tag:blogger.com,1999:blog-8231784566931768362.post8737587596527511855..comments2017-06-19T05:16:01.513-04:00Comments on MathNotations: Multiple Representations (Rule of 4) in Algebra 2 or PrecalculusDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-8231784566931768362.post-60018959626071106972011-08-11T11:56:32.821-04:002011-08-11T11:56:32.821-04:00I like the idea of keeping the "Rule of Four&...I like the idea of keeping the "Rule of Four" in mind as I teach. <br /><br />For example, in this problem, asking the students to explain (using words) the relationships between the expressions in numerator and denominator might be interesting. <br /><br />Understanding that the quotient could be of two positive numbers OR two negative numbers might be considered before actually solving anything, for example.<br /><br />This would also support the students' understanding of basic number properties. How would they know if multiplying both sides by the denominator affected the sense of the inequality, unless they knew that the denominator HAD to be positive.<br /><br />I guess I might withhold the information about the value of B at the start, so students could grapple with those ideas a little bit in the process.<br /><br />Thanks for the stimulating example.mike mchttp://www.blogger.com/profile/08819856702973076565noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-65397838424172160062008-05-08T09:27:00.000-04:002008-05-08T09:27:00.000-04:00I am not shure about which grade weare talking her...I am not shure about which grade we<BR/>are talking here but if students are <BR/>unable to investigate a problem with <BR/>the tools they've learned in class <BR/>(like simplifing an equation) how can <BR/>they learn something about math?<BR/>At best they will be good math <BR/>technicians right?Floriannoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-22877539595103131142008-05-08T08:05:00.000-04:002008-05-08T08:05:00.000-04:00A few thoughts, Florian...Whenever I make statemen...A few thoughts, Florian...<BR/><BR/>Whenever I make statements like "Many (or some) students would...", I recognize that it's foolish to generalize globally since students' responses are affected by their educational training in math. I base my statements on anecdotal evidence from listening to students in my own classes and classes I have observed from 37 years in education (35 in public education). Although, one would imagine that students would re-write the inequality fairly quickly, that doesn't always happen, even with above-average students.<BR/><BR/>The form of the division also has an effect. When presenting this problem, I would not recommend writing it in horizontal format as I did in the post. Vertical fraction format is better, but I didn't want to play with LaTeX.<BR/><BR/>Further, my primary goal in publishing this blog is to use somewhat challenging problems to demonstrate effective models of instruction, particularly the elusive art of questioning. The problem in this post was not intended to be challenging or even the focus, rather my intent was to suggest how the instructor can use multiple representations to promote conceptual understanding and problem-solving proficiency.<BR/><BR/>Making the connection to a linear inequality of the form "y ≤ ____" is not as obvious as one might suppose. Somehow, using variable names such as A and B lead "some students" to treat this problem purely as an arithmetic exercise!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-9884882170202636332008-05-08T02:57:00.000-04:002008-05-08T02:57:00.000-04:00I dont know but I think students would knowthat in...I dont know but I think students would know<BR/>that in order to make a fraction large the<BR/>divider needs to be small. Therefore its only<BR/>natual to plug in the least possible value<BR/>for B, 7. The least possible value of A<BR/>follows naturaly.<BR/><BR/>I would be surprised if students didn't know<BR/>that if they can't see a solution right away<BR/>they might want to try to simplyfy the equation<BR/>... which immeadiately leads to the solution.<BR/><BR/>To ask for the hightest possible value of<BR/>A I would change the inequality sign to the<BR/>other direction (=< instead =>) as well<BR/>as for B so there is an upper bound for A.Floriannoreply@blogger.com