mm-408 === Subject: Re: Matrix of Rotation of Coordinate : If I want to get the matrix representing for a coordinate rotation : about an arbitrary axis specified by the unit vector, say, (a,b,c); : how can I find that matrix out? How about ts... Form an orthonormal (right-handed) coordinate system: (a,b,c) (d,e,f) (g,h,i) using the Gram-Schmidt process or whatever is convenient. Then your matrix is: / a d g / 1 0 0 / a d g -1 | b e h | | 0 Cos -Sin | | b e h | c f i / 0 Sin Cos / c f i / The right-most matrix does a change-of-coordinates to the dard (1,0,0) (0,1,0) (0,0,1) coordinate system. The middle matrix does the rotation about the first vector in the coordinate system. The left-most matrix changes the coordinates back. -- === Subject: Re: radicals,quadratic equations etc <... Some rules, where here, a and b are integers: > (a*b)^(1/2) = a^(1/2)*b^(1/2), wch for radicals in square roots is > sqrt(a*b) = sqrt(a)*sqrt(b). Incorrect. Let a = -1, b = -1. -- -- === Subject: Re: radicals,quadratic equations etc Of course, positive integers. (The context is side lengths, always a positive number.) -- === Subject: Division Activities I am a new fifth grade teacher. We are working on long division (ugh!) in my class and I have several students who are having difficulty with the concept. What is the best way to teach ts concept? Are there any activities or manipulatives that help the students with ts? I have searched for information on the Internet and in my elementary math ed. textbooks, but find that the long division algorithm is not explained any differently from the way I have been attempting to teach it. Ts is with a two-digit divisor and a 3-4 digit dividend. Any help you could give would be greatly appreciated! === Subject: Re: Division Activities > I am a new fifth grade teacher. We are working on long division (ugh!) in > my class and I have several students who are having difficulty with the > concept. What is the best way to teach ts concept? Are there any > activities or manipulatives that help the students with ts? I have > searched for information on the Internet and in my elementary math ed. > textbooks, but find that the long division algorithm is not explained any > differently from the way I have been attempting to teach it. Ts is with a > two-digit divisor and a 3-4 digit dividend. Any help you could give would > be greatly appreciated! > Louisiana In addition to what Rich recommended, remember that long division is harder to learn if certain prerequisites are not first mastered. In ts light, here is a link: http://emintsteachers.more.net/lograssc/Math_files/55MathXD/division/divisio nLong.html Here's another link: http://www.jimloy.com/arith/longdiv.htm The concept is simple, in that we're just finding out how many divisors we can add together without going over the dividend. The remainder is just that dice between ts sum and the dividend. Long division can be painful with divisors of 2 or more digits because of all that estimation regarding multi-digit numbers. There is a way to eliminate most of the painful estimation, but it takes a lot longer sometimes. The idea to go up powers of 10 of a divisor without going over the dividend. Ts might be best reserved for really big dividends and divisors: Solve 6,591/29. The problem is to find how many 29's we can add together without going over 6,591. The remainder is the dice between ts sum and 6,591. Go up powers of 29, getting as close to 6,591 without going over 6,591, wch is 29*100 = 2,900, then subtract: 6,591-2,900 = 3,691 We repeat the procedure, going up powers of 29, ts time getting as close to 3,691 without going over 3,691, wch is 29*100 = 2,900, then subtract: 3,691-2,900 = 791 We repeat the procedure, ts time on 791 with 29*10 = 290: 791-290 = 501 Then on 501 with 29*10 = 290: 501-290 = 211 Then we do 211/29 the usual way, obtaining 211 = 29*7 + (remainder) 8. We add up all those factors of 29 that we used, 100+100+10+10+7 = 227, and we have our answer, 227 with remainder 8: 6,591 = 29*227 + 8. Like I said, it's longer than the usual way, and the usual way is used at the end anyway, but there's less of painful estimation involving multi-digit numbers. (Of course, if 211/29 were the problem to begin with, the above wouldn't be an option. The concept still would be to find how many 29's we can add together without going over 211, with the remainder being the dice between ts sum and 211.) -- === Subject: Re: Division Activities >I am a new fifth grade teacher. >I have several students who are having difficulty with the >concept. What is the best way to teach ts concept? Are there any >activities or manipulatives that help the students with ts? I have >searched for information on the Internet and in my elementary math Are you sure it is the concept wch they have difficulty? Or is it the method? You can demonstrate the concept on physical, countable items. No need for any internet search of special techniques or fancy manipulatives. The concept should come first; then indoctrinate for the method. You could try matcng a physical demonstration with the corresponding numeric symbol operation steps. Ts will help build the concept. An important fundamental objective is to see that division is the count of how many times a quantity is contained in another quantity. -- === Subject: Re: Division Activities >I am a new fifth grade teacher. We are working on long division (ugh!) in >my class and I have several students who are having difficulty with the >concept. What is the best way to teach ts concept? Are there any >activities or manipulatives that help the students with ts? I have >searched for information on the Internet and in my elementary math ed. >textbooks, but find that the long division algorithm is not explained any >differently from the way I have been attempting to teach it. Ts is with a >two-digit divisor and a 3-4 digit dividend. Any help you could give would >be greatly appreciated! Hm. I wonder if ts is one of those cases where maybe the answer should be use a calculator. I assume the students know what division is (conceptually and