tag:blogger.com,1999:blog-7718462793516968883.post4203076571962222126..comments2020-07-09T23:18:25.865-04:00Comments on MadMath: Proofs of Distributing Exponents and RadicalsDeltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-7718462793516968883.post-49601914859595361432017-07-27T09:13:17.169-04:002017-07-27T09:13:17.169-04:00I like it, good point!I like it, good point!Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-71311798704004273022017-07-25T03:57:39.862-04:002017-07-25T03:57:39.862-04:00In my opinion, this is good for a basic algebra cl...In my opinion, this is good for a basic algebra class. For more advanced courses that have been introduced to complex numbers though, you may want to touch on how this changes when moving out of R and into C.<br /><br />For example, sqrt(-9)*sqrt(-9) = -9, but NOT positive 9 because of order of operations. So distributing the square root over negative numbers doesn't work.Mac Hollisterhttps://www.blogger.com/profile/13493435577200541535noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-80643291549177580312017-06-13T11:40:47.046-04:002017-06-13T11:40:47.046-04:00Thanks for the information, that's good to kno...Thanks for the information, that's good to know! We've been having conversations about our department's Discrete Math class lately (or rather: how to prepare students better for it). Good to know it's in some book somewhere.<br /><br />And I've heard this before about the lack of direct proofs in U.S. math; in fact, that's specifically the weak point for our community-college math majors in discrete math. <br /><br />I saw a question on StackExchange a while back to wit, "Why do students always want to use proof by contradiction for everything?". Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-8582059777748696152017-06-13T00:22:38.856-04:002017-06-13T00:22:38.856-04:004 years too late, but.....I teach these intro proo...4 years too late, but.....I teach these intro proofs in discrete math. The book is Susanna Epp, Discrete Mathematics with Applications edition 4. Your proof on radicals is problem 59 in 4.1. The Epp book still does not provide a comprehensive foundation in doing these types of proofs, but it gets you started down the right path. In particular it actually presents direct proofs. Direct proofs seem to be non-existent in the general math education of american students. Anonymoushttps://www.blogger.com/profile/08334416409328597850noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-7819696976378829382013-01-17T09:10:55.372-05:002013-01-17T09:10:55.372-05:00As a follow-up, I've deleted this presentation...As a follow-up, I've deleted this presentation from my remedial Basic Algebra class (partly because there was a recent reorganization and we don't even test students on distributing exponents anymore). But I am doing it in my College Algebra class, where I'm personally committed to showing proofs for everything. <br /><br />Works pretty well and was a nice emphasis for students to tell me how to write the "because" statement for radicals in the middle, and thus really understand the meaning of radicals.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-11458400844786940092012-10-20T19:19:21.826-04:002012-10-20T19:19:21.826-04:00Well put, that adds some clarity. Granted that I w...Well put, that adds some clarity. Granted that I want to use this in the basic algebra class, I think I should avoid a dependence on explaining induction. Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-86645844882279626052012-10-20T19:16:56.954-04:002012-10-20T19:16:56.954-04:00You know, I presented this in 3 different sections...You know, I presented this in 3 different sections last week, and it actually went better than I would have guessed. Some anticipated benefits -- (1) it got us to review the definition of what radicals really mean, (2) it generated a discussion about how we can turn radical problems into exponent problems (so they really must follow all the same rules), and (3) it challenged and intrigued the more advanced, often bored, students (by asking them to complete the end of each line, a good idea from S. Thrun). <br /><br />For me, the protective work around stuff like sqrt(a^2+b^2) (exp/rad not distributing over add) is best done by (a) frequent verbalizing/quizzing on the issue, in words, and (b) emphasizing that we can only use rules, in the specific format established by a proof. Obviously, the real mathematical demonstration comes from a counterexample.<br />Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-58803120665015183172012-10-19T17:09:02.196-04:002012-10-19T17:09:02.196-04:00The ellipses in your proof mean that you are using...The ellipses in your proof mean that you are using induction; it's up to you to decide whether that implicit usage is sufficiently formal or whether you want to formalize it via induction.Davidhttps://www.blogger.com/profile/04635496839285401611noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-53223523829304208372012-10-18T09:05:05.017-04:002012-10-18T09:05:05.017-04:00I can't see why induction would be any better....I can't see why induction would be any better. (Subscribing to comments in case someone else sees it differently.)<br /><br />I don't think this would be illuminating for basic algebra students. They take these things for granted. Comparing the steps of your proof to the steps one might take trying to prove something that isn't true (let's try to prove that sqrt(a^2+b^2) = a+b) might be illuminating.<br />Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.com