Topic Author's Original Post - Feb 21, 2014 - 09:09pm PT

well finals are coming up next week and my math teacher gave me a review packet with 50 problems that I dont know how to do. sooo... if any of you guys know your math then post up how to do the problems or the solutions and work, anything will help!!! help whitemeat out with school!!!!!!!

Your test is easy. It should take you less than an hour with no calculator, your teacher probably 10 minutes, a college student in math or engineering less than that. If you had a TI graphing calculator, you could solve and graph the whole thing that way. Lacking that, try http://www.mathportal.org/calculators.php However, you're only cheating yourself - like bringing a ladder to the crags - except worse. Good luck. I'll take fries with that.

I am going to assume you are not trolling .. . . .
You can check your work by entering the equations into the browser www.wolframalpha.com. For $4.99/month you can see all the solution steps.

My son uses this site to check his Algebra homework.

I'd like to help, but I'm not sure how my solving all the problems and putting the solutions up here will get you through your test. It's unlikely that your teacher will use exactly the same problems... though your teacher is likely to use the same "ideas" behind the problems given.

So what would you propose?

Maybe if you start, and put up what you've done, and ask for checks, and perhaps hints, or if you're really having difficulty, working through a problem...

But you've got to take the initiative... to paraphrase Jimi Hendrix "You're the one who has to take the the test, when it's time to take the test..." not me.

...one other thing, I'll be out climbing this weekend, but there are a lot of others here who could help.

Which ones can you do?
1-2 look hard - these involve factoring a 3rd-degree polynomial.
9 looks easy. Can you do it?

Can you factor a second degree polynomial?
If not, learn how to do that first:

x^2 + bx + c = 0

factors as:
(x-r)*(x-s) = 0

where r+s = -b
and r*s = c

Basically you think of numbers r and s which multiply to get c and
add to get -b .
Then r and s are the values of x for which the polynomial is zero.
(Aka "roots").

You shouldn't have been able to do your homework problems without
knowing how to do this.

(2x+4)(2x+5) for number 9 clint??? thanks for all the replies, I know number 47 and a couple others but not much, well math is really easy for me to learn I just forget it REALLY fast...

keep them coming!!!

briham, I have a good enough grade to get a 0% and still pass... not going to happen though, I would drop out of school before giving up my summer yosemite dirt bag days!! we will crush briham!!!!

I could tell someone everything they need to know to do a handstand, but it would still take them several months of practice before they even begin to do one.

Many people could write down all the workouts you need to build to your first marathon, but if you try to do all the workouts in the week before the race---it ain't gonna work.

There are so many climbing analogies, and you're good enough at it to supply your own.

I don't know how you got to this sorry place, what you did or didn't do, but I don't think there's any kind of free get-out-of-jail card. If you really don't know how to do any of those problems, as you say, then looking at a bunch of solutions produced by people who do know how to do this stuff isn't going to help, at least not in the short term.

If that's the case, chances are your only option at this point is to retake the course with the intention of changing whatever conditions led to your current situation.

On the other hand, if you do know how to do them but just sorta forgot, then look at your old homeworks, your old tests, and the textbook and do some serious studying.

briham, I have a good enough grade to get a 0% and still pass... not going to happen though, I would drop out of school before giving up my summer yosemite dirt bag days!! we will crush briham!!!!

Woohoo!!!! I'm glad you are keeping your grades up. Can't wait for spring/summer!

Also don't ever drop out of school!!! I know you won't, but I have to be an "adult" sometimes and say things like that ;)

Not counting coup, but I liked Mr. Hartouni's snappy answers and sound advice ... Wasn't there something about inflection points too?

if it is any help whitemeat, i remember a math professor saying that he always felt some frustration before solving a problem, then relief when he found the answer.

Maybe check this site as a reward for every few problems you solve too.

No, that doesn't work (you do get 4x^2 + 20, but you don't get 0*x).
But you are doing the "guessing" process of factoring right.

9. 4*x^2 + 20 = 0

This one you don't have to factor, because there is only one term with "x" in it. Factoring is for when there are both x^2 and b*x .
So for 9, just use simpler algebra to get the one x term on the left side and everything else on the right. Then it's fast to solve for x.

In general, there are just a couple of tools for quadratic equations, and you try to use the one that is fastest.

1. a*x^2 + c = 0
2. b*x + c = 0
Only one x term - just solve direct for x as above.

3. x^2 + b*x + c = 0
A. First try factoring as (x+r)(x+s), by guessing integers.
If c is negative, r and s are opposite sign, right?
If b is negative, at least one of r and s is negative.
B. If your guesses don't work, use the quadratic equation (it is slower
but always works, and it also tells you if the solution is imaginary).

4. a*x^2 + b*x + c = 0
If everything divides by a, do that and try factoring.
Otherwise use the quadratic equation.

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But seriously, this is YOUR homework. Do the work. When you go climbing you don't ask people to haul your ass, do you?

Those problems are not that difficult. You can do it!

It's nice that Clint has gone into full tutoring mode. I'm not going to do that myself, for reasons that should be evident from what I've already posted.

