tag:blogger.com,1999:blog-7245208048685880741.post1520183717690873275..comments2023-05-19T10:32:11.137-04:00Comments on Rational Expressions: Slope and SunsetMichael Pershanhttp://www.blogger.com/profile/17046644130957574890noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-7245208048685880741.post-72122294519260925632013-04-03T14:33:43.050-04:002013-04-03T14:33:43.050-04:00I have not read all prior comments in great detail...I have not read all prior comments in great detail, so forgive me if this is repetitious.<br /><br />(1) Love the sunset/sunrise/day length tasks. Lots of interesting relationships to explore and (as at least one commenter pointed out) lots of good challenging questions to be asked by playing with the geography. One other thing to play with is same latitude, opposite ends of the time zone. Say, for example, Detroit and Boston.<br /><br />Also good to ask is "Why is there not exactly 12 hours of daylight at the equinox?"<br /><br />And I am curious how/whether you look at these tasks differently after being immersed in thinking about functions for two weeks.<br /><br />Oh, also, you suggest that it's not really proper to consider slopes of curved objects. But that's all we've got! And calculus demands that we do so. As long as everyone throws around lots of "approximately" and "about", everything's A-OK in my book.<br /><br />(2) In my experience, solving by substitution is sensible <a href="http://christopherdanielson.wordpress.com/2012/11/02/systems-of-linear-equations/" rel="nofollow">in "racing" situations</a>, and not so sensible in mystery-value situations. And, as <strong>Ms Billings</strong> suggests, understanding the equal sign to mean <em>equivalence</em> or <em>is the same as</em> is a key prerequisite. Christopherhttps://www.blogger.com/profile/07581128707831107828noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-34217443816926194492013-03-22T16:08:07.396-04:002013-03-22T16:08:07.396-04:00My kids were struggling with substitution, too. T...My kids were struggling with substitution, too. Then I started asking them what y=whatever really meant. They weren't really thinking about it as an equivalence statement that gives them interchangeable parts. I think they understood it better after that conversation.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-48322123506429516342013-03-06T11:02:05.576-05:002013-03-06T11:02:05.576-05:00Amazing links. I know these folks, but I didn'...Amazing links. I know these folks, but I didn't know these posts. Thanks again.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-80806296054859164032013-03-06T01:11:06.814-05:002013-03-06T01:11:06.814-05:00I personally hate substitution and use elimination...I personally hate substitution and use elimination using addition as much as possible. Mt students and I have compared substitution and elimination and they prefer elimination in most cases. We only do substitution when it is super easy and would be too much with to change the equations around. If both equations are in slope intercept form we substitute for example. Substitution just isn't efficient. Anonymoushttps://www.blogger.com/profile/18342366268871521998noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-19193875731916379282013-03-05T20:05:41.456-05:002013-03-05T20:05:41.456-05:00Loved this -- especially the idea that getting pra...Loved this -- especially the idea that getting practice with skills can be the same process as making sense of how geography affects our life. <br /><br />If you're not already reading Brian Frank, you might appreciate his ideas about <a href="http://teachbrianteach.wordpress.com/2012/03/18/initial-ideas-about-sun-moon-and-shadows/" rel="nofollow">teaching seasons</a> (and <a href="http://teachbrianteach.wordpress.com/2012/03/26/my-lack-of-knowledge-about-student-ideas-about-astronomy-and-the-earth/" rel="nofollow">what students may think</a> about them... these posts are at his new site). He's also got an excellent post about why <a href="http://teachbrianteach.blogspot.ca/2011/04/i-said-i-didnt-want-to-talk-about.html" rel="nofollow">some ideas are more "generative"</a> than others and why he avoids the common focus on "misconceptions" (this post is at his old site). <br /><br />Same goes for Michael Doyle -- who writes some of the smartest stuff about <a href="http://doyle-scienceteach.blogspot.ca/2011/10/universe-is-made-of-stories-not-of.html" rel="nofollow">nature</a> and <a href="http://doyle-scienceteach.blogspot.ca/2012/02/teachers-are-not-yet-professionals-ii.html" rel="nofollow">teaching</a> that I have the pleasure to read.<br /><br />My students have similar problems with slope. The time when they are most successful at teasing a meaning out of the story of a graph is when we graph something that has a sudden change in slope. In my world that's a diode's current vs. voltage curve. I don't talk about slope, but they start to struggle to describe it, because it's the most salient feature of the graph and because "something different happens." The LED glows way, way brighter all of a sudden. The current "shoots up." They struggle to describe the different parts of a graph -- they're both "straight", but they're straight in a different way. They find it hard to talk about slope when it's constant; it's too backgrounded. It's like fish having no word for water.