Sunday, January 31, 2010

Can Your Students Find At Least Three Methods? Odds and Evens Week of 2-1-10

I've been working on a new website which I will share with you when ready but I haven't forgotten my faithful readers who may have forgotten me!

There are so many issues in mathematics education that it would take forever to update you on all of them, however, I know that you are already aware of most of these.

Some Significant Current Issues in Math Ed

  • Moving Inexorably Towards Common Standards in Math
  • Teachers Need a Clear Curriculum Map/Content Guide rather than Standards!
  • Rapid Push Toward Including Several Open-Ended Questions on State or Common Assessments is Slowing Down. Can you think of the major reasons for this?
  • Joel  Klein's Education Equality Project whose goal is to close the Achievement Gap

 Of course, most of you have already skipped down to the Challenge Problems!

The first can be tackled by middle schoolers, although many high schoolers may find it interesting and fall into a trap if not careful. The wording is challenging but your students may benefit from working in small groups.

Challenge Problem #1
a, b, c, d and e are positive integers with a ≤ b ≤ c ≤ d < e.
If a + b + c + d + e = 143, what is the least possible value of e?

Is this merely a guess-test-revise question or is there a strategy/method your students can come up with? How would you extend this problem? change the "143" to a larger value? Change the set of integers to 4 values (a,b,c,d)? 6? k? This is an important issue. Otherwise students may see each problem as an isolated quickly solved puzzle!

The goal of the next question is to review geometry and algebra skills and concepts and to encourage a variety of approaches. I will give the answer -- the challenge for your students is to find AT LEAST THREE METHODS! The teacher may want to submit the best team's efforts to me for acknowledgment on this site.

Challenge Problem #2

P(5,1), Q(8,2) and R(a,b) determine an isosceles right triangle with point R above line PQ and ∠ PRQ the right angle. Determine the coordinates a and b. In your group, you must devise at least THREE methods! 

Answer (6,7)

"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
Note: These lyrics provoked considerable criticism back in 1949-50 but Rodgers and Hammerstein would not take them out. Do they still have relevance today?

Tuesday, January 5, 2010

If We're 'Packing", Are We Going Somewhere?

Fascinating article from today's New York Times. In 2-dimensions we talk about tessellating objects to fill the plane. Circles of course will always leave gaps. In 3-dimensions, equal spheres will also leave gaps when packed as closely as possible, but the question then becomes, "How do we arrange the spheres which would result in the densest packing. Turns out that the grocer's method of stacking oranges solves that problem! Equal cubes can be packed together without any gaps, so we can say that the densest packing for cubes is 100%, that is, identical cubes can be packed so that they use 100% of the available space.

But packing regular tetrahedrons (a pyramid whose 4 faces are congruent equilateral triangles) as tightly as possible has defied the best logical mathematical minds, including Aristotle's, for nearly 2000 years. Recently, significant strides have been made, not only by the best mathematical and scientific minds, but also by graduate students, like Ann Chen from U. of Mich.,, who have taken dozens of tetrahedral dice from Dungeons and Dragons games and are using a hands-on approach to build various configurations and then computing the density of the packing. For the past several months teams from different schools have published their latest and best attempts, but, as of this moment, Ann has found the densest packing at 85.63%. That's right, she's outdone the best theoretical mathematicians and scientists in this quest for the new Holy Grail of packing problems. Aristotle mistakenly asserted that there exists a perfect 100% packing for tetrahedrons, but this was shown to be false. Now it appears that the percent is far more than we thought. Professor Nash, we need you!

I strongly believe that there is a place for this kind of discussion in our math classes from the earliest years on. Let students know that solving mathematical problems often involves hands-on experimentation as in science! Besides, who's to say that there isn't some middle schooler out there right now who might sit in her room playing with these dice who will arrive at 86%!!


"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)

Friday, January 1, 2010

HAPPY 2 x 3 x 5 x 67! Let The "Problems" Begin!

May this new year and decade bring happiness and prosperity to each of you now that the 2KO's have come to an end! 

BTW, the italicized symbol in red is my submission for the name we should give to the past 10 years. What do you think of it? Let me know if you came up with one of your own. According to Time Magazine, no one has yet created a name which has caught on (and dozens were listed!).  Also, I will avoid debating those who strongly believe that the first decade of the 21st century ends a year from now!

As MathNotations begins its 4th year, it has become an annual tradition for math ed blogs to challenge their readers to discover interesting facts about the number symbol representing the new year, in this case, 2010, or Twenty-Ten, for those who are as committed to multiple representations as I am!

Those who know me can anticipate that I would recommend making this an exercise for our middle schoolers. Here are a couple of ideas:

"In your group, list as many observations as you can about the number, 2010. Your team's score will be based on both quality and quantity. For example, an observation like "2010 is even" would only earn 1 pt, whereas "2010 must be divisible by 3 because the sum of its digits is divisible by 3" would earn 2 or 3  points since it contains both a fact and an explanation."

Another idea might be to have students write interesting word/number problems involving 2010 for the class to solve. Of course, to obtain credit the student posing the questions must  provide correct answers and solutions!

Your turn...

A final note ---

Some of you may have noticed that I've enabled Comment Moderation due to the number of spam comments which have gotten through. I held out for as long as I could. I do check throughout the day, so, hopefully, this should not prove problematic for my readers.