## Sunday, January 7, 2007

### Warmup #3 for Mon 1-8-07 A Quickie??

The last one probably made some viewers nauseous, recalling that pit in your stomach when walking into math class, so I'll keep this one shorter and less technical...

This is a well-known problem but I've added some variations:

Version 1 (Grades 3-6):
Using the digits 1,2,3 and 4 exactly once and in any order, what is the largest possible product you can make? Also, list the factors used. For example, 312 x 4 = 1248 is one possibility, but that may not be the largest. You can of course use a 2-digit number times a 2-digit number.

Answer: 1312 (I'll leave it to the reader to determine the factors!)
Comments: There aren't that many combinations here so 'guess-check with the calculator' would be the method preferred by most students. Consider our role as educators when presenting such a problem, however. What can we do to maximize the benefit from problem-solving? Would having students work in teams be more effective or not? Is it a no-brainer here to allow the use of the calculator, allowing the student to focus on the ideas behind choosing factors and making discoveries? Should we recommend that students record each attempt to develop a more systematic approach? We know how most 8-11 year olds jump around and 'push buttons' hyperactively! What is our role in their development? Most educators give these 'fun' puzzle problems, but do we take the time to plan the best way to present them? Quick warmups don't allow that, now do they??

Version 2 (Grades 5-8):
Same as above, but use the digits 1,2,3,4 and 5 this time! Record all attempts.

Answer: 22,412 (if you find a larger one let me know).
Comments: Using 5 numbers dramatically increases the number of combinations, an interesting combinatorial problem in its own right! How would you extend this problem to make it richer? If we use any 5 digits such as 2,5,3,9, and 0 would there be a pattern students could recognize? Would it be worth assigning that for extra credit? Does this problem develop number sense, pattern-based thinking and estimation skills? Is the calculator a much more necessary tool for this version than for Version 1? Will I ever stop asking 'obvious' rhetorical questions?!?

REMEMBER: I'M DEPENDING ON YOU FOR FEEDBACK NOW THAT I'VE ENABLED EVERYONE TO COMMENT! IF YOU WANT ME TO CONTINUE ENTERING THESE MATH WARMUPS, LET ME KNOW. IF THEY'RE A TURNOFF, THAT'S OK TOO! IF YOU TRY THESE OUT IN CLASS, PLS COMMENT ON HOW YOU IMPLEMENTED THEM, STUDENT REACTION AND THE GRADE LEVEL...

Myrtle Hocklemeier said...

Hi Dave,

I'm not sure how I found your blog but I have a question about something you said,

"Most educators give these 'fun' puzzle problems, but do we take the time to plan the best way to present them? "

Is there anything like math olympiad problems for children younger than 10? Currently I'm using Math Olympiad problems with my son as a form of supplementing, but I also have a first grader and a kindergartner that could use a few fun puzzlers.

Currently I present the problem as is without commenting and let him struggle. For example, yesterday he worked on this problem:

1/A + 1/B = 1/3 where A and B are positive whole numbers. He's ten and needless to say he did NOT find the solution on his own. However, he seemed to enjoy us speculating about a more general problem: Can all unit fractions can be expressed as the sum of two smaller unit fractions?

Dan Meyer said...

Looks like someone beat me to that first comment. Glad it's up and running.

Dave Marain said...

Myrtle,
Any unit fraction can be decomposed as follows into unit or Egyptian fractions:
1/n = 1/(n+1) + 1/[(n(n+1)].

For example, 1/5 = 1/6 + 1/30.
There's a lot of theory here. A good reference is Mathworld:
http://mathworld.wolfram.com/EgyptianFraction.html

My comment about 'taking the time to plan' was not meant to be critical. We are all looking for engaging problems that stimulate student thought, review concept/skills or develop important ideas in mathematics. I've seen these used frequently as quick class openers to keep kids busy while checking homework, but then they just vaporize into the ether. I'm thinking about how these kinds of questions can be explored further without compromising the curriculum. I'll keep supplying the problems if others suggest ways to make this happen! I really appreciate your giving these questions to your son and letting him explore and struggle. That is so worthwhile. Monday's warmup is not intended for enrichment or for math contest training. It's intended for ALL students in upper elementary or middle sachool, although I plan on using it in my skills class period 5 tomorrow. I'll let you know how it goes!

Richard Colvin said...

Hi Dave....enjoyed working on this problem myself. and will share it with my son. I'm not a teacher as you know but in my experience as ajournalist writing about education, I've found that for many teachers "activities" are good in and of themselves rather than because they are consciously "educative" to use Dewey's term. It's the teacher who connects an activity to a larger concept or truth for the student.