<br /><br />Finally, Riley dreamt up <a href="http://larkolicio.us/blog/?p=787" rel="nofollow">an exercise that targets space/time graphs specifically</a> but slope more generally. Good luck!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-14843515367108815432013-03-05T10:59:02.822-05:002013-03-05T10:59:02.822-05:00They don't understand what's meant by subs...They don't understand what's meant by substitution.<br />They don't know how to solve for a variable with any reliability.<br />They don't understand why this technique would help them solve a system.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-85282395096275122552013-03-05T10:50:48.295-05:002013-03-05T10:50:48.295-05:00I'm curious about what your pupils find diffic...I'm curious about what your pupils find difficult about the substitution method. Could you elaborate, please?JJhttps://www.blogger.com/profile/16829561981417320165noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-42891488605069743842013-03-05T07:45:24.150-05:002013-03-05T07:45:24.150-05:00Wow. These are really solid questions and ideas. I...Wow. These are really solid questions and ideas. I particularly like the "Draw the graph for Miami" idea. I also like the whole-year graph idea.<br /><br />Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-56972395197463954762013-03-05T07:44:29.947-05:002013-03-05T07:44:29.947-05:00Thanks for flagging the fact that minutes per day ...Thanks for flagging the fact that minutes per day is difficult to grasp. I can nail that in the beginning of class today by giving them a NYC table (instead of a graph) and asking what they notice.<br /><br />And thanks for the substitution lessons. In general, some sort of concrete visual aid might be helpful, so I might steal your lessons. Thanks!Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-36321531434236202392013-03-05T07:41:39.483-05:002013-03-05T07:41:39.483-05:00Your lessons are solid, and we've done stuff l...Your lessons are solid, and we've done stuff like that -- Skype plans, plumbers and babysitters who charge for reservations and per hour, lemonade stands that charge for the cup, music subscriptions, visual patterns where they model the pattern, speed of two people biking, cup stacking ...<br /><br />And, would you believe it, they still don't really get the connection between the slope of a graph and a rate? They don't get that the steepness of a graph tells you something clear and quantitative about the way something is happening. Plus, they don't read graphs particularly well.<br /><br />I definitely don't think that your lesson is too simple. But I also think that we're at a point in this class where we've burned through a lot of the simple things, and these kids get bored or distracted by just anything. That's what's pushing me towards a more involved lesson on slope.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-82690153399365487752013-03-04T22:21:46.759-05:002013-03-04T22:21:46.759-05:00You're doing too much. Did you see my modeling...You're doing too much. Did you see my modeling post? I realize you probably think it's simple, but it's designed for intermediate algebra (A2). You could make it simpler for Algebra I. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-7185541382398654872013-03-04T21:51:23.783-05:002013-03-04T21:51:23.783-05:00I like the fact that the graphs are not linear - g...I like the fact that the graphs are not linear - gives you something to talk about. Could you place the ones you have done on a map and pick another location and create what you think the graph would look like for that location? Minutes per Day is kind of a funky rate to latch onto, though.<br /><br />I found that using pictures in place of variables, and eventually moving to variables makes both substitution and linear combinations much more intuitive.<br /><br />Here is an example or two for substitution:<br /><br />https://docs.google.com/file/d/0B0LlvF7Dr9chdy0yT3VGMkJ5U1k/edit?usp=sharing<br /><br />https://docs.google.com/file/d/0B0LlvF7Dr9chYjNHT2JKa2VCems/edit?usp=sharing<br /><br />And a bunch of stuff for systems in general:<br /><br />https://docs.google.com/folder/d/0B0LlvF7Dr9chWFNwSU11RmJ0WGs/edit?usp=sharing<br /><br />l hodgenoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-78454025577658761242013-03-04T21:41:29.237-05:002013-03-04T21:41:29.237-05:00Cool, Michael!
You should definitely give them so...Cool, Michael!<br /><br />You should definitely give them some graphs where the location isn't given and ask them to figure out what the location is. Some in the southern hemisphere!<br /><br />And/or give them graphs from NYC, Jerusalem, and Alaska, but don't tell them which, or from which month. Have them figure out the where and the when.<br /><br />Show them whole-year graphs of those three locations and ask them to sketch whole-year graphs for other locations including: intermediate latitudes, on arctic circle, the North Pole, on the equator, southern hemisphere locations.<br /><br />FWIW. Let us know how it goes!Justin Lanierhttps://www.blogger.com/profile/11386367931599418555noreply@blogger.com