Wednesday, December 31, 2008

'CHARMED ERA' = RADEMACHER - Our MathAnagram finally revealed!



Well, it's been up there in the sidebar for 3 months and I'm sure it's been long forgotten, but we do have a winner of our contest.

Hans Rademacher was one of the most brilliant and prolific mathematicians of the 20th century. His research had broad scope from mathematical analysis to number theory including such diverse areas as analytic number theory, theory of partitions, Dedekind sums, quantum theory and mathematical genetics! Perhaps, even more significantly, Prof. Rademacher was deeply respected by his colleagues and students at the U. of Pennsylvania and known for his kindness and "charm!"

And our winner is...

SEAN HENDERSON


Here was Sean's contribution:

(1) Hans Rademacher (I'm assuming you want the one born in 1892)
(2) (a)Is a direct mathematical descendant of Klein and Lindemann
(b)Developed a system of orthogonal functions called Rademacher functions. Before it's publication he had expanded this to a system of orthonormals, but was advised not to publish it.
(3) (a)http://www.genealogy.ams.org/id.php?id=7518
(b)http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Rademacher.html


I wil leave Prof. Rademacher's image in the sidebar for awhile. Keep looking for our first MathAnagram of 2009!

Tuesday, December 30, 2008

A Holiday Riddle: What do you call solving an equation twice on Jan 1st?

The first 5 who email me with the correct answer to this silly riddle will receive international acclaim!
Please do not post your answer as a comment!
Email me using the link below the post or at dmarain at geeeeemailllll dottt kom!

Also, include the following info:
(1) Your full name
(2) Your connection to math (student, educator, math enthusiast, etc.)
(3) How you found MathNotations (or if you're a long-time visitor)
(4) If you have your own silly math riddle for the occasion, pls share it!

Sunday, December 28, 2008

A Different 'View' of Sums of Cubes? An Algebraic "Proof Without Words!"

A well-known and intriguing formula usually proved by Mathematical Induction states that
13 + 23 + 33 + ... +n3 = (1+2+3+...+n)2 .

In words:
The sum of the cubes of the first n positive integers equals the square of the sum of the first n positive integers (or the square of the nth triangular number).

Students as early as middle school can investigate numerical patterns of sums of powers of positive integers and can be led to such discoveries. However, in this post we will look at a different kind of "proof." Proofs without words can be fascinating, challenging and can develop a student's spatial reasoning. Just as there have been many visual proofs of the Pythagorean Theorem (dissection type), mathematicians have sought visual arguments for many other numerical patterns and algebraic formulas. The Greeks of antiquity developed many classical arguments of this type, necessitated perhaps by not having our symbolic algebra available.

You will surely find other examples of this on the web (e.g., "Cut-the-Knot") but I thought it might be nice to bring it down to a middle school or Algebra 1 level by having students play with some particular cases of this general formula. I have always been intrigued by this topic, ever since I saw several visual proofs of the Pythagorean Theorem. Later on I was introduced to the genius of Sidney Kung and Roger B. Nelson (Google them!). Prof. Kung's extraordinary visual proofs were (and may still be) a staple of Mathematics Magazine, an MAA publication. You may also recall I have published a couple of other such proofs, one of which came from a student of mine. Look here.

Part I
Let's try to demonstrate that 13 + 23 = (1 + 2)2

Before displaying the visual we will begin with an arithmetic-algebraic approach:

Think of (1+2)(1+2) as a special case of the form (a+b)(a+b):
Thus, (1+2)(1+2) = (1⋅1) + (1⋅2) + (2⋅1) + (2⋅2)
Now for some creativity. Since cubes involve a product of THREE factors, we can introduce an extra factor of "1" in each term:
(1+2)(1+2) = (1⋅1⋅1) + (1⋅1⋅2 )+ (1⋅2⋅1) + (1⋅2⋅2).

Even without a visual, we can see the first term on the right is 13!!
It will take some work to show that the sum of the other three terms is 23. Ok, with this background, here is a
PROOF WITHOUT WORDS
















Do you think your students will "see" the proof?? My crude attempt at a graphic leaves a lot to be desired! It may be helpful to have manipulatives such as algebra tiles available or have students physically build these models. I would encourage that strongly!

So we are proving a numerical formula using a sum of volumes. You might say we turned squares into cubes!!

Do you think this investigation is through? Of course not -- I did all the work for you. Now here is the real test:

Part II


Show that 13 + 23 + 33 = (1 + 2 + 3)2
using a "Proof Without Words."

Ok, I'll give you a little hint although you don't need to use this:
Rewrite
(1 + 2 + 3)2
as ((1 + 2 )+ 3)2

Have fun! Just think, if we have a sum of 4th powers, we might need hypercubes!

Friday, December 26, 2008

Updates...

Because I'm never sure if the 400 or so subscribers to the MathNotations feed ever see revisions I've made to a post (which I do frequently!), I'm letting everyone know that the latest post displaying the calculus videos regarding the 'boring a hole through a solid sphere' problem has been updated a few times. I've restated the problem originally posted in the beginning of 2008 and I have annotated the video clips so that you can decide which ones you want to view. It is very long and may be at times as 'boring' as the title of the problem!! Hey, folks, it's a learning process for me and eventually I'll just post links on my blog to these videos which I will upload to YouTube.

