MathNotations Soaring With Eagles or Just For the Birds? Updates 5-28-10

NOTE: I added a new solution (see (e) below). Also, read the comments to see even more solutions. Thanks to Jonathan for pointing out my error in (d) of my results.

I'll get to that cryptic title in a moment (may be obvious to some)...

1.  Remember the challenge problem I posted in the tribute to Martin Gardner a few days ago? Well, we rec'd several excellent replies and I have an additional response from a very sharp high schooler as well. Here was the problem:

Can you form 95 using each of the digits 5-2-2-1-0 exactly once? No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc...  You may combine the digits to form numerals like 12 or 120.



Mr. Lomas: 5! - (2+2)! - 1 - 0   Perhaps the most elegant since it uses the individual digits in the given order.


Robot Guy: (21-2)*5+0


Nate (high schooler): 120-5^2   Oh, the simplicity of that one! Combining digits is not the first way I thought of...


Mine so far:


(a) 102 - (5+2)  Pretty simple but I wasn't thinking much of combining digits until I saw Nate's


(b) 120 -25 (Shameless plagiarism from Nate's but I couldn't resist!)


(c) (2^5)(2+1) - 0! (I posted this one already)


(d) 10^2 - 5 x (2 - 0!)   (I knew there had to be a way using 100 - 5)
NOTE: JONATHAN POINTED OUT MY ERROR HERE. SEE COMMENTS.


(e) A new one: (2 + 2)! x (5-1) - 0!  I felt I needed to atone for my error in (d)!


I suspect Mr. Lomas has even more! It was definitely the spirit of Martin Gardner at work here!

Keep these coming if you can find more. I'd like to see us get to 10 ways.

2. Remember the hens -a- layin' problem I posted a few days ago? The video on YouTube gave the answer for 6 hens in 6 days: 24 eggs.

The problem on the blog was:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Here the answer is: 6 eggs

Here's a black-box method, i.e., work shown but no explanation:

(2/3) egg per (hen⋅day) x 3 hens x 3 days = 6 eggs.

This is how most solutions are given online and in the literature. It has little to do with middle schoolers actually learning the underlying principles. See the video for details.

3. Now for something completely different as M.P would say!

I've decided for now to tweet a daily (SAT) Problem of the Day.  "SAT" is in quotes because you can use these in your class as regular warm-ups or students can try these on their own to prepare for the upcoming SAT on June 5th and beyond.

Answers to each question will generally appear the next day, just before I tweet the new question. I've posted two problems thus far and the answers are up there today. Today's question will appear shortly.

My Twitter address is naturally dmarain.
Get the RSS feed for this at Twitter/dmarain if you want to see the daily problems.
If you have a question about the problems or want more details about solutions, send me a Direct Message in Twitter or email me.

Follow me if you'd like. These questions will not appear on this blog, so you will need a Twitter account or subscribe to the RSS feed above. Let your students know about it as well if you'd like.

Let me know by commenting here or replying on Twitter (Direct Message) if you like these and want me to continue next fall. Last SAT Problem of the Day on Twitter for this school year will be 6-15-10.

Requiescant in Pacem, Martin...

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

Odds and Ends -- Week of 2-15-10

Catchup time...

• So where are the answers and followup discussion to the previous two problems I posted? They're coming. I plan to make a video of the investigation in the "143" problem. 
• I think it would be so useful to post videos of actual lessons that demonstrate how some teachers implement explorations in the classroom at all levels from K-12. I'm sure there are some of these already online but they seem to be few and far between and I'm really referring to "best practices" here to serve as models for others. We have NCTM Illuminations for example but I'm really looking for something else here.
• More important than investigations is to see models of daily lessons which incorporate the best of what we know about effective practice. Lessons which show HOW to blend procedural and conceptual understanding, help students develop skills mastery while engaging in rich problem-solving. Easy to do? Of course not, but if we want the US to be academically competitive, we had better move quickly in this direction and use international models to guide us.
• How are you, the mathematics teacher, dealing with all of the confusing and overwhelming issues in math education today? OR have you learned to ignore all the "noise", close your door and simply go about the business of teaching? If you're able to, that is! Unfortunately some of the decisions which are being made independently of your input will have significant impact on how well you will be able to do your job today and in the future...  Issues like
  
