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?


Sue VanHattum said...

Hmm, that site starts out with "A effective teacher..." An effective site manager would have made sure there were no typos like that.

Dave Marain said...

That is the first thing I noticed too!
Hopefully their message won't be lost.

Let me know what you think of the challenges and the rest of the post.

mathmom said...

In the first problem, is it intentional that the first 3 comparisons are less than or equal, and the last is strictly less than?

Dave Marain said...

Yes, mathmom, it was intentional! After all, the unicode for "less than" is a pain!

Jeff said...

Enjoy your blog! Couple of burning questions:

Do you subscribe to the theory that mathematicians are influenced by a collective consciousness?

Also, could you clarify a question I have on integers?

I realize that in Fermat's version, x^n+y^n=z^n has non integer solutions when n=2. But are there non-zero integers (a, b and c) that would solve the Pythagorean equation? I hope I'm making sense trying to convey this question.


Jeff said...

What I came up with on my aformentioned problem:

let's say x=2 and y=3
so...2^3 + 3^3=z^3=8+27=z^3
35^ 1/3=z

Dave Marain said...

I appreciate your clarification. I think it's valuable for students to see non-integer solutions of Fermat's general equation in order to appreciate how remarkable it is that there are no integer solutions for n greater than 2.

Can you define "collective consciousness"?? Mathematicians often disagree about issues in their own discipline, including the definition of what mathematics is!

Jeff said...

Dave, wasn't sure I was on the right track working out that equation. Does it look okay to you?

Dave Marain said...

The algebra looked fine but I'm not sure I understand what your goal is. Is there a reason why you're looking for non-integer solutions to Fermat's Equation?

Jeff said...

Dave, the question came up in a math explorations class I'm taking; it was centered on the proof discovered by Andrew Wiles of Fermat's x^n+y^n=z^n (having no non-zero integer solutions when n is greater than 3).

Today we were debating how an individual influences the development of math vs. how the culture influences its development. You wouldn't have all those great mathematicians in Hungary if it hadn't been for a supportive, nurturing culture.