However, it is worth mentioning, perhaps as much for others who are curious as the OP, that the first two problems require a little more knowledge than the other factoring problems, specifically the rational roots theorem, the factor theorem, and the ability to do "long division" with polynomials. Descarte's rule of signs is not essential but can be helpful in cutting down on the possibilities to be checked.

As the name of the first theorem suggests, these techniques will only work if the polynomial happens to have a rational root, which is of course the case for the given problems. But there are cubics with integer coefficients, e.g. x^3 + x^2 - 1, which have no rational roots.

The problem of finding the roots of cubic polynomials and then of higher-degree polynomials has led to an enormous amount of mathematics, including the discovery and use of complex numbers and the ground-breaking work of Evariste Galois that settled all the original questions and opened the door to research questions that are still very much alive today.

Some thoughts on math in general. I hated school and wasn't about to put an ounce of extra effort into it. Generally I skated by but math was a weakness, not becuase I couldn't learn. I was pretty fair at picking it up when I put effort into it. Sounds a lot like what you said earlier.

The problem with math is this. It builds up and if your foundation isn't solid you will get lost and blow out at some point.

Sadly for those who hate school this makes math rough even if they have a decent head for it. In math you HAVE to do the work. It can't really be cheated.

Oh well it seems life has chosen a different path for you. Based on your writing and math skills and clear interest in not putting your effort in school I'd say college is out for a while and climbing will be in.

Not bagging on you one bit, and I might be wrong here. You gotta follow your heart in life especially if it's burning with specific dreams. In this I suspect you are a very lucky young man, I think you've got some burning dreams and that in my short 40 years of life experience seems one of the truly valuable things to have in life.

So my advice , Be kind to all you meet, help everyway you can wherever you find yourself, Become an expert in and study what you love. Grab the opportunities that come along from time to time. If I were you I'd consider YOSAR for a couple seasons or more.

I was a math tutor for the Boys & Girls clubs last year. It had been at least 25 yrs since I had done any of the types of problems that were like what you show here. I tried using example problems and the textbook to get caught up before a tutoring session, but it was taking too long.

Some friends told me that the intardnet now has numerous great websites where you can go online and have a video tutor walk you through each type of problem. They give enough examples so that you can actually learn (or in my case relearn) the subject, it's a great resource, take advantage of it! I was able to prepare in 30 min. for a whole slew of different students at different levels of algebra and geometry.

Math (and all the other high school classes for that matter) are the "hauling" of the real world. Dirty, ugly, painful work that you gotta do if you want to be a bigwaller. Study, educate yourself while youre young, and climb until you're an old man. If you skip out and don't try hard and just go play all the time you'll be 40 years old working night shifts at Taco Bell. You've got moxy kid, we all know that. I hate math too, but get it done, don't cheat, do your own work, suffer a bit, and the summit will be within striking distance in no time.

Be good at math because there will always be someone trying to take your money.

I was horrible at math all the way through high school. Spent five years in the Navy and then went back to school. Now, my highest level of math is calculus 1. While I never use calculus now, I know bullshit when I hear it.

As a Middle School math teacher, my advice is skip Number one and two and focus your time on the rest of the test. When all else is done and double checked, if you have time, go to 1 and 2.

Instead of asking for help on these equations, they are no problems in math, just equations, visit https://www.khanacademy.org/ and learn how to do these. Videos are short and excellent.

I remember telling all of my teachers that I hated math....didn't want to learn math.....and that I would NEVER enter a career that required me to use math.

I ended up in a career that has me using extensive amounts and numerous forms of math ever minute of every day. Running my own business requires it constantly.

I didn't like school in general and hatted math even more. Avoided the subject most of my life. Like you, when I started my own business, I ended up using it a lot When I sold the business, I was pretty comfortable with numbers in general.

My first middle school teaching position, they needed a teacher to teach an Algebra classes, so I added it to my teaching schedule. That was 11 years ago, now, I teach math, occasionally something else.

Over the years, I've had the opportunity to observe a lot of math classes. I think one reason students don't do well or enjoy the subject is that Math is often assigned and not taught.

I was being observed by a math teacher once, a peer to peer evaluation. She was very traditional with lots of Skill and Drill. When we spoke afterward, she explained to me that my Math class was "To much like Science." My student did hands on activities, collected data instead of reading it from a text, and talked among themselves about the material. She meant it as a criticism, I took it as compliment.

Get students excited about the classroom, they will start to enjoy the subject matter, once they are excited about the classroom and enjoy the subject matter, you can really begin teaching.

For number 36: any 3rd degree polynomial function with real zeros of a,b and c will have 3 distinct linear factors of (x-a), (x-b) and (x-c).

So for your problem with given zeros of -3,0 and 5 the equation you are looking for is
f(x) = (x - -3)(x - 0)(x - 5), which is equivalent to f(x) = x(x + 3)(x - 5). It will be a polynomial with rational coefficients and a leading coefficient of 1. I'd multiply the rest out for you, but I generally charge $55/hr. for tutoring this stuff when I'm not teaching it in one of my Algebra 2 Advanced or Advanced Precalculus classes. (Never ceases to amaze me that parents will lay that kind of $$ down without a second thought, when their kid could get on Khan Academy, do a simple search and get excellent instruction for free.)

