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Brilliant logic! But you assume that all solutions to the equation are real. I would expect some bright middle school students to answer it for the real domain. Many middle school students know about complex numbers. I think that if you allow for complex numbers, I think the problem becomes too complex to solve in a few minutes - unless I have missed something. OTOH, if a and b can be assumed real, and they have been shown problems using similar tricks, the majority of bright students should get it.
I've answered those questions in this thread.
What's the absolute value of a complex number? I'm pretty sure the only solution is the real one. You lose i when you take the absolute value of a complex number so you'd have to lose it in the binomial expansion too which means every i that doesn't have a coefficient of zero must be squared. My guess is that the coefficients on the i's are zero and the solution is real but that's just my guess. I don't know the physical meaning of the problem so it's impossible to tell how many answers to expect. Since the author asked for a*b I and there is a real solution I'm going to say it's the only one. Don't you think they would have asked for more than one answer if they were expecting non real numbers to be included when there is a real solution?
ETA: Given that the absolute value of a complex number is the distance from the origin made by the real axis and the imaginary axis I would think you'd have to substitute in complex numbers for both a and b just to remove the absolute value sign. Then of course you'd have to do the same in the binomial so now we have four unknowns in one equation. The real portion associated with a and b and the coefficients on the imaginary portion.
I'm scratching my head as to why you'd even think that this has a complex solution that can be determined using algebra but you're more than welcome to try and find one and post your findings. I'd be very curious to see the solution.
Last edited by Ivorytickler; 04-03-2016 at 06:39 PM..
As a tutor, I've noticed that more and more students need tutoring, including some of the best and brightest students.
I've also noticed a trend towards studying more college-level material in high school, and to less free time for public school students in general.
I have some theories, that I have posted at
mod cut - removed soliciting.
I'd like some feedback. Do you agree with my logic? Have I left anything important out?
I recognize some of what I say is controversial, especially since I have little sympathy with what I think of as "The Educational/Industrial Complex", a set of special interests who seek to make obscene profits from the business of education.
Could you provide a link to that/those post/s? I haven't found one where you discuss the issue of students taking college-level courses, why they're doing that, how they feel about it.
I don't see that colleges are requiring much more to qualify for entry than they used to. It is probably more competitive to get into the top schools today than it was in the past but most colleges are begging students to apply. You can't turn on the TV and not see commercials for several colleges. The one that surprises me is why the University of Southern New Hampshire is advertising in Pittsburgh.
I have never seen a college requiring anthropology, world economics, (or any kind of economics) or any kind of geography. My three kids did not take any HS classes in these subjects. There also seems to be less requirements for foreign languages today to get into college. Many students are also taking the least amount of math that they can get away with in both HS and college. Unless a student is planning to major in math, pre-med or engineering, they are taking the minimal number of math classes.
I have three children who graduated from college, and eight nieces and nephews who I'm close to who are either in college or graduated. They represent schools like Pitt, Penn State, Duquesne, Carnegie Mellon, UPenn and some smaller liberal arts schools. I don't think any of them had volunteer work, impressive extra-curriculars, or a compelling personal story to give them an edge. They each played from one to three sports, and that was about the extent of their extra-curricular involvement. None had a job in high school. My son was accepted to UPenn without an interview and would've only written a personal essay if it was required. He did take seven AP courses and score 5s on six of them and a 4 on the other.
What I see are many students today who excel at gaming the system. They get good grades in high school but get to college and find they are not cut out for pre-med or engineering. A neighborhood girl fits that description. She was accepted to a five year physician assistant program and dropped out after one semester. It was too difficult despite a high GPA in high school.
I think our schools need to get back to teaching and get rid of the group projects, working with partners, posters, and all the other activities that take away from actual learning. If you don't know what I'm talking about, you need to visit a high school English, social studies, etc. classroom and see the art projects that are hung on the walls. The purpose is to be able to give grades to students who don't do well on tests.
