The problem with this, though, is that once they get accustomed to just memorizing facts instead of breaking numbers down, it is virtually impossible to convince them to do it in any other way than the one they are comfortable with. They are very, very resistant to going back and re-learning how to do it in another way that is less efficient at first, even if it is more efficient in the long run. I'd rather see kids working more slowly and with smaller numbers in the elementary grades, but learning mathematical strategies like this, than working faster through more material, but learning only fewer strategies. Edited: To me, teaching math facts as something to be memorized seems more like teaching words as something to be memorized, rather than teaching phonics rules so that children can figure out words that they don't know by sight. Yet I have been told by reading teachers that it was too difficult for children to learn phonics and that whole words were "better for them", and that if they needed phonics they could learn it later. As with math, children (and adults) who have learned to read inefficiently with whole words are frequently very resistant to going back and starting with phonics.

Sometimes it also works to just take a break for a bit, work onwards in other math, and then come back and try again. If your supplements don't work, you might consider this.

I would recommend strongly against it. While there IS review of those chapters, it is brief and assumes the student has already seen them and needs only to be reminded. For example, compare chapter 11 -- rational expressions -- to chapter 7 -- rational algebraic functions. In Alg 2/trig, there is a brief introduction and then it goes immediately to discontinuities and asymptotes. In Alg 1, it builds more slowly, going carefully through simplifying, multiplying and dividing, how to find LCM, how to add and subtract ... this is covered in far fewer sections in Alg 2/Trig. This difference is even more pronounced in chapter 13 -- inequalities. In chapter 14, quadratic functions are covered, which is a really essential algebra 1 topic. I would recommend instead going quickly through any sections in Alg 2/Trig that seem to be review, assigning just a few of the more difficult problems.

Beast Academy, Zaccaro's Challenge Math, Hard Math for Elementary School (http://www.technologyreview.com/article/524231/when-math-is-no-problem/), Cleo Borac's Competitive Mathematics for Gifted Students (http://www.amazon.com/-/e/B00DXHDM2A)

Looking for AP or just a course?

Isn't SM 6 a bunch of review for the year-end tests? A lot of people have skipped it and gone to pre-algebra.

While I totally agree that breaking down something like this seems unnecessary, the whole point of teaching kids to break down something like 9 + 6 (where it's easy) is so that they can break down problems like 199 + 106, and then continue to problems like 1979 + 176. Learning a new mathematical skill where the problems are very difficult and the numbers are bigger tends to lead towards frustration as well. Again, though, I think that if the teacher doesn't understand why they're doing this, or why the curriculum wants them to learn this -- if the teacher doesn't see where it's going later on -- it's going to be little more than a different kind of rote memorization for the students.

Just because the author says it doesn't make it true.

I still haven't decided on one.

More seriously, I don't like it as a side. It's too filling and distracts from the main dish. I prefer it with meat added and a side of vegetables -- salad or steamed veggies, whatever you want.

The whole point of p/f is that it doesn't count into the gpa at all. Are you using transcript generating software that calculates your grades for you? If so, you might have to count it as a graded course in order to workaround your software. If not, though, just divide by one fewer course. For example: Math: A English: B Science: B History: C Spanish: C PE: P There are five graded courses. GPA: (4 + 3 + 3 + 2 + 2) /5 = 14/5 = 2.8

It is legal as long as you remember at your final step that y cannot under any circumstances equal 1. For this one it does not matter, but if you were given a similar problem where your final solution ended up being "y = 1" you would need to recognize that it was not a real answer (because when you try to plug 1 in to check you end up dividing by 0) and cross it out.

Now I really want to make Mac and Cheese. THANKS.

Yes. And Liping Ma's book discusses the results of similar lack conceptual understanding when she discusses teachers who were asked how they would deal with a student who always forgets to move over the partial products when doing a sum such as 123 x 456. If someone really, deeply understands that the standard algorithm is really doing 3x456 + 20x456 + 100x456 and then adding up the results, they should never make this mistake. Yet many of the teachers said they would teach the students to put asterisks in the space because there were no numbers there, or that they would teach the students to put 0s there but remind them that the 0s weren't really there. Of course they are there! It is just shorthand to leave them out.

Yes. In order to put this on one line, you need to use parentheses in the denominator as follows: y/(y-1). If you write y/y-1, because of order of operations, it would look like this as a fraction: y -- - 1 y Now, as to what was wrong with your first attempt -- you had: 3/2 --------- 3/2 - 1 and attempted to cancel the 3/2. You cannot do this. When you cancel common factors from a fraction, what you really are doing is dividing the whole top and bottom by that number. For example, in 10/14, when you reduce it to 5/7 you are dividing the top and bottom by 2. This is legal -- it is the same as multiplying the fraction by 1/2 over 1/2, which is 1. If you try to cancel the 3/2 here, you are dividing the top and only PART of the bottom by 3/2. That's illegal.

