tag:blogger.com,1999:blog-5303307482158922565.post4313436691730334936..comments2021-04-05T17:28:31.416-07:00Comments on Math Mama Writes...: My Math Alphabet: F is for FactoringSue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-5303307482158922565.post-19254542599661762011-01-06T07:13:09.190-08:002011-01-06T07:13:09.190-08:00The coolest thing to me about factoring is that it...The coolest thing to me about factoring is that it turns an addition problem into a multiplication problem. Symmetrically, distributing turns a multiplication problem into an addition problem. This is almost mathematical alchemy!<br /><br />The value of turning addition into multiplication is that it allows us to use the "zero product property" to find solutions to polynomials which we can factor (if they are set equal to zero). It is also the basis for the "completing the square" process, which allows us to rewrite a general quadratic in vertex form and solve for the variable using square roots.<br /><br />So, without the ability to factor, the quadratic formula would not have been derived. But, I digress...<br /><br />Somehow, I just find it really neat that it is possible to rewrite an addition problem as a multiplication problem (or vice versa). This is a critical skill to be comfortable with as folks move along in algebra, as sometimes it is the only way to either collect like terms in an expression or "make this expression look like that one" (think equation forms, many trig identities, etc.).<br /><br />In real life, if students are using the quadratic formula without need for understanding why it works, then factoring is not a needed skill. For those students who like to understand how/why a process works however, I think factoring is a critical skill when studying surface area or projectile motion problems.<br /><br />http://mathmaine.wordpress.comAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-24770664975524553582010-12-25T05:53:31.304-08:002010-12-25T05:53:31.304-08:00And then the fun question is: Why do these methods...And then the fun question is: Why do these methods work?Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-30358229755782686752010-12-24T18:06:43.690-08:002010-12-24T18:06:43.690-08:00Check 2, 3, 5... good.
Now, use that 7*11*13 = 10...Check 2, 3, 5... good.<br /><br />Now, use that 7*11*13 = 1001 to check the next three. Divide the number into groups of three digits, and alternately add and subtract (sounds awful, example coming), leaving a 3-digit difference. Check that difference for 7, 11, 13. If one works, it works in the original number.<br /><br />Ex:<br /><br />83,123,432,798<br />83 - 123 + 432 - 798 = 406. 7 goes into 406 (420 - 14 = 406), so it goes into the original number. If 7 didn't work, we'd try 11, then 13.<br /><br />23,496<br />496 - 23 = 473. 7 doesn't go in (473 + 7 = 480. phhhpt). 11 does go in, so 11 goes into the original.<br /><br />Other tricks? For 11, add/subtract every other digit. If the difference is divisible by 11, so was the original.<br /><br />ex<br />2348<br />2 - 3 + 4 - 8 = -5. No good<br /><br />4381805<br />4 - 3 + 8 - 1 + 8 - 0 + 5 = 11. Yup, 11 goes into 4381805.<br /><br />for 7, separate the last digit, double it, and subtract from the rest.<br /><br />ex:<br /><br />347291<br />34729 - 2 = 34727<br />3472 - 14 = 3458<br />345 - 16 = 329<br />32 - 18 = 14<br />Yeah!<br />or 1 - 8 = -7<br />Yeah!<br /><br />littler number<br />1964<br />196 - 8 = 188<br />18 - 16 = 2<br />boo.<br /><br />Happy Holiday.<br /><br />JonathanAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-26316434348684694742010-12-23T12:31:18.913-08:002010-12-23T12:31:18.913-08:00I have got to watch those Vi Hart videos, haven...I have got to watch those Vi Hart videos, haven't I? I'll reply more thoughtfully here later. (I'm in a bookstore, trying to catch up just enough to not worry about my online life while I enjoy xmas.) ;^)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-5382199282890481352010-12-21T09:37:32.058-08:002010-12-21T09:37:32.058-08:00I found this link a few days ago: factoring and dr...I found this link a few days ago: factoring and drawing stars. Besides the topic, it's really beautiful. the other video I've seen by her is also really great...<br />http://www.youtube.com/watch?v=CfJzrmS9UfYmathmomhttps://www.blogger.com/profile/07887205622583099966noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-42295838379095278802010-12-21T06:30:58.776-08:002010-12-21T06:30:58.776-08:00nicely put, MBP.