I can answer that one!
Southern New Hampshire University has a student complement of 64,000. All but 4,000 of those students are online and/or continuing education students. Online students are far less expensive to serve than those on campus, and SNHU is a private college. Their endowment, at less than 17 million dollars, is frighteningly low, and they need to bring in more online students to continue operating their non-profit campus programs. They aren't trying to get those Pittsburgh kids to move to campus, but to choose them instead of the likes of University of Phoenix.
Most colleges are NOT requiring anthropology or world economics or anything other than the core courses for admission. They do look at other things to distinguish students from one another because there are just more students applying. Some require economics as part of the history sequence.
Colleges require years of English, 3 years of math (though some schools want 4 years and want kids to have calculus), 2 years of science (in some schools 3 or 4 years are preferred), 2 years of history and 2 years of a language (this is not required at all schools though).
Texas A&M requires:
Courses No. of cred More about this requirement
English 4 credits None
Lab Sci 4 credits At least one credit must be in Biology, Chemistry, or Physics
Math 4 credits Must include Algebra I, Geometry, Algebra II and a higher math
Soc Stud 4 credits Must include Economics
For Lang 2 credits Must be 2 credits in one foreign language or ASL
While it is not required for admission, a high school course in computing may provide the skills needed to fulfill the University's computer literacy requirement.
MIT requires:
High school physics 1 year
High school chemistry 1 year
High school biology 1 year
English 4 years
Math through Calculus so at least 3 years
History or soc science 2 years
Alfred University in upstate new york (my old school) requires
English 4
Math 2
Science 2
Foreign Language
Social Studies 2
History
Electives
TBH, not everyone has to go to a top school. There are plenty of less competitive schools where a student without all those APs and with a lower GPA can do well. Kids can also go to community colleges for 2 years and transfer to a 4 year school as long as they check out what the college they want to go to will accept.
Sure, that is what MIT 'requires', but accomplishing that in no way places you in the incoming freshman class. These 'minimum requirements', and those listed by most other non-community schools as well are simply the first stepping stones in the admission process. Those extra 'non-required' classes help push you higher up the ladder towards the promised land of educational nirvana.
It's not just my school. The problem is that kids do a brain dump after the test. I've even had parents complain that I can't grade their child on their writing because I'm not an English teacher (their writing is terrible but I KNOW they have been taught to write) or tell me it's unfair to take off points for math mistakes because my class isn't a math class. They don't take things from one class to the next and they and their parents cry foul if they are expected to know something they were taught in a previous class. Our kids (in general in the US) are in the habit of learning for the test and then forgetting what they learned and expecting a review of any material taught in a previous class they may need before they need it. .
No, the problem is that teachers today ONLY teach for the test, NOT to build skills for future classes. Many elementary, middle, and Jr. High classes are 'taught' without the use of textbooks, or with textbooks available only during class. In our schools, Pearson online has taken over from the textbooks, and the source materials DO NOT explain beyond the basic concepts, and certainly don't explain WHY the information is taught. In addition, it isn't possible to view what would have been a single page in a book, all at once online without scrolling up and down. So if a concept is not COMPLETELY explained within a short paragraph or two, it is impossible to view in my stomach entirety. Next come the canned homework problems. They start off mimicking the listed examples. Then the assigned problems leap from the type illustrated in the lesson to problems involving advanced abstraction not covered in any of the examples. Then the real fun begins. Let's say you are assigned 40 problems for homework. Every problem the students answers incorrectly triggers 3 additional problem which MUST be answered to complete the assignment. Answer 10 problems incorrectly? Your assignment is extended to 70 problems. Each extension provides that many more problems to do perfectly, or another increase. I can't help but feel this process was inspired by Disney's 'The Sorcerer's Apprentice'. And we know how great THAT turned out.
