MathNotations

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.

Here the answer is: 6 eggs

Posted by Dave Marain at 8:55 AM 7 comments

Labels: hens laying eggs, puzzle, twitter, update

Today and over the next few days, you will find, in the media and on many math and science blogs, many touching, almost reverential, tributes to the greatest puzzler of our generation. How I looked eagerly to the next edition of Scientific American when I was younger. We didn't have much money but my dad insisted on purchasing a subscription to this classic magazine, intended for those scientists and non-scientists who wanted to know what was happening in the forefront of modern science and mathematics. Of course, I turned immediately to the back page to tackle another set of Mr. Gardner's challenging puzzles. I was so proud of myself if I could solve even one of these! Many of his puzzles had an almost magical quality to them. Now you see it -- now you don't. My forte was the logic type of puzzle but I tried them all.

Martin Gardner died Saturday, 5-22-10, at the age of 95. (See the puzzle created below in dedication to Mr. Gardner).
By the way, 95 = 19x5, 94 = 47x2, 93 = 31x3.
It is only fitting that he left us at an age which is the largest 2-digit number with exactly two prime factors.

For you puzzlers out there, here is my conundrum dedicated to Mr. Gardner. Feel free to submit your solution, but only one, in the comments to this post. Our readers can choose which one they think is the most elegant. I found one way, but I'm certain there are others!

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.

He was not a mathematician, nor a professor, nor a scientist. Yet I feel strongly that he deeply influenced all of these groups as well as anyone who enjoyed the satisfaction of challenging the mind. Read about him in the Wikipedia article and in the many tributes. If you're too young to have experienced the sheer joy brought to so many of us then discover it for yourself by looking at the annals of Scientific American or reading one of Mr. Gardner's many books.

Martin Gardner was more than a maker of puzzles of course. He was also known as a debunker of quackery and pseudoscience. He was an amateur magician, a philosopher, a lover of knowledge, a true Renaissance Man - a man for the ages.

Dr. Gardner - thank you for making a difference in my life and the life of so many others. Now if only I could remember how to get the cherry out of the martini glass by moving two matches...

On behalf of all my fellow bloggers, my sincerest condolences to your family.

--------------------------------------------------------------------

"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

Posted by Dave Marain at 8:07 AM 14 comments

Labels: Martin Gardner, puzzles, tribute

Hey, these are precious moments and, anyway, I can't do more than one video a week! So to take a respite. here's another anecdote, and, by the way, his 6-yr old brother and 3-yr old cousin are now demanding equal time.

So, he turned three the other day and when he went to bed that night his mommy, aka my daughter, told him: "____, now that you're three, I'm just going to tell you a story, give you a hug and kiss and say goodnight" (rather than staying with him until he fell asleep).

Knowing how this young man's mind operates, what do you think his reply was?

"Mommy, I'm only three during the day, I'm still two at night!"


Good luck to my daughter and all of his teachers!


Ok, fair's fair...
When the 6-yr old brother who was then 5, got his new bed (full size like his parents) my daughter and son-in-law heard him pacing and sighing loudly outside their door after they had tucked him in. Finally, his mommy came out and asked him what was wrong. Here was his response:

"I don't want to sleep in that bed. I'm not ready to be married!"

Finally, my son's beautiful 3-yr old daughter went to Disney World a month or so ago. My wife asked her what her favorite ride was and she replied, "the rollercoaster." Then my wife asked if she was afraid. This was her reply:


"I was not afraid. I just screamed the whole way."


May their innocence remain forever...

--------------------------------------------------------------------------------
"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

Posted by Dave Marain at 8:25 AM 0 comments

Labels: children's anecdotes

The video below presents a more challenging 3-dimensional geometry problem which would be at the upper end of SAT I or SAT II - Subject Tests (Math I/II). The key here is to challenge students' assumptions about a quadrilateral being a square because it has 4 congruent sides, a common error. This question will also review a considerable amount of geometry: Pythagorean Theorem, Volume of cube, spatial reasoning, 45-45-90 triangles, area of a rhombus, etc.

Posted by Dave Marain at 8:14 AM 4 comments

Labels: 3-dimensional geometry, cubes, geometry, mathnotationsvids

The full version in one of its many many variations:

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.

Putting aside the silliness of the riddle, there really is some serious mathematics going in these kinds of rate/ratio/proportion problems. Rather than solve the "hen" problem for you, I'll leave it to my readers to solve it by their own favorite methods. By the way, the answer to this riddle is in the description of the video below on my YouTube channel. Sorry 'bout that!!

Instead, the video below, which appears on my YouTube channel, MathNotationsVids, presents a developmental approach to a more complicated ratio problem for middle schoolers and beyond. I'm far more interested in your thoughts about the teaching strategies than I am about the problem itself. Please understand, further, that I am not suggesting the method shown in the video is efficient nor would it make much sense for the upper level math or science student. See comments below the video for further discussion of this.


The Problem in the Video Below:


If 10 workers can build 3 houses in 60 days, how many workers are needed to build 5 houses in 40 days? Assume all workers build at the same rate.

[埋込みオブジェクト:http://www.youtube.com/v/P_VCYl0zdts&hl=en_US&fs=1&rel=0]


More Advanced and Efficient Algorithms


(1) We assume from the "constant rate" assumption in the problem that the number of houses (H) which can be built varies jointly as the number of workers (W) and the number of days (D).
Thus, H = kWD.

