyst for the U.S. Environmental Protection Agency in Washington, D.C. He is a national advisor to NYC HOLD, an education advocacy organization that addresses mathematics education in schools throughout the United States.

I just replied to Mr. Garelick's reply to my original comment. Unfortunately, I received the following automatic response from edspresso:

Your comment submission failed for the following reasons: In an effort to curb malicious comment posting by abusive users, I've enabled a feature that requires a weblog commenter to wait a short amount of time before being able to post again. Please try to post your comment again in a short while. Thanks for your patience.

I assume I will be able to re-post that comment in another day or so, but I thought MathNotations' readers may want to see a preview.

Here is my original comment to Mr. Garelick's post and his recent reply:

"There are also teachers who maintain a truly balanced approach and who, while rejecting the discovery-oriented and textbook-less programs being foisted on schools across the country, are admonished by their administrators to do as they are told."

Although now retired, I was one of these educators for the past several decades.I believe the Panel paid lip service to these educators. Mr. Garelick, just what benefit does this report have for this group of math teachers? There are many dedicated professionals who have always balanced the need for 'correct answers' with conceptual understanding. Educators who always knew that there must be mastery of essentials before one can move on in mathematics. Educators who continue to find creative ways to satisfy their administration and their personal integrity...

The problem is that it is just not easy to blend skill practice, mastery and rich problem-solving experiences and explorations when one has to essentially create one's own materials. Particularly when the rewards for going 'above and beyond' are purely intrisic in the teaching profession. Experienced math teachers know that computational proficiency is absolutely essential but, when confronted with problems that are not formulaic and require recognition of essential concepts and making connections, many of our students flounder. Yes, it is really hard to do the right thing, isn't it?

In your opinion how will textbook publishers respond to the Panel's report? IMO, skills-based texts that neglect exploration and more challenging problem-solving would be just as damaging to this next generation as many of the reform texts have been to the current generation. Perhaps such 'skills' texts will not be the response to the Panel's report from textbook publishers. Perhaps...

But that's ok, the most dedicated of our profession will compensate for whatever materials they are handed. They'll continue to write their own and do what's right, just as they always have.

Dave Marain
MathNotations

Posted by: Dave Marain | April 9, 2008 08:17 PM

Mr. Garelick's Reply:

Mr. Marain,

Thank you very much for your comment. I've seen your writing on MathNotations and am glad you wrote.

I am aware that there are people who hold that "problems which are not formulaic" are not well-addressed by teaching students the components of math and algebra delineated in the NMP's report. Such problems are the so-called "messy" problems that have a range of answers or are open-ended, and so forth. Problems such as the "work" problems and others in math textbooks are held in disdain and thought not to lead to problem solving skills. Proper presentation of the solution of say, work problems, however, opens the door to "rate problems" in general, and which generalize to the solution of a great many problems in engineering and science. In fact, many of the standard so-called "formulaic" problems in algebra and other math classes are widely generalizable and have their purpose as I can attest as one who majored in math and work in a field that requires knowledge of scientific and engineering principles.

Providing students the opportunity to solve non-formulaic problems does not in and of itself prepare them to solve problems. Analytic and procedural skills and knowledge of form, which generalize do in fact provide such preparation. I tend to think the term "balanced approach" is one that is not well defined. I used the term "true balanced approach" in my essay, meaning an opportunity for student-centered instruction (such as discovery) that makes use of prior knowledge, rather than the melange of "just in time" skills, procedures and concepts that some teachers, textbook writers and policy makers seem to think students will discover because they need them to solve a problem.

It is my hope that those teachers who use textbooks that are written topics presented logically, sequentially, with expectation of mastery, and which builds upon concepts, will not be punished for doing so. Vern Williams who I quote in the essay is one of those teachers. He gives students very tough "out of the box" problems that are not in the textbook necessarily, but he makes sure they have the requisite skills and information (which he imparts via instruction) before giving them such problems.

