kitchen table math, the sequel: SAT Math Problem

## Monday, November 28, 2011

### SAT Math Problem

If a square has a side of length x+4 and a diagonal of length x+8, what is the value of x?

You have 75 seconds. Go. No thinking first and then starting the clock.

This is a simple problem, but where is the shortcut?

ChemProf said...

It is a right triangle problem with two equal sides, so the diagonal is equal to the square root of two times the side. Therefore x + 8 = sqrt(2) x + 4sqrt(2) and x = (8 - 4sqrt(2))/(sqrt(2) - 1)

Although I do see Catherine's point about no scratch paper since many students would see that quickly if they could draw the picture but not if they couldn't.

SteveH said...

Oh, you want to see the choices? Is that fair?

A) 4
B) 8
C) 16
D) 4Sqrt(2)
E) 8Sqrt(2)

Mathematical aptitude apparently includes something other than math, like testing each answer.

In my review of real SAT tests, my comments were things like:

careful
scary simple
look for exactly what they want
dumb mistake creator
seconds count

I don't have any comments related to the need to know significant math. I also verified what others have said about practice tests that are not official SAT tests. Don't trust them. Timing and problem shortcuts are critical and you have to practice with the real thing.

SteveH said...

When I did it, off I went immediately with a regular mathematical equation. After about 30 seconds it finally dawned on me that there must be a shortcut because I finally (bad technique) decided to look at some of the answers. Then I spent precious seconds looking to see if I missed some way to have parts of the equation drop out. I tried to transform my math into one of the answers. FINALLY, after 75 seconds were up, I realized that I just needed to plug in the answers. I knew, however, that the answer had to have a radical in it. That saved a little time.

I guess what annoyed me was that I was charging ahead like a good mathematician and it was the wrong thing to do.

Michael Weiss said...

The answer must be irrational, because the side and diagonal of a square are incommensurable. Therefore A, B, and C can be eliminated immediately.

To determine which of the remaining options is correct, try them out one at a time:

(D) If x = 4Sqrt(2) then the side is 4Sqrt(2) + 4, and the diagonal must be ( 4Sqrt(2)+4 )(Sqrt(2)) = 8 + 4Sqrt(2), which is x + 8, so we have found a solution and can stop.

On the other hand if we tried (E) we would have x = 8Sqrt(2) then the side is 8Sqrt(2) + 4, and the diagonal is ( 8Sqrt(2)+4 )(Sqrt(2)) = 16 + 4Sqrt(2), which is NOT x + 8, so (E) is wrong.

You can call that "testing each answer" if you like but I certainly wouldn't call it "something other than math".

SteveH said...

"... but I certainly wouldn't call it "something other than math". "

Of course it's something other than math. We're not doing curve-fitting or mathematical modeling. This is simply a test skill. The test is trying to catch those who are doing the math. The 75 second limit just makes it worse.

Michael Weiss said...

As far as "charging ahead like a good mathematician" goes, here is how this mathematician would solve the problem without looking at the answers.

Introduce a new variable s = x + 4. So the side is s and the diagonal is s + 4. On the other hand the diagonal of any square is s*Sqrt(2). So the equation is s +4 = s*Sqrt(2). Solve for s = 4/(Sqt(2)-1). Rationalize the denominator by multiplying both numerator and denominator by the conjugate Sqrt(2)+1: The result is s = (4Sqrt(2) + 4)/1. Convenient how the denominator is 1. Since s = 4Sqrt(2) + 4, x = 4Sqrt(2).

Is that "something other than math"? Took me longer to type it than to solve it.

gasstationwithoutpumps said...

I'm with Michael on this one—it is about a 50–70-second problem if you do it the straightforward way (I didn't even bother introducing s), but a 20–30-second problem if you just test the 2 reasonable answers.

As multiple choice questions go, it is not a great one, as a tiny bit of knowledge is enough to get 50% right by guessing. But the SAT needs some easy questions, in order to separate the low scorers.

lgm said...

If you don't want to think, just plug/chug w/Pythagoras. It works out in about 45 sec if you write down every step since the numbers are so easy.

SteveH said...

"You can call that "testing each answer" if you like but I certainly wouldn't call it "something other than math"."

So now, in hindsight, you're changing your argument. Everything is easier looking back, isn't it. That's my whole point.

How did you know a priori that the substitution would help? You only came up with that because you were challenged. You still had to multiply by the conjugate.

Do the problem in 50-70 seconds without stopping to wonder whether this is what they wanted you to do? Not likely.

Having to test answers is not math.

Allison said...

2(x+4)^2 = (x+8)2
2(x^2 + 8x + 16 ) = x^2 + 16x + 64
2x^2 + 16x + 32 = x^2 + 16x + 64
x^2 - 32= 0
x = 4sqrt2

took me 55 seconds including typing
didn;t see solns.