Thursday, December 25, 2008

Boring Hole in Sphere Calc Video - Finally!!

Remember when I originally posted this problem back in January? Look here.

Here is the original problem:

A hole is drilled (bored) completely through a solid sphere, symmetrically through its center. If the resulting "hole" is 6 inches in height (or depth), show that the remaining volume must be 36π inches cubed.

OBJECTIVE: Motivation, explanation and application of method of cylindrical shells for finding volume of solid of revolution

TOTAL LENGTH: about 45 min


Please Note: These videos are not intended for students who want quick simple explanations for standard homework or typical exam items. This problem is above that level and the explanations are lengthy and very detailed!


Well, this 'video' is fragmented into 7 parts, the transitions are amateurish, it was composed over a few days (therefore different outfits!), cheap props and the quality is well, you know...

In spite of all the negatives, I'm hoping someone will find this helpful. Remember I'm doing this to cover a broad audience -- the Calc I/II student who wants understanding and clarity (not skipping steps!) to the AP student/Math-Sci-Engineering major who wants some theory and rigor. I'm also demonstrating some aspects of pedagogy here for the new calculus instructor who may have to prepare a similar lesson.

As mentioned in the video, there are many wonderful websites and videos which will provide better graphics, animation and quality. A couple of links are provided below. However, my purpose here to provide a highly detailed development of a classic calculus problem which reviews the method of cylindrical shells for volumes of solids of revolution.

Finally, my original intent was to find the volume that was removed by at least two methods and to generalize to a hole of depth h, but this is way too long as it is! Of course, I don't expect many views or comments but it will be out there for anyone who might have use of this for as long as this blog exists! I'm really hoping comments will look past the low-tech aspect and address the content and pedagogy.

Instructors
Please feel free to share this with your students or for whatever purpose you may have.
As stated above, the total length of all parts is about 45 minutes, the length of a typical hs class period, so it wouldn't make sense for the classroom. You might want to recommend students view this after learning the basic idea of the 'shell' method as reinforcement or after assigning this problem for hw or extra credit. These days students are savvy enough to locate, on the web, solutions and videos to most any problem we assign, so be careful! (You already knew that!)

Some Recommended Links
Volumes of Revolution - Cylindrical Shells
As mentioned in the video, patrickJMT is as good as it gets for clear, simple and mathematically accurate explanations.

Volumes - Cylindrical Shell Method
Wonderful explanations and excellent graphics and animation of the shell method (in Flash) from one of the best calculus sites on the web - utk (U Tenn Knoxville)

There are many other outstanding sites - I apologize in advance for omissions here. Just keep searching until you find the one that works for you!

As always, I am responsible for any errors - don't hesitate to point them out! At least we made it before XMAS 2008 ended!

The videos below are connected, so you might want to watch them in sequence.
However, the actual solution to the problem starts in the 5th segment below.
Read the descriptions of the segments to guide you in deciding where to begin. If you do not want a lengthy introduction, and already know the shell method, skip down to the 5th clip.


These first two video clips provide an overview for what I intend to cover.
Also the key relationship R2 - r2 = 9 is developed.

video



video

These next two segments motivate and derive the method of cylindrical shells.

video



video


The actual solution to the problem starts below!


video



video



video

Yes folks I know how drawn out this all was. I will try to improve on these but I will take an hiatus from my busy movie production schedule for awhile!

Happy New Year!

Get Ready for Happy 41*7^2

Let the amusement begin with all of the cutesy questions and math contest problems involving our new calendar year, 2009.

Shall we begin, looking for curiosities. Perhaps our students in grades 4-8 can discover their own. Why not post their best ideas or perhaps I may create a contest right here at MathNotations! Hmmm...

Ok, let's get started:

1) The difference of the units' digit and thousands' digit is 7, the smallest prime factor of 2009.

NOTE: An important benefit of these kinds of observations is that it helps students learn how to formulate and express their ideas using correct mathematical language. This is as hard for many high schoolers as it is for middle schoolers!

2) Who actually knows a divisibility rule for 7? (Proving it is another matter).
How about 200 - 18 = 182, then 18 - 4 = 14 which is divisible by 7, so 2009 is also!
No idea what I just did? You'll just have to research it, boys and girls! Ok, an excellent resource is the Math Forum of course. Look here.

3) When 2009 is divided by its units' digit, 9, the remainder is 2, the thousands' digit. Not surprising if you know about remainders when dividing by 9.

4) The product of the distinct prime factors of 2009 is 41x7 = 287, my favorite highway in NJ. This is probably not the curiosity I would be looking for from my students!!

Ok, enuf' of this silliness. I'll leave it to my astute readers to bring in the New Year in their own unique fashion. BTW, a useful site for a list of primes is here. Keep it handy and enjoy!

HAPPY 2009 (a bit early!)

Tuesday, December 23, 2008

The Number Warrior and the Mysterious Minds of Students!

