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He's now "got it," but I don't think he's "got" the fact that we are "unmultiplying" -- or that we're using the distributive property when we do these two steps:
3x(2x+1)+1(2x+1)
(3x+1)(2x+1)
I've found over the past couple of years that sheer repetition "naturalizes" a procedure in my mind, which feels like comprehension. But when I returned to the procedure for factoring a trinomial in which the coefficient of the squared term was greater than 1, I had forgotten how to do it -- and I couldn't figure out how to do it.
In other words, I didn't understand what I was doing well enough to reconstruct the procedure myself.
Here's the part that keeps stumping me.
6x^2+7x+2
solution:
6x2=12
factors 3 & 4 add to 7
therefore:
6x^2+3x+4x+2
3x(2x+1)+2(x+1)
(3x+2)(2x+1)
I'm going to sound dumb, but I am having trouble grasping why the factors of 12 work as the addends of 7.
I guess I have two questions:
- what is the best way to show that the final step above is a use of the distributive property?
- factors and addends - HELP!
"Reteaching" this year tends to mean "reminding" and "reviewing."
again with the answer keys
This weekend C's teacher sent home 4 pages of review problems to practice before taking today's test.
Fantastic!
On Saturday I realized he hadn't sent the answer key. [update 11-25-07: Teacher says he got the worksheets at a workshop - oy - and the workshop didn't supply the answer sheets. My tax dollars at work.] So I spent my weekend staring at a guillotine deadline (official publishing industry terminology) for my book and factoring trinomials on my son's review sheet so he can figure out whether he's actually doing the problems correctly.
Needless to say he was doing many of them incorrectly on Saturday. Some of his mistakes were "careless errors"; some were missing a step-type errors.
Both errors require marking and redoing, or at least analysis of what went wrong.
Speaking of which, C. is beginning to want to analyze his sequence of steps instead of simply starting over again. I think that's good, right? Often he'll say, "I see what I did wrong." Of course, he can't do this if he doesn't know he got the answer wrong.
Marking careless errors is particularly important in the case of trinomial factoring, I think, because along with learning how to factor a trinomial you need to learn how to check your answer mentally if possible. (Which I'm pretty sure C. has not learned...)
In the case of trinomials, students could check their answers by multiplying the factors they've ended up with, but I don't think they've been told to do that; they definitely haven't been given homework requiring them to do it. I managed not to figure this out until late last night, sad to say.
Still.
We need the answer key.
I will never understand this place.
on beyond help with homework
Meanwhile, the course is now beyond most parents' ability to "help with homework." Ed took engineering calculus at Princeton and has no memory of how to factor trinomials. Plus I find the textbook, Glencoe Algebra, to be mystifying in most instances. It's not like Saxon or Foerster or Dolciani, textbooks a parent can use to teach or reteach himself the lessons his child has been taught at school.
If I find it mystifying it's a cinch other parents will, too. Not to mention the students themselves.
(Btw, my experience thus far has been that Foerster is incredible when it comes to a parent having to rapidly "teach" herself a concept she's never before seen in her life so she can (re)teach it to her child in time for the test.)
So.....what is the thinking here?
By the time most students reach 5th grade math, their parents are no longer able to "help with homework." (The math teacher/tutor I know told me: "I get the call in 5th grade.")
How are the kids supposed to learn algebra?
Can you FOIL the answers?
20 comments:
It's been so long since I first learned this, that after reading this post, I realized that either I have internalized the method described here or I'm just a really good guesser when it comes to factoring.
Anyway, here's the proof, or at least a demonstration of why it works (a real proof would cite theorems and stuff) [sorry this is ugly; blogger doesn't like <pre> or <code> or <tt> tags]:
Start by going backwards. Assume your factors are (ax + b) and (cx + d).
(ax + b)(cx + d) = (ax)(cx) + (ax)(d) + (b)(cx) + (b)(d) [Distributive property]
= (ac)(x^2) + (ad)x + (bc)x + (bd) [Associative, commutative]
= (ac)(x^2) + (ad + bc)x + (bd) [Distributive]
Note that the product of the x^2 coefficient, (ac), and the constant factor, (bd), is (ac)(bd) = abcd [Associative, commutative]
Note also that the product of the two addends that form the x coefficient, (ad) and (bc), is (ad)(bc) = abcd also. [Associative, commutative]
This means that if a trinomial has real factors, you can find them using these steps:
1. Call the x coefficent S.
2. Multiply the x^2 coefficient by the constant and call the result P.
3. List the two-factor integral factorizations of P (1 x P, 2 x P/2, 3 x P/3, ...).
4. Find the sum of each pair of factors.
5. Choose the pair of factors of P for which the sum equals S, the x coefficent.
This method will work for awhile, because at this introductory level, the problems will probably all involve only integral factors. Later on, they may require fractional factors. At that point, step (3) will result in an infinite list, so this method breaks down at that point. Even later on, the roots of the trinomial, and hence the factors required, may be complex (i.e. they may have a component on the imaginary axis).
This method will work for awhile, because at this introductory level, the problems will probably all involve only integral factors.
Yes.
THANK YOU!
