tag:blogger.com,1999:blog-8850230383845297552.post5520182926816058088..comments2018-07-27T00:29:14.504-07:00Comments on Fail Early and Often: A different way of looking at AlgebraAndy Pethanhttp://www.blogger.com/profile/05159258049094512496noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-8850230383845297552.post-74919301689999421502014-09-01T08:26:44.228-07:002014-09-01T08:26:44.228-07:00Thank you! I did my best to fix some of the issues...Thank you! I did my best to fix some of the issues you noticed by reorganizing the number classification and creating a template form for transformations (and not working with points at all). That helped a lot. For clarification on purpose, I wrote it for future curriculum planning as a different way to think about the course with no intention of directly using it with students, but I'm not opposed to doing so if there is a good place to use it.Andy Pethanhttps://www.blogger.com/profile/05159258049094512496noreply@blogger.comtag:blogger.com,1999:blog-8850230383845297552.post-32705886118277545032014-08-28T05:36:02.985-07:002014-08-28T05:36:02.985-07:00wow, that is extensive and impressive! i'm cur...wow, that is extensive and impressive! i'm curious though -- do you intend this for your own clarifying and planning purposes, or is this something you would share with students? <br /><br />i have semi-extensive comments on the section on analyzing number types, which is part of my first unit with my students, so i work with this a lot. the first thing i notice is that the way you break it up into categories can be a little confusing for students, because while it is clear that whole numbers is a subset of integers, it's not at all clear that it's also a subset of real numbers. the way it's broken up it almost seems like integers aren't real.<br /><br />i worry that your definition of integers is too broad because couldn't you lose, for example, $2.50? wouldn't that imply that -2.5 is an integer? i think it needs to be clearly stated that integers are positive and negative *whole* numbers. (btw, as an interesting aside, internationally the set of natural numbers includes the number 0; there is no set called whole numbers. if you want what americans call natural numbers, that is Z+ or positive integers. i think it's nice to see how arbitrary things can be!)<br /><br />i think the concept of real numbers is actually a bit tricky so an actual definition of them is needed: any number that can be placed somewhere (even approximately) on the number line. that always begs the question what *can't* be put on the number line, which counterexamples, yay! <br /><br />the definition of a rational number is similarly challenging for students because they need to develop what kinds of numbers can be represented as a ratio, for example 7-->7/1 (not 7/7!!!), -11-->-11/1 (yes, it works for negatives too!), 0.75-->3/4, 0.3333->1/3 (hey, what about 0.4444, etc?), sqrt4-->2/1, -1.9--> -19/10. the question is therefore what kinds of decimal numbers are rational numbers? and therefore what kinds are irrational numbers? always nice to get an example like 0.1011011101111 as well as pi and sqrt7. <br /><br />i hope the above is helpful and not annoying. :)<br /><br />on another note, with regards to transformational geometry, for translations it seems like you're thinking of translating a function, ie translate y = 2x +5 to the right 3 units gives y = 2(x-3)+5 since you say subtract from x, whereas if you just had a point (17,4) and translate to the right 3 units, you'd add to get (20,4). but then when you talk about scale/stretch, it seems as if you're talking about points because if you have (4,3) and want to stretch horizontally by a factor of 2, you'd do as you say and multiply to get (8,3), but if you're working with a function and talking about scale, you don't want to multiply by the scale factor but rather divide. for example, if you want to stretch horizontally by a factor of 2, you'd take your equation y = 7x + 5 and do y = 7(x/2) + 5. so i found that a little confusing in your explanation because it seems like you switched midsection which things (points or functions) you were talking about. that also explains why y = 2x^2 is "thinner" -- it was stretched vertically [(y/2)=x^2--> y = 2x^2] and why y = 0.5x^2 is "fatter" -- it's been stretched horizontally by a scale factor of sqrt2 (woah, tough stuff!!). also with rotations, i think "take the negative of one of them" is pretty unspecific -- which one? how will the students know?<br /><br />all of my comments depend on what your plans for this are, of course. i'm super impressed with what you've done and the comments are only meant to help you develop it further! :) Anonymousnoreply@blogger.com