Jason over at Number Warrior, an excellent blog for math teachers, has a short but fascinating post on trying to analyze why students make careless errors when it comes to negative fractional exponents.

I hope he doesn't mind if I repeat my comment over here - I think it raises some important issues for all of us who are trying to help students overcome these apparently 'careless' errors. I also recommend you visit his blog - fascinating stuff...

Jason's post:

So why would a student incorrectly evaluate 16^{-\frac{1}{2}} to be -4 but manage to correctly get on the very next problem that 5^{\frac{1}{4}}\cdot5^{-\frac{9}{4}} is \frac{1}{25}?

I believe this is a case that the knowledge of negative exponents was stored somewhere back there, but because the first problem looked “easy” my students just went for the impulse answer. (Nearly everyone — even students who scored very high overall — got it wrong.) I wonder how I can get students to reach back there more often, because neither gentle admonishments nor fierce reminders seem to work.


My response:

Jason,
We can speculate about why students make errors, but I’ve learned there are usually several reasons. I found it helpful to simply ask them to explain how they got that result (if they can!).

Some thoughts:
Your 2nd example procedurally involved fractional exponents, but ended up raising the base to a negative integer, not a negative fraction. This is a minor distinction, one extra step, but you never know. Also, I found it helpful to encourage them to write the extra step or two rather than do it mentally. Thus, 16^(-1/2) = 1/(16^(1/2)) might help. in other words, when they have to cope with both the negative and the fraction, make them always do the negative first. Some individuals are simply not detail-oriented and have trouble with precise procedures. I believe left-brained people have fewer of these issues because they are wired to do step-by-step procedures!

Finally, although none will admit to this, some youngsters know how to study for a math test and some simply don’t practice sufficiently. The “I think I know the material” students who didn’t review enough usually get burned on these procedural problems that have that one extra step. Ok, I’m probably over-analyzing all of this - it’s just a darn common error! Happy Holidays!
Dave Marain


My gut feeling is that these kinds of issues which math teachers have to confront daily, beg for considerable dialog. I know I benefited from asking more experienced teachers for advice when so many of my students struggled with certain types of questions. Asking students themselves to analyze their own errors is rarely a waste of time in my opinion. We always want to encourage self-reflection and it's usually good practice to have students correct their errors after receiving their tests back. And, of course, this kind of dialog also serves as a window into their 'mysterious' minds!

I hope this generates some further discussion about 'careless errors' and what we can do to help students cope!

Happy Holidays Everyone!

Monday, December 22, 2008

New Feature! Blogs I Read Daily (Latest Updates)

Of course this is new for me as opposed to other Bloggers who have been using this feature for awhile!

Anyway, this gives my readers an opportunity to see the titles of the latest posts from some of the math bloggers who have been so generous in their support for MathNotations. These are also some of the bloggers for whom I have great respect and whose articles I particularly enjoy reading. Please note that this is a partial listing - I'm not done yet! I just felt the need to get started being so late into this. There are several other math or edublogs that I enjoy and highly recommend. They will be added in due time -- just don't be mad at me if you don't see yours listed yet!

Check out the sidebar to see this feature. You won't see this if you just read my feed so occasionally my subscribers may want to visit the site! For now I am still keeping my blogroll which is more extensive than the above. At some point I may revise the blogroll or even delete it.

Sunday, December 21, 2008

The Passing of Kyosito Ito - Farewell to a Modest Brilliant Mathematician

Does anyone remember our Mystery Mathematician from Feb 2008? None other than one of the greatest mathematicians of our era - Kyosito Ito.

Excerpts from a recent NYT article:

Kiyoshi Ito, a mathematician whose innovative models of random motion are used today in fields as diverse as finance and biology, died Nov. 10 at a hospital in Kyoto, Japan. He was 93.

Mr. Ito is known for his contributions to probability theory, the study of randomness. His work, starting in the 1940s, built on the earlier breakthroughs of Albert Einstein and Norbert Wiener. Mr. Ito’s mathematical framework for describing the evolution of random phenomena came to be known as the Ito Calculus.

“People all over realized that what Ito had done explained things that were unexplainable before,” said Daniel Stroock, a professor of mathematics at the Massachusetts Institute of Technology.

Mr. Ito’s research was theoretical, but his models served as a tool kit for others, notably in finance. Robert C. Merton, a winner of the Nobel in economic science, said he found Mr. Ito’s model “a very useful tool” in his research on the evolution of stock prices in a portfolio and, later, in helping develop a theory for pricing stock options that is used on Wall Street today. Mr. Ito, he said, was “a very eminent mathematician.”

Mr. Ito collected many professional honors and awards over the years. He was a foreign member of the national academies of science in the United States and France. He was awarded the Kyoto Prize, the Wolf Foundation Prize of Israel and the Carl Friedrich Gauss Prize of Germany.

To demonstrate Professor Ito's humility, here are his own words from the article posted in MathNotations back in February of this year:

"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."

Farewell, Dr. Ito. You have enriched our lives and those of future generations...

A Holiday Geometry Gift -- All Rolled Up For You!



