If the coefficients of the original polynomial ax^2+bx+c are all integers, then the coefficients in the factorization (dx+e)(fx+g) are either integers or they are irrational. They will never be fractions--if you start with integers, then step 3 always yields a finite list. If there are no solutions in step 5, then you can conclude that any solutions will be irrational and/or complex (and your next stop is the quadratic formula).
I don't understand why teachers refuse to give the kids the answer key. Can someone explain that to me.
Paul
Can this possibly be normal?
Things really are daffy around here.
Everything is so extreme -- what's that saying....
Can't remember.
It's the saying about a rational idea taken to its logical extreme...
Point is: I can see where you might not want kids to have the answer key for homework they're going to go over in class the next day.
I don't agree with it, but I can see it. (I don't agree with it if only because it doesn't build any kind of responsibility into the student. IMO 13-year old students should be learning to check their work and re-do if they got it wrong, not wait passively until the next day for the teacher to write stuff on the board.)
But when you take that logic to the extreme of giving students review sheets and test prep books without answer keys you've lost all track of the logic of your position.
This reminds me.
A former colleague worked with Andrea Dworkin. She was very fond of Dworkin, and once said, "Andrea's the kind of person who, when everyone else gets off the bus, she keeps right on riding to the end."
That's the situation around here.
No answers for homework AND no answers for state test prep books.
I can't resist pointing out that state test scores are no skin of my nose.
"We need the answer key."
Many math books I used had answers to the odd questions in the back. We were assigned both odd and even problems.
Catherine,
Very well said. I can see why the answer key isn't given, but I don't agree with this position either.
My 9-year old son is given "math challenges" and told to figure them out. If he gets the answers wrong, he has to bring it home so he can get my husband to help him. The teacher writes comments like, "don't know how you came up with that answer", "is there an answer here" on his papers. This does not foster responsibility, it fosters frustration.
Like Steve, I had the answers to the odd questions, and I remember being able to understand a question that stumped me by looking at the answer key.
Of course, we had actual math textbooks back then with answers we could rely on.
"don't know how you came up with that answer",
"is there an answer here"
Real quotes? There seems to be an implied "you idiot!" that follows.
So, they want kids to discover things, they don't give them much prior knowledge, and then they try to motivate them with comments like this? Did they learn that in ed school in Motivating Coments 101.
As I have mentioned before, we spend too much time talking about theory, assumptions, and pedagogy, when we really should be talking about competence. The ed school talk is just a smokescreen.
Yes, these were actual comments.
One of the math challenge questions was "how many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?"
My son had written all over the paper in his attempts to discover the answer. He did write 9+8+7+6+5+43+21=99, so technically he did get the answer, but his answer was lost on the page due to all the scribbling. On the side of the paper was the comment "is there an answer here?"
I can see why the teacher said it because his paper was one big scratch sheet, but I agree that this comment is not helpful.
Speaking of positive behavior, our county is presenting two half-day seminars on school-wide positive behavior support. It will be presented by Dr. George Sugai, co-director of the National Center on Positive Behavioral Interventions and Supports.
We are to call the Behavior Support Coordinator or the Special Ed. Coordinator if we are interested in attending. Maybe the Behavior Support Coordinator needs to come to our school do discuss Motivation 101.
So, they want kids to discover things, they don't give them much prior knowledge, and then they try to motivate them with comments like this? Did they learn that in ed school in Motivating Coments 101.
That's where we are.
The middle school is hugely negative.
Reading about Clicker Training while also dealing with middle school makes your head spin.
Check this out: there's all kinds of research showing that when you train horses through negative reinforcement, which is what is being used with our kids (negative reinforcement and in some cases outright punishment, which is different) horses get dumber and less creative.
This is horses we're talking about.
I forgot.
I got an email from the teacher saying that he got the worksheets at a workshop and doesn't have the answers.
... teacher doesn't have the answers.
Priceless!
Would it be possible for the teacher to solve the problems showing the work and actually create the answer key? The odd ones, at least?
... so how does the teacher correct them?
The teacher doesn't have the answers? You know, somehow this doesn't surprise me.
I think you have a typo on your example (which may be contributing to the problem ?) the second line should be:
3x(2x+1)+2(2x+1)
I was never taught that method of factoring trinomials, and I still turned out pretty well.
We were taught to factor the coefficient on the leading term and to factor the constant term and to fuss with it until it worked. So for the problem in the post we would do:
(2x )(3x ) [start by factoring leading term]
(2x 1)(3x 2) or (2x 2)(3x 1) [leaving room for the signs]
Noticing with the latter option you get 2x and 6x, which can never combine to give you 7x, you ignore it and try to see if you can get the signs to work out on the first one.
Since everything in the problem has + signs, try
(2x + 1)(3x +2)
and multiply out, and see that it works!
If that didn't work, we'd try
(6x )(x )
and keep trying until it worked (or we gave up and used the quadratic formula).
That's how we did it back in 1986.
"That's how we did it back in 1986."
That's how I was taught back in 1967. "fuss with it" I might have been taught the other way later on, but the fuss with it approach stuck. You can get pretty fast at picking out the correct combination of factors and signs. All I remember is that we did a lot of them.
Translation please!
Talking cats
Maybe they are discussing trinomials.
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