Allison said...

my shortcut was not to think in or take sqrts.

SteveH said...

Some problems on the SAT are designed so that using a normal mathematical route will take too much time. For this problem, using the sqrt(2) relationship takes time. The Pythagorean formula is another technique, but how would you know that the terms nicely drop out? You know that you are going to have a quadratic equation. How is that a better route than avoiding square roots? The SAT is notorious for trying to send you down the wrong path.

The goal of my post is not to get a bunch of mathy people to show how it could be done in 75 seconds. What happens on the problems where you didn't guess the proper route? Are you less of a mathematician? Do you have less aptitude? A study was done that showed that the best students were the ones most helped by a little extra time. Limited time is used to separate students even if the separation is not very meaningful.

Text Savvy said...

Thanks for this problem. A side effect I noticed while playing around with it is that it seems that when one has a square side length of x + p and a diagonal length of x + 2p, the value of x is simply 2p/sqrt(2).

I could be wrong. Still haven't figured out why. So don't tell me, even if I'm wrong. : )

lgm said...

With the Pythagorus route, I felt the numbers chosen were friendly to work with - all multiples of 2. Quadratics aren't scary to work with, I figure if it's on the test, it's solvable. I thought about substitution briefly but discarded that thought as a) it would take more time and b) mathy 7th graders do well on this test, and that technique is not taught in many Alg I classes. I was surprised to find out the NY saves the discriminant and the quadratic formula for Alg II/Trig, which answered a lot of my ??s which arose from hearing that a kid needs Alg II/Trig to do well on the SAT. Substitution hasn't been taught yet, but the Alg II/Trig year isn't over, maybe NY gets to that before Precalc.

ChemProf said...

I know it bothers you that there is strategy for working through these exams and separating students, but your son will need lots of these tricks (including knowing common triangles -- physics profs love those because they can give exams that are faster if you don't use the calculator). The strategies will vary, but it is unusual for the "right" approach to just plug away at the math. Many of the physics exams I took could not be done in the allotted time, and if you didn't remember some minor relationship the prof mentioned once, you'd have a problem you couldn't do.

For the record, once I came back and saw you had posted the answers, I went back to my solution, used the x^2-y^2 = (x+y)(x-y) relationship (and it turns out in lovely SAT fashion that the denominator becomes 1) and got the answer in about 40 seconds total. For a lot of SAT problems, the major strategy I found useful is:
1. always leave things in radical form (no calculators)
2. don't try to simplify until you check the possible answers, as you may not need to and it will let you know when to stop

Here, it may well have been faster to plug in the answers, but only if you see that you can drop it to two answers. Having seen how slow my students are with their calculators, I'd bet many of them would take 2-3 minutes to plug in all five possible answers.

Text Savvy said...

Ah, yes. Because an increase of diagonal length (+8, or 2p) represents an increase of 8/sqrt(2), or 2p/sqrt(2) of horizontal (or vertical) length of the square. Similarly, an increase of horizontal or vertical length (+4, or p) represents an increase of p*sqrt(2) of diagonal length.

That is, 2p/sqrt(2) = p*sqrt(2).

Michael Weiss said...

iThe goal of my post is not to get a bunch of mathy people to show how it could be done in 75 seconds. What happens on the problems where you didn't guess the proper route? Are you less of a mathematician?

Well, look, if your argument is that timed tests are not a good way of measuring mathematical aptitude, you're not going to get any disagreement from me.

One of my favorite essays is Henri Poincare's 1908 "The Foundations of Science", in which he describes working fruitlessly on a problem for 15 days. Then, after drinking black coffee late one evening, which kept him up unable to sleep, the ideas suddenly came to him, and in the morning they were fully formed. Then the work stopped again, until several weeks later, when just as he was getting on a bus a new idea suddenly popped into his head, fully formed. Poincare came to believe that his subconscious worked on the problem for weeks while his conscious mind was doing other things.

If that's what a normal mathematical route looks like then no, of course no timed test can possibly capture it. But schooling is all about batch-processing and standardization; it is fast food, whereas authentic mathematical work is more like slow-food.

SteveH said...

What if the side is x+2 and the diagonal is x+8? Are those nice numbers for the Pythagorean formula? The radical approach would be less of a dead end. When I used the radical approach, I stopped in the middle and couldn't help but think that there was a trick I was missing. That's when it dawned on me (in a non-mathematical fashion) to plug in the numbers. I would want to do it in the end anyway to check my answer.

"I know it bothers you that there is strategy for working through these exams and separating students, but your son will need lots of these tricks (including knowing common triangles -- "

I'm not questioning that it's a game. (Perhaps I'm whining a little.) That's my point.

"1. always leave things in radical form (no calculators)
2. don't try to simplify until you check the possible answers, as you may not need to and it will let you know when to stop."

