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I would never create two unknowns in a situation like this, where the two radii are not independent. Since the distance from R to S was given as 12, the radius of one circle made a good unknown, and the radius of the other was 12 minus that SAME unknown. Either circle would do, of course.
The length of the curve can then be expressed in terms of the one unknown for both semicircles. Using the left circle, and calling its radius r, the right has to be 12-r, so the two semicircles added together were,
= pi*r + pi(12-r)
= pi*r + pi*12 - pi*r
= 12pi
If I took part of my $100 and gave it to a friend, there would be only one unknown. Whether you made it the amount I gave him, or the amount that I kept, or the percent I gave him, or the percent I kept, or the difference in dollars or percent or fraction between what he got and what I kept, or the ratio of our money, or whatever, there is only one unknown. Everything else in such a problem can be expressed in terms of that one unknown, which usually makes the problem easier to manage.
THIS is what I was trying to do.
THIS is what I always do, if possible.
I don't know what the problem was.
Inflexible knowledge?
Heat prostration?
I'm half serious about the heat. I took the test outside in 85+ temp. All summer long I've had severe performance deterioration any time I work in the heat. One day, when the temperature was close to 100, I found myself unable to solve even the simplest of problems. I sat at the picnic table working the same problems over and over again in slow motion. Five, 6, 7 times. Or more. I'd crawl through the problem, check my (wrong) answer, then go back to the beginning and crawl through it again and then again until finally the correct answer appeared.
Then I'd go on to the next problem and do that one 6 or 7 times.
I love summer. Have to soak up the sun while I can.
12 comments:
Heat prostration?
I'm half serious about the heat. I took the test outside in 85+ temp. All summer long I've had severe performance deterioration any time I work in the heat. One day, when the temperature was close to 100, I found myself unable to solve even the simplest of problems.
This happens when I'm running, too. I distinctly remember somewhere around mile 9 of the half-marathon when I was no longer capable of figuring out what pace I was running at. And, strangely, I could think about just about anything else. In fact, some sorts of deep thinking seem to be enhanced by long distance running. But simple math? Out the window.
Although the problem could be heat, it is more likely to be the related problem of dehydration.
Although Glen would never create 2 unknowns, preferring r and 12-r to r and s, I often find it easier to create multiple unknowns when initially setting up the problem, then remove the unnecessary ones. In this case, it was easier to remove (r+s) as a single unit, and never worry about manipulating 12-r.
I think this probably depends on your algebra teacher when you first were taught the the process of turning a word or picture problem into algebra. It is a process that is taught like long multiplication and if my memory serves me there are many ways to do that efficiently and effectively.
The key idea is to reduce the number of variables either initially by holding the idea of (12-r) as a single entity or by writing s and noting the connection between r and s
I probably shouldn't have said "never", since there are cases, such as the dummy variable "u" in calculus (e.g. chain rule for composite functions), where so-called "dummy" variables are useful. And, there are tricks that use dummy variables to convert an apparently complex problem (a contest-type problem) to a recognizable form. I use them, too, if I'm intentionally using a simplification technique.
It's just that in an SAT-type question, taking care as you go along not to *mistakenly* create a new variable when you don't have a new degree of freedom tends to help keep the problem under control.
I like to use more variables than are needed because I find it easier to create correct equations. I know that I can always turn the algebra crank later without much thought.
r+s=12 is easy and I know that it's correct. I also know that the half perimeters are pi*r and pi*s.
I then look for enough equations to meet my unknowns. That is what's funny about this problem. You don't have enough information to directly solve for the answer before you look at the choices. There are not enough equations for the variables. Even if you use just r and (12-r), you have no equation, unless, that is, you plug in each answer.
I don't like problems like this, because my first reaction is that you don't have enough information. You do, however, if you look at the possible answers.
Also, why is there no variable in the answer? It's just a unique aspect of this particular problem. What if one of the semicircles is replaced by half of a square? You would have something like this:
4r + (12-r)*pi
for the perimeter. the variable does not disappear when the expression is reduced.
You can't trust what you think because problems try very hard to trick your understanding. You just have to follow the facts (equations) and see where they lead you. As I always say, let the math give you the understanding, not the other way around.
