mm-3819 === Subject: Ralston's review of California Dreaming The cited review can be read with an Acrobat reader at: My Letter to the Editor is attached. Dom Rosa To the Editor: Reforming Mathematics Education, pointed out that the New Math curricula were mainly the brainchild of university mathematicians. It is equally true, however, that many mathematicians criticized the excessive formalism and abstraction of the New Math. This is demonstrated by the memorandum, signed by 75 leading mathematicians, that was published in the March 1962 issue of The American Mathematical Monthly (On the mathematics curriculum of the high school, pp.189-193). Ralston also claimed that the new math failed most notably because teachers were not prepared to teach this mathematics nor were they given adequate support. This major canard continues to be repeated by those who are unable to accept responsibility for having demolished the traditional college preparatory mathematics curriculum in the United States. Ralston's comment that the New Math's trajectory spann[ed] little more than the decade of the 1960s is patently absurd. In fact, the New Math strand developed by E.G. Begle's School Mathematics Study Group (SMSG) had become completely institutionalized by 1970. This was accomplished through the Houghton Mifflin series of books, which were copied by describing how this occurred was published in the EVERETT SCHOOL NEWS, posted at: http://mathforum.org/epigone/math-teach/blerdzhiclon/ and was reprinted as a filler item in The American Mathematical Monthly, January 2002, page 12. The impeccable SMSG credentials of Dolciani and her co-authors are posted at: http://mathforum.org/epigone/math-teach/skypablimp/tj3dd0o2xmip@forum.mathfo rum.com Domenico Rosa -- tml === Subject: More on vectors (but not as wired) With so many posts on vectors (especially the wierd cyclohexane one), I thought I'd submit a problem of my own. I can't figure our this one which I got from Nrich. The (1,1,1) vertex of a cube of unit side is cut off by a plane that intersects the edges of the cube at the points (1,1,0), (1,0.5,1), (0.5,1,1). What is the angle between the normal to this plane (directed away from the origin) and the (1,1,1) direction? Also... How can you prove that for arbitary a, b and c a.(bxc)=(axb).c = (bxc).a where x is cross Any ideas would be great! MI -- tml === Subject: Re: More on vectors (but not as wired) > The (1,1,1) vertex of a cube of unit side is cut off by a plane that > intersects the edges of the cube at the points (1,1,0), (1,0.5,1), > (0.5,1,1). What is the angle between the normal to this plane > (directed away from the origin) and the (1,1,1) direction? (I'm tired so be sure to double check my calculations.) 3 points in the plane are given. A vector normal to this plane can be found by crossing any two vectors in the plane. How about: u = component vector, meaning initial point at origin having direction and length as that of the segment from (1,1,0) to (1,.5,1) = <1-1,1-.5,0-1> = <0,.5,-1> <---this is the terminal point v = same for (1,1,0) to (.5,1,1) = <1-.5,1-1,0-1> = <.5,0,-1> u cross v = = <(.5)(-1)-(-1)(0),-[(0)(-1)-(-1)(.5)],(0)(0)-(.5)(.5)> = < -.5 , -.5 , -.25> Any other vector parallel to this therefore also normal to u and v, so: w = <2,2,1> ...simply cleared fractions by multiplying each component by 4 then additionally by -1 which made the components positive, strictly for convenience. It's still normal to u and v. You can verify by dotting this with either u or v, or any other vector in the plane, and the dot product should be zero. This is by no means the only course of action, but it's the one I chose. You could, alternatively, simply find a vector normal to u by inspection, more or less, and the fact that dot product is zero for orthogonal(normal) vectors: for some w = u dot w = <0,.5,-1> dot = 0 = 0+.5w2-w3 = 0 = .5w2-w3 = 0 ==> .5w2 = w3, regardless of w1 let's see, w = <0,2,1> would do it, since the third component is half the second, and indeed that's fine since: <0,2,1> dot <0,.5,-1> = 0 My goal here is not to bombard you with many different ways of approaching the problem, but just to make you aware that there are indeed different angles of attack, which also gives you practice finding cross and dot products, realizing their implications, which makes you generally more comfortable (hopefully) dealing with vectors. For the actual question, what is the angle this vector makes when directed away from the origin (and BTW <0,2,1> is indeed directed away from the origin since it's in component form thus by definition has initial point at the origin) with a vector in the direction of <1,1,1>. Utilize the formula for the angle theta between two vectors: cos(theta) = (their dot product)/(product of their norms) |<0,2,1>| = sqrt(0^2+2^2+1^2) = sqrt(5) and similarly |<1,1,1>| = sqrt(1^2+1^2+1^2) = sqrt(3) <0,2,1>dot<1,1,1> = 0+2+1 = 3 and plug this all into the formula: cos(theta) = 3/[sqrt(3)*sqrt(5)] = 3/sqrt(15) then... theta = arccos [3/sqrt15 > Also... > How can you prove that for arbitrary a, b and c > a.