mm-2189 == === Subject: Re: Peer tutoring, basic mathematics I agree! You really have to let them try. There is a Chinese proverb: Tell me and I will forget. Show me and I may remember. Involve me and I will understand. > : Starting next Monday, I'll be peer tutoring a freshmen (in high school) > I think I'll throw in my two bits: > I am involved witht he Math Clinic at my high school. Very commonly, > I see students watching their tutor work the problems - and go away > feeling very good about the whole thing - but failing the test. > Try very hard to spend a lot of time watching your student work problems. > It may not feel like tutoring, but you should gain insight into the > difficulties. Sometimes the difficulty is simply that the student has > trouble getting down to work on their own. Having you watch may be > enough. But life is seldom that easy. When you DO help your student > with a problem, or show them how to do it, try to follow up with a > very similar problem for the student to do by themselves while you > watch. If they can do it - you know they are set. Often you will > discover that they missed the point you were trying to make. So try > again (and give further follow problems). > Robert > |)|/| || Burnaby South Secondary School > || |orewood@olc.ubc.ca || Beautiful British Columbia > Mathematics & Computer Science || (Canada) -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: quizzles www.cna.aleks.com and try 24 hour free trial of ALEKS. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Diffy Puzzle My son's teacher gave him diffy puzzles (specifically puzzle #3) to solve. Curious, if anyone knows of a comprehensive resource (text book) where I can get more of these puzzles? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Diffy Puzzle > My son's teacher gave him diffy puzzles (specifically puzzle #3) to > solve. Curious, if anyone knows of a comprehensive resource (text > book) where I can get more of these puzzles? Please give us more information about what a diffy puzzle is. What age or grade is your son? Where does he go to school? (I'm guessing somewhere in the USA from the .com address.) Since his teacher gave him the puzzle, he or she may be the best contact to find out where the puzzle came from. Unfortunately, diffy is a rather generic slang term used to abbreviate difficult, differential, and different, so is not much use in identifying the puzzle you mention. The only relevant things I found on the web were Name of activity: DIFFY Grade level: 2 - 8 Source of activity: Smith, Seaton E. , Jr. & Backman, Carl A. (Ed.). (1975). Games and Puzzles for Elementary and Middle School Mathematics: Readings from the Arithmetic Teacher. Reston, VA: National Council of Teachers of Mathematics. and Diffy--This game was created by Herbert Wills. Place a 2-digit number in each of the four corners of the Diffy game sheet as shown in Figure #. Subtract the two numbers from each other and place the difference between them. Continue in this manner until you get to all zeros or complete the entire game sheet. [The web page did not include the referenced figure.] ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: district choice Hi all, I'm in the process of applying to different districts to get ready for the 2005-05 school year. One district that I'm considering is hiring a lot of new teachers, and I took a look at their report card and they are failing when it comes to NCLB. All three high schools in the district are not meeting the minimum standards. How should I react to this? It looks like they cleaned house because of the test results, how likely are they to do that again? John -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: district choice > I'm in the process of applying to different districts to get ready for the > 2005-05 school year. > One district that I'm considering is hiring a lot of new teachers, and I > took a look at their report card and they are failing when it comes to NCLB. > All three high schools in the district are not meeting the minimum > standards. > How should I react to this? It looks like they cleaned house because of the > test results, how likely are they to do that again? Since there is no evidence that replacing all the teachers in a failing school system makes any improvement in test scores, it is quite likely that the school will continue to fail, and so continue to churn the teacher population. The theory behind NCLB is that if a school is failing, making a dramatic change is bound to be an improvement. Unfortunately, there is no experimental evidence to support this theory---things can, and often do, get worse. ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: district choice > Hi all, > I'm in the process of applying to different districts to get ready for the > 2005-05 school year. > One district that I'm considering is hiring a lot of new teachers, and I > took a look at their report card and they are failing when it comes to NCLB. > All three high schools in the district are not meeting the minimum > standards. > How should I react to this? It looks like they cleaned house because of the > test results, how likely are they to do that again? Why do you jump to conclusions? Why do you assume that they 'cleaned house' rather than voting additional support to meet the irrational NCLB goals? Why do you assume that they 'cleaned house' rather than suffering a particularly high number of retirements? Why do you assume that they 'cleaned house' rather than suffering a spike in enrollments? You see, without knowing the facts, you jump to conclusions, and could quite possibly be talking yourself out of a job which may, in the long run, provide you with both security and personal and professional satisfaction. Do your homework. Alan -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Completely worked out solutions, motivation, and good pedagogy Some have repeatedly questioned the providing of completely worked out solutions, claiming that it's bad pedagogy. Here's a reply. In the thread Re: Limit n^(1/n) = ? when n approaches +infinite solution): > This kind of complete solution tends to destroy any motivation > for completing the problem by yourself, as well as being of dubious > pedagogical value. Of dubious pedagogical value? Many disagree. Please read my ideas about student solutions manuals in the first two threads below, the second containing my reply to some arguments against them. And it destroys motivation? Again, many disagree. Please read below my ideas on how to transfer to the classroom what psychologists know about the workplace regarding creating happy, productive workers. Are student solutions manuals an answer to the math education problem? http://mathforum.org/epigone/k12.ed.math/thandfangcreu Why student solutions manuals are an answer to the math education problem http://mathforum.org/epigone/k12.ed.math/twyrfomcrex The affective domain and the Good Boss Principle http://mathforum.org/epigone/k12.ed.math/weldwirwhan In the context of a strict, unyielding guided discovery method, there are many, many students who are trying as hard as they can, yet they still struggle. And then they fail because they are never given the help they need, completely worked out solutions in a sufficient number of well-designed examples. And then they lose their motivation to continue to try to learn. Paul -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Completely worked out solutions, motivation, and good pedagogy I know that student solutions manuals were the only way I survived upper level math courses. Students tend to get stuck on the first step and therefore not bother with completing the problem. I teach lower level Algebra students. My solution has been to make my OWN solution manuel. I work the problems out, showing all of my steps. I try to have the work done before I assign the problems. Then I keep the assignments in a binder and have it available in my classroom for students to use as needed. The book stays at my round table at the front of the room. I found this method to be especially helpful. I have very needy students who want me to hold their hands through each step. Having my own little solutions manual has helped ease some of the neediness from the students who ultimately know what they are doing, but just need the reassurance that they are doing it right. Just my input on this discussion. Jenny Kruger Northeast High School Alegbra 1A/Algebra 1B/Geometry >Some have repeatedly questioned the providing of completely worked out >solutions, claiming that it's bad pedagogy. Here's a reply. >In the thread >Re: Limit n^(1/n) = ? when n approaches +infinite >solution): >> This kind of complete solution tends to destroy any motivation >> for completing the problem by yourself, as well as being of dubious >> pedagogical value. >Of dubious pedagogical value? Many disagree. Please read my ideas about >student solutions manuals in the first two threads below, the second >containing my reply to some arguments against them. And it destroys >motivation? Again, many disagree. Please read below my ideas on how to >transfer to the classroom what psychologists know about the workplace >regarding creating happy, productive workers. >Are student solutions manuals an answer to the math education problem? >http://mathforum.org/epigone/k12.ed.math/thandfangcreu >Why student solutions manuals are an answer to the math education >problem >http://mathforum.org/epigone/k12.ed.math/twyrfomcrex >The affective domain and the Good Boss Principle >http://mathforum.org/epigone/k12.ed.math/weldwirwhan >In the context of a strict, unyielding guided discovery method, there >are many, many students who are trying as hard as they can, yet they >still struggle. And then they fail because they are never given the >help they need, completely worked out solutions in a sufficient number >of well-designed examples. And then they lose their motivation to >continue to try to learn. >Paul -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Completely worked out solutions, motivation, and good pedagogy Having the solution, not just the answer, to a portion of the problems is beneficial to the student. It can prove a helpful reminder to parents trying to help their teens with long-forgotten Algebra! > I know that student solutions manuals were the only way I survived > upper level math courses. Students tend to get stuck on the first > step and therefore not bother with completing the problem. I teach > lower level Algebra students. > My solution has been to make my OWN solution manuel. I work the > problems out, showing all of my steps. I try to have the work done > before I assign the problems. Then I keep the assignments in a binder > and have it available in my classroom for students to use as needed. > The book stays at my round table at the front of the room. I found > this method to be especially helpful. I have very needy students who > want me to hold their hands through each step. Having my own little > solutions manual has helped ease some of the neediness from the > students who ultimately know what they are doing, but just need the > reassurance that they are doing it right. who want you to come and check their work as they are doing each > problem! Just my input on this discussion. > Jenny Kruger > Northeast High School > Alegbra 1A/Algebra 1B/Geometry >Some have repeatedly questioned the providing of completely worked > out >solutions, claiming that it's bad pedagogy. Here's a reply. >In the thread >Re: Limit n^(1/n) = ? when n approaches +infinite >solution): >> This kind of complete solution tends to destroy any motivation >> for completing the problem by yourself, as well as being of dubious >> pedagogical value. >Of dubious pedagogical value? Many disagree. Please read my ideas > about >student solutions manuals in the first two threads below, the second >containing my reply to some arguments against them. And it destroys >motivation? Again, many disagree. Please read below my ideas on how > to >transfer to the classroom what psychologists know about the workplace >regarding creating happy, productive workers. >Are student solutions manuals an answer to the math education > problem? >http://mathforum.org/epigone/k12.ed.math/thandfangcreu >Why student solutions manuals are an answer to the math education >problem >http://mathforum.org/epigone/k12.ed.math/twyrfomcrex >The affective domain and the Good Boss Principle >http://mathforum.org/epigone/k12.ed.math/weldwirwhan >In the context of a strict, unyielding guided discovery method, there >are many, many students who are trying as hard as they can, yet they >still struggle. And then they fail because they are never given the >help they need, completely worked out solutions in a sufficient > number >of well-designed examples. And then they lose their motivation to >continue to try to learn. >Paul -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Completely worked out solutions, motivation, and good pedagogy >Having the solution, not just the answer, to a portion of the problems is >beneficial to the student. It can prove a helpful reminder to parents trying >to help their teens with long-forgotten Algebra! A portion, yes, but such discussion is fruitless in general. Consider ....exactly how many solutions does it take before the student grasps some sort of pattern, and what happens when there is some slight variation in the question without leaving the topic being studied? I've seen students do 100 type physics problems, and fail their exam. How many times have we seen the joke of the structural engineer asked to build a bridge who says he can't because it doesn't look like one he's seen? What do you call understanding of a topic? At what point is that understanding reached? Teachers are there to teach students, not their parents, and they should decide how many examples with variation are sufficient to indicate general method. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Completely worked out solutions, motivation, and good pedagogy I'll reply to some of Paul Tanner's remarks, below, in the midst of those remarks. Paul Tanner: > Some have repeatedly questioned the providing of completely worked out > solutions, claiming that it's bad pedagogy. Here's a reply. > In the thread > Re: Limit n^(1/n) = ? when n approaches +infinite > solution): Paul Tanner, quoting me, without attribution: > This kind of complete solution tends to destroy any motivation > for completing the problem by yourself, as well as being of dubious > pedagogical value. Paul Tanner, mistakenly setting up an argument against some idea that Paul has that someone has advocated NO help at ANY time: > Of dubious pedagogical value? Many disagree. Please read my ideas about > student solutions manuals in the first two threads below... Joseph Sroka: I certainly have never argued against student solution manuals, and I know of no one that has in k.e.m. To the contrary, Paul, I have even mentioned the fact that the back of the book contains *some* -- BUT NOT ALL -- of the answers. See my previous posts, where I have used the term teacher's prerogative regarding what problems will be assigned vis-a-vis what problems have solutions provided to the students. Paul Tanner: > In the context of a strict, unyielding guided discovery method Joseph Sroka: I certainly have never argued in favor of a strict, unyielding guided discovery method, and I know of no one that has in k.e.m. Here is what I posted to k.e.m. on September 28: In general, answer books in back of the text and solutions manuals are commonly available, and provide either answers or complete solutions to half of the homework problems in the text. It is the instructor's judgement call as to how many assigned homework problems will be covered by the solutions manual or answer book. Providing (via anonymouus students' pleas on k.e.m) complete answers to stock, standard textbook problems interferes with that instructor's judgement and prerogative. *************** It is where, when, and how the complete answers are delivered that is really more to the point. Not whether any should be given at all. -- Delete the second o to email me. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Geometry Textbook Suggestions for Gifted Math Program? I wonder what you would suggest as a best textbook for a first high-school-level geometry course for talented students? I'm trying to gather information to make a suggestion to a local mathematics program. I am aware of one program that formerly used Gene Murrow and Serge Lang's textbook Geometry (Springer-Verlag) and in recent years has used Michael Serra's textbook Discovering Geometry (Key Curriculum Press). Another textbook I have never seen, but have heard of being used at gifted magnet schools is Geometry for Enjoyment and Challenge (McDougall Littell), which has VERY MIXED reviews on Amazon.com http://www.amazon.com/exec/obidos/tg/detail/-/0866099654/ What do you think? If you were recommending a textbook for a class of students selected for high math ability taking a first secondary-level geometry course, what would you recommend? What do you like about your preferred textbook? -- Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 Learn in Freedom (TM) http://learninfreedom.org/ remove .de to email -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Geometry Textbook Suggestions for Gifted Math Program? FWIW this is the book that was used in every school I've had a preclinical in. I've taught out of it for an honors class, and have found it to be a great book. The only problem I may have with it is the format of pages, it's all in black and red. To me that is pretty boring, I like colorful textbooks. However, content wise it's a great book. John > I wonder what you would suggest as a best textbook for a first > high-school-level geometry course for talented students? I'm trying to > gather information to make a suggestion to a local mathematics program. > I am aware of one program that formerly used Gene Murrow and Serge > Lang's textbook Geometry (Springer-Verlag) and in recent years has used > Michael Serra's textbook Discovering Geometry (Key Curriculum Press). > Another textbook I have never seen, but have heard of being used at > gifted magnet schools is Geometry for Enjoyment and Challenge (McDougall > Littell), which has VERY MIXED reviews on Amazon.com > http://www.amazon.com/exec/obidos/tg/detail/-/0866099654/ > What do you think? If you were recommending a textbook for a class of > students selected for high math ability taking a first secondary-level > geometry course, what would you recommend? What do you like about your > preferred textbook? > -- > Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 > Learn in Freedom (TM) http://learninfreedom.org/ > remove .de to email -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Geometry Textbook Suggestions for Gifted Math Program? >> I wonder what you would suggest as a best textbook for a first >> high-school-level geometry course for talented students? I'm trying >> to gather information to make a suggestion to a local mathematics >> program. I am aware of one program that formerly used Gene Murrow >> and Serge Lang's textbook Geometry (Springer-Verlag) and in recent >> years has used Michael Serra's textbook Discovering Geometry (Key >> Curriculum Press). Another textbook I have never seen, but have >> heard of being used at gifted magnet schools is Geometry for >> Enjoyment and Challenge (McDougall Littell), which has VERY MIXED >> reviews on Amazon.com >> http://www.amazon.com/exec/obidos/tg/detail/-/0866099654/ > FWIW this is the book that was used in every school I've had a preclinical > in. and Challenge book? > I've taught out of it for an honors class, and have found it to be a > great book. The only problem I may have with it is the format of pages, it's > all in black and red. To me that is pretty boring, I like colorful > textbooks. Well, I like four-color illustrations on slick paper as much as the next guy, but they had better be doing something to advance the understanding of the content in a math textbook. Some of the illustrations in the Serra textbook, although pretty, are smarmy wastes of time and ink. > However, content wise it's a great book. If you are referring to Geometry for Enjoyment and Challenge (aren't you?), does that book include content on transformations and other more modern topics of high school geometry? P.S. As in my previous reply to N. Silver, my apologies for the trash message posted with a spoofing of my name. -- Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 Learn in Freedom (TM) http://learninfreedom.org/ remove .de to email -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Geometry Textbook Suggestions for Gifted Math Program? Yes, the one listed here on the Amazon link. > I wonder what you would suggest as a best textbook for a first > high-school-level geometry course for talented students? I'm trying > to gather information to make a suggestion to a local mathematics > program. I am aware of one program that formerly used Gene Murrow > and Serge Lang's textbook Geometry (Springer-Verlag) and in recent > years has used Michael Serra's textbook Discovering Geometry (Key > Curriculum Press). Another textbook I have never seen, but have > heard of being used at gifted magnet schools is Geometry for > Enjoyment and Challenge (McDougall Littell), which has VERY MIXED > reviews on Amazon.com > http://www.amazon.com/exec/obidos/tg/detail/-/0866099654/ >> FWIW this is the book that was used in every school I've had a >> preclinical >> in. > and Challenge book? >> I've taught out of it for an honors class, and have found it to be a >> great book. The only problem I may have with it is the format of pages, >> it's >> all in black and red. To me that is pretty boring, I like colorful >> textbooks. > Well, I like four-color illustrations on slick paper as much as the next > guy, but they had better be doing something to advance the understanding > of the content in a math textbook. Some of the illustrations in the > Serra textbook, although pretty, are smarmy wastes of time and ink. >> However, content wise it's a great book. > If you are referring to Geometry for Enjoyment and Challenge (aren't > you?), does that book include content on transformations and other more > modern topics of high school geometry? > P.S. As in my previous reply to N. Silver, my apologies for the trash > message posted with a spoofing of my name. > -- > Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 > Learn in Freedom (TM) http://learninfreedom.org/ > remove .de to email -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Geometry Textbook Suggestions for Gifted Math Program? > I wonder what you would suggest as a best textbook for a first > high-school-level geometry course for talented students? I'm trying to > gather information to make a suggestion to a local mathematics program. > I am aware of one program that formerly used Gene Murrow and Serge > Lang's textbook Geometry (Springer-Verlag) and Lang has written upteen texts. This is one I have not read, but the reviews warn me off this one for gifted students. > in recent years has used Michael Serra's textbook Discovering Geometry > (Key Curriculum Press). In my opinion, this text is inappropriate. It's designed for struggling students. > What do you think? If you were recommending a textbook for a class of > students selected for high math ability taking a first secondary-level > geometry course, what would you recommend? What do you like > about your preferred textbook? Look at: Elementary Geometry from an Advanced Standpoint, Third Edition [FACSIMILE] by Edwin Moise. It's the one I would use. Read the reviews on Amazon.com. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Geometry Textbook Suggestions for Gifted Math Program? >>I wonder what you would suggest as a best textbook for a first >>high-school-level geometry course for talented students? I'm trying to >>gather information to make a suggestion to a local mathematics program. >>I am aware of one program that formerly used Gene Murrow and Serge >>Lang's textbook Geometry (Springer-Verlag) and > Lang has written upteen texts. This is one I have not read, > but the reviews warn me off this one for gifted students. I have the book at hand, and it looks not too bad to me, maybe because Murrow (the co-author) moderates Lang's touch in how the content is expressed. It is a fact, however, that one program I know of has DROPPED this book after formerly using it. A good review of Murrow and Lang I have seen is by Professor Hung-hsi Wu of UC Berkeley, http://math.berkeley.edu/~wu/pspd3d.pdf who describes that textbook as the rock bottom minimum of what a high school geometry teacher should know about geometry. But of course that review implies that the book is suited for a teacher-training course at a university, and doesn't endorse it for gifted preteens and teenagers who later might be scientists or mathematicians. >>in recent years has used Michael Serra's textbook Discovering Geometry >>(Key Curriculum Press). > In my opinion, this text is inappropriate. It's designed for struggling > students. Yeah, that is my dismaying impression too. I am by no means opposed to a discovery approach in presenting mathematics, but the content of Serra's textbook seems too easy for a course of the kind I am asking about, certainly not enough geometry for a last look at secondary geometry before a strong calculus sequence. I am told, by program participants, that most of the students in the program where the Serra text is now used think that the textbook talks down to them. Maybe the Serra text has to be used at a VERY young age, if at all, among stronger students--it looks all right to me as a middle-school-geometry book, although it definitely presupposes study of algebra. >>What do you think? If you were recommending a textbook for a class of >>students selected for high math ability taking a first secondary-level >>geometry course, what would you recommend? What do you like >>about your preferred textbook? > Look at: Elementary Geometry from an Advanced Standpoint, Third Edition > [FACSIMILE] by Edwin Moise. It's the one I would use. Read the reviews > on Amazon.com. I have that book at hand too, and like it. I think I bought it originally because of the Amazon reviews that were posted as of early 2001. It's approach is quite different, because it starts out looking a lot like Landau's Grundlagen der Analysis, and gets into geometry proper (mostly from an analytical point of view rather than synthetic) only after laying a foundation of the ordered field properties of the real numbers. I would love to have anyone I know take a course in geometry based on that textbook: I wonder if the program I know of would dare to try it out? P.S. As you read this thread, do you see a second post, apparently in my name, that has garbage content? Either my newsreader is SERIOUSLY buggy, or somone on Usenet is hijacking threads. I thought this k12.ed.math archive still shows the junk message. My apologies to you and to the moderator for something that I didn't intend to do and didn't knowingly do. If this happens again, I will probably switch to a different newsreader. [moderator responds to PS: Karl, the nonsense post posted under your email address is not through any action of your own. I have just may not remove the nonsense message, due to a variety of issues with how Usenet works. If you have further questions about this, please contact me privately. Although the newsgroup is moderated, there are ways to get around the moderation procedure, and the forgers of this nonsense message have done exactly that.] -- Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 Learn in Freedom (TM) http://learninfreedom.org/ remove .de to email -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Geometry Textbook Suggestions for Gifted Math Program? >> Look at: Elementary Geometry from an Advanced Standpoint, Third Edition >> [FACSIMILE] by Edwin Moise. It's the one I would use. Read the reviews >> on Amazon.com. > I have that book at hand too, and like it. I think I bought it > originally because of the Amazon reviews that were posted > as of early 2001. It's approach is quite different, because it > starts out looking a lot like Landau's Grundlagen der Analysis,... LOL. You're right. This text might be a bad gamble. > and gets into geometry proper (mostly from an analytical point > of view rather than synthetic) which is what the course should be about?! > only after laying a foundation of the ordered field properties of the real > numbers. I would love to have anyone I know take a course in geometry > based on that textbook: I wonder if the program I know of would dare to > try it out? 1.) Here's another stab: Geometry by Ray Jurgensen, Richard G. Brown, which you can look inside at Amazon.com. http://www.amazon.com/exec/obidos/ASIN/0395977274/ref=sib_rdr_dp/104-1790268 -8707900 2.) Here's a reference to an old-fashioned, dry book by Welchons, Plane Geometry. 3.) Another classic is: Basic Geometry by George David Birkhoff, Ralph Beatley, which you also can look inside at Amazon.com. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Peer tutoring, basic mathematics can anyone help me solve a math problem or two please! -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Closure principle I dislike books that have poor indexes and absent glossaries, but the book I'm chomping through on teaching elementary math has a convoluted sentence with the gist that the closure prinicple of addition and multiplication for whole numbers is extended to include the closure principle of division when applied to fractional computation. mostly philosophical websites, a few religious web sites and even some mathematical web sites that didn't define or give an example of the closure principle, but assumed you knew the meaning already. It isn't in my chemistry or biology dictionary (big surprize). Help. TIA blacksalt OT: Thrill of the day: I met a woman named Xanthippe today! I also met someone trying to commit suicide via voodoo...i.e. hanging an effigy of herself. They were not one and same people. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Closure principle >I dislike books that have poor indexes and absent glossaries, > but the book I'm chomping through on teaching elementary > math has a convoluted sentence with the gist that the > closure prinicple of addition and multiplication for whole > numbers is extended to include the closure principle of > division when applied to fractional computation. Closure is a property of an operation on a given set (of numbers). Examples of some (but not all) operations are addition, subtraction, multiplication and division. The set {1, 2, 3, ...}of natural numbers is closed under addition, because when you add any two natural numbers, their sum again is a natural number. The set {1, 2, 3, ...}of natural numbers is closed under multiplication, because when you multiply any two natural numbers, their product again is a natural number. Is the set {1, 2, 3, ...}of natural numbers closed under subtraction? No, because, for example, 3 - 5 = -2. Their difference, -2, is not a natural number. In order for a set to be closed under an operation (say) *, m*n also must be in the set for every number m and n in the set. This brings us to ask about the closure of the naturals under subtraction; i.e., how many more numbers do we need in order that every two naturals has its difference in the bigger set? The answer is the set of integers {...-3, -2, -1, 0, 1, 2, 3,...}. Putting in zero and negative integers will do the trick. In *your* example, there are no negative numbers, because it is elementary school and/or we are not considering subtraction. We only know about the closure of the set of whole numbers under addition and multiplication. Probably the set of whole numbers W = {0, 1, 2, 3, 4,...}. I say probably, because although it's not a standard set in higher math, that's the usual meaning. It looks like, in *your* text, though, W = {1, 2, 3, ...}, without 0, because we cannot meaningfully divide by zero. We see this last set W = {1, 2, 3, ...} is not closed under division, because there are an unlimited number of examples such as 1 divided by 2 = 1/2, where the quotient of two whole numbers is not in the set W. The question becomes, what is the closure of W under division? The set (not the operation) needs to be enlarged to include all possible quotients of two whole numbers. The enlarged set is called the set of positive rational numbers, all positive fractions that have whole numbers in the numerator and denominator. All this is hopelessly pedantic (especially the way I explained it) and should not be the focus of an elementary school child, of which I have one and last year had two. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Closure principle > All this is hopelessly pedantic (especially the way I explained it) So when would I, as the non-child, ever look at a problem or situation and say eureka, I see the closure property is involved and it would help me? I understand I might not be at a level to easily comprehend the answer. I assumed, foolishly perhaps, that the reason this book suddenly swooped in this sentence about the closure property being extended to fractional division because it was something useful. It has not been a book where the authors term drop (along the lines of name dropping), but it is NOT well edited. blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Closure principle > All this is hopelessly pedantic (especially the way I explained it) > and say eureka, I see the closure property is involved and it would > help me? > I understand I might not be at a level to easily comprehend the answer. > I assumed, foolishly perhaps, that the reason this book suddenly swooped > in this sentence about the closure property being extended to fractional > division because it was something useful. I would not agree with Nat and say never or that it is hopelessly pedantic, because I can think of many instances, and one problem or situation in particular where I think there has to be the comment eureka, I see the closure property is involved. It involves a problem so common, it provokes many questions as to why it is true. This problem is why division by 0 is not allowed - not allowed in the context of a field. And so closure is quite useful. (To those who might not know: Division by 0 is allowed outside the context of a field. It just so happens that the set of rational numbers is a field. [And so for the sets of the real and the complex numbers.] Which is possibly why the textbook mentions the closure property at the point where it talks about fractional division.) Whether we realize it or not, the closure property (axiom is an acceptable alternative) is always lurking in the background as a necessary precondition for everything we do in our usual number systems. Like I said, a very important example has to do with division by 0, and here's how (You can treat all this below as sort of an index if you want - it's the spirit in which I offer it. And I think you are at a level to easily comprehend it. It's actually not hard to comprehend, just possibly a little involved): Division, a/b, is defined as multiplication by a multiplicative inverse, a(1/b). The inverse property under multiplication is that for some x in a field like the rational numbers, there is some 1/x such that x(1/x) = 1. One of the most important theorems in a field is that 0x = 0 (a couple of proofs below if you want them) for all x in that field. So think about it: Can there be any 1/0 in the given field such 0(1/0) = 1? No, there isn't, because of the aforementioned theorem. So 0 has no multiplicative inverse in that field, which means by the above definition of division that there is no division by 0 - in that field. Notice I kept explicitly pointing out every step of the way that we were in a field, showing how the closure axiom (I prefer this alternative to property to separate this and other properties that are axioms from properties that are theorems) is always lurking in the background as a necessary precondition for everything we do in our usual number systems. Normally, this would not be repeatedly made explicit, for obvious reasons. And so nothing would work in our usual number systems without the closure axiom. Proof 1 of 0x = 0 using only the field axioms: 0x = 0x + 0 = 0x + (0x + (-(0x)) = (0x + 0x) + (-(0x)) = (0 + 0)x + (-(0x)) = 0x + (-(0x)) = 0 Proof 2 of 0x = 0 by appealing to the uniqueness of the additive 0x = 0x + 0 0x = (0 + 0)x = 0x + 0x Look at the last expression in each line. We can conclude that 0 = 0x either by appealing to the uniqueness of the additive identity and be done or by setting them equal (since both are equal to 0x) and using And just in case you'd like to see them: Proof of the uniqueness of the identity (for both the additive and multiplicative identities using # as a substitute operation): Suppose e and e' were both identities. Then e = e # e' = e' by applying the definition of the identity on e and e' one at a time. expressions by the same inverse. Paul -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Closure principle >> All this is hopelessly pedantic (especially the way I explained it) > and say eureka, I see the closure property is involved and it would > help me? Not to be facetious, the short answer is never. Closure is important in more abstract settings. Here's the idea. If a given set, (you can think of, say, whole numbers) with given operations (think of addition and multiplication) on the elements of the set, satisfies certain algebraic properties (think of closure, commutative, distributive props., etc.), the set and the operations together may be said to have certain algebraic structures (with names like group, ring, field, or vector space). Then all theorems proved about, say, groups apply to the given set, which means you know a lot about it. To recap, suppose you have a set with operations that you want to understand more about. If you can show it conforms to a standard algebraic structure, then all the theorems in the literature come into play and you know much more about it. Extending a set and operations to a certain algebraic closure may allow one to solve problems in a more general way, which is always a goal of the research mathematician. In fact, that is how complex numbers came into being. > I understand I might not be at a level to easily comprehend the answer. > I assumed, foolishly perhaps, that the reason this book suddenly swooped > in this sentence about the closure property being extended to fractional > division because it was something useful. Authors are constrained by publishers, who listen to reviewers. Reviewers have to write something to earn their money. (I have reviewed math texts.) Standardized (and state) tests may contain questions about closure. So, it's a topic that must be in the math curriculum of a (responsible) district. And, most of all, to get an adoption, publishers listen to district representatives. > It has not been a book where the authors term drop (along the > lines of name dropping), but it is NOT well edited. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Closure principle > Authors are constrained by publishers, who listen to reviewers. > Reviewers have to write something to earn their money. (I have > reviewed math texts.) Standardized (and state) tests may contain > questions about closure. So, it's a topic that must be in the math > curriculum of a (responsible) district. And, most of all, to get an > adoption, publishers listen to district representatives. This is book for education majors, on how to teach math to kids under the age of 13. If they are to toss in terms like this, they ought to bloody well have it in the index, and a glossary would have been helpful as well. I cannot believe, knowing the education majors I've known, and watching a few of the Annenberg tapes streaming online on education, the average reader of this text would know what the closure property is. Even the text itself yaks on about how math is the weakest subject of the average education major. However, the sample problems, what the authors seem to be really in to, are great. I suspect the fact the whole cover is filled with authors names (at least it seems this way) has hampered the readability. However, onward and upward. blacksalt > It has not been a book where the authors term drop (along the > lines of name dropping), but it is NOT well edited. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Boy do I feel foolish (WAS:Re: Closure principle it. I was so busy making sure I was using the right principle (vs. principal) I missed the fact I had the wrong word completely. So sorry, off to dope-slap my effigy. blacksalt > I dislike books that have poor indexes and absent glossaries, but the > book I'm chomping through on teaching elementary math has a convoluted > sentence with the gist that the closure prinicple of addition and > multiplication for whole numbers is extended to include the closure > principle of division when applied to fractional computation. > mostly philosophical websites, a few religious web sites and even some > mathematical web sites that didn't define or give an example of the > closure principle, but assumed you knew the meaning already. It isn't > in my chemistry or biology dictionary (big surprize). Help. > TIA > blacksalt > OT: Thrill of the day: I met a woman named Xanthippe today! I also met > someone trying to commit suicide via voodoo...i.e. hanging an effigy of > herself. They were not one and same people. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: help can you help me do this equation 2x-4y=16 -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: help Hello! Is this a normal equation, or a diophantine equation? Bence Sz=E1sz > can you help me do this equation 2x-4y=3D16 -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: help > Hello! > Is this a normal equation, or a diophantine equation? It is an equation that one must do. Isn't that enough? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: help > Hello! > Is this a normal equation, or a diophantine equation? > It is an equation that one must do. Isn't that enough? So, we have an equation: 2x-4y=16 Now order y to the right side, and anything other to the left side: 0.5x-8=y This is a simple linear function [http://www.apaczai.elte.hu/~08amszbb/func.PNG]. Now make a product from the left side: 0.5*(x-4)=y Examine the domain (D) and the range (R) of the function. The domain is the set of real numbers, and the range is the set of real numbers evenly divisible with 0.5. So there are infinity x values, because there are infinity real numbers evenly divisible with 0.5, and because of this, there are infinity y values, and because of this there are infinity pairs of x and y values, and because of this there are infinity solutions of the equation. I hope I helped, my english is poor especially on terminologies Bence Szasz -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: help > can you help me do this equation 2x-4y=16 No, because an equation is a statement, not a command. I can't do an equation any more than I can do a statement like paper is stiff. Perhaps you meant plot the equation or solve the equation. The first is easily done, since it is a simple linear equation in x and y. The second is still not well defined, since there are two free variables in the equation. I could solve for y in terms of x or x in terms of y, but I can't find a unique solution for both x and y. Indeed the set of solutions (x,y) is the straight line that we would get if we plotted the equations, so there are infinitely many solutions. ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: what is a hyperbola vertex ? HELP ! I have no access to a math book over the Holidays and am trying to help my son before his math test Tuesday. I am an ex-electrical engineer and have the math background but its rusty. What are the vertices of a hyperbola ? Are they the points on each hyperbola closest to each other, e.g. at the 'peaks' of the hyperbolas ? And is there a formula for determining them ? When the two closest points are on the x axis or y axis I remember how to find them by setting y or x equal to zero and solving for the intercepts. But what if the hyperbolas are not vertically or horizontally aligned along the axis but rather offset ? tutorial on parabolas ? Bob -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: what is a hyperbola vertex ? Found out what a hyperbola vertex is (what I thought). But does anyone have the formula for finding the vertex (if the vertices are not on the X or Y axis) ? Bo b -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: what is a hyperbola vertex ? > HELP ! I have no access to a math book over the Holidays and am > trying to help my son before his math test Tuesday. I am an > ex-electrical engineer and have the math background but its rusty. > What are the vertices of a hyperbola ? Are they the points on each > hyperbola closest to each other, e.g. at the 'peaks' of the hyperbolas > And is there a formula for determining them ? When the two closest > points are on the x axis or y axis I remember how to find them by > setting y or x equal to zero and solving for the intercepts. But what > if the hyperbolas are not vertically or horizontally aligned along the > axis but rather offset ? > tutorial on parabolas ? In the standard equation of the hyperbola, the center is located at (x0, y0), the foci are at (x0+/-c, y0), and the vertices are at (x0+/-a, y0). http://mathworld.wolfram.com/Hyperbola.html -- It is only those who have neither fired a shot nor heard the shrieks and groans of the wounded who cry aloud for blood, more vengeance, more desolation. War is hell. --William Tecumseh Sherman In war, there are no unwounded soldiers. --Jose Narosky The urge to save humanity is almost always a false front for the urge to rule. --H.L. Mencken -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: what is a hyperbola vertex ? > HELP ! I have no access to a math book over the Holidays and am > trying to help my son before his math test Tuesday. I am an > ex-electrical engineer and have the math background but its rusty. > What are the vertices of a hyperbola ? Are they the points on each > hyperbola closest to each other, e.g. at the 'peaks' of the hyperbolas > And is there a formula for determining them ? When the two closest > points are on the x axis or y axis I remember how to find them by > setting y or x equal to zero and solving for the intercepts. But what > if the hyperbolas are not vertically or horizontally aligned along the > axis but rather offset ? > tutorial on parabolas ? Try googling vertices hyperbola which gives several web sites, including the definition of the vertices of a hyperbola, applets for exploring hyperbolas, interactive tutorials on hyperbolas, ... If you want parabolas as well, they are equally easy to find. ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: what is a hyperbola vertex ? > HELP ! I have no access to a math book over the Holidays and am > trying to help my son before his math test Tuesday. I am an > ex-electrical engineer and have the math background but its rusty. > What are the vertices of a hyperbola ? Are they the points on each > hyperbola closest to each other, e.g. at the 'peaks' of the hyperbolas Yes. > And is there a formula for determining them ? That depends on how the hyperbola is given; polar equation, cartesian equation.... What have you got? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: what is a hyperbola vertex ? >HELP ! I have no access to a math book over the Holidays and am >trying to help my son before his math test Tuesday. I am an >ex-electrical engineer and have the math background but its rusty. >What are the vertices of a hyperbola ? Are they the points on each >hyperbola closest to each other, e.g. at the 'peaks' of the hyperbolas Yes. >And is there a formula for determining them ? Too much math to try to write here in ASCII, especially if rotation is involved. Perhaps some kind soul has hands on the formula, but there's more at stake here. If your son does not know [you *are* raising the question] that the above is true, then a wider knowledge, which should be necessary for a reasonable test, might be out of reach if relying on cramming for a few days. I'd see no depth of understanding being reached there, when any quite reasonable variation on a test question would simply put it out of reach. I'd strongly suggest having your son get together with a serious classmate, who does have a text handy, for his studies. They can feed off each other. Sometimes peer help is best in an emergency. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: HELP: Question about Integers Which positive # less than a 100 cannot be expressed as a sum of a series of consecutive pos. integers? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: HELP: Question about Integers > Which positive # less than a 100 cannot be expressed as a sum of a > series of consecutive pos. integers? Answer: none or all. 1) Technically, a series is an infinite sum, and any series of positive integers will not converge. 2) If you just meant some number of consecutive positive integers, then all positive integers can be so represented, as a sum of precisely one positive integer (itself). You probably meant something different, like what positive number less than 100 cannot be expressed as a sum of 2 or more consecutive positive integers? It is very important in math to be precise in your questions, or the answers you get may not make sense. Or perhpas you meant what positive numbers ...? The difference between the singular and the plural is important---do you need all the numbers that have that property, or just one example? You can answer the question fairly easily by considering that numbers congurent to 1 mod 2 (2n+1) can be represented as n+(n+1), numbers congruent to 0 mod 3 (3n) can be represented as (n-1)+n+(n+1) numbers congruent to 2 mod 4 (4n+2) as (n-1)+n+(n+1)+(n+2), numbers congruent to 0 mod 5 (5n) as (n-2)+(n-1)+n+(n+1)+(n+2) numbers congruent to 3 mod 6 (6n+3) as (n-2)+(n-1)+n+(n+1)+(n+2)+(n+3), numbers congruent to 0 mod 7 (7n) as (n-3)+(n-2)+(n-1)+n+(n+1)+(n+2)+(n+3) numbers congruent to 4 mod 8 (8n+4) as (n-3)+(n-2)+(n-1)+n+(n+1)+(n+2)+(n+3)+(n+4) ... If you continue in this way you will cover all the positive integers, except those whose values are so small that the first terms of the sum are not positive integers. (For 2n+1, need n>0, for 3n, need (n-1)>0, for 4n+2, need (n-1)>0, ...) For a limit as low as 100, it is probably easiest to make a sieve, crossing out the numbers that CAN be represented as a sum of 2, 3, 4, ... consecutive integers. ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Finding A, B, and C ABC I have to find the values of A, B and C where A>B>C - CBA _________ = CAB -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Finding A, B, and C Sylvain Croussette answered on alt.math.undergrad: > ABC I have to find the values of A, B and C where A>B>C >- CBA >_________ >= CAB Assuming A,B and C are single digits: Rewrite as: CAB +CBA -------- ABC so it's impossible, unless there is a carry from the 1st; so it's 1+A+B=B+10, because A+B produced a carry in the 1st column. Rewrite these equations: 1) A+B-C=10 2) A+B-B=9 3) A-2C=1 Clearly from equation 2, A=9, so from equation 3, C=4 so B=5. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Finding A, B, and C > ABC I have to find the values of A, B and C where A>B>C > - CBA > _________ > = CAB Hint: Looking at the leftmost digits A - C = C means either that A = 2C (nothing borrowed from A to make the middle digit subtraction work) or A = 2C + 1 (borrowed a 1 from A on the middle digit subtraction) The given, A > B > C means that the second one (A = 2C + 1) has to be true, since there's obviously borrowing going on reading the digit subtractions right to left. Since A,B, and C are all single-digits, that leaves five possibilities based solely on A = 2C + 1 : (1,0) (3,1) (5,2) (7,3) (9,4) All but the last two can be eliminated because they do not lead to B digits between A and C. (A > B > C) -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Finding A, B, and C > ABC I have to find the values of A, B and C where A>B>C > - CBA > _________ >= CAB You did not say, but I assume that A, B, and C are digits of a 3-digit number, so the equation is really 100*A+10*B+C - (100*C+10*B+A) = 100*C+10*A+B which can be easily rearranged to 89*A -B - 199*C = 0 We also have 9>=A>B>C>=0 You can use trial and error to get the solution from here, but it helps to notice that A/C has to be approximately 199/89. ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: language of division The text I'm using on teaching children math forbids the use of goes into when talking about division. They say it is meaningless. For the problem 427/62, they advise the child think aloud along the lines of 1) can 6 tens and 2 ones be subtracted from 4 tens and 2 ones? No. Can 6 tens and 2 ones be subtracted from 4 hundreds and 2 tens? Yes. How many times? etc.. OR 2) How many groups of 62 can I make out of 427 objects? Is this proper or farfetched? If the above is farfetched, is goes into still commonly used, or, if not, what is used? I am not clear from the text how the child goes about answering the 'how many times' or 'how many groups' question. Trial and error? Estimation and best guess first? TIA blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: language of division The way a child determines 'how many times' or 'how many groups' is by determining how many times the smaller number goes into the larger number. > The text I'm using on teaching children math forbids the use of goes > into when talking about division. They say it is meaningless. For the > problem 427/62, they advise the child think aloud along the lines of > 1) can 6 tens and 2 ones be subtracted from 4 tens and 2 ones? No. Can 6 > tens and 2 ones be subtracted from 4 hundreds and 2 tens? Yes. How many > times? etc.. > OR > 2) How many groups of 62 can I make out of 427 objects? > Is this proper or farfetched? If the above is farfetched, is goes into > still commonly used, or, if not, what is used? > I am not clear from the text how the child goes about answering the 'how > many times' or 'how many groups' question. Trial and error? Estimation > and best guess first? > TIA > blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: language of division Another way of looking at division is to think of division as successive subtractions in the same way as you can think of multiplication as successive additions. You can think of it as how many 62's can you take extensive explanation of this. The text I'm using on teaching children math forbids the use of goes > into when talking about division. They say it is meaningless. For the > problem 427/62, they advise the child think aloud along the lines of > 1) can 6 tens and 2 ones be subtracted from 4 tens and 2 ones? No. Can 6 > tens and 2 ones be subtracted from 4 hundreds and 2 tens? Yes. How many > times? etc.. > OR > 2) How