Whether you view this as an SAT-type problem, a geometry challenge or just another investigation, I hope you will enjoy this in the true holiday spirit of giving! So Happy Chanukkah and Merry XMAS!


OVERVIEW

Math Standards

  • 3-dimensional objects, spatials sense, volumes of cylinders
Target Grades:
Although this investigation appears to be aimed at secondary students taking geometry, cylinders are introduced in middle school and even earlier. There's no reason why middle schoolers shouldn't be able to tackle this or perhaps a modified version. Are 7th and 8th graders expected to know the formula for the volume of a cylinder, or at least, the general form: Area of Base x Height?


THE PROBLEM
In each figure, a 3x5 rectangle is rolled to form a cylindrical shape (a "can" without a top or bottom). In Fig. I, we "identify", i.e., paste edge AD onto edge BC. For Fig. II we reorient the same rectangle and paste edge DC onto edge AB. Even though these cylinders do not have a 'bottom', assume they are sitting on a flat surface and we will be "filling them up" to determine their volumes.

For the Instructor: Provide 3x5 index cards for the students. Students, working in pairs, should physically form each of the cylindrical shapes and refer to these while working the problem.

Investigation/Questions

1) Without calculating, make a conjecture: Which cylinder would have the greater volume or would they be equal? (For instructor: Record the results of these conjectures on the board)

2) Compute the volumes of each cylinder - no calculators! Leave results in terms of π. Was your conjecture accurate?

3) Compute the ratio of the volume of cylinder I to the volume of cylinder II. Any surprises?

4) It wouldn't be an investigation if we didn't generalize! But this time, YOU have to write the generalization, state the conclusion and prove it! Remember, the true spirit of the holidays is to give, not only receive!!

HAPPY HOLIDAYS
FROM
MATHNOTATIONS

Saturday, December 20, 2008

A Traditional Algebra 2 Rational Equation for the Holidays or ...

Solve for x:

\frac{4x-1}{2x}+\frac{2x}{4x-1}=2

Comments:
(1) The traditional procedure is time-consuming but part of the rites of passage for Algebra 2 students. This is the kind of exercise that "trains the brain!"
(2) Note that the form of the equation is a + (1/a) = 2. Does this suggest anything to you?
(3) Would this question be appropriate for SATs? Algebra 2 End of Course Exams?

Wednesday, December 17, 2008

Santa Knows Linear Programming!

Oh No! The economy has caught up with Santa too. With one week to go, with huge cost overruns and unable to obtain any credit, Santa is forced to now charge parents $10 for each toy for children over the age of 5 and $5 for each toy for children 5 and under. He has determined there are 560 elf-hours available for making the toys and 100 hours available for quality control ("checking them twice"). Each toy for the older children requires 1.5 hrs to produce and 15 minutes to check, while each toy for the younger children requires 1 hr to produce and 12 minutes to check.
How many of each should be produced to maximize Santa's revenue?
Why will the mathematical solution lead to unhappy children 5 and under? Will Santa follow his mathematical mind or his heart or Rudolph?

(BTW, my wife just read this and her suggestion is that Santa should consider outsourcing or visiting the local dollar store where one can buy $5 toys for a buck - she is definitely more practical than I am!).


Comments:
Anyone (including us!) who hasn't worked with these recently or doesn't have much experience with these kinds of problems often struggle to get started: choosing variables, organizing all of the information, setting up the inequalities, etc. The issue of the best way to introduce this topic is a separate issue. The problem above does not represent an introductory problem to this topic! Furthermore, I threw in some complications (units, rounding issues, etc.). Again, there are pedagogical issues which I will not address for now (maybe in the comments). Have Fun!

Here's the setup:
x = the number of toys to be made for the older children
y = the number of toys to be made for the younger children

Objective Function: Revenue (R) = 10x + 5y

Constraints (defining the polygonal region):
x ≥ 0
y≥ 0
1.5x + 1y ≤ 560
0.25x + 0.2y ≤ 100


Monday, December 15, 2008

WSJ Article Calls for National Standards, Complete Overhaul of Public Education

If you haven't had a chance to read Louis Gerstner's compelling piece from the Dec 1 Wall Street Journal, entitled Lessons from 40 Years of Education 'Reform,' here is the link. Mr. Gerstner is the former CEO of IBM and Chairman of the Teaching Commission from 2003-2006.

I'll leave it to the readers of this blog to decide if some of Mr. Gerstner's major suggestions echo the positions taken by this blog for the past two years.

Mr. Gerstner's article is not just another set of opinions -- it is a call for drastic change and immediate action. It will surely be controversial. I do not necessarily agree with all of his statements and/or recommendations but I agree with the 'core.'

Perhaps, the time for change has come to American Education...

Here are some excerpts -- I strongly urge you to read the entire piece:

I believe the problem lies with the structure and corporate governance of our public schools. We have over 15,000 school districts in America; each of them, in its own way, is involved in standards, curriculum, teacher selection, classroom rules and so on. This unbelievably unwieldy structure is incapable of executing a program of fundamental change. While we have islands of excellence as a result of great reform programs, we continually fail to scale up systemic change.

