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?
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3 comments:
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I'm not a math instructor or even a math student, but the method that pops out at me for the million and googol problems is first to express the number as 10^k, then as (2⋅5)^k, and finally as (2^k)⋅(5^k). Then you can say that the largest odd factor is 5^k.
The million and googol problems are probably easier than the 90 problem, because they'd actually have to factor 90 (and its factorization is not as easy as (2^k)⋅(5^k)), or repeatedly divide by 2.
(Argh, no <sup> allowed!)
Thanks, Susan. I think these problems were not viewed as very interesting but I do see their potential for developing prime factorization concepts as well as other related ideas.
I agree with your solutions and comments. I can tell you love the reasoning and logical parts of mathematics. You're a problem solver!
The idea of starting with a million and having students divide by 2 repeatedly until an odd factor remains is an interesting way to get middle schoolers to analyze a situation. After they try this with a few large even numbers, they should begin to understand why the method works. Of course, we don't have to restrict the numbers to products of powers of 2 and 5.
Another interesting variation is play the "million" game on the calculator as early as 4th or 5th grade.
The teacher (or parent) can play against the child or children can compete against each other. Let's say I go first:
1000000/2 = 500000
The the child goes:
500000/5 = 100000
Play alternates with each player dividing by either 2 or 5 each time. The player who produces a result of 1 loses!
Most will figure out after awhile that the player who goes first will always win after 12 divisions. An interesting way to develop the idea of prime factors I think. Students will tire of starting with a million each time so change the the prime factors to say 2,3, and 7 and of course change the starting number to something like (2^4)(3^3)((7^2). Since this number will result in 9 plays, the person who goes first loses!
What do you think?
I solved this problem in my head exactly the same way as Susan.
I don't know for sure the best way for students to do it. Experimenting with a calculator sounds good to me for starters.
In general I tend to favor a solution that is efficient and "professional". By that I mean that if people who know math well tend to solve a problem a certain way, then that's a "professional" way to solve it.
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