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?
Friday, July 3, 2009
Taking Middle Schoolers Beyond Procedures To The Next Level...
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16 comments:
Dave,
Good stuff.... and how about one more that asks something like, "List all pairs of numbers (x, y) so that LCM (x,y) = 24.:
Nice, I really like this. I teach GCD and LCM using prime factorization, so if they understand why that works, they should be able to do problems like this easily, and conversely the problems should help cement their understanding. I'll have to mark this to use next year!
Pat, Mathmom--
Thanks for the kind words. Your comments are what make these efforts worthwhile.
One can Google lcm and get some interesting resources but it's not always easy for teachers to find examples of more open-ended or nonroutine problems of good quality. Certainly math contests are a source but often the questions are a couple of levels beyond the typical students in our classes.
Pat, I really like your variation. I'm trying to imagine how most youngsters would approach this. I think many would try some random pairs of factors of 24 and hopefully realize that one of the numbers would have to be 24 or 8. Ultimately the instructor might want to demonstrate a systematic approach using prime factorization. This method is not far removed from the procedure for counting the factors of a positive integers using its prime factorization. Very cool stuff! In the UK, do they use the same abbreviations (lcm, gcf or gcd)? I do know that my spelling of factorization is different!
There are "word problems" that exercise LCM that students should get used to recognizing as well. These do show up in places like MATHCOUNTS. Such as the type where one clock chimes every 15 minutes, another every 20 minutes, and another every 35 -- if they've all just chimed together, how long until the next time they do so, etc...
Once students have a grasp of the concept, maybe you can have them create their own examples for others to solve. This is a great way to reinforce the material, and it makes it kind of fun for them to be creative.
I put up prize puzzles occasionally. This one is one of my favorites:
A, B, and C are positive integers.
The Greatest Common Factor of A and B is 5
The Greatest Common Factor of B and C is 2
The Greatest Common Factor of C and A is 3
The Least Common Multiple of A and B is 900
The Least Common Multiple of B and C is 2100
The Least Common Multiple of C and A is 630
What is A?
Curmudgeon, you should have asked "What is C?" since it is a very famous answer ;-)
True, but how many students are gonna know THAT?
Probably more students know "that" than can actually solve the problem ;-(
You mentioned that even for following step by step procedure some student still struggle to be able to do it. Have you had any experience that students perform better on open ended problem than on procedural problem?
Steve-
I think it's a great idea to have students invent their own challenges for their classmates, not to mention that it increases the pool of questions you can use for next year's classes!
Curmudgeon-
Mathmom's tongue-in-cheek Douglas C. Adams reference notwithstanding, I enjoyed your challenge! And I don't think you threw that problem together by randomly choosing 3 numbers! The understanding of the prime factorization method together with puzzle-type logic makes the question outstanding. In keeping with Steve's comment, I think it would be highly instructive for your students to be asked to write one of these on their own. They not only would learn much about lcm, gcf and prime factorization but they might gain a new appreciation for how hard it is to write a good question! One more thing -- you've provided many teachers out there with a wonderful type of problem to deepen student thinking. Not to mention that students loves "extra-credit" challenges and the satisfaction of solving a puzzle (if they can!).
Watchmath-
I suspect you might already know my response to your insightful question! I have observed that students who struggle with procedures because of weak background skills or because they are not "detail-oriented", often display highly creative thinking for certain types of puzzle problems, particularly those that require spatial/mechanical sense or commonsense. They might struggle to write out the details of their thought processes but could contribute to a group effort. This is a good reason for giving students a variety of assessments.
Thanks, all, for enriching this discussion...
Once you give out all that information, Curmudgeon, the problem becomes pretty simple. After all, gcd(a,b) lcm(a,b) = ab for positive integers a and b. So, students can calculate AB, AC, and BC.It's easiest for them to multiply all three together to get A²B²C², take the square root to get ABC, and divide by AB, AC, or BC to get C, B, or A. I'd use a calculator.
The unfortunate thing about the problem is that one doesn't need to calculate gcd's or lcm's.
I've been wondering about other exercises for students. Perhaps this is why Lewis Carroll invented (or popularized) word ladders, where one transforms a word into another word by replacing its letters, one at a time: CAT-COT-COG-DOG. Many problems we discuss are solved in he same way
When introducing a new topic, if I have a good analogy where the slower students can easily follow the logic, I usually use it. I had to make one up for LCM using prime factorization. Can you improve it or replace it with something better?
Part 1. Mix and Match Store
I had to buy a baby stroller and a push cart. At Mix and Match store, I can buy the parts for any item and easily assemble them.
Baby Stroller parts – front wheels, back wheels, baby seat, roof, handle
Push Cart parts – front wheels, back wheels, basket, handle
The nice thing is, I can use the same part for more than one item. For example, I could just buy one set of wheels and use them for both the stroller and the cart. To save money, I’ll buy the fewest parts possible. What parts would I need to buy? What parts do I not need to duplicate? The parts I bought contain both the stroller and the cart.
Part 2. Prime Factors Game
A math game uses the same idea as the Mix and Match store. I have to make two products using its parts, in this case the parts are the prime factors of the product. I can buy as many prime number cards as I want. Each card contains one prime number.
The activity card I picked said to multiply my prime numbers to form the product 18. I can use all my prime number cards once for this activity. I also picked an activity card that said to multiply my prime numbers to form the product 24. I can also use all my prime number cards once for this activity.
What are the prime factors of 18? What are the prime factors of 24? Since each prime number I buy costs points, I want to buy the fewest numbers possible. What prime number cards do I need to buy? What is the product that I can form using all the prime number cards I bought? What is the relationship between the original numbers 18 and 24, and the product of all my prime numbers? All the prime numbers I bought “contain,” in a multiplication sense, both products 18 and 24.
I wanted to try an analogy. This would be followed by some practice and the kinds of higher order questions listed here. I am not sure this analogy would be worth trying. What do you think?
Burt--
I like the Prime Factors game - very creative! It reinforces prime factorization and the "least" concept is modeled nicely by minimizing the number of purchases. The game could also be modified for gcf and other prime factor problems. I might restrict it to just a few primes like 2,3,5, and 7 at first. Also, include numbers like 15 and 28 which involve no duplication of primes so that they can also discuss the case of 'relatively prime' numbers. Terminology can be introduced when appropriate after they've enjoyed the game for awhile. Then you can have them construct the numbers themselves which produce different lcm's and gcf's. So when are you patenting this 'prime' deck!
Eric--
Elegant solution as always! I wouldn't assume however that most students would know this relationship between lcm and gcf nor would most be familiar with the multiplication of equations method. These are typically math contest approaches. However, I strongly believe the relationships should be developed in middle school through activities such as these. "When will the lcm equal the product of the numbers?" is the kind of question which leads students toward the discovery of the lcm-gcf relationship. Thanks for reminding us of this important idea!
I cannot explain how much I have enjoyed this blog and the subsequent dialog. The largest question it elicits for me is; as a new middle school math teacher, what are the blogs and web resources which this community might recommend to ensure my (hopefully) imparting math excellence and true conceptual understandings of mathematics?
JD--
Thank you for the support.
There are so many excellent math blogs and resources out there, it would be difficult for me to rank the best ones. However, for Middle School Math, you can't do much better than Denise's Let's Play Math blog. Her posts are all about enrichment and her links to resources will get you going in the right direction. Of course, you can check the right sidebar of my blog for other top-notch blogs and websites.
Best of luck as you embark on this exciting journey! Come back and visit again.
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