Personally, I admire your resourcefulness getting on the Taco for this, and cannot imagine what it must be like to sit in high school math classes having experienced the Shield. Good luck with that. Had I climbed El Cap before graduating high school, I never would have finished.

If you do choose to throw out the test and go climbing, I recommend that you not become one of those Valley lifer egomaniacs that thinks all Americans are stupid.

Micronut considers the length (l/s)of slack in the rope and how it will affect the vector(v) and kinetic energy(ke) involved in a potential swing off the pitch approaching Crescent Ledge on Fairview's Regular Route.

If a gallon of water weighs 8.34 lbs.,
and you paid twelve cents per gallon to buy Evian in the largest size,
did you get a deal?

Who can you pay to haul that tonnage that you estimate you might need
to climb the Aquarian solo at comparable wages?

We all know about Chongo. He usually works for free, when he works.

He taught himself to do the math.

You can be like Chongo, a voice in the wilderness, write books and sit back and enjoy "the life of Reilley," a builder who knows all the angles, or like norwegian, who knows all the angels.

Trip on that, young man. Then get to studying--you'll be more likely to become a John Gill of the Walls.

Unless you become a math teacher, or physicist or something, you'll forget all this sh#t in 10 years anyway. Doesn't mean you don't need to learn it.

Math was easily my best subject, I'm a working civil engineer, and these days I couldn't have completed your test without a book to consult. Couldn't even remember the quadratic formula without minutes of thinking about it and writing it down to see if it looked right.

One college professor told us "the best thing you'll learn in school is HOW TO LEARN". He was exactly right. In college there's no hand holding, you sink or swim, and quickly learn how to teach yourself from whatever resources available..fellow students in study groups, books, and these days online resources like Khan (you whipper snappers don't realize how lucky you are to have these modern tools). If I needed to use those math concepts today, I'd pull down a book, reacquaint myself in a couple minutes and do it. That only works if you learned and understood it the first time around.

So before you blow off school to live in the dirt, think about this:

Would you rather have a pimped out sportsmobile, hot wife, not think twice about dropping cash at restaurants concerts or gear, and be able to retire from work, while living in cool cities and taking trips to climb in the alps, himalaya, patagonia...or would you rather be digging ditches at 50 yo, crippled from a lifetime of bullshit manual labor and bad nutrition, with 3 rotten teeth left in your head, having climbed the same 50 routes over and over because you're too poor to go anywhere and your partners are tired of supporting your bum ass, and have not been laid since you were 25?

Educate yourself, don't blow it off because it isn't fun. Whether it's for white collar desk work, or craftsman/trades work, or anything else, apply yourself and get GOOD at what you do. Find something that you like and become the best you can be. And not one of these bs things everyone thinks is the ticket to happiness...photography, guiding...that ain't it, too many people competing for too few $$$ means sh#t wages and a bare few who rise above that, much like trying to be an NBA player...millions try, 50 make it.

People - employers, potential mentors, etc will judge you on your past record, including school. They will judge you on your ability to speak and write, not whether you can stand in aiders and play with widgets...a look at the climbing bum contingent will quickly show you any moron with 3 brain cells can climb up some etriers and swill bad malt liquor.

Unless you become a math teacher, or physicist or something, you'll forget all this sh#t in 10 years anyway. Doesn't mean you don't need to learn it.

It's not about the math...it's about training the mind to think about problem solving and looking at multiple variables and figuring out how to solve a problem...regardless if it has math or not. When "we have a problem Houston" ... And all that "stuff" was thrown on the table, I would suggest that the type of thinking and discipline of mindful problem solving did a great deal to get them home....it wasn't raw math but the ability to engage in mindful and rigorous problem solving. Mathematical and scientific thinking are excellent for training the mind to see things in a problem solving mode. Even if you don't go into those disciplines the mind training is invaluable. If you forget the "arithmetic" the ability to mindfully problem solve across multiple contexts and situations will remain. Especially if you ever have to self rescue.

That only works if you learned and understood it the first time around.

+1

Do you have a crystal ball so you can say you will NEVER need any math?
No.

Even carpenters need some.

Or are you just going to dig ditches?
Wait a minute, even ditch diggers have to avoid buried lines.
So they have to figure out where those lines are.

I recently had to find the minima & maxima of a curve that is the sum of cosines.
I don't use math enough that I could just write down the derivative.
But I have enough of a background that I knew that I could figure it out.
And I did, although I had to work it out step by step.

"the best thing you'll learn in school is HOW TO LEARN".

But seeing the comma OUTSIDE the frown and the smile means that, automatically, post-haste, you gotta spend the night in the (figurative) Taco walk-in/lock-in or give up those points.

Live & learn, or learn to live, as I do.

