tag:blogger.com,1999:blog-30226356.post6826291879549215784..comments2022-03-26T22:04:19.090-07:00Comments on The Exponential Curve: Showdown!Dan Wekselgreenehttp://www.blogger.com/profile/08696028020767073620noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-30226356.post-20645326677702844392010-11-22T15:42:17.275-08:002010-11-22T15:42:17.275-08:00The captain is doing the work as well. Their addi...The captain is doing the work as well. Their additional job is to pay attention to when the others seem to be finished or stuck, and to handle the cards. But it is not assumed that they are better at the skills than the other students in the group.Dan Wekselgreenehttps://www.blogger.com/profile/08696028020767073620noreply@blogger.comtag:blogger.com,1999:blog-30226356.post-77560456484061284672010-11-22T14:26:53.173-08:002010-11-22T14:26:53.173-08:00What is the captain doing while everyone else is w...What is the captain doing while everyone else is writing?Julia Tsyganhttps://www.blogger.com/profile/04354702485097004759noreply@blogger.comtag:blogger.com,1999:blog-30226356.post-5188112922293560192009-11-19T04:50:52.984-08:002009-11-19T04:50:52.984-08:00When I teach this I do something similar to the &q...When I teach this I do something similar to the "magic square" but with less mysticism involved. <br />Consider: 2x^2 + 7x + 5 We take the 2 and multiply it to the 5, and then pretend the number infront of the x^2 is a 1 when finding factors of (the new) c that add (or subtract) to be b. However, when we create 2 binomials the 2x^2 gets broken into 2x and 2x. This is clearly wrong as 2x * 2x = 4x^2, but since we broke the rules in our first step we need to undo it. We can do this by dividing one of the binomials by what we multiplied in the start. So...<br />2x^2 + 7x + 5 becomes<br />2x^2 + 7x + 10 and factors of 10 that add to be 7 are 2 and 5 so<br />(2x + 2)(2x + 5) But then we divide the first binomial by 2 and get<br />(x + 1)(2x + 5)<br />Sometimes you have like 6x^2 in the start and will have to divide one binomial by 2 and the other by 3 to get a total of 6.VB teachernoreply@blogger.comtag:blogger.com,1999:blog-30226356.post-76079327866422283912007-01-14T16:26:00.000-08:002007-01-14T16:26:00.000-08:00I think we use a version of the magicky square thi...I think we use a version of the magicky square thing. Example:<br /><br />34x^2 + 13xy - 15y^2<br /><br />We need to break 13xy into the sum of two terms. Their sum is 13 (I just said that), their product will be (-15)(34) = -510<br /><br />Search:<br />(14)(-1) = -14<br />(17)(-4) = -68<br />(23)(-10) = -230<br />(30)(-17) = -510<br /><br />Break the Middle term:<br />34x^2 -17xy + 30xy - 15y^2<br /><br />Factor by grouping<br /><br />17x(2x - y) + 15y(2x - y) <br />(2x-y)(17x+15y)<br /><br />Why does it work?<br /><br />(ax + by)(cx + dy) =<br /><br />acx^2 + (ad + bc)xy + bdy^2<br /><br />So ad + bc will break the middle and allow us to factor by grouping. But ad + bc is the middle coefficient and (ad)(bc) = (ac)(bd) = the product of the first and last coefficient.<br /><br />Since all the math teachers in my school are on the same page, we reinforce this nicely<br /><br />JonathanAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-30226356.post-492946886696757842007-01-07T11:57:00.000-08:002007-01-07T11:57:00.000-08:00n Algebra 1, we don't really deal with that situat...n Algebra 1, we don't really deal with that situation much (the book we have has some crazy scheme to list lots of factor pairs and guess and check stuff); we hold off for Algebra 2. <br /><br />I taught Algebra 2 for the first time last year (I plan the honors class, and another teacher plans the regular class - we use a shared prep model at our school). He came up with a "magic square" technique, where you find the factors of <i>ac</i> that sum to <i>b</i>, and then use those as the "cornerstones" of the magic square, and then do some other mystical stuff. <br /><br />So I had to use that in my regular classes, and I agreed to try it in honors also. It was a good experiment, but it did not work out so well. Students didn't understand why it worked and couldn't use it consistently. And then, when I taught "completing the square" in honors, they kept mixing up the terms and I heard nightmarish phrases like "oh, don't we have to use the completing the magic square thingy?"<br /><br />This year, we decided to do the more traditional factoring by grouping, which I believe is exactly what you are referring to. It seems to be going pretty well so far, but I think my students still need more practice at it. <br /><br />I also wanted to get them started on factoring by grouping now, so later in the semester they can handle factoring expressions like x^3+5x^2-9x-45 into (x^2-9)(x+5) into (x-3)(x+3)(x+5).Dan Wekselgreenehttps://www.blogger.com/profile/08696028020767073620noreply@blogger.comtag:blogger.com,1999:blog-30226356.post-36318294292539777902007-01-07T09:22:00.000-08:002007-01-07T09:22:00.000-08:00How do you teach them how to factor trinomials in ...How do you teach them how to factor trinomials in the form ax^2 + bx + c when the gcf = 1 ?<br /><br />In the next week or so I will post about the factoring unit we've written (and used for 5 years). <br /><br />In short, we factor things like<br /><br />10ab - 15b + 6ac - 9<br /><br />a whole lot, first, then we move to "breaking the middle"<br /><br />What do you use? What have you tried and rejected?<br /><br />JonathanAnonymousnoreply@blogger.com