It is true that *some* of the problems require some level of understanding of geometry, and that hyperbola is typically mentioned in Algebra 2, yet we should distinguish between studying a topic and simply mentioning it in the course of earlier study, or situating a problem *within* a geometrical context, even if there is little geometry per-se in the problem. For example, hyperbola could have been mentioned earlier and one really doesn't need to know anything about it except that it represent a graph of xy=a, as is stated in the problem. All the rest can be easily derived as a part of the problem solution. Similarly, one does not need to formally study the triangle inequality to observe in a classroom -- typically in much earlier grade -- the nature of this inequality. (Incidentally, the Common Core places it in grade 7, standard 7G2.) Similarly, while proving that the center of a circle lies on a perpendicular bisector of a chord may be an early Geometry theorem studied in 9th grade, any student who studies any type of construction with straightedge and compass in earlier grades should be familiar with this non-proven fact. (Common Core defers all constructions to high school while, for example, California pre common-core expected them in grade 7, standards 7.MG.3.1).
What makes these problems often seem complex is that they expect the student to follow the logic of few basic facts -- which he is typically supposed to be familiar with -- to solve a 2-3 step problem, paying attention to the math as clearly defined *within* the problem. What our students are most often trained to do (pun intended) is to recognize a problem as one of a class and recall the technique for its solution, rather than follow the math as defined in the problem itself wherever it leads. THAT is what makes them difficult, not whether the student ever heard the word "hyperbola" before. The graph could have been called a "sausage" for all it matters, defined as y=a/x.
On 1/18/2013 12:07 PM, Ed Wall wrote: > > Dennis > > Thanks for this. > > That seems to dovetail with my 'ancient' middle school teaching > experiences. However, there has been talk since then of pushing > Algebra 1 down into middle school and I know that in some places > Algebra and Geometry are, insofar as textbooks are concerned, somewhat > integrated. Thus I was wondering. > > Ed > > On Jan 18, 2013, at 2:50 PM, email@example.com > <mailto:starcap50%40aol.com> wrote: > > > Sorry for the double link to the 2010 Virginia Standards of Learning > (SOL) > > 8th Grade Mathematics Test. > > > > The first one does not work. > > > > The second one does. > > > > All The Best, > > > > Dennis > > > > > > In a message dated 1/18/2013 2:39:04 P.M. Eastern Standard Time, > > firstname.lastname@example.org <mailto:starcap50%40aol.com> writes: > > > > > > > > > > Hello Ed, > > > > My reason for assessing the sample problems as "advanced" is that their > > level of difficulty and prerequisite knowledge for solving them are far > > beyond the standards for any middle school math curriculum I have ever > > worked > > with in the state of Virginia since my retirement in 2005. > > > > In the state of Virginia, students who,are enrolled in middle school > are > > usually enrolled in Grades 6,7, and 8. Students usually begin high > school > > in Grade 9. > > > > In Virginia, the "average" student takes Algebra 1 in the 9th grade, > > Geometry in the 10th grade, and Algebra 2 in the 11th grade. Above > average > > students usually take Algebra 1 in the 8th grade (the last year of > middle > > school), Geometry in the 9th grade, and Algebra 2 in the 10th grade. > > > > A small minority of highly advanced students can take Algebra 1 in > the 7th > > grade, Geometry in the 8th grade, and Algebra 2 in the 9th grade. At > the > > middle school where I taught before I retired in 2005, however, > Algebra 1 > > was not yet available for 7th grade students. > > > > In the set of problems presented, problem #42 involves the hyperbola, > > which > > is not taught in the state of Virginia until Algebra 2, which most > > students take in either the 10th or the 11th grade in high school. > Having > > tutored > > students privately for the past several years, I have not seen the > > hyperbola presented in any math course curriculum until Algebra 2 is > taken. > > > > Problem #41 involves the construction of a circle within given > parameters, > > which is a topic covered in Geometry. Most students take Geometry in > > either the 9th or 10th grade in high school. Only a small minority of > > students > > take Geometry in 8th grade, which is the last year of middle school > in the > > state of Virginia. Problem #39 involves the Triangle Inequality > Theorem, > > which is also not covered until Geometry. > > > > The following is a link to a pdf copy of the 2010 Virginia Standards of > > Learning (SOL) Test for 8th Grade Mathematics: > > > > > __http://www.doe.virginia.gov/testing/sol/released_tests/2010/test10_math8.p > > d_ > > > (http://www.doe.virginia.gov/testing/sol/released_tests/2010/test10_math8.pd) > > > f_ > > > (_http://www.doe.virginia.gov/testing/sol/released_tests/2010/test10_math8.p > > df_ > > > (http://www.doe.virginia.gov/testing/sol/released_tests/2010/test10_math8.pdf) > ) > > > > If this link opens, you will see a fair representation of the level of > > problems which are taught in the 8th grade, which is the last year of > > middle > > school in the state of Virginia. > > > > Speaking from the trenches, and based upon my nearly 30 years of > teaching > > and tutoring combined, for whatever it may be worth, the sampling of > > problems presented in the "Five Triangles" blog website more closely > > resemble > > advanced problems on the high school level than on the middle school > level. > > > > Best Wishes, > > > > Dennis > > > > > > In a message dated 1/18/2013 12:21:42 P.M. Eastern Standard Time, > > email@example.com <mailto:_ewall%40umich.edu>_ > (mailto:firstname.lastname@example.org <mailto:ewall%40umich.edu>) writes: > > > > > > > > > > Dennis > > > > While I haven't taught middle school since 1995, some of these problems > > seem to call on knowledge which isn't in the repertoire of the > average 8th > > grader and, in some cases, the average 9th grader. I'd be interested in > > what > > others, currently involved in middle school mathematics teaching, > think. > > Putting all that aside, these problems do seem accessible to a > student who > > is versed in Algebra I and Geometry, so I'm wondering why you think > them > > for advanced students (I don't say they aren't, by the way). Some of > these > > problems do require some careful thinking and, perhaps, even more > > importantly an investment of solution time. Is this what you are > pointing > > at? > > I looked through the archives, by the way. The quality of problems > vary as > > to difficulty and - this is a personal judgement - appeal. I wish I had > > known about it before as I teach course for teachers-to-be and some > of the > > problems are nice. I rather liked the geometric construction in the > > original > > email as I just spent a semester grumbling about student performance on > > such. > > > > Ed Wall > > > > On Jan 17, 2013, at 10:36 PM, email@example.com > <mailto:__starcap50%40aol.com>_ > > (mailto:firstname.lastname@example.org <mailto:_starcap50%40aol.com>) _ > > (mailto:email@example.com <mailto:_starcap50%40aol.com>_ > (mailto:firstname.lastname@example.org <mailto:starcap50%40aol.com>) ) wrote: > > > >> Yikes! I taught middle school mathematics for 22 years (1983-2005) > and I > > > > > >> have never seen any middle school math problems quite this difficult > >> before. Are these for advanced middle school aged students? > >> > >> I now do private tutoring for high school students taking Algebra 1, > >> Geometry, and Algebra 2, and these problems appear to be on the > advanced > > level > >> for even these subjects. > >> > >> Dennis > >> > >> >
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