Sponge-Bob Bobcat from Merced

Honest Abe, the autodidact with tact intact, quaintly studying his new guide to sharpening his wit. Splitter knows...

Credit: mouse from merced

Fail to plan, plan to fail, in math, or anything except good old Secks. That should involve "spontaneity."

O! the magic of cell division
It oft requires no decision.

Agree with the posts above. Learning to learn is the most important skill you develop in any situation. True in school, work and climbing. The more you arm yourself with a solid foundation, the better chance you have of success.

One advantage of learning is it opens new doors. You can choose to have a career or you can choose physical labor. At least you have a choice.

After high school, I took a year off and lived in the Valley. I think most people call them "gap years" now. At the end of the year, decided school wasn't for me, took a year living in the car. Eventually, when I was 30, I headed off to college. Now I have a career that is rewarding and I can afford to travel to various climbing areas around the globe.

Unfortunately I have set vacation dates, so some areas I will never get to. Well, unless I take another GAP YEAR soon.

Cheers from Mongolia where the weather is warming up and it's almost time to go climbing outdoors again. Just need to buy a drill to put up some new routes.

Don't listen to all these naysayers, Whitemeat. The World is ruled by powerful people who are clueless yet somehow talk an army of lackeys into doing all their dirty work. This thread is proof positive that you are destined for greatness, though you still have a ways to go in perfecting your technique.

Think about it, Clinton talked Hillary into tackling health care, Madelaine Albright into tackling diplomacy, and his Willy into tackling Monica Lewinski. The dude makes bank now giving speeches, and you can bet your favorite haul bag that he doesn't write them.

Math, shmath, that's what calculators and Internet forums are for.

The World is ruled by powerful people who are clueless yet somehow talk an army of lackeys into doing all their dirty work.

If they were clueless, they wouldn't know how to talk others into working for them...

The lackeys who are not clueless can become powerful people, when they learn how to talk others into working for them. But when they end up working for others, maybe it's just because they need the money.

Knowledge is power. Learning how to learn can yield knowledge.

One of the many problems in math education is the tendency to treat the subject as procedural. Here's how you do this, here's how you do that. This boils down to viewing the learners as automata that need to be programmed to carry out certain tasks. The machines do not need to understand any of the issues surrounding or raised by the tasks, they just have to know how to accomplish the tasks.

The result can easily be Pink Floyd's Another Brick in the Wall (Part 2)

We don't need no education,
We don't need no thought control!

Given this situation, I don't find it at all surprising that lots of people hate something they think is math, and are anxious to tell others about the irrelevance of the thing they think is math. I don't find it surprising that people forget all the procedural mumbo-jumbo as soon as possible, even when they are going to need it in a few months for the next math course.

Modern life takes place embedded in an ether of mathematics, an ether that is invisible to most people, but which shapes and, by and large, enhances their lives and small and enormous ways every day. It may well be that only a relatively few wizzards need to understand the workings of this matrix, but it does seem appropriate, at the beginnings of education, to at least start people on a path that will allow them to become wizzards some day, if the calling beckons.

Math grad students make lots of money tutoring people who thought they'd never need to know any math, but who now find their career opportunities and future dreams obstructed by their lack of mathematical knowledge. True, these people may be a small minority, but I can tell you from the other side of the tutoring relationship, that it sucks to be them and they are painfully aware of it.

For those who, like climbers with climbing, see something compelling where others see something repulsive, mathematics is far more than the hidden rods and levers of the utilitarian ether mentioned above. It is, much like climbing, a limitless world of intricate problems and solutions, interesting for their own sake, with no concern for potential utility.

A number of years ago, there was an ad in one of the climbing magazines that mentioned a seminal mathematical event. A mathematician by the name of Andrew Wiles had recently invested almost all of the waking hours of seven years in a successful effort to prove a theorem (the Taniyama-Shimura Conjecture) which has as one of its consequences a famous unsolved problem called Fermat's Last Theorem. The ad showed a bunch of boulderers staring up at a problem, recounted the fact that Wiles had spent seven years working on and finally "sending" his "project," and ended with, "Can you relate?"

Yeah, so I think climbers, more than almost anyone, can relate to mathematics.

One of the many problems in math education is the tendency to treat the subject as procedural. Here's how you do this, here's how you do that. This boils down to viewing the learners as automata that need to be programmed to carry out certain tasks. The machines do not need to understand any of the issues surrounding or raised by the tasks, they just have to know how to accomplish the tasks.

The main reason I was a failure at the subject thru elementary and High School.

When I started my electrical apprenticeship it became clear that AC theory was all math, especially trig.

So, I tried to get some classes at the local JC only to be told I'd have to basically repeat everything with the same methods that had been a failure before.

So I found out you could take optical design without the math prerequisites. Realizing that this was going to be mostly applied trig, I signed up.

Three or four semesters later I'd mastered all the trig I needed plus, matrices of simultaneous equations, and started programming in fortran.

Yeah, teaching math in a vacuum isn't the way to do it for most of us.

When I went back to school in my late 30's I was dreading calculus and statistics. I was in a ULV remote class at So Cal Edison's HQ and most of my classmates were women that had been the high school math whizzes.

