## Thursday, June 10, 2010

### A Call for Problems

This post is solely a request and includes absolutely no insightful, bland, or even obtuse (no pun intended...ok, pun intended) repartee.

With that disclaimer out of the way, what is/are your favorite math problem(s)? I'm specifically looking for questions that are be accessible for middle schoolers (doesn't mean they have to be easy for middle schoolers).

Non-math-teacher-types, sorry if you're feeling left out but still feel free to contribute.

Thanks!

Oh, and just so you don't think I'm slacking, I'll contribute one that I like.
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I'm not sure why, but I know that 1+2+3+4+5+6+7+8+9+10=55.
Find 2+4+6+8+10+12+14+16+18+20.

EDIT: Ok, I just decided that I was getting too into the middle school spirit by asking for the most bestest favoritest problem of, like, all time. Share a problem you like. The one I shared isn't my favorite problem; I'm not even sure I could come up with a true favorite.

1. Four 4's Puzzle. Using four 4's and any operations, can you make each of the numbers from 1 to 100? I still haven't done them all myself, but a colleague has. I've gotten everything up to 32, and all the evens to 100, I think.

Decimal Point Pickle Game

Crossing the River. 5 adults, 2 kids, a boat that can hold 1 adult or 2 kids. How many trips to get them all across?

Path Walking Puzzle. (go here and then it's a pdf)

Are these along the lines you'd like?

2. "Are these along the lines you'd like?"

Thanks! There really is no right answer to this. The last problem is new to me and I must say...I like it.

3. I think you'll find a lot of problems you like at http://mathteacherscircle.org . They're aimed at middle school teachers; some of them translate very well to working with students, and others take some work to transform into something that you would want to use in the classroom.

4. Math/Science: How many 2 liter bottles of soda would it take to completely fill our classroom?
-justify your answer: if you measure something, give its exact measurements. if you estimate, explain the reasoning behind your estimate. if you decide to ignore certain features of our classroom's shape, explain.

I arm them with two meter sticks per group and some chart paper and markers, and off they go. Great for volume, understanding of metric units, conversion within the metric system, and measurement and arithmetic practice.

5. The path-walking puzzles are cool!

6. Here's a question for you, btw.

I was a good math student, and I like puzzles, but I have very low "task-commitment" to use a term I learned at a gifted ed conference. Certain types of problems in certain contexts will totally get me working doggedly long past when anyone else cares. But other types of puzzles bore me (once I realize there's a trick/strategy/etc., I don't care that much about figuring it out in detail, even if I can't actually completely solve it). And, I'm ashamed to admit, I also sometimes get bored when I hit a wall and have run through everything I can think of.

And I think I grew up in a family that appreciated puzzles and math and had a pretty good math education for the most part. So...

How do you build stamina for such things in your students? Or is it just a mismatch between problem and personality? Or are these actually not such great problems, and the really great ones DO promote commitment? Or am I just lazy?

7. Find a fraction you can multiply by itself to get 2.

It's mean, I know. I'm a stinker.

8. Problem A
It's pretty easy to cut up two 1X1 squares into pieces to get a square of area =2. That could be the starter problem. Now figure out how to cut up/ rearrange the pieces of three 1X1 squares to get a square of area =3. There are at least two solutions I know of. One's famous, and both (I think) generalize.

Problem B
Take the golden ratio. Multiply it by itself. Simplify. Repeat. Find a formula for the Fibonacci numbers.

Problem C
You have a laser gun and you are trapped between two mirrors in a Vee. Which way do you shoot in order to kill the space louse on the back of your head?

9. Ms V: I agree that the path problems are cool. I like the idea of part of the problem being to figure out the problem. I'm stuck though trying to prove that the last 2 are impossible (or find a solution).

Also, I wanted to let you know that I'm not ignoring your "why should I give a hoot" question. In fact, I think it deserves a post which is in the works.

Kate: That is cruel, but I think it's important (and would actually be good for morale) for students to work on unsolvable problems. I had a whole unit on unsolvable problems when I worked in a school with no curriculum.

bnachumi: Thanks for the contributions. I love the first problem, but I'm not sure I could successfully give this as a task to middle schoolers. It has multiple solution methods (well, I'll have to take your word on this because I haven't found them), but I'm struggling to find multiple entry points. Maybe I haven't worked on it enough, though. I think the second problem reveals a fascinating result, but that as a problem it's pretty contrived. Don't mean to be hating on your contributions...it's probably just because we're friends. :) If it makes you feel better, in retrospect I don't like my problem as a task as much as I originally did...I think it works better as an assessment problem.

10. This relates to conversations Avery and I have been having about precision of language.

I noticed that Kate's problem said "fraction", not "rational number". So, what's wrong with sqrt2/1 * sqrt2/1?

Though I graciously concede your point about the benefit of working on unsolvable problems, Avery. ;)