Tuesday, January 23, 2007

Problems for 1-24-07

8 comments:

Totally_clueless said...

For (2), I might have preferred the question to be the largest possible value for the largest number. This can be arrived at directly via reasoning rather than enumeration.

(3) is quite interesting, and requires visualization and recognition of the figure obtained.

Dave Marain said...

point well taken, totally! however, here was my thought process:
1. finding the largest possible is a well-known SAT question; I was looking for more analysis
2. although somestudents will engage in random listing, this problem requires some organization but more importantly, the teacher's role is to guide a more analytic approach and share students' strategies:
i. once we determine that 8 is the largest possible value, do we change the 4 smallest values or do we ask the opposite question: can 7 be the largest? can 6 be the largest? 5? demonstrate it!
dave

Jonathan said...

Dave, I am going to steal #3 and rewrite it for a teacher-level challenge.

I like #5, quite a bit. Do you know the probability (related, but not on topic really) that two arbitrary numbers will be relatively prime?

For #2, I like that after finding the max and min values, the students still have to be careful not to just subtract.

Dave Marain said...

Answers to 1-24-07 Problems:
1) 122
2) 3 (#'s are 8,7 and 6)
3) 14
4) 14

Hopefully, these are correct!
Jonathan, I like #3 too - just something slightly different from the routine rectangle problem. It encourages a diagram and some reasoning about where to put the 4th point and the nature of the quadrilateral. How many students/teachers would calculate the area of the rectangle first, namely 90, subtract from 117, leaving a right triangle with an area of 27 and a height of 9. Therefore, the base is 6 and x = 8 + 6 = 14?? OR do you think the majority would just 'recall' the formula for the trapezoid! OR ???

I love the relatively prime question! Since number theory was my focus many many years ago, I am rusty but I do remember using this to tell my students that pi appears in the most unexpected places. I suppose if one thoroughly grasps why the sum of the squares of the reciprocals of positive integers is pi^2 divided by 6, one could really feel they know math! For your question, the answer is the reciprocal, 6/(pi^2).

My purpose in writing questions about 'Lowest terms' is to get at more number theory in middle school but also to get students to think carefully - it's very easy to miss one of the primes up to 50, but even easier to forget that 49 is one of the possibilities and in fact the only one that is not prime (other than one of course).

BTW, I keep wondering if anyone other than the 4-5 of you are actually trying these problems or using them in the classroom? The other 50-60 visits a day? I have no idea but if some are enjoying these, I'll keep going for awhile. I may have to take a few days off to recharge and get caught up but I'll try to keep it going. Does anyone think it would be worth compiling these questions (with solutions) and making them available in a book (down the road of course)? My wife says I always talk about that but never get around to doing it!

Totally_clueless said...

I am not an educator, so I don't use these problems, but I do enjoy them immensely.

I tried to get my kid, who should be taking the SAT in a year or so, to start doing these for practice. However, the same excitement about neat math problems does not seem to be shared by my progeny 8-(

Dave Marain said...

totally--
thank for the support - I like these problems too but I'm not supposed to say that!
don't feel bad - i show these questions to my children and they roll their eyes! i do believe that these challenges go just beyond SATs and that repeated exposure to thinking at a higher level does pay off but only a few of my students are willing to pay their dues. In fact, the ones who try these are those who already enjoy pitting their minds against brain-teasers. I believe our textbooks should have these kinds of questions interspersed throughout and stop labeling them as standardized test questions or challenge problems. That provides an easy excuse for not assigning them or not doing them. Further, I believe, that educators should begin to incorporate these into assessments, first as bonus questions, then make them count. I guarantee that over time these students will perform better on college entrance exams.
BTW, your screen name belies the fact that you enjoy math and are obviously very good at it. I have to believe you took more math in school than you're willing to admit. You may not be certified to teach, but, if you're a parent, you're definitely a teacher or at least 'certifiable'!
dave

MathMom said...

Number 4 is useful for encouraging students to learn their primes as well. I think it's useful for them to know the primes under 100, and this will get them halfway there. I forgot 1 as a numerator when I tried it at first! ;-/

mathmom said...

Number 2 reminded me of Cross Sums puzzles (now trendily re-named Kakuro). Look them up if you've never seen them. Once you do the "multiply the mean by the number of entries to get the sum" thing, the problem is very much a cross-sums one where you have to find ways of making a given sum using a certain number of digits (to fit into the puzzle) all of which must be different. I used various levels of difficulty cross sums puzzles with my upper elementary and middle school kids, and they really enjoy them while flexing those numeracy and logical thinking muscles.