manipulatively). If so, why do long division by hand any more? Can they do division with a one digit divisor using long division format? -- === Subject: Re: Division Activities > I am a new fifth grade teacher. We are working on long division (ugh!) in > my class and I have several students who are having difficulty with the > concept. What is the best way to teach ts concept? Are there any > activities or manipulatives that help the students with ts? I have > searched for information on the Internet and in my elementary math ed. > textbooks, but find that the long division algorithm is not explained any > differently from the way I have been attempting to teach it. Ts is with a > two-digit divisor and a 3-4 digit dividend. Any help you could give would > be greatly appreciated! > > Louisiana There are two resources I can recommend. The first is a draft of a chapter on whole number arithmetic from an upcoming book by Hung-Hsi Wu, a math professor at Berkeley who has done a lot of work in teacher education, Here is the link: http://math.berkeley.edu/~wu/EMI1c.pdf The second is a book by a woman named Liping Ma called Knowing and Teacng Elementary Mathematics. It is far and away the best book I have ever read on the subject of teacng elementary mathematics, even though that is not its primary mission. Here is a link: http://www.amazon.com/exec/obidos/search-handle-form/104-9982144-7739143 Hope ts helps. Rich -- === Subject: Re: Division Activities > There are two resources I can recommend. > The first is a draft of a chapter on whole number arithmetic from an > upcoming book by Hung-Hsi Wu, a math professor at Berkeley who has done a > lot of work in teacher education, Here is the link: > http://math.berkeley.edu/~wu/EMI1c.pdf > The second is a book by a woman named Liping Ma called Knowing and Teacng > Elementary Mathematics. It is far and away the best book I have ever read > on the subject of teacng elementary mathematics, even though that is not > its primary mission. Here is a link: > http://www.amazon.com/exec/obidos/search-handle-form/104-9982144-7739143 > Hope ts helps. > Rich for your help. Your information helped solidify my belief that I should show my cldren how to divide by each digit in the dividend individually rather than looking and 2 and 3 digits at a time. I tnk that ts will help them better underd placement of the digits in the quotient. -- === Subject: Word Problem help Hey..please help me with ts word problem: The auditorium at Centennial gh Schoool has one thousand seats. numbered from 1 to 1000, One day each seat was filled and the 1000 people followed these directions: First, each person stood up. Next, every second person, including the person in seat two sat down. Then every trd person including the person in seat 3, changed to the opposite. That is, if the person was ding, he or she sat down. If the person was sitting, he or she stood up. Following ts, every fourth person, including the person in seat 4, changed to the opposite. Then, every fifth person, including the person in seat 5, changed to the opposite, and so on. Finally, the one thousandth person changed to the opposite. After ts last change, was the one thousandth person sitting or was that person ding? Questions: 1. Was the person in seat 1 sitting or was that person ding? 2. For wch of the seats 1-20 were people sitting? ding? 3. Was the person in seat 1000 sitting or was that person ding? 4. For wch of seats 1-1000 were people ding? -- === Subject: Re: Word Problem help > Hey..please help me with ts word problem: > The auditorium at Centennial gh Schoool has one thousand seats. > numbered from 1 to 1000, One day each seat was filled and the 1000 > people followed these directions: First, each person stood up. > Next, every second person, including the person in seat two sat down. > Then every trd person including the person in seat 3, changed to the > opposite. That is, if the person was ding, he or she sat down. If > the person was sitting, he or she stood up. Following ts, every fourth person, including the person in seat 4, > changed to the opposite. Then, every fifth person, including the > person in seat 5, changed to the opposite, and so on. Finally, the one > thousandth person changed to the opposite. After ts last change, was > the one thousandth person sitting or was that person ding? Questions: > 1. Was the person in seat 1 sitting or was that person ding? > 2. For wch of the seats 1-20 were people sitting? ding? > 3. Was the person in seat 1000 sitting or was that person ding? > 4. For wch of seats 1-1000 were people ding? For the moment, forget that there are 1000 seats, and consider just the first, say, 20. Work out a few steps, then see if you can see a pattern of some sort, or work out what is happening. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 (step) u u u u u u u u u u u u u u u u u u u u (1) u d u d u d u d u d u d u d u d u d u d (2) u d d d u u u u d d u u u d d d u u u d (3) u d d u u u u d d d u d u d d u u u u u (4) u d d u d u u d d u u d u d u u u u u d (5) and so forth. Some questions you might ask: * What makes a person change from ding (u) to sitting (d) or vice versa? That is, how must the person's seat number relate to the step number for m/her to change? * For any given seat, how many times will the person change from sitting to ding or vice versa? * If a person changes position n times, what is s/her final position? * What are the conditions that will result in a final ding position? What are the conditions that will result in a final sitting position? It's a bit hard knowing how much of a nt to give without just giving the whole tng away. Even if these questions don't make much sense, at least do the full 20 steps for the first 20 seats and look for a pattern. If you still can't see it, do a few more seats and steps ... perhaps up to 25 or 30. -- === Subject: I Am Desperate I am having trouble with line segment congruencies and being able to fold a sheet of paper to come up with an exact measurement. Ts is still part of the basics so it is essential that I learn ts right away. --