Substituting, H=3, W=10 and D=60, we obtain:
3 = k(10)(60) or k = 1/200. Note that the units of k are Houses/(Workers x Days).
We can interpret k to mean that 1/200 of a house can be built by 1 worker in 1 day. Thus, k is not only a constant but actually represents a rate. Another way of expressing this rate is
(1 House)/(200 Worker-Days) or the reciprocal version:
(200 Worker⋅Days)/(1 House)

Substituting the new set of values into the relationship H = (1/200)WD, we obtain:
5 = (1/200)(W)(40) or W = 25 workers.

(2) This can be made even more efficient using the "factor-label" (dimensional analysis, etc.) format:

(200 Worker⋅Days)/(1 House)) x (5 Houses)/(40 Days) = 25 Workers!

(3) I could also exploit the inverse variation between W and D, but that's for my readers to bring up or for another video!

I see these efficient methods as "black box" methods for some students. Developing a deeper understanding of direct and inverse variation is far more important for the younger student.



-------------------------------------------------------------------------------------------------------------------------------

"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)

Posted by Dave Marain at 2:37 PM 3 comments

Labels: average rates, math videos, mathnotationsvids, middle school, proportions, ratios, work problems

I've decided to post a few headlines from various sources (Ed WQeek, NYTimes, other blogs, etc.) over the past few weeks. I will not post links to most of these or make comments at this time. I believe they speak for themselves...

• Stand and Deliver Teacher Dies of Cancer
• Is Teaching A Marathon or a Sprint?
• 
  
  A Letter For Anyone Concerned About K-12 Mathematics Education
• U.S. Needs Better-Trained Math Teachers To Compete Globally
• U.S. Falls Short In Measure of Future Math Teachers -- American college students earned a C on a new test comparing their skills with their counterparts in 15 other countries.
• Teacher Training No Boon For Student Math Scores -- A major federal study concludes that intensive teacher professional development in math did not immediately improve test scores.
• The Boys Have Fallen Behind
• Merit Pay For Students Fails to Raise Score, Study Finds -- Research suggests that payments can boost achievement if they reward behavior conducive to learning, rather than test scores themselves.
• Both Value and Harm Seen in K-3 Common Standards -- While some view the proposed expectations as valuable guidance, others worry that they are inappropriate for youngsters.
• What They're Watching -- The Ten Most Popular Course Lectures Available on You Tube
• An Open Mind -- Putting free courseware online was a first step in reimagining education. What now? Wiki U’s, smart courses and, maybe, learning

-------------------------------------------------------------------------------
"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

Posted by Dave Marain at 12:51 PM 2 comments

Ok, so I don't have another anecdote from my grandson today so I'll have to get back to mathematics -- problem-solving, teaching and learning.

The video below deals with an algebraic equation in 2 variables which should be straightforward for your stronger Algebra II or Precalculus student. But will it be? I invite you to predict how many in your classes will answer it correctly, then try it out. After all, it is multiple choice, so statistically some should get it right by some means or other!

Hopefully, the purpose of the problem and the video will become clear to all of you. If we want our students to demonstrate better reasoning and an understanding of important ideas in math, we need to feel comfortable in teaching for meaning and understanding. This doesn't mean we stop teaching algorithms and procedures, however. Exactly what all this means and how to do it is the reason for this blog. I certainly never claimed to know the answers or any other mystical secrets. I only know that I never gave up trying. Sometimes my efforts failed miserably, but I hopefully learned from these attempts.

It would mean a lot to me if you share your thoughts here or on my You Tube channel, MathNotationsVids, where you will find my other videos.

[埋込みオブジェクト:http://www.youtube.com/v/xGKmPHfsUaU&hl=en_US&fs=1&]


Note: Another subtle point I should have made in the video---
y(x-4) = 0 → y = 0 OR x = 4
It is important for us to stress this point and distinguish it from "AND" logic. If the equation were in the form: y² + (x-4)² = 0, we would have (y = 0) AND (x = 4), whose graph would be the single point (4,0). Another instance where an exercise on the board can lead to a rich, fruiful and profound discussion. If all of this is seen as taking too much time away from content, remember this is precisely the kind of change in curriculum and instruction that Prof. Schmidt has been trying to tell us about for over 15 years! Well, I'm preaching to the converted, aren't I...
-------------------------------------------------------------------

"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

Posted by Dave Marain at 5:17 PM 3 comments

Labels: algebra, instructional strategies, math videos, SAT-type problems, video lesson

With all the material I'm getting from my 2-year old grandson, I may as well give up my math blog and start a new blog featuring "anecdotes about grandchildren"! I betcha' some of these would go viral like no math article ever would!

Anyway, while I am preparing my next "exciting" video, I decided to share with you what my grandson said to his grandmother (better known as "Mimi" and also known as my wife) this morning.

His mother (my daughter) and grandmother (my wife) took him this morning for his first blood test (for allergies). My daughter needed "Mimi" to hold her son as she thought she would pass out! So, my wife had him sit on her lap. When he realized that the needle was for him, the screaming began and he tried to jump out of "Mimi's" arms. Fortunately the whole procedure took only 30 seconds. My grandson jumped off my wife's lap, wiped his tears away and in a serious tone stated to Mimi, "You're fired!" My wife was stunned, never expecting to be fired from being a grandmother! There's definitely some Donald Trump in that boy.

Of course, a few minutes later all was forgiven and they all lived happily ever after.

I realize grandparents think all their grandchildren are special and anything the children say must be a gem worth repeating a thousand times. But, I do believe this young man is quite a character. By the way, my daughter has no idea where he got the expression "You're fired!" from, unless of course he is watching the Apprentice on the sly!

---------------------------------------------------------------------------------------

"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

Posted by Dave Marain at 6:21 PM 0 comments

Labels: humor

Posts

Posts

All Comments

All Comments