tag:blogger.com,1999:blog-5964889903484807623.post8769456546831584342..comments2021-10-13T13:00:30.262-07:00Comments on Questions?: Equations Three WaysDavid Coxhttp://www.blogger.com/profile/06277427735527075341noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-5964889903484807623.post-83183242499113288392010-10-01T11:30:56.445-07:002010-10-01T11:30:56.445-07:00Alison
We played around with graphing the expressi...<b>Alison</b><br />We played around with graphing the expressions set equal to y. We plugged them into GeoGebra and just looked around for things they thought were interesting. They were like, "hey, the x value for the point of intersection is our <i>answer</i> and the y value is the number we get when we know our guess was right."<br /><br /><b>Grace</b><br />I couldn't agree more. Sorry I wasn't more clear in the post. <br /><br />The modeling comes first as a way of just playing with the idea of equivalent expressions. They solve without really "solving." <br /><br />The guess and check gets them to simplify expressions without worrying about solving and has checking embedded. (I have always felt like it was a fight to get kids to check their solutions)<br /><br />As a class, we kind of decided that 0 was a good first guess because it's easy to use (also sets the table for finding y intercepts later) and 1 is a pretty good second guess because it gives us the rate of change between the answers. Once they have locked in on rate of change, they only need a third guess. <br /><br />Normally, when I teach guess and check, I encourage guessing too high and then too low (or vice versa), but the type of information we gain from two guesses one unit apart is too important to pass up. <br /><br />Once we have those two dialed in, we can go to construction and kids can get all kinds of crazy (As <b>Andrew</b> suggested) with their equations and guarantee that the solution is an integer. It also helps formalize the "steps to solving an equation."<br /><br />I didn't quite plan it out as well as it seems, but after some reflection, I can't wait to get to equations with my 7th graders. In fact, we have already started playing with modeling.David Coxhttps://www.blogger.com/profile/06277427735527075341noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-82409998964896915322010-10-01T07:56:50.595-07:002010-10-01T07:56:50.595-07:00While I really like guess-and-check as an introduc...While I really like guess-and-check as an introductory strategy, it makes me wonder how we can help students codify the intuition that they develop in using it; even if students are able to get the right answer fairly efficiently once they start to develop a sense of how it works, I'd want them to be able to explain <i>why</i> their guesses are good guesses rather than just saying "it felt right." <br /><br />I'm not clear from your post, but it feels like this set of activities, in the order in which you describe them, is actually a strong sequence of scaffolding as students progress from easier equations to more challenging ones and can revert back to other methods and graphing to check their work (and to reinforce that there's more than one way to solve a problem).gracehttps://www.blogger.com/profile/09629147659164801681noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-54099597225748768402010-09-30T17:16:40.144-07:002010-09-30T17:16:40.144-07:00The way we teach our kids with calculator accomoda...The way we teach our kids with calculator accomodations is to put one side of the equal sign in Y1 and the other side in Y2, graph the lines, and find the intersection. The biggest weakness (besides not learning how to manipulate equations) is that they have to learn how to manipulate the graphing window.<br /><br />Example: Y1=x+7<br /> Y2=9Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-29762698028256370572010-09-30T08:47:57.480-07:002010-09-30T08:47:57.480-07:00At the very beginning, I teach solving by graphing...At the very beginning, I teach solving by graphing. They make a table for y = 4x + 5 (or whatever) and graph it. Then we look at the graph and I ask "What is y when x is 2?" questions for a while, then switch to "What is x when y is 1?"<br />I motivate other methods of solving by "there's got to be a better way to get the answer to that question..."<br /><br />I like doing it this way because it really stresses variables as representing all different values, rather than having students think that x always represents a single number, which you just have to find.Unknownhttps://www.blogger.com/profile/02915824676653296316noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-86224953115441670632010-09-29T10:53:39.704-07:002010-09-29T10:53:39.704-07:00Nice call, Andrew.
Dave,
Yeah, we could use nega...Nice call, <i>Andrew.</i> <br /><br /><i>Dave,</i><br />Yeah, we could use negatives but it gets a bit confusing when you need to "take away negatives" from one side when there aren't any on the other. You'd need to add "zeroes" in the form of pairs of positives and negatives. That's not quite as intuitive as I'd like.David Coxhttps://www.blogger.com/profile/06277427735527075341noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-39921789131405464342010-09-28T18:27:59.190-07:002010-09-28T18:27:59.190-07:00For modeling, the negatives could be balloons?For modeling, the negatives could be balloons?CalcDavehttps://www.blogger.com/profile/14039458440867020542noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-22993371613446516502010-09-28T17:40:07.144-07:002010-09-28T17:40:07.144-07:00I have had some success with "construction to...I have had some success with "construction to deconstruction" by asking students to make up problems to try to stump their colleagues. This prompt motivated at least some of them to try to cook up very complicated things.Andrewhttps://www.blogger.com/profile/00605171970083004552noreply@blogger.com