Friday, July 3, 2009
Taking Middle Schoolers Beyond Procedures To The Next Level...
Typical Classroom Scenario?
We're introducing the idea of least common multiple of two positive integers and after defining the terminology and illustrating several examples most students are catching on to some procedural method of which there are many:
Listing common multiples of each
Prime Factorization
The "upside down division method" you saw at a conference...
Yes, we are all very good at demonstrating step by step procedures and having students practice repetitively until they catch on and can reproduce this with some speed and accuracy. We feel this is a worthwhile skill (they'll need it for common denominators, clearing denominators in rational equations, useful for solving certain types of word problems, etc), it's in the curriculum and the standards, it will be tested in various places and the lesson plays out. Some students pick up the method(s) quickly, while others struggle, particularly those who haven't learned their basic facts.
BUT how can we raise the bar to stretch their minds? Can the above scenario be restructured to enable students to gain a deeper understanding of the concepts of lcm and gcf? Perhaps we can start the class off with a more open-ended type of question and ask them to work in small groups to solve it. Perhaps, we can ask a different type of question after teaching some standard procedure. A nonroutine, higher-order question that is not in the text...
What resources are available for more open-ended or nonroutine questions to enable our students to delve beneath the surface and actually think about what they are doing? Well, I can't answer all these questions but here are a few thoughts...
1) Write two examples for which the lcm of two numbers is their product.
2) Write two examples for which the lcm of two numbers is not their product. The numbers in each example must be distinct (different).
3) The lcm of 12 and N is 24.
a) What is the greatest possible integer value of N?
b) What is the least positive integer value of N?
These are just a few samples to start you off. You could probably come up with better ones or you've read some excellent ideas in some publication. Please share...
To see a more challenging version of the examples above, click Read more...
You might want to give the following for homework or an extra practice problem in class. Do you think students will require a calculator? How about telling them they cannot use it!
The lcm of 100 and N is 500. What is the least positive integer value of N?
Posted by Dave Marain at 7:30 AM 16 comments
Labels: higher-order questions, lcm, middle school, more, number theory, teaching for understanding