This is a complex problem, but countless experiments and analyses have clearly indicated we need to do four straightforward things to bring fundamental changes to K-12 education:

1) Set high academic standards for all of our kids, supported by a rigorous curriculum.

2) Greatly improve the quality of teaching in our classrooms, supported by substantially higher compensation for our best teachers.

3) Measure student and teacher performance on a systematic basis, supported by tests and assessments.

4) Increase "time on task" for all students; this means more time in school each day, and a longer school year.


Therefore, I recommend that President-elect Barack Obama convene a meeting of our nation's governors and seek agreement to the following:

- Abolish all local school districts, save 70 (50 states; 20 largest cities). Some states may choose to leave some of the rest as community service organizations, but they would have no direct involvement in the critical task of establishing standards, selecting teachers, and developing curricula.

- Establish a set of national standards for a core curriculum. I would suggest we start with four subjects: reading, math, science and social studies.

- Establish a National Skills Day on which every third, sixth, ninth and 12th-grader would be tested against the national standards. Results would be published nationwide for every school in America.

- Establish national standards for teacher certification and require regular re-evaluations of teacher skills. Increase teacher compensation to permit the best teachers (as measured by advances in student learning) to earn well in excess of $100,000 per year, and allow school leaders to remove underperforming teachers.

- Extend the school day and the school year to effectively add 20 more days of schooling for all K-12 students.


Sunday, December 14, 2008

Teaching Probabilities and Strategies Via Games!

NCTM Teaching Standards:

  • Develop and evaluate inferences and predictions that are based on data
  • Understand and apply basic concepts of probability

  • Target Audience: Grades 7-12

    Tools Needed:
    Graphing calculator with a random integer generator or an online random number generator (look here for example)

    Classroom Organization
    (After demo mode): Students working in groups of 3 (two opponents, the 3rd calls out the numbers and keeps score; roles are rotated)

    Sample Classroom Scenario
    Who thinks they can beat me at a game of chance? I will demo the game, then I will play against an opponent. If you beat me two out of three, you are the new champ and you can pick your opponent. After 10 minutes, you will be playing in small groups and recording the results.

    The Play
    Using a random integer generator we will generate random digits, one at a time, from 1 through 9, inclusive (no zeros). The object is to build a 5-digit integer which is greater than your opponent's by placing each 'called' digit into one of the five place-value positions. Once you place a digit you cannot change it!

    Let's try it... Ok, Marissa, turn on the random integer generator, press Enter and call out the first integer. FOUR!
    Ok, I'll place it here: ___ ___ 4 ___ ____
    Call out the next integer: SIX!
    I'll place it here: ___ 6 4 ___ ___
    Next: TWO!
    ___ 6 4 ___ 2
    Next: FOUR!
    ___ 6 4 4 2
    Last digit! FIVE!
    5 6 4 4 2
    How did I do? Could I have used a better strategy? Do you think you could have beaten me?
    Who wants to play! To win, you have to beat me two out of three. Ok, Dimitri, I will work on my paper and you work on yours. Remember, you cannot change a digit's position once you place it...

    Brief Discussion of Strategy Based on Probability Arguments:
    Suppose the first two digits called are 3 and 6 in that order. Would it be better to place the 6 in the thousands' place or the ten thousands' (leftmost) position? If you place the digits here:
    ___ 6 ___ 3 ___, what is the probability that at least one of the next three digits chosen will be 6,7,8, or 9. (Otherwise, your strategy would have backfired). To compute this, we look at the complementary condition, i.e., we determine the probability that the next 3 digits chosen will all be in the range 1 through 5. The probability of this is (5/9)(5/9)(5/9) or approximately 17%, so the probability that our strategy works is about 83%, odds that seem worth playing! Experienced game players often compute these probabilities mentally or have seen these situations so many times they know these probabilities by heart!

    Notes
    (1) Students may not know there is a Random Integer generator built into many graphing calculators. For example on the TI-84, press MATH, then PRB, then 5:randInt(.
    From the home screen, Enter randInt(1,9), ENTER.
    Each time you press ENTER another "random" digit will be displayed. The person calling these out must be instructed to announce only ONE digit at a time!
    (2) Why 5-digit numbers? This seems to make the game fairly interesting and moving at a good pace. Expect ties of course!

    Perhaps this is a good activity before the holidays. Have fun and let me know how it goes!

    Thursday, December 11, 2008

    Instructional Strategy Series for Middle School and Beyond: Developing Direct & Inverse Ratio Concepts

    Three beagles can dig 4 holes in five days. How many days will it take 6 beagles to dig 8 holes?




    Standard Assumptions

    Note: It may be highly instructive to ask the students what natural assumptions (stated below) are being made here before you tell them!
    (1) All beagles work at the same rate. (If you understand beagle behavior intimately, you might question this). Seriously, it's the underlying assumption of constant "rate of work" that is so fundamental here.
    (2) All holes are the same size.

    Instructional Commentary

    Well, at least, I didn't ask the classic: "How many eggs can 1.5 hens lay in 1.5 days (my all-time favorite word problem)!