Online lessons and today's teachers both rarely explain WHY something is learned, or satisfactorily describe what is meant by the terminology in real world terms. Here are a few examples. If you don't teach the students that '7/4' is the exact same thing as '7 divided by 4', they never understand what they are doing with fractions. If they don't know that '7 + 5 = __' is exactly the same as 'find X where 7 + 5 = X', they will never actually UNDERSTAND algebra. Far too many students have never been taught to think of a negative number as a deficit, and not just another type of number. The concept of '-7 apples' makes no sense to them at all. To a teacher, it may be self-evident that if you perform a mathematical operation on one side of an equation you must perform an identical operation on the other side as well, but to many students, this is an alien concept.
Worst, as far as math goes, is the use of ridiculous equations that would have no possible real world use, mixing forms of expression, randomly grouping operations and numbers in pointless combinations meant only to delight the wonk who wrote the question. How about this: Write an equation which actually MEANS something in a real world sense, in both words and mathematical notation, and teach the students to understand that they are EXACTLY THE SAME. Teach them WHY they should learn the concepts of absolute value, logarithms, trigonometry, calculus, and a thousand other ideas that mean nothing to someone who has lived their life just fine so far without them.
In the past, many of these functions were learned through rote memorization, which was good enough to allow a student to progress to the next level. When a student has not received the information that I have listed, he/she is bound to require tutoring if they or their parents care about their academic future.
I view tutoring as a positive. If you go back half a century, receiving tutoring carried a sort of a stigma. A student didn't want to admit they were "dumb". Sometimes I wonder how different my life would've been if I would have sought tutoring for some of those difficult university level math courses, instead of just walking away.
No, the problem is that teachers today ONLY teach for the test, NOT to build skills for future classes. Many elementary, middle, and Jr. High classes are 'taught' without the use of textbooks, or with textbooks available only during class. In our schools, Pearson online has taken over from the textbooks, and the source materials DO NOT explain beyond the basic concepts, and certainly don't explain WHY the information is taught. In addition, it isn't possible to view what would have been a single page in a book, all at once online without scrolling up and down. So if a concept is not COMPLETELY explained within a short paragraph or two, it is impossible to view in my stomach entirety. Next come the canned homework problems. They start off mimicking the listed examples. Then the assigned problems leap from the type illustrated in the lesson to problems involving advanced abstraction not covered in any of the examples. Then the real fun begins. Let's say you are assigned 40 problems for homework. Every problem the students answers incorrectly triggers 3 additional problem which MUST be answered to complete the assignment. Answer 10 problems incorrectly? Your assignment is extended to 70 problems. Each extension provides that many more problems to do perfectly, or another increase. I can't help but feel this process was inspired by Disney's 'The Sorcerer's Apprentice'. And we know how great THAT turned out.
Online lessons and today's teachers both rarely explain WHY something is learned, or satisfactorily describe what is meant by the terminology in real world terms. Here are a few examples. If you don't teach the students that '7/4' is the exact same thing as '7 divided by 4', they never understand what they are doing with fractions. If they don't know that '7 + 5 = __' is exactly the same as 'find X where 7 + 5 = X', they will never actually UNDERSTAND algebra. Far too many students have never been taught to think of a negative number as a deficit, and not just another type of number. The concept of '-7 apples' makes no sense to them at all. To a teacher, it may be self-evident that if you perform a mathematical operation on one side of an equation you must perform an identical operation on the other side as well, but to many students, this is an alien concept.
Worst, as far as math goes, is the use of ridiculous equations that would have no possible real world use, mixing forms of expression, randomly grouping operations and numbers in pointless combinations meant only to delight the wonk who wrote the question. How about this: Write an equation which actually MEANS something in a real world sense, in both words and mathematical notation, and teach the students to understand that they are EXACTLY THE SAME. Teach them WHY they should learn the concepts of absolute value, logarithms, trigonometry, calculus, and a thousand other ideas that mean nothing to someone who has lived their life just fine so far without them.
