tag:blogger.com,1999:blog-7245208048685880741.post3623568763779491657..comments2023-05-19T10:32:11.137-04:00Comments on Rational Expressions: Exponents Without Repeated MultiplicationMichael Pershanhttp://www.blogger.com/profile/17046644130957574890noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-7245208048685880741.post-53660777987555961212015-05-30T21:08:10.429-04:002015-05-30T21:08:10.429-04:00This and Snap Hotel have definitely inspired me to...This and Snap Hotel have definitely inspired me to invest in unifix cubes for the middle school classroom! Thanks.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-79128666116264552852014-05-07T20:37:21.294-04:002014-05-07T20:37:21.294-04:00Don't give up on the geometric model too fast!...Don't give up on the geometric model too fast! It is possible to continue without going into the 4th dimension, and in fact it solves one problem that I had with the <a href="http://rationalexpressions.blogspot.com/2014/05/they-need-to-talk-about-powers-before.html" rel="nofollow">fractal model</a>. You've already shown that:<br />3^0=single block=1<br />3^1=line of 3 blocks=3<br />3^2= square=9<br />3^3=cube=27<br />Now look at the cube as a new "single" block:<br />3^4=row of cubes=81<br />3^5=square of cubes=243<br />3^6=cube of cubes, which becomes the new unit for the next steps:<br />3^7=row of cube of cubes, etc.<br /><br />The problem this solves (for me) is that it's very easy to think of negative exponents as cutting the block into pieces, so we have an intuitive way of understanding negative exponents. With the fractal model, I didn't see an intuitive way to continue the 5-pattern into fifths and fifths of fifths. But with the geometric model, it's easy to imagine continuing the pattern: slicing a single block into square thirds, and then slicing that into row-like thirds of thirds, and then cutting that into block-like thirds of thirds of thirds, and so on forever.<br /><br />I'm still waiting for a good way to help my students make intuitive sense of fractional exponents. Denise in ILhttps://www.blogger.com/profile/11928843626113889088noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-11241533938474678202013-11-22T18:57:08.094-05:002013-11-22T18:57:08.094-05:00Have you taken a look at Vi Hart's latest vide...Have you taken a look at Vi Hart's latest video on logarithms? It does a good job (IMHO) of laying out the relationship of fractional exponentiation. (And I think it would be fairly straightforward to extend the ideas to negative exponents, simply by adapting the existing knowledge about negative numbers on the number line into the log-space).<br /><br />link: http://www.youtube.com/watch?v=N-7tcTIrersHaohttps://www.blogger.com/profile/02348974241652264510noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-26158261784364293552013-11-12T19:55:50.082-05:002013-11-12T19:55:50.082-05:00Thanks for this, Max. I should say, this post is p...Thanks for this, Max. I should say, this post is pretty explicitly intended as a response to Danielson's. It's his suggestion that we look for something beyond repeated multiplication that I'm really running with here.<br /><br />I think that doubling isn't a great model for exponents. First, because we have a fairly limited vocabulary in English for things of that sort. We have doubling, tripling, quadrupling, and then things get kind of hairy. I think that we're going to want to be able to talk about powers of all sorts of numbers, and not want to limit ourselves to just a few special cases.<br /><br />I'd say that Griffy isn't using the recursive model because there's no connection here to anything having to do with powers. Kids can follow doubling patterns with relative ease. The trick is to use that to build a network of knowledge about powers. <br /><br />Or, maybe I'm still sort of confused. Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-20622065565288894142013-11-12T18:03:06.033-05:002013-11-12T18:03:06.033-05:00Er, I meant Recursive Model not Recursive Definiti...Er, I meant Recursive Model not Recursive Definition.Maxhttps://www.blogger.com/profile/16935784635103701185noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-6400688262012118912013-11-12T18:02:36.754-05:002013-11-12T18:02:36.754-05:00I think I've pointed here before, but now I ge...I think I've pointed here before, but now I get to do it with a new question: http://christopherdanielson.wordpress.com/2013/04/11/rational-exponents-third-grade-style/ -- in this story, is Griffy using the recursive definition?<br /><br />I wonder what he already knew about lines, squares, and cubes.<br /><br />I wonder if doubling is special in the minds of children (i.e. easier to access) as compared to tripling, quadrupling, etc.<br /><br />I wish I had some 3rd and 4th graders to hang out with this year...Maxhttps://www.blogger.com/profile/16935784635103701185noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-65940823277373887522013-11-12T15:36:05.704-05:002013-11-12T15:36:05.704-05:00Can it be a bloody one?Can it be a bloody one?Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-22561443646277093012013-11-12T14:44:34.546-05:002013-11-12T14:44:34.546-05:00Michael
Lovely stuff here. I shared it out to my M...Michael<br />Lovely stuff here. I shared it out to my Middle School team and my Alg I and Alg II teachers. Let's start a revolution here!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-37950611990986434832013-11-12T12:53:57.567-05:002013-11-12T12:53:57.567-05:00Bah, you're right. I'll fix this and updat...Bah, you're right. I'll fix this and update the post.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-29579866245345085832013-11-12T12:17:16.701-05:002013-11-12T12:17:16.701-05:00Great ideas about how to teach exponents, and how ...Great ideas about how to teach exponents, and how to start getting them in earlier.<br /><br />I think you have a typo in one of your images: it goes 1, square, cube instead of 1, line, square, cube. And probably the 1 should be a point?Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.com