MathNotations

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:

Posted by Dave Marain at 8:24 AM 2 comments

Labels: contest, history of mathematics, mathanagram

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!

Posted by Dave Marain at 7:11 AM 0 comments

Labels: riddle

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)<sup>2

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

Posted by Dave Marain at 8:20 AM 7 comments

Labels: algebra, patterns, proof without words, spatial sense

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.

Posted by Dave Marain at 9:19 AM 0 comments

Labels: update

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.

[フレーム]



[フレーム]

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

[フレーム]



[フレーム]


The actual solution to the problem starts below!

[フレーム]



[フレーム]



[フレーム]

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!

Posted by Dave Marain at 2:18 PM 3 comments

Labels: calculus, hole in sphere problem, method of shells, video lesson, volume

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!

Posted by Dave Marain at 7:58 AM 6 comments

Labels: fun problems, investigations, middle school, primes

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.

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

Posted by Dave Marain at 5:47 AM 7 comments

Labels: careless errors

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.

Posted by Dave Marain at 1:14 PM 1 comments

Labels: update