Posted by: Barry Garelick | April 14, 2008 11:06 AM

Here is my latest reply of 4-18-08 which was temporarily rejected:

Dear Mr. Garelick,

Your quote:
"I am aware that there are people who hold that "problems which are not formulaic" are not well-addressed by teaching students the components of math and algebra delineated in the NMP's report."

My quote:
"The problem is that it is just not easy to blend skill practice, mastery and rich problem-solving experiences and explorations when one has to essentially create one's own materials..."

I never suggested that direct instruction of powerful models such as rate problems is not desirable, nor do I hold these traditional rate problems in disdain. In fact, in my classroom, I always used the same RATE X TIME = DISTANCE chart that I learned in high school, several decades ago. The emphasis on my blog is on rich open-ended investigations that one normally does not see in most textbooks. Problems that teachers have to search for in most cases. Most of my regular readers know that success with these problems require a strong base of skill and knowledge of traditional algorithms. I've stated this repeatedly.

Consider what happened when the AP Calculus Exam changed dramatically a few years ago. The tried-and-true methods still were needed and still were assessed. But the test changed dramatically from an emphasis on technique to an emphasis on application and deeper understanding of fundamental concepts. The decline in scores was predictable and occurred until textbooks and instruction were revised to address this shift in focus. The most capable students were able to 'generalize' from formulaic problems - they always will. However, the scores on the AB Exam initially reflected that many others could not. It has been my experience that students perform well on non-routine problems when they have that strong base of skill AND experience with many nonroutine problems that require more than a superficial understanding of content. When presented with such problems, students need TIME TO EXPLORE. Direct instruction initially will usually fail in this case. After giving a reasonable amount of time to grapple with the problem and discuss it, the solution is explained clearly and thoroughly, with alternate methods presented if time permits. One cannot rush this process, that's why it is called a process...

That's my definition of 'balance.' There's nothing fuzzy about it. I've never suggested that students be given challenging problems and left entirely to their own devices to invent something out of nothing. I have suggested they need to know their trig values, their trig identities, their differentiation and integration formulas well. After they have demonstrated this KNOWLEDGE, they can be challenged with problems they have never seen before. Exploration does not mean one is "a blind man, searching in the dark for a black cat." (Luv that Escalante quote!). Exploration means that one is PREPARED to explore and then given the OPPORTUNITY to EXPLORE!

Me. Williams and I share many common beliefs about teaching. But he doesn't represent even a majority of math teachers out there. This is why I repeatedly emailed the Panel asking for more frontline teacher representation on the NMP (and, of course, I was politely dismissed). There was not a single high school math teacher represented, not to mention an underrepresentation of research mathematicians.

I appreciate your thoughtful reply, but you attempted to pigeonhole my beliefs. No one yet has been able to do that and I'm afraid you did not succeed either! I welcome an opportunity to discuss this further with you, perhaps in another venue. How about a debate on my blog? We'll probably discover we have far more in common than one might think.

Oh well, nothing has changed. Does anyone out there really get my balanced perspective, other than my faithful readers of course!

Discussion of Algebra 2 End of Course Exam from ADP/Achieve/Pearson - Continued...

Unfortunately, I did not receive permission from Pearson Educational Measurement to reproduce any of the released items from the Achieve website. I will respect their wishes. However, they understand that I plan to discuss some of the items indirectly without specifics. For this to make sense to my readers, you will need to download the pdf document as suggested in an earlier post and have that in front of you as I refer to individual items. Here is the link to the Achieve website that contains the released items (in the sidebar).

For this post, however, I plan to discuss the implications of this exam and related issues.

I would like to invite comments about the issues raised by a common exam that will be administered to students in 14 states (up from the original 9). I consider this to be a highly significant development in the movement toward more standardization for all students. Up to this point, the only similar kind of interstate standardized test covering Algebra 2 topics has been the SAT Subject Test - Math I. Several states now give their own end of course exams in Algebra I and Algebra 2, but the Algebra 2/ADP exam from Pearson is impacting on students from many states and this is just the beginning of this trend. For example, a similar exam for Algebra I is already under development (if not already completed) and, here in New Jersey, it will become operational shortly.