If he's concerned about not getting full aid, he can enroll for 3 credits of courses such as PE (they offer many options at most universities), Health, or other courses that are usually 1-credit courses which do not require excessive time outside of class, in addition to 9 credits of academic classes. Even many engineering schools require some sort of lifetime fitness classes for gen eds, so he could get that done.

Frankly, for you, your student, and your situation, it sounds like a good idea. It is very common for underprepared engineering students to start out at a community college. I would recommend getting through calc 1 before transferring, and then repeating calc 1 at the engineering school for a firmer foundation. While he is at it, he could do algebra-based physics (he will need calc-based for engineering school, but again, this will help his foundation) and work on gen eds to lighten his load once he gets there. Other class recommendations would depend on where he places and how long he needs to spend at the CC to be ready for calculus, but I would recommend economics as a social science gen ed and intro to programming as an elective.

Was it written as a fraction? Like y ---- y-1 ? Because if it was written as y/y-1 (in one line, just like that), then the actual answer would be "none of them".

I absolutely agree that for some kids beginning with procedural fluency (the algorithm) and then figuring out how they got there later is a far superior method to trying to front-load the concepts. I definitely also agree with you about the difficult task the public schools have. Don't forget, also, to add in teacher competence -- because the conceptual curricula, while I much prefer them, are extraordinarily terrible with a teacher who does not understand math at a fundamental level. If I were choosing curricula for an elementary school with teachers whose understanding of math was shaky themselves, frankly, I would use Saxon (even though I do not care for it) because I think that the heavy emphasis on procedural fluency would be much better for the students than a half-understood conceptual curriculum. It is much easier to teach concepts later to students who are procedurally fluent than to try to teach concepts and procedures to students who have understood little and retained less.

He's got some very good points. Some in particular: "My impression is that too many K–8 teachers in this country, and consequently too many students as well, see mathematics as lists of disjoint, disconnected factoids, to be memorized, regurgitated on the proximal test, and forgotten, much like the dates on a historical time-line. Moreover, and more disturbing, since the isolated items are not seen as coherent and connected, there does not seem to be any good reason that other facts cannot be substituted for the ones that are out of favor." This viewpoint is definitely prevalent and very worrying. It is especially prevalent among developmental students, and trying to train them to see the bigger picture, rather than attempt to memorize separate algorithms for each type and sub-type of problem, is a huge issue. I also strongly agree with him about the lack of precision in definitions. Again, this is not something that I'm applying to advanced undergraduates or as a mathematician, but the fundamental lack of understanding I see of students in developmental math classes. They'll look at an equation like "2x = 14" and not understand whether they're supposed to subtract 2 from both sides or divide by 2 on both sides, because they don't really understand what 2x means. Another comment: "For the common stages of school mathematics, students must practice with numbers. They must add them until basic addition is automatic. The same for subtraction and multiplication. They must practice until these operations are automatic. This is not so that they can amaze parents and friends with mathematical parlor tricks, but to facilitate the non-verbal processes of problem solving. At this time we know of no other way to do this, and I can tell you, from personal experience with students, that it is a grim thing to watch otherwise very bright undergraduates struggle with more advanced courses because they have to ï¬gure everything out at a basic verbal level. What happens with such students, since they do not have total fluency with basic concepts, is that – though they can often do the work – they simply take far too long working through the most basic material, and soon ï¬nd themselves too far behind to catch up." If someone is attempting to learn algebra, and wants to factor something like x^2 - 17x + 42, but cannot reliably factor 42 multiple ways without a calculator, this will be a tremendous hindrance. The fluency with basic calculations is very necessary for higher-level performance in any sort of math. It is possible for an extraordinarily bright person to achieve without it, but it is far more difficult.

Pretty sure she meant y/(y-1). You are correct that the equation as written would evaluate to (y/y) - 1 = 1-1 = 0 for every nonzero value of y (which would mean there would not be a solution), but that interpretation wouldn't make sense (imo) with the question asked. Therefore I answered the question that I believed she meant to ask rather than the one she actually asked.

Does the answer key say 3/2? If y = 3/2, then y-1 = 1/2, and 3/2 divided by 1/2 is 3.

I think she'll be much more competitive with solid knowledge of alg 1 - geom - alg 2 - precalc than she would be with rushing through 5 credits.

Especially this. I'd prefer to grade based only on either examinations or (for more advanced courses) fewer, but more comprehensive problem sets. The issue is that if I grade on only examinations, the majority of students will do no studying until possibly the week of the examination, and then they will all fail.

Agree on this. FTR, I wouldn't have let him fix it either, if I were teaching. A quiz is a very low-stakes way to learn to read the directions and check both sides of the paper. Learning this BEFORE College is a good thing.