    The focus of this post will be on the first two stages of concept development using a concrete numerical example. You may take strong exception to the approach below of combining both direct and inverse variation in the same lesson, but, remember, the goal here is concept development, not proficiency with an algorithm! The algebraic stage will be deferred or left to the reader. The algebraic relationships are extremely important and worthy of extended discussion but that needs to be a separate discussion.

    Stage I: Building on Intuition

    Before developing a strict mathematical procedure involving direct, inverse or joint variation I feel it is critical for students to trust their "math sense." Encourage this with comments like:
    "Forget calculations here, boys and girls, just think about this problem, use commonsense, and you might be able to arrive at the answer in less than 10 seconds!"
    Don't think they can? No harm in trying...

    I believe that when we tell them to trust their intuition, some will arrive at the correct answer of 5 days. Encourage those who "see" it to share their reasoning: WHY will the number days not change! This will vary according to the ability level and confidence of individuals in the group but, even more importantly, according to the environment you create in the classroom (accepting non-judgmental climate leads to greater risk-taking).

    When review of homework, content coverage and time for guided practice (before the assignment is given) are the highest priorities of our lessons, then it is natural to question the wisdom of the above strategy.
    This is obvious from typical comments like:

    "Very nice, Dave, but who has the time for that, it's not going to be tested on the State Test and, moreover, I'm not teaching gifted kids like you must have had."


    I won't react to my own Devil's Advocate arguments. Those you who know my philosophy of education know what my response would be!

    Stage II: Beyond Intuition - Developing Proportionality Concepts via a Systematic Approach
    "Well, boys and girls, now that we believe the answer is still FIVE days, let's try to approach this more mathematically, that is, more logically and systematically, in case the answer cannot be 'guessed' so easily."

    I have found over the years that the following TABLE or matrix approach is a powerful model for devleoping proportionality concepts before the student sees a single algebraic relationship:

    EVERY DOG HAS HIS DAY!

    Beagles


    Holes


    Days



    3
    4
    5
    3
    8
    ??
    6
    8
    ???

    Note how this approach avoid changing both the number of holes and the number of dogs in the same step! By keeping one quantity fixed, the student may better be able to focus on the relationship between the other two. Thus, in the second row I kept the number of dogs constant, changing only the number of holes:
    "Boys and girls, if the number of dogs stays the same and we double the number of holes, then what will happen to the number of days ?"(they will double).
    [Note that I asked for the effect on the the number of days before I asked for the actual number of days, namely 10 days.]

    This approach develops the idea of direct variation before we express the relationship algebraically: As one quantity increases, so does a second quantity proportionately.

    Now that we have filled in the second row (replace the ?? with 10 days), we can move on to another relationship:
    "Boys and girls, look at the 3rd row. What quantity (variable) did we not change (keep constant)? What quantity did change? If we double the number of dogs, what should happen to the number of days needed to dig the same number of holes?"
    (Yes, some will think 'double', since direct variation is often the initial reaction of many students).

    Thus we are literally constructing direct and inverse variation via numerical computation before we develop any general relationships. Yes, this is time-consuming, but hopefully you will see the payoff in comprehension.

    Stage III: Expressing Relationships Algebraically
    Not in this post!
    Important Note:
    Normally, we would be very reluctant to mix both types of variation in one lesson, choosing to develop mastery of just direct variation first, then inverse much later on. Yes? Therefore you might feel that combining these will lead to confusion on the part of most students in most classes. Remember, though, the intent here was to develop a strong intuitive base for different types of variations before attempting to formalize any of this! You may not agree, but I'm proposing it anyway. I have done this with good results. Once the concept foundation is laid, students can take off with all the formulas!

    Wednesday, December 10, 2008

    A Different "Approach" to 0.99999999...??

    An investigation for middle schoolers? Precalculus students? Calculus students? Anyone who is fascinated by patterns and an understanding of the infinite and infinite processes?? Enjoy this at any level or depth you wish...

    Take out your calculators folks....
    Determine the first dozen decimal places, then the exact decimal for each of the following:

    1 - 1/9

    1 - 1/99

    1 - 1/999

    1 - 1/9999

    1 - 1/99999

    Continue this pattern until the denominator has a string of 9 nines.

    Questions:
    (1) Describe any patterns you observe. What if the denominator had a string of 100 nines? A string of N nines?
    (2) What does all of this suggest (not prove) about the meaning of 0.999999... (repeating)?
    (3) Oh, and by the way, you may also want to examine the decimal expansions of 1/9, 1/99, 1/999, 1/9999, ... How would you describe the exact decimal representation of 1/9999...9, where the denominator has 100 nines? N nines?

    Is there anything new under the sun here? OR just another view of well-known facts about infinite repeating decimals, sums of infinite geometric series, limits and real numbers???
    Your thoughts...

    Monday, December 8, 2008

    Odds and Evens - FEATURED BLOGS and Update for Dec 2008

    Some thoughts and loose ends...