Catherine,
Steve's answer is close to what I would have said. But if I were you, I'd try to find the forest a bit more, and forget the trees.
You're getting advice from experts. You aren't an expert. Part of the problem you're having is you're doing cargo cult math--trying to use the trappings of problem solving to solve a problem, when you don't really have good intuition for WHY those things work.
You shouldn't be trying to say "two unknowns! two equations!" at all. That shorthand is only meaningful when you've already understood that the problem is asking for you TO SOLVE FOR TWO UNKNOWNS.
But in this case, you don't need to know the value of either radii. You don't need to solve for either unknown. That's why you need to see how to factor--and why keeping the problem as r+s is useful. But to see that, you need to know how to write down what the problem says succinctly, not start trying to solve a problem not posed.
Likewise, everyone here trying to show you how we think about the problem, and we are already skipping so many steps that a novice can't skip--how do you turn the first sentence into a math statement, how do you immediately turn the segment RS into a statement about length, how do you factor, etc.
In any non-SAT context, I'd be suggesting you back up and stop trying to jump straight to techniques for solving the problem. Maybe in the SAT context, you feel you don't have time to learn that. But in 400 days, you do have time to learn to think mathematically. So stop worrying about techniques to solve problems, and just get to where you can turn sentences into math.
This happens when I'm running, too. I distinctly remember somewhere around mile 9 of the half-marathon when I was no longer capable of figuring out what pace I was running at.
Funny thing is, it takes me a while to realize that my performance has fallen apart.
it is more likely to be the related problem of dehydration
why is that?
(that's what I was thinking...but I realize I don't know very much about dehydration & heat...)
I LOVE the gas station on your blog, btw!
It's just that in an SAT-type question, taking care as you go along not to *mistakenly* create a new variable when you don't have a new degree of freedom tends to help keep the problem under control.
right --- we've got confounding variables in this discussion.
As I mentioned, when I did two of the 3 math sections on the train (in air conditioning!) & untimed, I missed only 1 question on each section. (Had to bring that up.)
I suspect I would have solved this problem if I'd encountered it on the train, not outdoors at the picnic table.
One of the issues with trying to work 'tricky' questions in a small space of paper under time pressure is exactly what you say: you have to keep the problem under control.
I'm working on that issue with C. & his friends ---- how to write their work in the space of one SAT-item square so that you've still got room to maneuver if your answer isn't amongst the choices.
You also need to be able to look at your work and instantly SEE where you put things.
Alfred North Whitehead has a nice passage about symbolic notation & 'seeing' the problem all at once.
Another thing to track down and post.
stop worrying about techniques to solve problems, and just get to where you can turn sentences into math
I don't think a person who is teaching herself can follow this advice.
Since I am already able to turn sentences into math, I don't know what the standard is by which to judge myself.
How will I know when I am able to turn sentences into math by an expert's standards?
And how will I teach myself to do this?
What materials will I use?
How will I assess myself?
I'm pretty sure this isn't doable -- not unless there's a specific curriculum I can use.
Well, I'm repeating myself, but you start with Polya. So you've got materials.
re: assess yourself: You follow a template for answering the questions, and you'll know you've improved if you can quickly answer those questions on the template (yes, there's a chance you'll get them wrong, but for now, just answering them quickly at all is an improvement.) Then, once you've done the template, you solve the problem as stated in the test book or whatever.
Let's take the probability problems for an example. The template: ask yourself 1. What is the Sample Space? 2. What is the event I'm being asked to find the probability of? 3. What model does this problem correspond to? Those answers should be enough for you to then compute the answer. If you get the wrong answer, you go back and assess: did I get the model wrong? The event wrong? The sample space wrong?
In a geometry problem, the template is a bit different, but still related: 1. Draw the figure. 2. label all of the things I know on the figure (elements which are congruent, various angles, lines parallel, etc.) 3. Ask self "how can I write what I know in terms of what I'm being asked?"
use above to solve problem. If you get the problem wrong, go back and assess: did I fail to write it in terms being asked? did I forget/miss relevant labels on the figure? Did I incorrectly draw the figure?
Each of these templates is really very similar. They are suggesting that you take each sentence given in the problem and turn it into the most compact representation that helps you find the answer.
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