• "Algebra for All?" So where has that experiment gone?
• States joining consortia to develop a common standards and assessments in math
• How many consortia should there be? Will most eventually merge into one or two?
• Will common math standards ultimately lead to more consistency in content  -- i.e., that which is actually taught in the classroom?
• Are math ed departments in colleges and universities adapting rapidly enough to prepare preservice teachers for the paradigm shifts which are occurring? 
• Will "methods" courses in ed schools increase focus on actual content, e.g.,
  
  "You will all prepare a lesson on the effect of changing the parameter "b" in a quadratic function. Your lesson should utilize multiple representations and include a carefully planned series of Socratic questions which develop meaning and conceptual understanding for the algorithms. Specify what actions you took to balance procedural learning with conceptual understanding. Also, be prepared to answer the essential question: "WHAT ACTIONS DID YOU TAKE AND HOW DID YOU ASSESS THAT LEARNING TOOK PLACE?"
  
• How can we all use emerging technologies to enhance our teaching repertoire. Regardless of whether you "tweet", "Buzz" or Facebook, or all of the above, how can networking and sharing make us more effective teachers? This is sort of obvious, but how many of us are using these on a daily basis in our planning and in our classrooms? 25% 50% More?
• What are the most effective online learning tools for mathematics, particularly middle and secondary math? Many excellent bloggers have researched and compiled excellent lists of resources, but specifically, what are the best online video sites (interactive or not)? This is crucial for me and it's something to which I want to dedicate myself.
• I choose not to get into NCLB or Charter Schools debates at this time...
• So where is my new website I've alluded to? It's taking forever to set up, but I want to do it right. More to follow...

Ok, so we have to have a little problem for your students to think about. This is part of a well-known genre of "puzzles" which frequently travel across the web and are always intriguing for students and adults alike. You've probably seen it...
For this blog, the essential question is, how we can make this a teachable moment in our math classes?

MIND GAME

2% or 98%

This is strange...can you figure it out?

Are you the 2% or 98% of the population?

Follow the instructions! NO PEEKING AHEAD!

* Do the following exercise, guaranteed to raise an eyebrow.

* There's no trick or surprise.

* Just follow these instructions, and answer the questions one at a time

and as quickly as you can!

* Again, as quickly as you can but don't advance until you've done each

of them ..... really.

* Now, scroll down (but not too fast, you might miss something).

Think of a number from 1 to 10

Multiply that number by 9.

If the number is a 2-digit number, add the digits together.

Now subtract 5.

Determine which letter in the alphabet corresponds to the number you

ended up with

(example: 1=a, 2=B,  3=c,etc.)

Think of a country that starts with that letter.

Remember the last letter of the name of that country

Think of the name of an animal that starts with that letter.

Remember the last letter in the name of that  animal.

Think of the name of a fruit that starts with that letter.

Are you thinking of a Kangaroo in   Denmark  eating an Orange  ?

I told you this was FREAKY!! If not, you're among the 2% of the

population whose minds are different enough to think of something else.

98% of people will answer with kangaroos in  Denmark  when given this

exercise.

COMMENTARY

Is there some basic number theory here? 

What would you want your middle schoolers to do with this after they play it a few times?

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

A Middle School Coin Puzzle To Start The Year

I have an equal number of pennies, nickels and dimes. I also have some quarters which have the same value as the pennies, nickels and dimes combined. If I have no other coins, what is the fewest possible total number of coins I could have? What is the value of all the coins?

Comments
(1) An opening day problem?
(2) Would you have students working alone or in small groups?
(3) Would you allow the calculator?
(4) Appropriate for prealgebra students? Students below grade 6?
(5) Is zero a possible answer?
(6) Wording too confusing for most students? Is it ambiguous or clear?
(7) Do you feel there are important underlying concepts and ideas embedded here or is it just a fun puzzle to engage students?
(8) Do students have difficulty in separating number of coins from their value?


REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates:
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

Light Humor and More on a Math Puzzle

We all need a break from the seriousness of math, so here are some (actual) amusing headlines. Before I get nasty comments, remember these are real!

1. Teachers Strike Idle Kids
2. Eye Drops Off Shelf
3. Miners Refuse To Work After Death
4. Hospitals Are Sued By 7 Foot Doctors
5. British Left Waffles On Falkland Islands

Ok, enough of the silliness...

I received several emails and comments about the math puzzle from the other day. Essentially, they all indicated that the problem was readily solved using technology.

First, here is the solution:
123456789 = 10821 x 11409

Aniket S., a computer science major from Solapur University, India, sent me the solution and the code she used in C++ to find the factors.

Both Mike Croucher and Pat Ballew, who emailed me, used the powerful new computing engine Wolfram Alpha to find the factors. As Pat pointed out, "the times they are a changin'...

In fact, I found the prime factors of 123456789 on the TI-89:

factor (123456789)
3^2 x 3607 x 3803

A few additional thoughts...
Is there anything special about the number 123456789? If we randomly permute the digits to, say, 586421739, we obtain the prime factorization 3^2 x 107 x 608953, which cannot be rearranged to form two 5-digit factors due to the 6-digit prime. Can you find a permutation of 123456789 which can also be factored into two 5-digit numbers?

If we make one change to the original number we can solve the puzzle:
123456784 = 10406 x 11864.

Have fun computing...

A Puzzle To Start the Week

Number puzzles always intrigued me and, perhaps, they are one way we can invite our students into the wonderful and exciting world of mathematics. Oh, alright, maybe that's a bit of a stretch, but, I suspect that if you give the following famous puzzle to your students in Grades 5 and up, they will try it even if you don't offer food or a 10 point bonus! Yes, calculators are allowed but after a few minutes of frustration they will be begging for a hint.

(Oh, and if you give them this problem at the beginning of class, you may as well forget the lesson!)

Find two 5-digit numbers whose product is 123456789.


If you solve it, don't post your answer immediately. I will probably publish a hint or the answer in a day or so. You can always email me with your solution at "dmarain at gmail dot com."

Click Read more for a hint and comments...


HINT: Rather than pressing random numbers into the calculator as some would do, encourage them to find the prime factors of 123456789. It's easy to show that this number is divisible by 3 and 9, but find finding the other factors will be challenging. I'll post another hint if you request it...

COMMENT: This beautiful puzzle was invented by Y. Yamamoto and has intrigued many puzzle enthusiasts for awhile now. Is there some profound meaning behind the solution or is it just a curiosity? Perhaps we'll have to wait for Dan Brown's next book to unlock the mystery! I will probably post the answer if I don't get a response within 24 hours. Probably...

If any of your students solve it, email me at "dmarain at gmail dot com" and let me know if I can post their names.

...Read more

Another New Feature - KENKEN is Mind-Bending and Instructive!!

KENKEN® Puzzles

Invented by Japanese math teacher, Tetsuya Miyamoto, KENKEN® allows you to test your puzzle acumen and improve your math skills at the same time.
Exclusively on NYTimes.com, updated with 6 new puzzles daily.

This extraordinary new math-logic puzzle started appearing in the NYTimes about a month ago. Link here to start playing. I also placed a link in the sidebar so that you can play every day. It will take you a few minutes to catch on to the rules and then you will give up Sudoku, Kakuro, your daily crossword puzzle and Jumble! I just started playing and I'm hooked. Most importantly, it will reinforce and develop basic arithmetic skills for your students and/or children!

I also recommend that you read the Times article which introduced this new feature. The creator is a talented and unique math teacher. Here is some fascinating background from the article:

KenKen was invented in 2004 by the Japanese educator Tetsuya Miyamoto, who founded and teaches at the Miyamoto Math Classroom in Tokyo. Students attend his class on weekends to improve their math and thinking skills. Mr. Miyamoto said he believes in “the art of teaching without teaching.”

