Showing posts with label prime factorization. Show all posts
Showing posts with label prime factorization. Show all posts
Friday, September 12, 2008
The Largest Odd Factor of 90? Too Easy? How About A Million? A Googol!
Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:
PRINCE? NAH! E-ROI!
We would hope that by grade 5, most youngsters would be able to answer the first question in the title fairly rapidly and without a calculator. Or are you thinking many would incorrectly blurt out '9' as the answer?
Well, why should anyone care about finding the largest odd factor of some positive integer? Will it lead to a better understanding of the origins of the universe? Perhaps not, but these questions may deepen student understanding of
(a) Factors
(b) Prime factorization (particularly of powers of 10)
(c) The important concept that an even number may have odd factors but an odd number can never have any even factors!
(d) Other ideas...
Ninety is not a very large number so students will usually see that the answer is 45. However there are so many ways of looking at this simple result. So many important methods -- so little time! Further, a method that works effectively for 90 may not be as effective for 1,000,000 or a googol.
My suggestion is to give middle schoolers the 'million' problem, let them work with a partner, allow the use of a calculator and see what happens.
Here are some thoughts:
(a) Which of the following is more instructive, more important conceptually?
Writing 1,000,000 as 26⋅56, etc.,
OR
Having students, on the calculator, divide 1,000,000 by 2, then the quotient by 2 and so on, until an odd result occurs
(b) Do these 2 approaches reinforce/develop the same concepts/skills or different ideas?
(c) Which method is most reasonable for 90? for 1,000,00? for a googol?
(d) Does the calculator enhance or not enhance understanding here? Does it depend on the number we start with?
Well, why should anyone care about finding the largest odd factor of some positive integer? Will it lead to a better understanding of the origins of the universe? Perhaps not, but these questions may deepen student understanding of
(a) Factors
(b) Prime factorization (particularly of powers of 10)
(c) The important concept that an even number may have odd factors but an odd number can never have any even factors!
(d) Other ideas...
Ninety is not a very large number so students will usually see that the answer is 45. However there are so many ways of looking at this simple result. So many important methods -- so little time! Further, a method that works effectively for 90 may not be as effective for 1,000,000 or a googol.
My suggestion is to give middle schoolers the 'million' problem, let them work with a partner, allow the use of a calculator and see what happens.
Here are some thoughts:
(a) Which of the following is more instructive, more important conceptually?
Writing 1,000,000 as 26⋅56, etc.,
OR
Having students, on the calculator, divide 1,000,000 by 2, then the quotient by 2 and so on, until an odd result occurs
(b) Do these 2 approaches reinforce/develop the same concepts/skills or different ideas?
(c) Which method is most reasonable for 90? for 1,000,00? for a googol?
(d) Does the calculator enhance or not enhance understanding here? Does it depend on the number we start with?
Posted by Dave Marain at 2:51 PM 3 comments
Labels: factors, prime factorization, reasoning
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