    It's long overdue but better late than never - I've decided to feature two math websites/blogs for this month. I will try to do this on a monthly basis. It's important to me to acknowledge the outstanding work of other math bloggers particularly those who have been kind enough to support my efforts over the past two years. There are so many excellent sites to choose from and I don't want anyone to think I'm ranking these by this selection. I will acknowledge several others but I had to start somewhere.

    FEATURED MATH BLOG/WEBSITES FOR DEC.


    Maria Miller's HomeSchool Math blog.

    In Maria's own words:
    I love teaching and I love math. This blog is my way of reaching out and helping you to teach it too.

    Maria has hundreds of posts covering many topics in math and math education as can be seen from the extensive Labels section in the right sidebar. Maria particularly enjoys demonstrating the bar model method from Singapore Math. Well worth reading...

    Maria also has a comprehensive math resource site with links to free materials as well as other excellent products. In particular, I'd like to bring the following page to your attention: Problem Solving, Word problems and Math Projects - Free resources on line for K-8 students and beyond. Maria is very thorough and writes a short review of each. Make no mistake -- these resources are for public school parents, students and educators as well. Maria's enthusiasm, dedication, and knowledge are incomparable.

    Finally, Maria has developed a series of highly regarded workbooks, her Math Mammoth series, for Grades 1-5. Affordable, easy to follow, a complete curriculum... Enjoy!


    MathNEXUS
    MathNotations was the featured website of the week on this excellent site. I would strongly recommend this site for all math educators. Jerry Johnson who maintains it somehow manages to update frequently with useful links to just about everything going on in math education. His weekly features include Problem of the Week, Quote of the Week, Statistic of the Week, Humor of the Week, Website of the Week (MathNotations made it!), Calendar Events and Famous Birthdays to name a few! This site receives thousands of visits each week. You will not be disappointed.

    Talk about weird coincidences. Jerry found my blog and somehow remembered me from a presentation I gave back in the 90's at the Annual NCTM Convention in San Diego. Turns out that he was the program chair!

    UPDATE
    1. Anyone remember a MathNotations post from awhile ago regarding the classic boring of a hole through a sphere problem. This well-known Geometry/Calculus conundrum continues to tantalize and intrigue readers. Here's the idea of the puzzle if you forgot it:

    Imagine you have a bowling ball with a diameter of 8.5 inches and you drill a hole exactly through the middle so that the remaining part of the ball is 6 inches high. Now imagine boring a large hole though a giant bowling with a diameter of 8.5 feet so that the part of the ball left behind is still only 6 inches high. Incredibly, the volumes remaining of these two drilled-out bowling balls are equal
    ! Can you picture this? The scientific approach here might be to physically drill through two such objects of equal density and compare the volumes remaining by weighing them, but mathematicians usually try to avoid such 'hands-on' experiments, choosing to prove the result using methods from geometry or calculus. From the number of views this puzzle has received, I'm working on a video demonstrating the result using elementary calculus (volumes by cylindrical shells or disks). Because the writing on the board I'm using tends to be small, it will be difficult to read all the details but I'll do my best. I recognize that these videos are not of the quality you can find elsewhere on YouTube but I still think that working through such a fascinating problem makes it worthwhile and somewhat unusual. Considering that it reviews fundamental integral methods it might also be helpful for students who will soon be getting to this topic (or reviewing for exams). If you watched any of my other videos you know I explain the details thoroughly, trying to avoid skipping any key steps. That guarantees that the hole in the sphere video will be quite long (I'll probably break it up) and it will take some time to bring it to fruition. Despite the technical issues, if you want to see more of these, let me know. I'm also considering a few dedicated to the lower grades, e.g., how to develop algebra sense in middle schoolers (from numerical patterns to algebraic expressions). Any interest?

    2. The problem from the other day regarding the angles of an isosceles triangle received some nice responses but the question would not have made a good contest or test problem. There's nothing worse than an apparently challenging question that students can answer correctly using incorrect reasoning! Here again is the problem:

    Exactly two of the sides of a triangle are congruent and one of the equal angles is known to have degree measure greater than 50. How many integer values are possible for the measure of the remaining angle (the one that is different from the base angles)?

    I had indicated there were a couple of sticky points one had to be wary of (my readers caught on to the fact that the equal angles do not have to have integer measure and that the triangle is not equilateral), but, if the student overlooks both of these, the correct answer can still be obtained! Thus, if the student starts with a base angle of 51°, one obtains a vertex angle of 78°. Reasoning quickly, one arrives at 78, the correct answer, as the number of possible values (all integers from 1 to 78). Two errors which nullify each other and produce a correct result! That's why it's always preferable to have assessments in which students are required to explain their methods but it's simply impractical to have standardized tests with 45-50 such questions. I should have caught that but it does demonstrate the challenges of writing high-quality short-answer (objective-type) questions!

    3. By now you've probably all forgotten about the MathAnagram for Oct-Nov-Dec.
    Check the link if you're interested. Here it is again:
    CHARMED ERA





    Thursday, December 4, 2008

    FRACTIONS, FRACTIONS, FRACTIONS EVERYWHERE - A MIDDLE SCHOOL ACTIVITY

    What middle schooler doesn't get that warm and fuzzy feeling when we tell them we're going to play with fractions! Here's a small investigation to capture that mood of euphoria...