Furthermore, textbook publishers such as Pearson have already begun to publish texts (e.g., Pearson, Algebra I, 2007) that correlate with the American Diploma Project's Algebra standards (download this). Imagine that! Instead of inserts in the text that correlate to fifty different sets of state standards, we are now going to see some consistency. Glory Hallelujah!

Here are just a few of the issues that each state will have to confront as these exams proliferate:
1) How will a student's grade be determined in the course? Will the exam be worth a percentage of the final grade? What if the results do not come back in a timely fashion?
2) If a student falls below the minimum level of proficiency on the exam (great euphemism/edu-jargon for 'failing'!), how will this be recorded on their transcript?
3) Will the exam be required for graduation just as current graduation tests do in some states?
4) Can a student re-take the exam if they don't make it the first time?
5) Should the Algebra I exam have been developed and implemented before this exam?
6) Will different groups and consortia now compete to develop curricula and assessments independently as a result of the recommendations from Achieve/ADP, the National Math Panel and NCTM's Curriculum Focal Points? More splintering?
7) Will there be more than one administration of these exams each year? For the Algebra 2 Exam, the answer is found on the Achieve website:

The exam will be administered at the end of fall—December and January and at the end of spring—May and June beginning with the 2008-2009 school year.

Ironically, while this exam improves consistency of curriculum, there might not be as much consistency about how these issues will be addressed. Of course, one can hope...

Are these exams intended to impact on instructional methods, emphases, strategies, techniques just as currently occurs on the AP Calculus Exam? If you read through the pdf document and examine the Released Items document (p.4), the answer is clear for the Algebra 2 Exam (the following is an image which will appear blurry);













Your thoughts...

Joseph Liouville - Our Mystery Mathematician Was One of a Kind...

Two winners this week in identifying Joseph Liouville (1809-1882), a brilliant mathematician whose work continues to have an impact today in so many branches of math and physics.



Here is an excerpt from an excellent site for biographies of mathematicians:

Liouville's mathematical work was extremely wide ranging, from mathematical physics to astronomy to pure mathematics. One of the first topics he studied, which developed from his early work on electromagnetism, was a new topic, now called the fractional calculus. He defined differential operators of arbitrary order D^t. Usually t is an integer but in this theory developed by Liouville in papers between 1832 and 1837, t could be a rational, an irrational or most generally of all a complex number.

Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33. Having established this in four papers, Liouville went on to investigate the general problem of integration of algebraic functions in finite terms. His work at first was independent of that of Abel, but later he learnt of Abel's work and included several ideas into his own work.

Another important area which Liouville is remembered for today is that of transcendental numbers. Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove that e is transcendental but he did not succeed. However his contributions were great and led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number

0.1100010000000000000000010000...

where there is a 1 in place n! and 0 elsewhere.

His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations. This theory, which has major importance in mathematical physics, was developed between 1829 and 1837. Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.

Liouville contributed to differential geometry studying conformal transformations. He proved a major theorem concerning the measure preserving property of Hamiltonian dynamics. The result is of fundamental importance in statistical mechanics and measure theory.

In 1842 Liouville began to read Galois's unpublished papers. In September of 1843 he announced to the Paris Academy that he had found deep results in Galois's work and promised to publish Galois's papers together with his own commentary. Liouville was therefore a major influence in bringing Galois's work to general notice when he published this work in 1846 in his Journal. However he had waited three years before publishing the papers and, rather strangely, he never published his commentary although he certainly wrote a commentary which filled in the gaps in Galois's proofs. Liouville also lectured on Galois's work and Serret, possibly together with Bertrand and Hermite, attended the course.

And our winners are...

Vlorbik:

okay ... liouville ...

most notable in *my* mind for having founded

"liouville's journal" ... major mathematicians

who were also great *editors* are very few

(& as a former self-publisher, i have a soft spot

for people involved in getting the word out).

best *known*, i suppose, for having proved

that a bounded entire function on {\Bbb C}

is a constant function ...

the first time i looked at the picture, i thought,

"oh, gee ... another one of those photos

i've seen a hundred times in textbooks and so on;

beats me who the devil it actually *is*" ....

but just now, i checked back (looking for comments

on your math-panel post) and at a glance thought,

"hmm ... liouville. maybe i'll just go verify that

at google images." ... funny the tricks one's mind plays ...