It had me worried at first, but it turned out to be completely unfounded. Algebra and trig were part of my daily life, they hadn't applied it since high school. After a couple of weeks they were all coming to me for help.

Yeah, teaching math in a vacuum isn't the way to do it for most of us.

Many people don't learn how to watch their blood sugar until they develop Type 2 diabetes.

Fundamental skills are fundamental skills. 'Tis a shame that "most of us" 'Mericans won't listen to the lessons learned by the people who have struggled through life ahead of us.

Math grad students make lots of money tutoring people who thought they'd never need to know any math, but who now find their career opportunities and future dreams obstructed by their lack of mathematical knowledge (rgold)

Forty four years ago, as a PhD grad student, I drove weekly to Bell Labs in Denver to teach a refresher math course to senior engineers. I think we all learned something.

The fundamental skills can be taught applied to the real world and NOT in a paper vacuum.

Of course.

But they can also be taught in a vacuum. Math is math -- it's application independent.

Real-world examples make it more pertinent, but the fundamentals don't change.

The idea that it's only relevant if one can immediately apply it is the problem. That was my point: if students will listen to the people who have been-there-done-that and suck it up to learn the fundamentals, then they will have the fundamentals before they need them.

Or, they can wait until they need them.

Kinda like practicing first aid and self-rescue before the emergency, no?

Back in High School they sent me to the county math contest. Wut? I was a stoner but I went anyway. I had always been really good at whatever math classes threw at me. I am not competitive. I have been a cook for most of my adult life and I consider my knowledge of mathematics as a strong part of how I approach challenges in my chosen field of endeavor.

Don't be a dumbsh#t, whitemeat. You are way cooler than that.

It's not just about learning it before you need it. It's simply easier to learn those esoteric problem solving and structural concepts when your brain is in the roughly 11-16yo range.

Glad you found Kahn academy, that's exactly the kind of site I was talking about. In my case, the makers of the actual textbook, that the students were using, had videos online where an instructor worked through the example problems for each chapter. You might check with your instructor and see if that is an option with your particular textbook. Good luck!

The post that Elcapinyoazz made upstream is very insightful.

The value in learning math is not the procedure, but the way it trains one to think. But that has to start somewhere. It's easy to say that when it is taught as a set of procedures that it becomes meaningless. But the reality is if you try to teach a 1st grader how to think mathematically without procedure, most will be lost (and perhaps even intimidated by the subject for a life time.) If you teach a person procedural skill, then it can be applied to developing a way of thinking. Developing this way of thinking is a lifelong pursuit for me.

Making math engaging, by (for example) having the kids take measurements to perform math on, is an education school myth. For one thing, most kids don't get too fired up about "real world" data because today they have so much at the touch of a keypad that it just doesn't impress them very much.

Also, the value in math is its ability to describe abstract things, things that cannot be described otherwise. As kids get older and increase their proficiency in math, it is undesirable to try to make it too "concrete". The ability to think abstractly is the goal, and math is the perfect vehicle for training the brain to think in this challenging way.

These platitudes are becoming a horrible bore. He wants to climb. His story is timeless. I wish him the best of luck. He will follow the patterns that have been set for him by his family and only he can change them, if he wishes, in due time, possibly never.

Making math engaging, by (for example) having the kids take measurements to perform math on, is an education school myth. For one thing, most kids don't get too fired up about "real world" data because today they have so much at the touch of a keypad that it just doesn't impress them very much.

As math teacher, I couldn't disagree more. Currently, my eight graders are doing a "Bungie Baribie" project. (Algebra I) They are engaged and excited. Actually, we even added some technology components to it so students can use video cameras and their tablets.

Same is true when my seventh graders were learning about scale. We learned the basics, but they spent over a week mapping various parts of the school grounds to scale. We then took the various parts of the school and put them together into a large collage for a full layout of the school done to scale.

In my experience, students do enjoy technology added to the material whenever possible, but they also enjoy getting out of their seats, moving aorund and using hands on tools to do real math.

In two weeks, my seventh graders will explore slope. We'll use our stairs. Some of us might even meet at the local ski area with GPS and altimeter to figure the average slope of the ski area. (Extra credit)

Math procedure do need to be part of the instruction, but having students take those procedures one step further helps them enjoy math more. If I can get students excited about coming to class, they are easy to teach.

if you try to teach a 1st grader how to think mathematically without procedure, most will be lost (and perhaps even intimidated by the subject for a life time.)

Sure, but this is a false dichotomy. Moreover the situation of upper-grade high-school students is different.

You are a hero. I think there are a constellation of factors that make learning math possible: an engaged teacher and making math relevant seem the most obvious to me. I go back and forth about dialing back the difficulty and making the problems easier. Sorry if I've missed or restated previous posts. I tuned out of this thread when Clint actually started helping OP with the concepts. Clint you too get a hero badge.

