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Is that correct?
I think it's correct, but I have no idea whether what I think is correct is correct.
If it is correct, I'm looking forward to briefing Christopher on this. Depending on what mood I catch him in, I'll get enthusiasm or eye-rolling, either one. Doesn't matter, though; he's going to like this idea whether he rolls his eyes or not.
Boys love it when their moms teach them math.
............................
Unfortunately, I don't remember which website published the "factoring is unmultiplying" line, so I can't credit it.
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I guess you might say that factoring is "unmultiplying."
Factoring, of course, usually means finding ALL the factors, so while "6 x 5 = 30" is multiplying, the "unmultiplying" of 30 would be "Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30."
Or, if you like, 1 x 30, 2 x 15, 3 x 10, 5 x 6.
Factoring could be simple "unmultiplying, like this:
3x + 9 = 3(x+3)
or it could be the reverse FOIL technique:
x^2 + 4x - 5 = (x-1)(x+5)
I don't think of this type of factoring as unmultiplying, but it might help for some kids.
As a side issue, I don't like the FOIL technique because it's not general enough. You can't apply it to:
(3x-1)(x^2 - 2x +1)
I liked to tell students that you need to take each term in the first factor (x-1) and multiply it with each term of the second factor. This works even for:
3(x+3)
This brings up other points that I think will help students, especially when the expressions or equations get more complex.
1. All expressions and equations can be divided into terms that are separated by a plus sign, a minus sign, or the equal sign if you have an equation.
2. ALWAYS include the sign (add or subtract) with the term that follows the sign. If the term doesn't explicitly have a sign in front of it, then the sign is a plus. I used to have students circle the terms of equations. That is the first breakdown of a complex equation.
2. All terms can be seen as fractions or rational expressions. If a term doesn't have a denominator, then always imagine that you could divide it by 1 if you want. It sounds stupid, but it can be very important.
3. Each term can be broken into its factors; those things that are multiplied together. There may be factors in the numerator and there may be factors in the denominator. Have students circle each of those.
For example, one could have:
5x
---
x+1
In the numerator, 5 is a factor and x is a factor, and (x+1) is a factor in the denominator. It's a good idea for students to put parens around factors if it's not clear. In this case, it keeps them from trying to cancel the 'x' in the numerator with the 'x' in the denominator. You can only cancel (or combine) factors.
4. Each factor has an exponent. If you can't see one, then the exponent is 1. Always remember that it's there. In the above example, you could imagine that the factor (x+1) has an exponent of 1.
5. You can move any factor in a term above or below the dividing line by changing the sign of the exponent. Remember, all factors have at least 1 for an exponent. Also, don't let fractional or decimal exponents bother you. Exponents are just numbers. Something raised to the 2.375 power might seem odd, but it can happen. All of the algorithms or rules work exactly the same no matter what form the exponent is in. You can even have an exponent that is a variable, but you can worry about that later.
I never liked changing factors with fractional exponents to the square root (cube root, etc.) form. I guess you need to know how to do this, but most engineers and scientists stick with the fractional (or decimal) exponent. Square root (etc.) signs are usually used only for constants, not variables.
The expresssion above could then be written as:
5x(x+1)^-1
I don't know why you would want to do that, but "don't want" is not "can't".
6. complex rational terms work just like simple fractions when you want to add/subtract, or multiply/divide them. Think about adding the following two fractions
1/(3*5) + 3/(5*7)
where you aren't allowed to first multiply the 3 times 5 and the 5 times 7. Your common denominator could be (3*5*7) This is what you have to do when you need to add complex rational expressions.
Finally, getting back to the problem of multiplying terms, if you keep the preceeding sign with the term, then you won't have any problem multiplying terms. You just have to remember the rules about multiplying signs.
There are a lot of these things I learned the hard way. I mentioned before that I didn't feel that I really "knew" algebra until after my junior year trig course. Part of the problem is that it takes time to put all of the pieces together. But some problems would never have happened in the first place with the proper instruction.
i can't go along with mr. person
on what "factor 30" oughta mean.
in my mind, the correct response
to such a prompt would be "30=2*3*5",
i.e., one should break down
the given number into its prime factors.
of course, listing all the factors
of a given number might very well
be useful in some contexts:
i suggest that "list all the factors"
is the appropriate prompt in such cases.
wow!
cool!
thank you!
The other key aspect of factoring--as in factoring a polynomial--becomes apparent when the thing being factored is equal to zero. In these cases, factoring gives you the "zero product form." For example,
x^2 + 5x - 6 = 0
could be factored to
(x + 6)(x - 1) = 0
which is in zero product form.
The value of this factoring is that whenever you have two quantities multiplied together to get zero, one or the other of those two quantities must be zero.
So, in our example above, either x + 6 = 0 or x - 1 = 0. These are easily solved to find x = -6 or x = 1. That is the motivation for factoring polynomials: you turn an addition equation into a multiplication equation, which you can solve.
Dan K.
You guys are a bunch of math geeks. I love it.
p.s. I would go with anon on the factoring thing... Factoring 30 would give you 2 x 3 x 5, I wouldn't think to include 1 and 30.
Then again, the Greatest Common Factor (GFC) could include 6 and 15.
Anyways, if factoring isn't unmultiplying, its close enough not to quibble about.
I'll mention that I put up a couple of quadratic factoring worksheets on the old KTM.
If you want practice problems with answer sheets, they should provide enough for a while.
"p.s. I would go with anon on the factoring thing... Factoring 30 would give you 2 x 3 x 5, I wouldn't think to include 1 and 30."
You have to keep factoring and prime factorization apart. You can use both methods to find GCF.
When I am too lazy to find factors, I use these nifty sheets: http://factorzone.tripod.com/101-150.htm
Getting back to the original question, for what it's worth, I would say that division--not factoring--is unmultiplying. When you've got an equation like
3x = 15
you need to find a way to get just x on the left side. So, you want to undo the multiplication by 3 that has been done to x. To undo that multiplication, you divide.
Similarly, subtraction is unadding; taking a square root is unsquaring; and applying the inverse sine is unsining.
Dan K.
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