Kevin:

Dave: Joseph Liouville. My personal favorite Liouville theorem: Conformal mappings for E^n, and S^n, n > 2 are restrictions of moebius transformations. Probably all anecdotes, well told, are in Lützen, J. Joseph Liouville 1809-1882: Master of Pure and Applied Mathematics. But I observe that his "mathematical" path has crossed that of some other of the "mystery mathematicians": Kaplansky, the previous mystery mathematician, wrote a small and elegant monograph "An Introduction to Differential Algebra" which discusses Liouville extensions and in particular the example: y'' + xy = 0 which is not integrable in finite terms ("the solutions of this equation cannot be obtained from the field of rational functions of x by any sequence of finite algebraic extensions, adjunction of integrals and adjunction of exponentials of integrals"). Liouville wrote a number of items in the first volume of the journal he founded in 1836 . Included in that first volume is an article by Jacobi - another week's mystery mathematician. According to MacTutor History: they also share fellowship in the Royal Society, fellowship in the Royal Society of Edinburgh, and they each have a lunar crater named after them. Lastly, they share the Jacobi-Liouville formula in dynamical systems.
Best regards, Kevin

Released Items for Upcoming End-of-Course Exam for Algebra 2- Achieve/ADP

With the first operational test running from May 1-June 13, 2008, I'm sure that the participating districts are being given regular updates and materials to help their students prepare for the exam. Just in case you haven't checked the exam website recently (the released items appeared in March I believe) or you are curious about seeing some sample items, just click here.

You will need to click on Released Items in the right sidebar and a pdf document will be downloaded to your desktop. This is a well-designed 45-page document providing a wealth of useful information for teachers, administrators and students. It is far more than a collection of sample exercises. In particular, page 45 provides an actual breakdown of the exam in matrix form, showing how many of each of the 3 types of questions (Multiple-Choice, Short Answer and Extended Response) there are for each module tested. A total of 76 raw score points are possible broken down as follows:

Multiple Choice: 46 questions - 1 pt. ea.
Short Answer: 7 questions - 2 pts. ea.
Extended Response: 4 questions - 4 pts. ea.

Further, the questions are broken into 3 cognitive levels with the majority of questions at Level 2 which "requires students to make some decisions as to how to approaqch the problem or activity."

Here's a quick overview of the content covered in both the core exam and the optional modules:

The Algebra II end-of-course exam will consist of a common core, which will be taken by students across all participating states. This core will cover a range of algebraic topics that are typically taught in an Algebra II course, and fall into five content standards: 1) Operations and Expressions 2) Equations and Inequalities 3) Polynomial and Rational Functions 4) Exponential Functions and 5) Function Operations and Inverses.

In addition, seven optional modules will be available to states to enrich the core with content that is important to colleges and employers alike. These include: 1) Data and Statistics 2) Probability 3) Trigonometric Functions 4) Logarithmic Functions 5) Matrices 6) Conic Sections 7) Sequences and Series.

Initial Reaction from MathNotations

Since I've always been a firm believer that required exams have a major impact on what is covered and how a course will be taught (e.g., the AP Calculus Exam), the released items in this document will be scrutinized by instructors, supervisors, etc. From my experience with many other standardized tests (e.g., state tests), released questions tend to reflect the more challenging aspects of the test.

That being said, my immediate feeling was that the questions reflected considerable traditional content.

However, the impact of reform was felt strongly in the extensive discussion following each item. In addition to traditional approaches, solutions were provided that demonstrated the use of multiple representations, solutions by graphs and tables (using a graphing calculator). The discussion following each item is the most important part of this document, IMO.

NOTE: In its stated calculator policy, Achieve recommends the use of a graphing calculator. They were discreet in not requiring it, however, as that would get into equity issues. Advanced QWERTY-type calculators (such as the TI-89) are not permitted.