    RULES OF THE GAME:

    • No Calculators - No decimals!
    • All fractions must be expressed in lowest terms
    On a number line mark off 0 and 1 and approximately locate points A and B whose coordinates are 1/4 and 1/3 respectively.

    1. Write the fraction that divides the segment between A and B into two equal parts. How can we verify that this point satisfies the desired condition.
    Complete: This fraction is the _______ of 1/4 and 1/3 and is ____-way between 1/4 and 1/3. The corresponding point is the _________ of segment AB.

    2. Write the two fractions that divide the segment between A and B into three equal parts.

    3. Do the same for four, five and six equal parts.

    4. Reach/Extension: Describe a general procedure for dividing the segment AB into any desired number of equal parts (mathematically speaking, we would say n equal parts).


    Comments:
    (1) What prerequisite skills do students need to have in order to attempt this investigation? When planning a lesson like this, I found I had to consider this question first and review those needed skills. This avoided many issues that would otherwise slow down the lesson. I always tried to avoid the "You don't remember this?" comment. Sometimes this took superhuman effort on my part!

    (2) Students should be organized into pairs or teams of 4. They can "divide" up the labor.

    (3) There are several approaches to these problems. Many confident students with strong foundation skills (ok, this narrows it down to one student in the back of the room), recognize that a common denominator approach makes the most sense. You might see some very clever resourceful methods coming from your youngsters.
    Note that 1/4 = 3/12, 1/3 = 4/12. In order to place a fraction in the 'middle', rewrite 1/4 = 6/24, 1/3 = 8/24. Don't be surprised to see some students invent similar methods for the other divisions.

    Wednesday, December 3, 2008

    Geometry Problem Requiring Critical Reading & Thinking

    Exactly two of the sides of a triangle are congruent and one of the equal angles is known to have degree measure greater than 50. How many integer values are possible for the measure of the remaining angle (the one that is different from the base angles)?

    Comments:
    (1) This is not intended to be a significant challenge. Rather it is meant as a warmup or for a slightly more extended discussion.
    (2) Since middle school students know the sum of the angles of a triangle and can be told the basic fact about the base angles of an isosceles triangle, this problem is appropriate for them too.
    (3) How would you expect most students to approach this? Do you think the majority would start by plugging in 51, 52, 53, etc.?
    (4) Does this type of question promote important problem-solving skills and strategies? Do students recognize the significance of the 'boundary values' 50 and 90, values that are not in the domain of the base angles yet can be critical for the analysis?
    (5) There are at least two 'traps' set in this problem that are intended to help students become more critical thinkers and not jump too quickly to conclusions. After all, what is a trap? If one is circumspect, details are not so easily overlooked.

    Tuesday, December 2, 2008

    The Product of a 40-Digit Integer and a 60-Digit Integer has ___ or ___ Digits. A Problem for the Calculating Middle School Mind!

    Too ambitious as a WarmUp for the 6th or 7th grader? Would they immediately employ the "Make it simpler and look for a pattern" strategy? Is the calculator appropriate for this activity? Is this really an activity/investigation?

    Since I'm already regarded as an anachronism, I guess it wouldn't hurt to play word games here:

    Is this problem un'characteristic' of MathNotations!?!

    Hey, there's a whole generation (or more) who may have no idea what that means! If you do know, you can always say you heard about it from your great-grandfather who carried around his slide rule! Hey, anyone have their Keuffel & Esser handy?

    Monday, December 1, 2008

    Just Another SAT-Type Combinatorial Problem

    How many 3-digit positive even integers have at least one digit equal to 2?

    Comments:
    (1) Talk about intense information overloading! Students need considerable exposure to these short but concentrated exercises. There are at least 6 pieces of information packed into these 14 words! What % of students do you think would miss or misinterpret one or more of these 'clues'?
    (2) Do they really put questions like this on the SATs? Ask any student who recently took the PSAT!
    (3) There are many strategies possible here. Many students (if they fully comprehend the question) will start listing 102,112,120,... but how successful will they be using this approach? There is a powerful approach for the "at least one" types of counting problems. I strongly advocate this starting in middle school.
    (4) I have published many other similar problems (look under combinatorial math in the index). Do these get easier with practice over time? I think so but there always needs to be clarity of thinking and a careful organized approach. The quick clever student often falters when detail is required. This may help that student to mature!
    (5) Are you thinking I'm making too big a deal over SAT-types of questions? What if your students won't even take these tests, choosing ACTs instead? Hopefully, you will come to believe that my purpose is to use these kinds of problems simply as a vehicle for taking students to a higher level of thought. What would be the harm of using these for the occasional class opener (aka Warm-Up, Problem of the Day, Do Now, etc.). In fact I would encourage this at least once a week!

    Addendum
    Another compelling reason to discuss more than one method of solution for combinatorial problems: One is rarely 100% certain of the accuracy of one's answer without doing the problem by an alternate method and getting the same result!