Hey, I have a question. Why bother with factoring? Why not just start with the quadratic equation? Way back in the day, the guessing involved in factoring made me very suspicious and confused about the whole field. Then the teacher might, after the fact, reverse engineer the factored equation. I guess for pre-calculator times, the factored equation is much more informative.

Rgold, you suggested the dichotomy with the following statement upthread:

One of the many problems in math education is the tendency to treat the subject as procedural.

What I am saying is the thinking in math begins for most with the procedure. After the procedure, then questions such as "why doesn't this work in all cases" might be asked and addressed. Yes, it is different for upper high school aged kids than first graders. But when teaching trigonometry to talented math students, the first thing I do is show where all trig values can be found (the unit circle), and then we start discussing other ways of thinking about the trig functions, and why different ways are helpful in different situations. First the more-procedural stuff, then the thinking can take place, in my opinion. . although I understand that many folks get turned off by the procedure.

Which is why we do a disservice by selling it as "fun". (No offense Guangzhou. If you're turning kids onto math by taking a week to introduce slope by walking up and down stairs and measuring, and they love it, you are engaging in some worthy teaching. I don't have a week in the curriculum that I am required to teach to do so.)I don't disagree that an interested student is easier to teach than a disinterested student. It's just that by saying it should be "fun", that detracts from the long-term goal of math which is to develop a way of thinking. That long term process is often anything but "fun", and I find it to be insincere to say otherwise.

Hey, I have a question. Why bother with factoring? Why not just start with the quadratic equation? Way back in the day, the guessing involved in factoring made me very suspicious and confused about the whole field.

The quadratic equation is super-calculation intensive, and when I see students bust it out on a quadratic that can be reasonably factored, they often make mistakes.

Plugging numbers into the quadratic formula is procedure. Getting a quadratic to factor is more along the lines of thinking. Also, some other kinds of polynomials can be solved by formula, but those formulas are not practical because they are very complicated. Solving many 3rd degree polynomials by factoring is far easier than using a cubic formula for doing so.

The quadratic equation is super-calculation intensive, and when I see students bust it out on a quadratic that can be reasonably factored, they often make mistakes.

In the real world, nothing is factorable, nothing of matter is even calculated by hand.

The calculation and modeling tools NASA (for example) uses to fly sh#t to the moon are tested and verified to no end by highly specialized and small PhD-been-doing-this-and-only-this-forever kinds of teams.

They then sell a computational product (Ansys, Matlab, Ansoft, etc) to engineers who will iterate, optimize and match to laboratory test results to no end until a working model of the system of concern is verified. ONLY THEN, the magic calculation is made.

Quadratic Formula = kindergarteners learning to wipe their butt.

One of the many problems in math education is the tendency to treat the subject as procedural (rgold)

Well, yes and no, Rich. One can easily go too far in the other direction. In the mid 1960s I taught at Murray State in Kentucky and we adopted a college algebra text that began with axiomatic set theory and field theory and required students prove some of the basic properties of algebra. This was the New Math and it didn't fly well there, to say the least. Perhaps with a class of A-level students it might have.

I agree there should be a balance between procedural and theoretical, but the former will get most students further than the latter.

On the other hand you may be talking about challenging word problems, which do foster the ability to think, sometimes well outside the traditional box.

Actually Reilly, I am constantly amazed that our education system works at all in a society in which more than 50% of married people get divorced, where poor people get marginalized in every conceivable way, and where success is defined as the ability to get more and more materialistic stuff than any single person could need. I think you're wrong.

The expectations we have of our schools to babysit, feed, socialize, moralize, inspire, remediate and yes, even educate our children far exceeds the total amount of resources we put into the endeavor. A free public education is the best bargain a person could ever get.

We tend to measure math education using standardized tests taken by the top students in the world. I would put money on some of the kids in my school competing against the world's best, although admittedly I don't teach at an "average" public high school.

Hey, I have a question. Why bother with factoring? Why not just start with the quadratic equation? Way back in the day, the guessing involved in factoring made me very suspicious and confused about the whole field. Then the teacher might, after the fact, reverse engineer the factored equation. I guess for pre-calculator times, the factored equation is much more informative.

Well, factoring applies to far more than quadratic equations. Moreover, for quadratic equations, factoring is equivalent to finding roots, but for higher-degree polynomials, factorizations do not necessarily involve linear factors and so is not about finding roots in the same sense.

The process you learned isn't guessing, properly understood. The point is that if a polynomial with integer coefficients can be factored, then there are a finite number of possibilities, and by trying all of them you either discover a factorization or you determine that the equation is irreducible. Of course, in order to make a valid conclusion of irreducibility, you have to have a factoring method that really does look at all possibilities, and the subject may or may not be taught in a way that provides students with the tools to have that assurance.

One can prove in general that any polynomial with integer coefficients (this includes polynomials in several variables!) is either irreducible or else can be factored in a finite number of steps. Contrast this with the fact that general formulas for roots analogous to the quadratic formula are only available for polynomials of degree less than or equal to four.

Factoring integers and polynomials has a vast array of applications in mathematics. The concepts and techniques can't even remotely be replaced by the quadratic formula, useful though it is.