The released questions included many that I would rate of average to above-average difficulty.
They also include some challenging items along the lines of Math I from the SAT Subject Tests. I've contacted Pearson, asking for permission to reprint some of these items for discussion purposes. I'm awaiting their response...

I urge my readers to download the document and share their reactions. Again, specific items should not be stated verbatim in your comments as I do not yet have permission for this.

1,2,2,3,3,3,4,4,4,4,... What is the 2008th term? SAT-type Questions vs. Math Contest Problems

Don't forget to submit the name of our Mystery Mathematician. Contest ends around 4-15-08. Thus far, only one correct submission (emailed of course!).

The problem in the title is the math contest version. Knowing the formula for the nth triangular number would be helpful (so might a calculator). Do all middle school and hs students become familiar with triangular and other figurate numbers? Should they? My vote: Yes!


The SAT -type would be:

What is the 56th term of the sequence 1,2,2,3,3,3,4,4,4,4,..., in which each positive integer N occurs N times?

Comment: This is considerably easier than the contest problem as one could do it by listing with or without a calculator. It also may reveal your strong number sense students who will see the idea fairly rapidly. Try it as a warm-up in class!

Another SAT-type (probability, counting) to help students prepare for the May Exam:

Let S be the set of all 3-digit positive integers whose middle digit is zero. If a number is chosen at random from S, what is the probability that the sum of its digits is even?

Note: I wrote this question to demonstrate basic principles of probability and counting. Although one could use the Multiplication Principle (aka, Fundamental Principle of Counting), students should also be encouraged to make an organized list and, by grouping, see why the answer is 1/2.

Balancing the Equation: MathNotations Comment on edspresso

While I'm waiting for edspresso to approve the comment I posted last night, I 've decided to post it here first.

I was commenting on Barry Garelick's well-written post re the recent report from the National Math Panel: Living in a Post-National Math Panel World. This blog will eventually do a more in-depth analysis of the report but for now here is my comment...

Quoted from the edspresso post:
"There are also teachers who maintain a truly balanced approach and who, while rejecting the discovery-oriented and textbook-less programs being foisted on schools across the country, are admonished by their administrators to do as they are told."


My Comment:
Although now retired, I was one of these educators for the past several decades. I believe the Panel paid lip service to these educators. Mr. Garelick, just what benefit does this report have for this group of math teachers? There are many dedicated professionals who have always balanced the need for 'correct answers' with conceptual understanding. Educators who always knew that there must be mastery of essentials before one can move on in mathematics. Educators who continue to find creative ways to satisfy both their administration and their personal integrity...

The problem is that it is just not easy to blend skill practice, mastery and rich problem-solving experiences and explorations when one has to essentially create one's own materials. Particularly when the rewards for going 'above and beyond' are purely intrinsic in the teaching profession. Experienced math teachers know that computational proficiency is absolutely essential but, when confronted with problems that are not formulaic and require recognition of essential concepts and making connections, many of our students flounder. Yes, it is really hard to do the right thing, isn't it?

In your opinion how will textbook publishers respond to the Panel's report? IMO, skills-based texts that lack depth and neglect exploration and more challenging problem-solving would be just as damaging to this next generation as many of the reform texts have been. Perhaps such texts will not be the response to the Panel's report from textbook publishers. Perhaps...

But that's ok, the most dedicated of our profession will compensate for whatever materials they are handed. They'll continue to write their own and do what's right, just as they always have.

Dave Marain
MathNotations

What I should have added is that there was only one currently practicing teacher on the Panel. I have no evidence to indicate that the balanced approach to curriculum and instruction was represented at all on this commission. If that is the case, it would seriously detract from the credibility of this report. However, I will withhold further judgment until I've had a chance to thoroughly analyze the detailed recommendations.

In the long integer 36912151821...9999, what is the 1107th digit?

The title question would be a medium level math contest problem.

The following could be the SAT version:

369121518...99
The integer above is formed from consecutive positive integer multiples of 3 from 3 to 99 inclusive. What would be the 50th digit in this number?

A student could theoretically take the time to list all the digits up to the 50th, however, this would be more time consuming than using logic.