(bxc)=(axb).c = (bxc).a > where x is cross ...and . is dot product. Well, you could always use brute force if for no other reason just for the practice: let a = b = c = Find axb, and bxc. Then calculate a.(bxc), then (axb).c, then (bxc).a and see if they all three are the same. When tired of practice, a few generalizations are in order: . = ad+be+cf = da+eb+fc = . ...that equates the first with the last so all that's left is to equate the middle with either the first or last, not necessarily both. You may want to think of the cross product definition (the formula I gave for it above) in terms of a determinant (the method of evaluating a determinant is an easy way to remember the cross product). The value of a determinant is multiplied by -1 if two rows are interchanged. After two such interchanges, the value of a determinant will be unchanged. See if you can use this knowledge to show that these triple scalar products are equivalent. -- Darrell > Any ideas would be great! > MI -- tml === Subject: Re: More on vectors (but not as wired) > With so many posts on vectors (especially the wierd cyclohexane one), > I thought I'd submit a problem of my own. I can't figure our this one > which I got from Nrich. > The (1,1,1) vertex of a cube of unit side is cut off by a plane that > intersects the edges of the cube at the points (1,1,0), (1,0.5,1), > (0.5,1,1). What is the angle between the normal to this plane > (directed away from the origin) and the (1,1,1) direction? First find a vector that is normal to the plane that contains the points. (1,1,0), (1,0.5,1) and (0.5,1,1) Do this by first finding two vectors in the plane: a=<1,1,0>-<1,.5,1>=<0,.5,-1> and b=<1,1,0>-<0.5,1,1>=<0.5,0,-1> and finding a vector that is perpendicular to a and b c=a x b = <-0.5,.-0.5,-0.25> (check my numbers!) Well this vector is pointing toward the origin kind of so lets take -c Now what is the angle between <1,1,1> and -c=<0.5,0.5,0.25> Use: A . B =|A||B| cos(t) or cos(t)=A.B/(|A| |B|) cos(t)=<1,1,1>.<0.5,0.5,0.25>/(sqrt{<1,1,1>.<1,1,1>} sqrt{<0.5,0.5,0.25>.<0.5,0.5,0.25>}) cos(t)=0.96225 t=15.793 deg You should check my numbers > Also... > How can you prove that for arbitary a, b and c > a.(bxc)=(axb).c = (bxc).a > where x is cross > Any ideas would be great! Just do it: let a= b= c= Plug in and simplyfly! > MI -- tml === Subject: Sources for Math Applications Problems What are the sources for formulas or functions suitable for Algebra 2 applications problems? Just as one example, a logarithm problem refers to the frequency of a piano note as a function of its position on a keyboard as F(n) = 440 * 2^(n/12). Are there source books with lists of usable relationships like this? Or does each one have to researched separately? Any help will be appreciated. -- tml === Subject: Re: Sources for Math Applications Problems > What are the sources for formulas or functions suitable for Algebra 2 > applications problems? Just as one example, a logarithm problem refers > to the frequency of a piano note as a function of its position on a > keyboard as F(n) = 440 * 2^(n/12). Are there source books with lists > of usable relationships like this? Or does each one have to researched > separately? Any help will be appreciated. Atmospheric pressure varies according to P(h)=14.7(.5)^(h/3.6). You could ask at what height a certain pressure is found. David Moran -- tml === Subject: Re: Subtraction is evil. It must be stopped. > I teach Saturday school, in addition to my other classes. I love > Saturday school since there are *no discipline problems* But, working > with 11th and 12th graders (rather than 10th and 9th graders?) has > brought to my attention how important an understanding of signed > numbers really is. > I have students who can do all sort of complex math (well for high > school) but who still are confused about the meaning of 3 - 4 > They really really want it to be 1. Why? It's not dyslexia. Because subtraction is at first taught as being the difference between two numbers because that's an easy way of visualizing it when you're just dealing with individual digits. The difference between 3 and 4 is the same the difference between 4 and 3. Or maybe they just really like the commutative operations. As far as I can tell, Elementary School arithematic tends to gloss over a lot of the finer points with the hope that they'll be corrected when they get to Algebra. > I've started to wonder if the problem really is the teaching of > subtraction as an operation. Subtraction effectively disappears once > you get to algebra. You just add negatives. I remember a high school > teacher telling me this and I was so shocked. It was all just a lie! > Why do we teach children to subtract? Why not just teach adding > negatives then introduce the notation without the plus sign later. Basically, I think it is that numbers can be really hard to conceptualize, so schools try to avoid this by sticking to natural numbers as long as possible, moving into the more complex stuff (pun slightly intended) only when absolutely neccesary. I am not really sure of the finer points of education, (being a 12th grader myself) but I think it might be worth it some of the time to introduce negative numbers early, although I'm not certian how well students would be able to take it. -- tml