In the past, many of these functions were learned through rote memorization, which was good enough to allow a student to progress to the next level. When a student has not received the information that I have listed, he/she is bound to require tutoring if they or their parents care about their academic future.
I disagree. My students certainly were and are taught material that is intended to be built upon. They just don't remember it when they get to me. I love unit 8 in chemistry. They get to the problem set and they're stuck. They ask me about molar masses and I tell them they learned it in unit 6. They puzzle over going from a chemical name to a formula and I remind them that was unit 5. They balk at writing an equation given just the reactants and I remind them that was unit 7. Seriously, *I* taught them the material in units 1-7 and when I write problems that utilize what they learned before they don't know what to do. I KNOW they get taught the material they need in the future and I also know they don't remember it when they get there. I don't know why. They seem to think that once they've been tested on something they're done with it. It's a ticket punched now move on mentality. I see it over and over. I teach them something this week, test them next week and they can't use it the following week. Then mom and dad start calling and complaining that I'm testing over previous material. Sigh.
Actually, being able to group and ungroup in math is critical to solving algebraic equations. This is a weakness in my students. They struggle with isolating a variable even though they've been taught it multiple times and given the opportunity to practice repeatedly. The problem is they don't do the practice which means we have to keep coming back to the same topic over and over and over instead of moving on to something more interesting. As with anything you have to build the foundation before you can build the house.
The math book I use has plenty of problems that have real world applications.
My students certainly were and are taught material that is intended to be built upon. They just don't remember it when they get to me. I love unit 8 in chemistry. They get to the problem set and they're stuck.
If it was a very small number of students, you can blame it on them. But if it is true of many students, you must blame it on the system or how they are taught.
I won't go quite as far as "There are no bad students, only bad teachers" - but I will say there are not a great many bad students. Most kids (and most people) LOVE to learn, if you don't make it too unpleasant.
So how can you fix the problem?
That physics teacher who taught us Quantum Field Theory had a method of briefly reviewing material. If I remember right, each definition, method and theorem had a number or a name. So when he needed a prior definition, method or theorem, a long time after it was mentioned before, he wrote something like (changing to chemistry)
Recall: Definition 3.1 of Avogadro's number (N0) - The number of atoms or molecules such that one mole of C12 atoms weighs exactly 12 grams. N0 of anything is called one mole of that thing.
Theorem 3.2: N0 atoms or molecules of (average) molecular weight W weighs exactly W grams. Define those W grams to be the molar weight of those N0 atoms or molecules.
(N0 should have said N subscript 0, and C12 should have said C subscript 12 - I don't know how to write that on this board.)
(Chemists say "weighs" when physicists say "has mass". As long as your scale is calibrated using standard weights, it doesn't matter.)
You do not need to repeat the original proof or the original examples. (Since you gave the original Definition and Theorem numbers, students who don't quite understand the review can go back to the earlier notes from your class, and review the original context.) Instead you go on to use the theorem in the new application.
By using very concise definitions and statements, you can repeat them without taking much time. This form ("Recall" or "Review"...) is not much longer than saying "If you don't remember the idea of molar mass, go back and re-read unit 6", but gives them more information, and serves to create a review.
Brevity isn't just "the soul of wit" - it is the soul of good learning.
This method works best if:
1. You have taught them all the units.
2. You are well organized, and created a well organized set of numbered notes before you began teaching the course.
I certainly hope part one is true. But even if not true, such brief reminders (without definition and theorem numbers) fix all problems, without forcing you to spend hours on them.
BTW, this is the way I would write my New-And-Improved-Schaum's-Outline-like ideal textbooks.
If you are clever, you can sneakily use these very brief reviews to force them to remember everything important you taught in the class - without using zillions of words to achieve that important goal. After all, almost all subjects worth learning are progressive - you need what came before.
Last edited by grunes; 04-05-2016 at 08:34 PM..
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