He provides the tools for students to learn at their own pace using their own trial-and-error methods. If these tools are engaging enough, he said, students are more motivated and learn better than they would through formal instruction.

About 90 minutes of class time each week is set aside for solving puzzles, usually designed by Mr. Miyamoto. The most popular one has been KenKen.

NOTE: The Math Problem of the Day for Wed 3-4-09 was deleted because I deemed that the context was inappropriate for younger readers who may visit. I'm assuming this was an aberration and this feature will resume shortly.

A Very Simple Alphametic Message to Mom

OX
XO
--------
MOM

Please make allowances for the spacing and my crude attempt to produce an alphametic for Mom's Day. The 2nd letter 'M' is supposed to be aligned under the 'X' and 'O', etc. For those unfamiliar with the definition and rules of alphametics, here is some information I copied from Mike Keith's wonderful site:

An alphametic is a peculiar type of mathematical puzzle, in which a set of words is written down in the form of an ordinary "long-hand" addition sum, and it is required that the letters of the alphabet be replaced with decimal digits so that the result is a valid arithmetic sum. For an example one can do no better than the first modern alphametic, published by the great puzzlist H.E. Dudeney in the July 1924 issue of Strand Magazine:

SEND
MORE
-----
MONEY

whose (unique) solution is:

9567
1085
-----
10652

There are two fairly obvious (but worth stating) rules which every alphametic obeys:

1. The mapping of letters to numbers is one-to-one. That is, the same letter always stands for the same digit, and the same digit is always represented by the same letter.

2. The digit zero is not allowed to appear as the left-most digit in any of the addends or the sum.

You may recall that on Pi Day, I linked my readers to Mike Keith's extraordinary opus, Poe, E.: Near A Raven. Mike is more than a Poe and Pi devotee, however, as the link above demonstrates. Another excellent site providing numerous examples of alphametics is Truman Collins' fascinating page.
I strongly urge my readers to visit both of these sites. There are enough puzzles there to keep you busy for decades!

A 'Boring' Volume Problem or "If You Find Yourself in a Hole, Stop Digging!"

Important Note: It took forever but I finally posted the detailed video explanation of this problem here.


Please don't gag on my feeble attempt at humor in the title (my wife actually had bought a sign with that quote -- it's hanging on the dining room wall).

There are a couple of classic volume problems in calculus which have always been my favorites:

• The Volume of the Torus Problem (using 2 methods: cylindrical shells and by disks)

• The Hole in the Sphere Problem (also by 2 methods)

I always assigned one or both of these to my BC Calculus classes, most often as Extra Credit problems. Because of the extra points they could earn, most students tried these and submitted solutions. My feeling is that if a student could do both of these by both methods, they really understood disk and shell!

In this post we will focus on the 2nd problem as it always seems to generate curiosity and interest. I'm guessing that most of you know the puzzle version of this question that was answered by Marilyn vos Savant in her Ask Marilyn column over a decade ago. It's just possible that some calculus student in some second semester class is feeling some anxiety over this problem!

Here's one version of that famous conundrum. There are many approaches here, even the clever mathematical approach of assuming that the problem is well-defined and therefore independent of the radii involved (I expect at least one of our readers to do it that way!).

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.

That's right, the answer is independent of the radius of the sphere and the diameter of the hole! The total volume of the sphere and the volume removed however do depend on the radii. Note that the volume removed is a cylinder with two spherical caps.


The original problem was worded ambiguously in Marilyn's column and then clarified somewhat. My version is not perfect but hopefully you'll get the 'picture', although a real picture would be far better. I will probably do a video presentation of the solution and a discussion of the problem because the diagram and the math expressions are cumbersome and it's not worth the time to play with Draw programs or LaTeX right now. I plan on presenting in detail the disk-washer and cylindrical shells method using a general depth of h inches for the hole.

For now, have fun playing with this. This is a well-known problem and therefore searchable on the web but try it yourself first. Try to use calculus to set up the integral and if you're brave you'll evaluate those integrals without Mathematica or the TI-89! Can you see why the answer for the volume remaining depends only on the depth of the hole?