These are the things I'm trying to find. I also leave things in non-reduced fractional form. I also look for a least common multiple when I decide to plug in a sample number instead of using percents. The hope is that the fractions will disappear. I'm teaching my son to immediately be able to switch from one side to another for common triangles. And, obviously, be able to deal with radicals very quickly.

There are so many non-math ways to screw up the test. When I look at almost any SAT math question, I know immediately that I can solve it. Quickly is another thing.

The other thing I've noticed is that some problems seem like a critical reading test. One (I'll have to dig it up) I couldn't figure it out at all.

Also, I didn't know that "between" meant > and <, not >= and <=. It seems like it has to be that way, but I've never been forced to figure it out.

Well, I guess I'm whining a lot. I'm just trying to reduce a whole lot of SAT stuff into a minimal set that my son has to worry about. He asked for Mathematica for Christmas. I'd rather do that.

pckeller said...

And for students who are not fluent at manipulating radicals, this is one of those times where a ti89 can help...

solve(sqrt(2)(x+4)=(x+8),x)

..but you use the radical symbol where I have typed "sqrt".

Part of SAT prep should include knowing the tools you have available and when using them is worth it.

SteveH said...

Thanks! I keep thinking that the TI-89 is not allowed. I think Catherine has also talked about the value of the table feature. It's interesting how I continue to use my ancient calculator (at least I'm not using my even older slide rule) while my son graphs on his TI-83.

Text Savvy said...

I have to wonder. What made you write, "This is a simple problem" in the first place when it most certainly isn't a simple problem?

1) I solved it the way ChemProf did, except that I multiplied numerator and denominator by (sqrt(2) + 1), after which the correct answer pops right out in the simplification. This didn't take 75 seconds.

2) If you solve it this way, then changing the problem so that the sides are (x+2) and (x+8) doesn't change the difficulty at all.

3) It *is* a simple problem, much more so without a calculator.

4) Everyone, or at least everyone taking the SAT, should be familiar with at least the 1-1-sqrt(2) (45-45-90) and 1-sqrt(3)-2 (30-60-90) right triangles.

Debbie Stier said...

Do you know how many times I've said "what's the big rush" about the SAT?

If I had 10 more minutes per section, I could do sooooo much better.....and I have confirmed this theory.

It's the race against the clock that kills me every time.

What is the big rush?

If I need 10 more minutes per section, is that such a big deal in the real world? Should I be shut out of opportunities because it takes me a little bit longer to think about these problems than it does other people?

Lucky for me, I've survived life so far without being able to answer these questions under the gun.

And lucky for my son, he seems to be much quicker than me at answering these questions FAST (especially when it comes to the math).

The other day we were doing a practice section and he decided that he DIDN'T want me at the table with him (a first) -- so I went down to my office (right below him) -- and I could hear him WALKING AROUND BEFORE THE TIME WAS UP. I nearly stomped up the stairs and started yelling at him to CHECK HIS ANSWERS!

But I didn't want to squander my own time (even if it was just pretend). And of course, he did way better than me, despite the fact that he finished five minutes early.

Though he didn't do perfectly, I might add, and I keep telling him, unless you're perfect, you need to take the full time.

Oy. Some of us just take a few more minutes.....and what is so wrong with that????

Allison said...

--If I need 10 more minutes per section, is that such a big deal in the real world? Should I be shut out of opportunities because it takes me a little bit longer to think about these problems than it does other people?

Yes, you should.

A week has a finite number of hours in it. If you need 10 more minutes for these 40 problems, you need more than that on hard problems you'll be getting in college. That doesn't translate linearly to other kinds of problems, but in college, it means it's going to take you 10 hours to do a problem set that the average kid takes 8. If you take 4 courses, you're going to have to do 8 more hours of work a week just to keep up. When the going gets tough, and the lab or the research requires 20 hours, you're going to need, say, 30. But you aren't going to have 30. You already have 8 hours less than the others because you're spending more time on the average work than they are.

So you will be slower. And you will learn less. And over a term, you are unlikely to put in the extra time needed to keep treading water with the others, so you will be more likely to be a lower grade student.

Am I exaggerating the effect? I don't know, maybe. But here's a good example: the best lecturer of the first course in the CS major at UC Berkeley had this rule of thumb: if he could do the whole midterm in under 60 seconds, then the average student could do it in 1 hour, and then we gave 2 hours to everyone.

If it took him just a few seconds longer, the class COULD NOT FINISH the test in time. In fact, if it took him 90 seconds to do the test, less than 10 in 200 would finish in two hours. Small deviations in expert understanding become BIG deviations in novice understanding.
A student who needs ten more minutes on the SAT math section compared to the rest of the 200 kids is not an expert in the stuff they need to be ROTE at in order to move on to the higher level stuff in a CS course midterm. They are not going to be able to do that midterm in the allotted two hours.