Nice platitude JLP, but, uh, we were speaking in the context of math class; not rocket science.

No, the context here is academia vs the real world - go learn to factor polynomials so you can be successful - go learn to think by solving math problems - or else you'll be boiling fries - it's a farce, a huge cultural myth - and for most people it's just plain wrong.

What makes it even funnier, as I mention, is that nobody actually uses this crap in the real world.

Reality is that most complex theories and equations came from trying to explain why some weird sh#t blew up in a lab somewhere. So they blew some more up and eventually fitted a few equations to it. Galileo, Newton, Bell, the stories are classics - reduced to mere numbers crammed down the throats of today's youth with near zero context.

Math teachers like you - frankly - are perpetuating a reality that does not exist in the real world - and for people like whitemeat, it all just may not sit well for this reason - a lie never does - and maybe he just doesn't know why yet.

The best engineers took their toys apart. The worst ones didn't. Etc.

I have finished the packet and have all correct answers!!!!!!!

think I am lying? ask me the hardest question on there and I will give you the answer, I also understand the stuff!!!!

PS: I did it on my own rather copy :)

If all of a sudden you can do all the problems, it looks like you've been up to a wee bit o' trollin' WM. But no matter---well done.

I can't say what the hardest problems would be, but the ones involving the most advanced technique are the four requiring factoring a cubic; that would be 1, 2, 43, 44.

What makes it even funnier, as I mention, is that nobody actually uses this crap in the real world.

That is indeed a knee-slapper.

But even funnier is the fact that in typing that hilarity, displaying the results on your screen, sending it over the internet, and then having it displayed on all of our screens with all our different browsers and operating systems, this humorous tidbit used hundreds if not thousands of pages of pure and applied mathematics from contemporary results all the way back to Euclid in 300 BC.

A vast amount of purportedly useless stuff with a lineage going back through the ages was required to transmit and display the message of its own uselessness. Now we're talkin' real humor.

And speaking of factoring (which we recently were) every time you use a credit or debit card you are relying on mathematical results about factoring the product of two large prime numbers.

Just because you can't "see" mathematics at work doesn't mean it isn't there. Just sayin.

By the way, I for one am not arguing one way or another that any one individual "needs" to know this stuff, or that "success" in life requires the ability to, say, factor cubic polynomials.

I aint no troll!!! I used that kahn thing, teacher, and students, and hours of work, now I get it!!!!

#1. 1, 2

#2. -2

#43. -2, 6 plus or minus the square root of 32 all over 2

#44. -4, 4 plus or minus the square root of 48 all over 2

bahahaha!!!!!

Well, then, even better!

Your answers are correct, but you didn't need me to tell you that. The results for #43 and #44 are not, however in the simplest possible form. This might matter for two reasons. (1) if your response is supposed to be written, your teacher might make deductions for not having simplified the result. (2) If the answers are multiple-choice, then it is likely that only the simplified results will appear as options, and you might not recognize that your answers are included (in another form) in the list of choices.

So: in #43, you need to write sqrt{32} as 4 sqrt{2} and then simplify the fractions you have. This yields 3 +/- 2 sqrt{2} as the answer. You'll have to do something analogous for #44 as well.

2x+4)(2x+5) for number 9 clint??? thanks for all the replies, I know number 47 and a couple others but not much, well math is really easy for me to learn I just forget it REALLY fast...

he gave the troll away here. Though it wasn't much of a troll. He knows how to do the work, or at least he did at one point, he has just forgotten and needed a refresher. I was that way in high school and university. Just talking about it sometimes brings back the memory. At least for me. though as I got older I found that what I hadn't committed to memory was lost without a whole lot of effort to bring it back.

Yes, plenty of software out there to help people with math. We even recommended a couple free internet sites here that will solve and give you step b step instructions.

On the other hand, I find it amusing when I see adult use their cellphone calculator to do some very simple math in stores or restaurant. Sometime, punching in the numbers takes longer than just doing thing in your head.

Will we use the quadratic formula or factor polynomials on a regular basis, probably not. I'm honest with students about how useful the material is. In my view, it's about being "math literate."

You can argue you've learned all the math you need by 6th grade if you want. Other will argue all you need is a 6th grade reading or writing level. Maybe it's true in some fields.

Somewhere above, I think it was John, mentioned a textbook use in Kentucky. I agree, heading to far in one direction or the other isn't good. Can't imagine being a student or teacher where everything was done through discovery learning.

The state of Education, berate it all you want. Education is an easy institute to bash for sure. I think the biggest problem with education is that politicians, press and even parent are constantly speaking ill of it. When kids, AKA students, keep hearing how bad education is, it's hard to take school serious.

I teach in Asia, I don't see a huge difference between Asian kids and western kids. I do see a lot more hours in programs outside of school in Asia than America. Asian student spent two, three sometimes four hours a day after school in cram School, jewkues or what ever other names they gie these "institutes."

As a math teacher, I hate parent conferences where the parents point out that "Math is the most important subject." I think you can get a very good education, liberal arts, where math is the most important.