Possible approach for SAT-type:

369 uses 3 digits.
This leaves 50-3 = 47 more digits.
Starting with 121518... there are two digits for each multiple of 3, so we divide 47 by 2, producing 23 with remainder 1. We now need to determine the 23rd multiple of 3 beginning with 12:

12 + (23-1)3 = 78. [At what point should students know this formula for arithmetic sequences?]

The remainder of 1 means we need to look at the first digit of the multiple of 3 that comes after 78: 81
So the answer would be 8, but, of course, I could have erred in my logic or calculation, so pls check that!

One could now assign the title problem for extra credit or as an extension.

Comment: How many readers believe that middle schoolers on up tend to rely on their calculator to find remainders. This has been discussed previously on this blog and a couple of algorithms were given for computing the remainder from the decimal result given by the calculator.

Monthly Math League Challenge: List all ordered triples of positive integers (x,y,z) whose product is 4 times their sum and x>y>z.

The problem in the title is another wonderful challenge for our readers or for students. Questions like these are powerful tools to develop student reasoning and problem-solving prowess. Once again I have received permission from the directors of the Math League to publish this question on MathNotations. This was the last question from the first contest this year.

Please cite the question in the title as:
Copyright Mathematics Leagues Inc 2007. May not be reproduced without permission of the copyright holder.

You can learn more about Math League Contests at the Math League website.

Now you know that I'm not going to simply copy a problem and just leave it at that!
How can we make this more of an enrichment experience for our algebra students who may not quite be ready for the contest level. Can you guess? Scroll down...


Well, what makes the contest problem particularly formidable is the use of three variables. In fact, two numbers is already a difficult problem for most! So we use the "Let's Make It Simpler" strategy:

Find all ordered pairs (x,y) of positive integers whose product is four times their sum and x>y.

I'll begin some analysis, starting with our basic equation:
xy = 4(x+y).

Before one starts the traditional solving for y in terms of x ritual (then let's go to the graphing calculator), we need to remind our students that this is a positive integer problem, which allows for a somewhat different kind of approach.
I also frequently suggest to students to consider the case that x = y even though the restrictions do not allow this.
If x=y, then x² = 8x leading to x=8, y = 8 (or ____?). Even though this is not allowed, it could suggest other solutions. Those who enjoy graphical solutions will also appreciate that this solution is one of the two points of intersection of the graph of our basic equation with the line y = x. I'll leave it to our readers to find the other.

Note that from xy = 4x + 4y we can see that neither x nor y can equal 4. For example, if x=4, we'd obtain 4y = 16+4y, which is impossible.
Similarly, neither x nor y could be less than 4. This is more readily proved algebraically. I'll omit the details.

Note: Another important tool for students in solving these kinds of problems is to consider symmetry. The basic equation is symmetrical in x and y. Symmetry can be very useful.

Ok, while most students are using guess-test methods (they would call it 'plugging in'), we will solve for y in terms of x (skipping a few steps):
y = 4x/(x-4).
There's a well-known algebraic device students need to see here when finding integer solutions, which is equivalent to long division. Rewrite the previous equation as:
y = (4x-16)/(x-4) + 16/(x-4). I'll let you guess why I subtracted 16 in the first fraction then added it back in the second. This is equivalent to:
y = 4 + 16/(x-4).

Now we use the integer condition (and the fact that x > 4 and x > y) to find solutions:
x-4 has to be a factor of 16 which implies that x-4 could equal 16, 8, 4, 2, or 1. In fact we can show that only 16 and 8 are possible. I'll leave the rest to you...

Mystery Mathematician Week of 4-1-08

It's that time again. Now I know some of our experts will find this 19th century legend within a few nanoseconds but try to dig for some interesting anecdote about him. His mathematical pursuits were so far-reaching, you should be able to ferret out some fascinating facts. Also, can you guess a couple of reasons why I chose him?



Don't forget to email me with your solution at dmarain at gee-mail dotcom and please include MysteryMath 10 in the title! Do not post your solution in a comment to this post!
This edition of the contest will end around 4-15-08.