here's john d. cook's r...nicely put, MBP. <br />here's john d. cook's recent<br /><a href="http://www.johndcook.com/blog/2010/12/07/cascading-needs/" rel="nofollow">maybe you only need it<br />because you have it</a>.<br /><br />but let it be said: these are "weeder" classes<br />and this has been a *very* effective technique for<br />treating people like weeds. why ask "why?"?<br /><br />OT... only formally anonymous<br />(because "word verification" is<br />already enough screening <br />without further logs-in)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-22675941308729825272010-12-20T20:25:34.713-08:002010-12-20T20:25:34.713-08:00The best that I can do to defend our emphasis on f...The best that I can do to defend our emphasis on factoring polynomials is that it enables us to talk about a deep analogy between polynomials and integers. The zero-product law is an expression of the fact that the ring of polynomials is an integral domain, and factoring allows us to talk about that.<br /><br />At the same time, the more I think about it the less justification I have for including factoring polynomials in our curriculum. First, a survey of the defenses of factoring that I've seen on ze intertubes:<br />1) Factoring leads to deeper mathematical topics<br />2) Factoring can be used to solve fun puzzles.<br />3) Factoring is necessary for more math.<br />(And of course, those who have a problem with factoring because it isn't real-world-useful are partially missing the point.)<br /><br />I'm dismissive of 2). All that pain isn't worth a few parlor tricks. And I have a hard time agreeing with 3), since I don't remember factoring anything in between graduating high school and teaching.<br /><br />My complaint with 1) is that there are so many deeper and richer mathematical ideas that we could be teaching our students that if the richness of factoring is what justifies its place in our curriculum, I'd gladly give it up. Give me an extra two days to work on the Monty Hall Problem or to convince students that there are higher infinities, or to prove the irrationality of 2 or Zeno's Paradoxes. <br /><br />One of our goals, as math teachers, is to convince some of students that math is worth learning more about. We're salesmen for all sorts of fields where math is key, and I doubt that factoring is the deepest, most exciting mathematics that we could bring to the high school level.<br /><br />Sadly, the best justification for the presence of factoring is the instrumental one; you need factoring for so much of the high school curriculum that our students need a good factoring background. But this just passes the buck to those other topics (simplifying rational expressions, solving quadratic equations, domain issues with rational functions, some limit problems, etc.).Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-70980332061799390512010-12-19T14:32:04.091-08:002010-12-19T14:32:04.091-08:00@KFouss, Thank you! That's the one I was looki...@KFouss, Thank you! That's the one I was looking for!Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-27910280427271426862010-12-19T14:23:46.053-08:002010-12-19T14:23:46.053-08:00@Hao, I don't think anyone intends WCYDWT to a...@Hao, I don't think anyone intends WCYDWT to apply to all of math. My impression is that it's a way to make application problems more relevant.<br /><br />@Owen (anonymous), I love your post on factoring. I vaguely remember reading it before, but enjoyed it just as much when I read it this morning. (The joys of a bad memory!)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-31861484435639639122010-12-19T14:06:07.525-08:002010-12-19T14:06:07.525-08:00This is a post that Sam Shah wrote about factoring...This is a post that Sam Shah wrote about factoring... includes a nice chart showing how many polynomials of the form x^2 + bx + c will/won't factor.<br /><br />http://samjshah.com/2009/08/13/factoring-schmactoring/KFousshttps://www.blogger.com/profile/04493982153040173831noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-73160490233918458842010-12-19T06:42:54.032-08:002010-12-19T06:42:54.032-08:00old factoring thread at jd's.old <a href="http://jd2718.wordpress.com/2007/08/19/teaching-factoring-should-we/" rel="nofollow">factoring thread</a> at jd's.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-37230305008457473382010-12-19T04:17:50.835-08:002010-12-19T04:17:50.835-08:00The Polynomial Remainder Theorem is a neat result ...The Polynomial Remainder Theorem is a neat result that can be used to evaluate polynomials quickly.<br /><br />As for WCYDWT, I worry that taking that philosophy to the extreme means a student tries to find applicability to everything and neglects to appreciate learning for its own sake. If one wanted a purely economical view of time, it would be more efficient to teach students how to earn money and simply inform them that there will always be a supply of highly-educated, but poor graduate students to do your math for you.Haohttps://www.blogger.com/profile/02348974241652264510noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-62673777052187640552010-12-19T04:08:07.747-08:002010-12-19T04:08:07.747-08:00i saw the dice problem in grad school
and solved i...i saw the dice problem in grad school<br />and solved it by trial-and-error (or,<br />if for some reason we want to sound<br />like schoolteachers, by "guess and<br />check"). the book i saw it in used<br />*generating functions* to display<br />a more systematic approach... a<br />topic i knew nothing about (except<br />that it gave me the heebie-jeebies;<br />i'd seen it at the tail end of a<br />probability course in a big hurry<br />and learned nothing except <br />"generating functions are scary").<br />anyhow, it's a great problem<br />*because* you can solve it<br />without high-tech (not even<br />polynomials!)... there are,<br />after all, only a small number<br />of cases to examine...*and* because<br />it also lends itself to more <br />advanced treatments.<br /><br />on factoring polynomials:<br />i'll just take this opportunity<br />to ask other algebra teachers<br />to (obsessively) mention *what<br />number set* we're allowing<br />as coefficients. it would be<br />very helpful in my guess to<br />introduce notations... Z[x]<br />for "the set of polynomials<br />(in the variable 'x')<br />having integer coefficients",<br />for example. Z[x] can then<br />be considered as a subset<br />of Q[x] or C[x] (polys with<br />Rational or Complex coefficients<br />respectively). one then has<br />the (so-called) fundamental<br />theorem of algebra: all<br />polynomials factor in C[x].<br />also, rather a surprising result<br />("gauss's lemma"): if a Z[x]<br />poly factors in Q[x], it factors<br />in Z[x].<br /><br />i ranted about x^4 + x^2 +1<br />almost exactly two years ago<br /><a href="http://vlorbik.wordpress.com/2008/12/18/why-live/" rel="nofollow">here</a> (and had a great<br />deal of fun doing it... the only "real<br />world" reason i know or want for<br />understanding more about polys<br />["but how can we use music in<br />*real life*, mister mozart?"]).<br /><br />love the "alphabet", by the way.Anonymousnoreply@blogger.com