many groups of 62 can I make out of 427 objects? > Is this proper or farfetched? If the above is farfetched, is goes into > still commonly used, or, if not, what is used? > I am not clear from the text how the child goes about answering the 'how > many times' or 'how many groups' question. Trial and error? Estimation > and best guess first? > TIA > blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: language of division > Another way of looking at division is to think of division as successive > subtractions in the same way as you can think of multiplication as > successive additions. You can think of it as how many 62's can you take > extensive explanation of this. > The text I'm using on teaching children math forbids the use of goes > into when talking about division. They say it is meaningless. For the > problem 427/62, they advise the child think aloud along the lines of > 1) can 6 tens and 2 ones be subtracted from 4 tens and 2 ones? No. Can 6 > tens and 2 ones be subtracted from 4 hundreds and 2 tens? Yes. How many > times? etc.. > OR > 2) How many groups of 62 can I make out of 427 objects? > Is this proper or farfetched? If the above is farfetched, is goes into > still commonly used, or, if not, what is used? > I am not clear from the text how the child goes about answering the 'how > many times' or 'how many groups' question. Trial and error? Estimation > and best guess first? > TIA > blacksalt The problem I see with my students using the successive subtractions is with long division. If the problem is 578 divided by 41, they don't stop to see that 41 will go into 57. That can make this method more time consuming than necessary. In addition, my students (in a remedial class) tend to make subtraction errors, especially when they must borrow. Lee in SC -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: language of division Just a personal opinion, but I think it's rubbish. Goes into is synonymous with divides into. This comes from and can then be related to the fact that 6 goes into 48, or 6 divides 48 into 8 equal parts, each of size 6. ...and so on . Any attempt at trial and error by subtraction is utter rubbish. Proof to the contrary will take long enough for these young people to know they deserved a better education as they are imminently unsuccessful later on. But it is the later teachers who will be held to blame no doubt. >The text I'm using on teaching children math forbids the use of goes >into when talking about division. They say it is meaningless. For the >problem 427/62, they advise the child think aloud along the lines of >1) can 6 tens and 2 ones be subtracted from 4 tens and 2 ones? No. Can 6 >tens and 2 ones be subtracted from 4 hundreds and 2 tens? Yes. How many >times? etc.. >2) How many groups of 62 can I make out of 427 objects? >Is this proper or farfetched? If the above is farfetched, is goes into >still commonly used, or, if not, what is used? >I am not clear from the text how the child goes about answering the 'how >many times' or 'how many groups' question. Trial and error? Estimation >and best guess first? >TIA >blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: language of division > The text I'm using on teaching children math forbids the use of goes > into when talking about division. They say it is meaningless. For the > problem 427/62, they advise the child think aloud along the lines of > 1) can 6 tens and 2 ones be subtracted from 4 tens and 2 ones? No. Can 6 > tens and 2 ones be subtracted from 4 hundreds and 2 tens? Yes. How many > times? etc.. > OR > 2) How many groups of 62 can I make out of 427 objects? > Is this proper or farfetched? If the above is farfetched, is goes into > still commonly used, or, if not, what is used? > I am not clear from the text how the child goes about answering the 'how > many times' or 'how many groups' question. Trial and error? Estimation > and best guess first? I don't know what elementary teachers now teach (mine tried hard to avoid goes into 40 years ago, so it isn't a new prejudice). Actually there is nothing inherently wrong with the goes into operator. You can define it easily and unambiguously: a goes into b =def b / a It is not meaningless, just non-standard. So far as I can tell, this operator is still used in speech (usually pronounced guzinta around here), but not in writing---there is no standard symbol for it. The long-division algorithm has always suffered from the need to guess how many times the divisor goes into the current part of the dividend. If you guess low, you'll end up with a remainder that is too big, and have to increment your guessed digit. If you guess high, you'll end up with a negative remander and have to decrement your guessed digit. Unfortunately, the algorithm is usually presented as if people always guessed perfectly, losing a great apportunity to teach how to recover from mistakes---a useful skill later on when dealing with more complicated algorithms. ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Lowest common denominator endangered? The book I'm reading gives the algorithms for adding and subtracting fractions: a/b+c/d=ad+bc/bd and similarly for subtraction. It notes that this does not necessarily give you the LCD. They do mention 4 methods for obtaining the LCD if desirable. That if desirable phrase appears three times. It never mentions *how* desirable finding the LCD is, and the whole section on it seems to be an afterthought. Given the emphasis on calculators in the book, I'm wondering if the LCD is still an issue, or is it considered nice to know, but don't spend alot of time having child memorize how to find this. What is the current thinking on LCD and does it come up on the standardized testing everyone is barking about? I have no memory of being taught fractions, and, when I put myself to a set of them, I found I used a combo of good guess and reduction after calculating, with a pinch of prime numbers sprinkled over the good guess. Not a method given to methodical instruction. TIA blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Lowest common denominator endangered? > The book I'm reading gives the algorithms for adding and subtracting > fractions: > a/b+c/d=ad+bc/bd > and similarly for subtraction. It notes that this does not necessarily > give you the LCD. They do mention 4 methods for obtaining the LCD if > desirable. That if desirable phrase appears three times. It never > mentions *how* desirable finding the LCD is, and the whole section on it > seems to be an afterthought. > Given the emphasis on calculators in the book, I'm wondering if the LCD > is still an issue, or is it considered nice to know, but don't spend > alot of time having child memorize how to find this. What is the current > thinking on LCD and does it come up on the standardized testing everyone > is barking about? > I have no memory of being taught fractions, and, when I put myself to a > set of them, I found I used a combo of good guess and reduction after > calculating, with a pinch of prime numbers sprinkled over the good > guess. Not a method given to methodical instruction. > TIA > blacksalt I teach remedial math to 7th and 8th graders. I used this method for the first time this year. Students had better success finding the correct answer using this method than the LCD method. There were not as many steps to remember and compute. The difficulties came when they needed to add more than two fractions. They had to add the first two, simplify, and add the answer to the subsequent fractions. Lee -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Lowest common denominator endangered? >The book I'm reading gives the algorithms for adding and subtracting >fractions: >a/b+c/d=ad+bc/bd >and similarly for subtraction. It notes that this does not necessarily >give you the LCD. They do mention 4 methods for obtaining the LCD if >desirable. That if desirable phrase appears three times. It never >mentions *how* desirable finding the LCD is, and the whole section on it >seems to be an afterthought. >Given the emphasis on calculators in the book, I'm wondering if the LCD >is still an issue, or is it considered nice to know, but don't spend >alot of time having child memorize how to find this. What is the current >thinking on LCD and does it come up on the standardized testing everyone >is barking about? >I have no memory of being taught fractions, and, when I put myself to a >set of them, I found I used a combo of good guess and reduction after >calculating, with a pinch of prime numbers sprinkled over the good >guess. Not a method given to methodical instruction. >TIA In algebra, it occurs either by devise for practice, or from some physical problem that denominators like, keeping it simple, (a+b), (a - b), and (a^2 - b^2) occur simulataneously. It is quite handy to know about LCD, and in fact that is the sole reason for its invention [discovery] and use ...to keep claculations simple. It is a much better practice to be on the lookout and to simplify as soon as and whenever possible, rather than wait. Students learning fractions, or attempting to re-learn them according to the current curriculum guideline, as often as not wind up with large values in numerator and denominator that could have been reduced much earlier with far less effort than waiting until the end ...when they seldom see any sort of common factor in any case. In algebra, it is far easier to reduce by the use of LCD rather than wait to reduce soem horribly complex expression later on. It's simply good practice and technique. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Lowest common denominator endangered? > The book I'm reading gives the algorithms for adding and subtracting > fractions: > a/b+c/d=ad+bc/bd > and similarly for subtraction. It notes that this does not necessarily > give you the LCD. They do mention 4 methods for obtaining the LCD if > desirable. That if desirable phrase appears three times. It never > mentions *how* desirable finding the LCD is, and the whole section on it > seems to be an afterthought. > Given the emphasis on calculators in the book, I'm wondering if the LCD > is still an issue, or is it considered nice to know, but don't spend > alot of time having child memorize how to find this. What is the current > thinking on LCD and does it come up on the standardized testing everyone > is barking about? > I have no memory of being taught fractions, and, when I put myself to a > set of them, I found I used a combo of good guess and reduction after > calculating, with a pinch of prime numbers sprinkled over the good > guess. Not a method given to methodical instruction. > TIA > blacksalt > -- This is my preferred main method (always along with some LCD method) for teaching the adding of two fractions (or ratios if you wish), since sooner or later, some thinking middle school (or later) student will ask about adding such as 2/pi + 3/sqrt(5) and you will really be stuck if you need to find the least common multiple of pi and sqrt(5) (the LCD), because there is no such thing as the least common multiple of the denominators when the denominators are arbitrary reals or even arbitrary rationals. This is because even when restricted to the positive side, any positive rational is a multiple of any other positive rational. Same for the positive reals. That is, where a and b are arbitrary positive rationals, ax = b always has a solution x = b/a where b/a is a positive rational. Again, same for the positive reals. When for instance we restrict ourselves to the positive integers, we can have least common multiples because it is not so that any positive integer is a multiple of any other positive integer. That is, where a and b are arbitrary positive integers, ax = b does not always has a solution x = b/a where b/a is a positive integer. Note that this connects to your other question in another thread of why should there be any talk about the closure axiom, about whether it has any usefulness. (I prefer axiom to property here to make the a distinction between properties treated as axioms to properties treated as theorems.) There, I showed how it was necessary to use it show how division by 0 is not allowed in the context of a field such as the rational, real, or complex numbers. Here, it is necessary to use it to see whether there is such a thing as a LCD. It is quite important to be able to add fractions (or ratios if you wish) when the denominators are not just arbitrary integers, since I'm always surprised at how great a percentage of even high school and college students can't add such as the above. They've always been taught only some LCD method. I recently was helping a trigonometry student (in a solving of a trig equation context) who could not deal with a starting expression of the form ((a/b) + (c/d))/(e + f) with all variables here denoting trig functions, even when I recast this expression in non- trig function form. She never saw the theorem a/b + c/d = (ad + bc)/(bd) that you share with us. Unfortunate, since such as this theorem or one of its proofs (as an algorithm) is necessary to deal with the numerator. I say one of its proofs because, when extended to a sum of any number of fractions ratios, the one that I have in mind is usable as an algorithm on a sum of any number of fractions. This algorithm I think is quite important to be able to bring back to mind, since adding more than two fractions is quite common. A point in adding fractions is to reduce the sum of a number of fractions to a single fraction. For what it's worth, the above theorem and this algorithm are two examples showing that it is not the case that we have to find a common denominator before we can reduce to a single fraction, contrary to what some think. Here's the algorithm generally applied to three fractions, keeping in mind the definition of division as multiplication by the multiplicative inverse, x/y = x(1/y), and keeping in mind that when applying this algorithm, some steps can be skipped: a/b + c/d + e/f = (a/b + c/d + e/f)(1) = (a/b + c/d + e/f)bdf(1/(bdf) = (abdf/b + cbdf/d + ebdf/f)(1/(bdf) = (adf + bcf + bde)(1/(bdf) All we are doing is multiplying the sum by 1 in the form of the product of the denominators divided by itself, distributing this product through the sum, and then dividing out. We then have a single fraction or ratio. And note that we can do the same algorithm in a LCD context: Just replace the product of the denominators with the least common multiple of the denominators when there is such a thing as the least common multiple of the denominators. It's quite important to be able to do this modification of this algorithm in a LCD context, because in algebra dealing with fractions ratios where the variables are polynomial functions, it is much more convenient - the algebraic manipulations are easier to deal with. Paul -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Lowest common denominator endangered? > The book I'm reading gives the algorithms for adding and subtracting > fractions: > a/b+c/d=ad+bc/bd I *hope* they get the parentheses right: a/b + c/d = (ad+bc)/(bd) > and similarly for subtraction. It notes that this does not necessarily > give you the LCD. They do mention 4 methods for obtaining the LCD if > desirable. That if desirable phrase appears three times. It never > mentions *how* desirable finding the LCD is, and the whole section on it > seems to be an afterthought. This seems right to me---the essence of adding fractions is in finding a common denominator, which the product of the denominators always is. Making it the *least* common denominator is pretty, but not essential. > Given the emphasis on calculators in the book, I'm wondering if the LCD > is still an issue, or is it considered nice to know, but don't spend > alot of time having child memorize how to find this. What is the current > thinking on LCD and does it come up on the standardized testing everyone > is barking about? I have no idea what the test writers think of LCD, but it would not surprise me at all to find out that they were really gung-ho about it. It is the sort of trivial thing that appeals to test writers. > I have no memory of being taught fractions, and, when I put myself to a > set of them, I found I used a combo of good guess and reduction after > calculating, with a pinch of prime numbers sprinkled over the good > guess. Not a method given to methodical instruction. Using the product as the denominator is generally a good algorithmic method that is easy to teach. If you want to teach LCD, you can teach Euclid's algorithm separately to find the GCD of the numerator and denominator after doing the standard addition. With a bit more effort, you can compute the least common multiple of the denominators ahead of time and convert both the initial fractions into having that LCM as the denominator For those who like factoring, you can try factoring the denominators before you start and computing the least common multiple that way, though Euclid's algorithm is a lot simpler when dealing with large, difficult-to-factor numbers. ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: instructions for prime drag math game I have an old math game called prime drag. It looks interesting. The directions are a bit unclear. If anyone has played it I'd like to know if the leagl moves need to pick up a card or just conitnue rolling the dice. Also how do you know to pick up form the prime or composite pile? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Tutoring Rates I live in a big city in the south east and have tutored primarily students from the private schools here charging $40 per session and sessions are typically 45 minutes. Most all of my students have been high school. All of my business comes by word of mouth and therefore the recommendations make all the difference in the price you are asking--if you come recommended, people will know you are worth the rate you are asking for. Also, find out what the learning centers near you like sylvan are charging...they charge quite a lot and I have found they aren't nearly as effective as private specialized tutoring. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Phyeagone theory Find the lengths of the legs of a right triangle whose perimeter is 56m if the hypotenuse is 25m long. hint: invloves sloving two equations. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Phyeagone theory > Find the lengths of the legs of a right triangle whose perimeter is > 56m if the hypotenuse is 25m long. units) x + y + 25 = 56 or x + y = 31 or y = 31 - x . . . . . . (1) x^2 + y^2 = 25^2 putting in y from (1) gives x^2 + (31 - x)^2 = 25^2 or 2*x^2 - 62*x + 336 = 0 or x^2 - 31*x + 168 = 0. Factorize this, if you can, as (x - 7)*(x - 24) = 0 otherwise use the quadratic formula: 31 +/- sqrt(31^2 - 4*168) x = ------------------------- 2 31 +/- sqrt(289) x = ---------------- 2 31 +/- 17 x = --------- . 2 So x = 24, and from (1), y = 7 or x = 7, and from (1), y = 24 . These can be checked against the original data. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Phyeagone theory > Find the lengths of the legs of a right triangle whose perimeter is > 56m if the hypotenuse is 25m long. > hint: invloves sloving two equations. There are two equations you can develop. You can develop an equation from the perimeter, and you can develop an equation from Pythagoras' right angled triangle rule. I have defined the 2 legs as variables a and b. Perimeter: 56 = 25 + a + b 31 = a + b a = 31 - b Pythagoras's right angled triangle rule: a^2 + b^2 = 25^2 Substituting from the previous equation: (31 - b)^2 + b^2 = 625 961 - 62b + b^2 + b^2 = 625 2b^2 - 62b + 961 = 625 2b^2 - 62b + 336 = 0 b can either be 24 or 7 If b = 24 then: a = 31 - 7 = 24 If b = 7 then: a = 31 - 24 = 7 The legs of the right angled triangle are therefore 24m and 7m. James Midolo -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Phyeagone theory >Find the lengths of the legs of a right triangle whose perimeter is >56m if the hypotenuse is 25m long. >hint: invloves sloving two equations. Got it! There are several approaches to solution of these two equations. What have you done so far? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: circle equations A circle passes though the vertices of the right triangle A(-2,-4), B(-2,3), and C (7,3). What is the equation of the circle? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: circle equations > A circle passes though the vertices of the right triangle A(-2,-4), > B(-2,3), and C (7,3). What is the equation of the circle? The centre of the circle is in the centre of the line segment BC which is (5/2, -1/2). The radius of the circle is half the length of the line segment BC which is sqrt(85)/2. The equation is therefore (x - 5/2)^2 + (y + 1/2)^2 = 85/4. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: circle equations > A circle passes though the vertices of the right triangle A(-2,-4), > B(-2,3), and C (7,3). What is the equation of the circle? D has sent us 3 of his/her homework problems, with no indication that any attempt has been done at any of them. I'll give one hint for this problem and ignore the rest. Hint: the triangle is a *right* triangle. What do you know about central angles and inscribed angles? ------------------------------------------------------------ Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Affiliations for identification only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: circle equations >A circle passes though the vertices of the right triangle A(-2,-4), >B(-2,3), and C (7,3). What is the equation of the circle? One approach: Substitute into the general equation for the circle. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: parabloas A point P moves in such a way that it is always the same distance from the point (-6,2) as it is from the line x=-2. Determine the equation in standard form. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Marginal and Joint Gaussian Densities can you send me joint gaussian documents. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: [media] Spam Punishment Doesn't Fit the Crime never seek equality with women. K.S A short excerpt from The Myth of Male Power by Warren Farrell There are many ways in which a woman experiences a greater sense of powerlessness than a man. She may fear pregnancy, aging, rape, date rape and criminal assault. She may feel greater pressure to marry and, without regard to her own wishes, interrupt her career for children. She may feel excluded from an old-boy network. She may resent having less freedom to walk into a bar without being bothered. Fortunately, most industrialized nations have acknowledged these experiences (as we have in these forums). Unfortunately, they have acknowledged only female experiences - and concluded that while women have the problem, men are the problem. A man, of course, has a different experience. He can see marriage become divorce, and often finds that shared financial burdens become alimony payments, his home become his wife's home and his children become support payments who have been turned against him. A man who finds himself in these situations feels as if he is spending his life working for people who hate him. He feels desperate for someone to love, but fears that another marriage may ultimately leave him with another mortgage payment, another set of children turned against him and a deeper desperation. When such a man is called commitment-phobic, he doesn't feel understood. When men try to keep up with payments by working overtime and are told they are insensitive, or try to handle the stress by drinking and are told they are drunkards, they don't feel powerful but powerless. When they fear a cry for help will be met with an instruction to stop whining, or that a plea to be heard will be met with yes, buts, they skip attempting suicide as a cry for help and just commit suicide. Men have remained the silent sex and are increasingly becoming the suicide sex. What feminism has === Subject: Partitive vs. measurement division problems The text I'm using to learn about teaching math to kids http://www.bestwebbuys.com/books/compare/isbn/0130322741/isrc/b-search-other stresses two kinds of division problems: partitive (fairsharing) and measurement (repeated subtraction). Their examples of the two, in order, is: Maria has six oranges. She puts an equal number of oranges in 3 bags. How many oranges in each one? AND Maria has six oranges. She puts oranges in each bag. How many bags does she end up using? The equation for both problems was identical: 6/3=2 I read on as it was a busy chapter, but now that I'm up to division with fractions, they stress how important it is go to back over these two types of division problems with the students before starting division with fractions. Having never taught kiddos division, does this distinction really help or is it an oddity of this book? My adult mind sort of rolls right over it, I don't recall any teaching on different kinds of story problems...you were just thrown into them after the section with the rote problems, and you sank or swam. If this distinction is important, why, does anyone have a more riveting way that some tired oranges and some paper bags as example, and finally, in order to teach these types to children, do you use the partitive/measurement names or the fairsharing/repeated subtraction. Both sets seem a little big, considering the book also devotes time to having the teacher us children's language for operations. TIA blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Partitive vs. measurement division problems >The text I'm using to learn about teaching math to kids >http://www.bestwebbuys.com/books/compare/isbn/0130322741/isrc/b-search-othe r >stresses two kinds of division problems: partitive (fairsharing) and >measurement (repeated subtraction). Their examples of the two, in order, >is: >Maria has six oranges. She puts an equal number of oranges in 3 bags. >How many oranges in each one? >AND >Maria has six oranges. She puts oranges in each bag. How many bags does >she end up using? You have to say how many she put in each bag. I'll presume you mean two. If she put them all into one bag, the answer would be 1. If she put one into each, it would be 6. >The equation for both problems was identical: 6/3=2 Not really, and that's not an equation, but a statement of equivalence. To find the number of oranges, you'd have 6/3 = 2, but to find the nunber of bags, again assuming two each, you'd have 6/2 = 3. >I read on as it was a busy chapter, but now that I'm up to division with >fractions, they stress how important it is go to back over these two >types of division problems with the students before starting division >with fractions. They are possibly trying to stress the manipulative skills needed when dealing wit hrational quantities, and how they can be learned at an early age without actually mentioning them as such. >If this distinction is important, why, does anyone have a more riveting >way that some tired oranges and some paper bags as example, and finally, >in order to teach these types to children, do you use the >partitive/measurement names or the fairsharing/repeated subtraction. Do you have any more riveting examples to offer yourself? You have not indicated that you yet see the purpose of the exercise. Have you talked to any of the teachers to see if you can see why they do this in the way that they do? Either way, do not expect the teachers of the young to be mathematicians. They do an awesome job, having to teach the fundamentals in all subject areas. Math can be particularly difficult, trying to please all of the people all of the time. Perhaps you can offer them some advice and assistance. I taught high school math, and at a much more advanced level than what is being discussed here, and am here to tell you that I admire the public school teachers for what they do, even if students did come insufficiently unprepared at times. Some of the needs are not obvious to them, and they must strictly follow the texts and curriculum guidelines or face the wrath of a thousand parents. There is sense to what they are doing here. The kids just wouldn't realise it until they met me [and other HS math teachers], and my expectations. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Partitive vs. measurement division problems > They are possibly trying to stress the manipulative skills needed when > dealing wit hrational quantities, and how they can be learned at an > early age without actually mentioning them as such. Ah, that is possible, as the book stressed manipulative skills a great deal. > Do you have any more riveting examples to offer yourself? You have > not indicated that you yet see the purpose of the exercise. I thought I indicated that I don't see the purpose of differentiating partitive division problems from measurement problems. The books seems to spend a fair amount of time on them, but does not say why. They do, e.g., explain why reciprocals of fractions can be useful (helpful later on for understanding division with fractions), but not why this issue is brought up more than once. > Have you > talked to any of the teachers to see if you can see why they do this > in the way that they do? Either way, do not expect the teachers of > the young to be mathematicians. They do an awesome job, having to > teach the fundamentals in all subject areas. Math can be particularly > difficult, trying to please all of the people all of the time. > Perhaps you can offer them some advice and assistance. I was speaking of the authors of the text I put a link to. There are no live breathing teachers, only these print teachers who are the authors of a book for people seeking to teach elementary and middle school children math. Since I might, possibly, maybe, something of a long shot, be in that position myself in the upscoming years, I'm trying to learn how to teach children math, most specifically my son. I have discovered that the journey shook loose alot of unpleasant memories on the math education I had, and have found significant gaps in my understand, even on the basic level. I can crank through the rules, but don't have a firm grasp of what underlies them. I am used to sounding all educated and grown up, but am stumbling along rather clumsily on this group, not even sure if the questions I ask have relevance, let alone an answer I can comprehend at my level of knowledge. Pardon my mathematical immaturity. blacksalt -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Partitive vs. measurement division problems >I taught high school math, and at a much more advanced level than what >is being discussed here, and am here to tell you that I admire the >public school teachers for what they do, even if students did come >insufficiently unprepared at times. A typo: should be insufficiently prepared. But that is still not a derogatory remark against the generally great efforts of these teachers of the very young. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Why integrated math I think the idea is an asinine one to begin with. Our district used [and TMK, still uses] this outdated math series. It has completely failed...and...3 years out of high school, I do not remember anything from this series...which I blame on the haphazard way the book bounced from subject matter to subject matter. I had an argument my Sr. year with a instructor in the math department over this--he was unwilling to hear my complaints about the series. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: parent needs math skills brush-up http://www.quickmath.com/ The above site might prove useful for solving specific problems. Lee > I'd like to brush up my math so my kiddo isn't left lost and alone over > math like I was. I never got beyond geometry (out of math-dread and some > really bored/boring teachers), but did well in college physics by > memorizing just what I needed to know out of trig, and I believe I have > a decent aptitude because I got 86%ile in math on the GRE (2 decades > ago), again without anything past 9th grade geometry and working my way > through a book I recall was titled Essential Mathematics for College > Students. I'm also old enough to not let a little math scare me ever > again. > That's an interesting background. Try my FAQ on math education for > bright youngsters http://www.artofproblemsolving.com/Forum/viewtopic.php?p=33140&highlight=#33 140 > and see if that helps. There are several useful books mentioned there. > You could learn a LOT of thinking about math through the math > textbooks published in Singapore. As mathematician W. W. Sawyer has > written, The proper thing for a parent to say is, 'I did badly at > mathematics, but I had a very bad teacher. I wish I had had a good > one.' W. W. Sawyer, Vision in Elementary Mathematics (1964), page 5. > -- > Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 > Learn in Freedom (TM) http://learninfreedom.org/ > remove .de to email -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: a math riddle Eggs don't lay chickens. > If 1.5 chickens lay 1.5 eggs in 1.5 days, how many eggs lay 9 chickens > in 9 days ? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html