As a math teacher, I hate parent conferences where the parents point out that "Math is the most important subject." I think you can get a very good education, liberal arts, where math is the most important.

Totally agree with you on this Guang. I work in a community that has some very high-powered math people at CU, and at other various research institutions such as NOAA and NCAR. I constantly hear that "math is the most important subject", to which I reply "no, it is one of many important subjects." Personally I believe foreign language and music are often underemphasized, which is a real shame.

I think one of the biggest problems with not understanding the math and just using today's "black box" approach is that you don't realize when you've punched a wrong button or used some software incorrectly, and then just trust the answers that come out.

I do a lot with signal analysis in my work, and I have seen people that have total faith in their pre-programed boxes happily basing results and interpreting artifacts that were generated using the software incorrectly.
If you understand how the stuff is calculated, often those artifacts stick out like a sore thumb.

If you understand how the stuff is calculated, often those artifacts stick out like a sore thumb.

IMO, an intuitive understanding the underlying physics trumps all software and measurement tools and equations.

In grade school, the underlying intuition building seems to stop at counting sticks and apples in 1st grade - then follows 11 years of mindless numerical procedures on conveniently contrived (~factorable) equations.

The only reason for these numerical procedures is that, unlike intuition building exercises, the numerical procedures are gradable. The kids need to be lined up and ranked on a totem pole for The Man.

and since we now live in a world where all the toys are virtual and cannot be touched, let alone, taken apart.

We're in trouble.

I don't agree with this. A friend and his son together recently picked out a bunch of hardware on newegg and built a PC. The OS wasn't from a canned distribution, it was downloaded and built directly from kernel.org. Radio Shack in Boulder tells me they are near #1 in the country for Arduino kits. We also have Spark Fun in Gunbarrel - doing quite well. Our company helps sponsor science fairs, robot building contests, etc - also doing well and popular.

The only reason for these numerical procedures is that, unlike intuition building exercises, the numerical procedures are gradable. The kids need to be lined up and ranked on a totem pole for The Man.

You are just another jaded bore, JLP. Your comments on every thread seem to be those of a guy that hasn't gotten laid in a very long time. Sorry about that, buddy. It must be miserable.

Mathematical intuition-building is a lifelong pursuit. It may start at first grade, but it certainly doesn't end there. I've been lectured many times by parents of students who speak a lot more intelligently than you write, and who have some pretty amazing credentials in math. With a little conversing, most get the distinction between learning math vs. using it in a professional setting. Using math in your so-called real world where you don't get laid is very different than teaching it to developing minds.

You seem to think you're a math tough guy. I've got kids I teach that could run circles around you in the subject, and at least part of the reason is because of their school experience. I'd urge caution bragging about yourself in this subject in Boulder. Good math people are a dime a dozen in the area, and they are mostly far more articulate than you.

The math principals and the intuition go hand in hand. They are part of the same thought process that allows you to use the good ole "scientific method" to reason things out. Even if the answer isn't "intuitively obvious", you still know the right questions to ask, which will put you on the right track to figuring it out.

I've got kids I teach that could run circles around you in the subject, and at least part of the reason is because of their school experience.

You take credit for this?

Do you also occasionally converse with people who climb 5.15 that you'd like to throw at me as well? LOL.

Those kids come from homes where the parents have carefully chosen their toys, their activities, their examples - for their entire lives before showing up to your math lecture. Your effect is nothing in comparison.

But I bet fat stacks you flunked the sh#t out of English Comp. The spelling alone probably cost you 50 pts and cost your teacher three red ink pens and a migraine.

I probably got a B on the route we were working...

...very slick water polish, dribs of water here and there (it is spring, after all)

4 pitches up, had to reroute the 4th pitch but not without a fight...

after Eric came down from his attempt I got to try.

Starting from the belay, the first move was a prelude to all those that followed. Pimping off of very slick holds, even the incuts seemed frictionless. But a series of face moves, each delicately executed, gets to the left facing corner with a thin-hands to hands crack, straight in. But even the jams were slick, usually that size would be secure, and the angle wasn't very steep, but without much friction all of it was cautious dancing.

Up to the roof. The crack split the right hand side, but the roof itself was arched like a breaking wave causing the crack to scimitar back towards you. The right hand side of the corner had a few desperate suggestions for footholds, all of them similar in quality to the rest of the pitch, frictionless.

After three moves I could get a great jam in at the point of the breaking wave in that crack, but with no feet and the crack flaring gloriously above the roof, there wasn't a lot in my tool box to solve this particular problem.

I tried to get a chicken-wing into the flare, but with no feet and my left wrist as a pivot point, I couldn't get the necessary leverage to lift up and in.

On another attempt I jammed the right hand in half thinking I could get a foot up and in, channeling Dick Cilley... no go.

The left hand jammed again and the right foot on an impossibly tiny point on a narrow edge was too much for me.

Six different ways and not even getting close... I bailed off a single nut leaving it and a biner.

We definitely thought that this was 5.11 something... not really for this route...

but who knows, we might go back up and top rope it and unlock the mystery of that passage.