Making a Challenge Math Problem the Springboard for Concept Development

My problem-posing creativity peaks in the hours between 6-9 am so I'd better publish this fast!

Here's a sample of an SAT-type problem although it's verging on a math contest offering...

Disclaimer: Remember, the problem or its solution is never the objective of this blog. It's merely a framework for helping our students learn to think mathematically while developing concepts in a collaborative setting that builds self-esteem. Wow, where did that rhetoric come from? Uh, me...

If the sum of 2 positive integers is 2^16 and their greatest possible product is 2^k, then k = ?

Answer: 30

Reflections...

1.  What are some strategies you want your students to use for these types of questions? And what type is this!

2.  What do you see are the "big" mathematical ideas embedded in this problem? Are "exponents" a big idea?

3.  What are the prerequisite skills needed for success with this type of question? Would you review these first?

4.  How would you utilize this problem with middle schoolers? High schoolers? Algebra 2 vs precalculus?

5.  Would you begin with an easier problem first, then build up to this or let them struggle with it as is?

6.  How would you assess that students grasped the ideas here?  Make up 10 similar questions for homework? Give them another one to try in class? Include one of these on the next test?

7.  Would you ask students to generalize this problem? First demonstrate what "generalize" means?
[One possibility: If the sum is 2^n, then the greatest possible product is 2^(2n-2).]

8. How do you create an environment of making connections in and/or applying mathematics?
[One possibility: Relate this question to the problem of finding the rectangle of maximum area for a given perimeter!]

So, have I successfully killed off all potential comments!




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