tag:blogger.com,1999:blog-8231784566931768362.post7823324587189860189..comments2018-04-30T02:03:49.242-04:00Comments on MathNotations: Video Solution and Discussion of Twitter SAT Probability Question from 8-25-10Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-8231784566931768362.post-8577592757477873532010-09-07T23:23:28.425-04:002010-09-07T23:23:28.425-04:00So let them, while they list, or maybe better afte...So let them, while they list, or maybe better after they list, let them create a list of what's left over. Tell them to. And if they notice nothing, just let it go. But odds are, one kid's going to ask why they didn't just start with the things we don't want, the easier list.<br /><br />And then you've got them thinking about complements.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-59120437822441545572010-09-07T13:58:34.435-04:002010-09-07T13:58:34.435-04:00Jonathan,
As usual, your comments are insightful a...Jonathan,<br />As usual, your comments are insightful and on target. You are right in stating that making an organized list does not come naturally for most students, but it is a skill that can be practiced. For those individual who are not analytical sequential thinkers this is difficult but one can improve. This process should start in the elementary grades!<br /><br />Yes, your "complementary" thinking is a well-known and powerful construct. However, students more often see it and use it in the context of "What is the probability of getting at least one six in 5 rolls of a die?" Your approach actually changed the experiment to selecting TWO cards instead of the original FOUR. This seems as obvious to you as 6C4 equals 6C2 but the equivalence is not transparent IMO. I do think that once it's explained to a group, most would really like it and even prefer it to other methods.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-20393546755934832752010-09-04T03:57:13.290-04:002010-09-04T03:57:13.290-04:00So, I do appreciate that making a list is a far be...So, I do appreciate that making a list is a far better strategy than many of us acknowledge.<br /><br />But 2 qualifications: <br /><br />Listmaking, that is, organized listmaking, is as much a skill as many of the other things we teach, and some kids have a knack for it, and others struggle to stay organized... Because it seems simpler to us does not mean it will actually be simpler for the kids who have the hardest time.<br /><br />And my razzle-dazzle hid what I was doing: finding the probability of the complement. Conceptually tough? Absolutely. But it so often makes a hard problem easier that it is really worth knowing, teaching, practicing.<br /><br />My dialogue with myself often looks like:<br />Hmm, probability, cool!<br />Oops, that looks involved. Is the complement simpler?<br /><br />I get to it that quickly. And fairly often. And it has that indirect feel that almost translates to other branches...<br /><br />JonathanAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-70422915640939461972010-08-31T09:31:56.649-04:002010-08-31T09:31:56.649-04:00Nice, Jonathan! In Part II, I was going to demonst...Nice, Jonathan! In Part II, I was going to demonstrate other methods using multiplication of probabilities but none of them comes close to the simplicity and elegance of your thinking. At this point I doubt I will do a Part II.<br /><br />In the end, I'm trying to help students who struggle in math learn to approach these questions using sound principles and basic methods/strategies. Listing the possible outcomes may not be practical for the more advanced problems but it usually suffices on the SAT. For students, this is a matter of survival. Many math teachers are appalled when students are shown test-taking strategies which circumvent the "real" math in the problem. They need to remember what it felt like when they were confronted with a math problem on a standardized test and they felt overwhelmed and lost...Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-15021811793320395682010-08-31T08:21:08.205-04:002010-08-31T08:21:08.205-04:00So, you want 2 and 2? I can help. I'll grab a...So, you want 2 and 2? I can help. I'll grab a red card and another red card, and you can take the rest. <br /><br />Hmmph. If I were grabbing at random, what would the probability of both cards being red be?<br /><br />4 out of 6 for the first, times 3 out of 5 for the second.<br /><br />So 12/30 of the time I'll leave you with 2 and 2.<br /><br />(it's hard for me not to do. I was not taught to do this as a counting problem)<br /><br />JonathanAnonymousnoreply@blogger.com