## Monday, February 01, 2010

### Algebra Conundrums

Yes, it's that time again. I have four algebra problems to present. Two of them are word problems. I hate algebra word problems....unless I can draw a diagram. If I can draw a diagram, I'm usually able to figure out the problem. If I can't diagram it, I'm at a loss. Thus, the famous following famous household quote was born....
Me: "What do you have?"
Smurf: "A drawing of a helicopter and a boat. What do you have?"
Me: "A helicopter and a boat."

The two word problems I'm presenting don't even have helicopters, boats, cars, or trains that can be drawn. (Ok, one of them does involve distance and driving times, but really, even I'm not desperate enough to draw a little car by the problem.)

The other two problems I'm presenting I can work when I do them one way, but get the wrong answer if I take a different approach. My question is, why is the "incorrect" approach incorrect? This is needed because Sweetling took the latter approach. I can show her the way I did the problem, which led to the text book answer, but I feel that's rather valueless without also being able to help her understand why the approach she took didn't work. Otherwise, math becomes a rote memorization of steps, and not a critical thinking exercise.

All four problems I'm presenting are from the same lesson on "mixed expressions". We have been learning how to multiply and divide polynomial fractions. We've learned how to find the LCD of polynomial fractions, to rewrite them as equivalent fractions using the LCD and then to add and subtract polynomial fractions. The textbook given objective for this lesson is "to write mixed expressions as fractions in simplest form.

So, c +5c = c2 +5c

(On an unrelated side note, its amazing what the absence of the first backslash in a short series of html tags does. I'm still not quite happy with how my fractions look, but its better than staight typing I suppose.)

And, (x) + (5x + 2x - 1) - (7x - 1)
Eventually equals x +5.

No, I'm not showing my work. I'm having enough headaches with just the html tags for the problems, and I still don't like how they look.

To give myself a break from my sorry html, here are the two word problems:

It took Jan y hours to drive 200km. If she had increased her speed by 10 km/h and driven for 2 h less, how far would she have gone? (Hint: make a chart. Answer in terms of y.)

OH MY GOSH! As I was typing that in, I realized what I missed. Ok, I can do that. You're welcome to do it for fun if you'd like. (If you're wondering, I totally missed the "answer in terms of y." I think that should have been part of the problem, not part of the hint underneath the problem that I didn't pay close enough attention to. Here's the second one, though if they just want an answer in terms of a variable, I can totally do that. I was thinking that they actually wanted me to solve the problems. My attempts to come up with a numerical solution were extrodinarly circular and frustrating. Now I know why. Duh.

Ted bought n rolls of film for a total of \$40. He then sold all but 2 of them for \$1 more per roll than he paid. How much did he receive for the rolls of film that he sold?

Now that I've embarrassed myself with the word problems, I'll move on to the second set of algebra conundrums.

Here's the problem:

( a+ba - 1)(ab +1)

If we add the fractions in parenthesis first and then multiply, we get the textbook answer. If we FOIL first and then add fractions, we do not. Is this just a matter of order of operations?

Here's the next problem:

(9- 1x2) / (3x-1)

When the problem is rewritten as:

(9x2-1x2) (13x-1)

The correct answer can be reached after some factoring and simplifying.

But, when the problem is rewritten as:
(9x2-1x2)(13x - 3x3x)

--in other words, when the individual terms of the second part of the expression are inverted rather than the whole term being inverted-- The correct answer is not reached. So, the question is, why should the whole term be inverted rather than inverting the terms and then combining them. "Don't do it that way; do it this way," isn't a very satisfactory answer.

To recap, I know longer need help on the word problems, but you are free to work them for yourselves for fun if you'd like. But, I really do need some clearly worded explanations for the last two problesms. I know how to do them, and did them correctly, but I don't know how to explain why the second approach for each problem didn't work. That's where I need help.

Finally, here are the answers for all four problems.

Don't read this far if you want to work the problems first ;)

The answer to the first word problem is:

distance = 10y2 + 180y - 400

The answer to the second word problem is:

\$ (n2+38n-80n)

The answer to the third problem is:

a+ba

The answer to the fourth problem is:

3x+1x2