This doesn't even address the fact that the goal of a test is to discriminate the population of test takers. If you all cluster as "above average", the test is useless. That you personally don't agree with their methodology or the metric by which the test measures fitness is a different issue.

SteveH said...

"I have to wonder. What made you write, "This is a simple problem" in the first place when it most certainly isn't a simple problem?"

From a mathematical standpoint, it is simple. You should be able to write it down immediately:

sqrt(2)*(X+4) = (X+8)

This is a simple relationship for a 45-45-90 triangle. Just turn the crank to solve.

What if it was something like:

2*(X+4) = (X+8)

It's "simple" in terms of mathematical understanding. With the sqrt and the need to get an answer in radical form, it becomes tedious, not difficult. That's where the 75 second limit rears its head. The SAT is a speed test in many cases, not an understanding test. They think the speed factor and the shortcuts will show aptitude, but it really shows preparation for the test. I don't think they worry too much about it as long as the effect is to separate students.

SteveH said...

"This didn't take 75 seconds."

Is this a justification for the validity of the question and the time limit, especially since it's just turning the crank?

Michael Weiss said:

"... if your argument is that timed tests are not a good way of measuring mathematical aptitude, you're not going to get any disagreement from me."

This is the argument that has been going on at KTM over a number of threads.

Obviously, my position is that the math SAT has become an arms race between the preparers and the College Board. I look at virtually all of the SAT math questions and the only things I think of (as I said before) are:

careful
scary simple
look for exactly what they want
dumb mistake creator
seconds count

The SAT may be a good way to sort people on hard work and preparation, but for math or aptitude, they should allow students to substitute the SAT II.

Allison said...

Steve,
Hard work and preparation are good things to correlate against college success, right? and if you're going into a math/sci/eng specialty, you'll take the sat II. So it's just an "and" instead of an "or". What's the problem?

I promise this is the last time I ever talk about this test. This topic is so boring. But here's one more thing:

It's Just A Test.

It's not a measure of someone's integrity or self worth or soul's value.

The weirdest thing about these threads is that the people most obsessed with the test and the ones complaining about how it discriminates its test takers are the ones IMPARTING THE MOST VALUE on the score!

That someone needed ten more minutes to do better does not mean that that person has less value as a human being. It just means your score is lower. This score, just like your salary, does not determine how much you are worth. It might determine how much a certain segment of society values you. But if you can't see the difference between how others value you and you value you or God values you, you're going to have more problems than just this exam.

Our world is filled with tests. You want to be a lawyer, doctor, or engineer? You'll take exams in college, in post-bac, and for your licenses. You'll take exams if you want to be a plumber or electrician or beautician. That's just how our society is set up. Railing that it's unfair or an arms race isn't adding anything.

Every test will make some people score higher than others. Acting like this is a defining moment of one's life is a ludicrous form of materialism.

I feel bad for children who have parents obsessed with this test. They deserve to have perspective on this exam, and they will find it hard to get that at home.

SteveH said...

Debbie said:

"Should I be shut out of opportunities because it takes me a little bit longer to think about these problems than it does other people?"

Allison replied:

"Yes, you should."

I disagree 100%.

"Am I exaggerating the effect? I don't know, maybe."

Wait until you start preparing your kids for the SAT. Then you will be able to calibrate the effect. The speed and stress factors are soooo annoying. And this is for "simple" math. If you work to get to the speed needed for the SAT, that speed doesn't translate into speed for advanced math.

How does my fast ability with 30-60-90 triangles help me with calculus? How does my really fast ability with odd/even problems help me with differential equations? The speed skills of the SAT don't translate. Only the preparation skills translate.

As I've said on other threads, college is more like a marathon and factors like character and determination are far more valuable. How many kids do well on the SAT but end up partying their way right out of college.

"This doesn't even address the fact that the goal of a test is to discriminate the population of test takers. If you all cluster as "above average", the test is useless. That you personally don't agree with their methodology or the metric by which the test measures fitness is a different issue."

That IS the issue of this thread - that the metric is too much tilted towards speed for a very limited domain of material. However, the argument doesn't mean that someone is unwilling to play the game.

Even though colleges do take into account the entire student, his or her SAT scores weight very highly.

SteveH said...

"This topic is so boring."

"Every test will make some people score higher than others. Acting like this is a defining moment of one's life is a ludicrous form of materialism."

Debbie Stier said...

@Allison First of all, I love this test, but I believe it measures a certain type of intelligence.

And, I'd argue that speed is but one of many variables that make for a successful student, and while some (i.e. me) may be slower at completing these problems than others, I think there are other traits that are just as necessary (if not more so) as speed, to make for a successful student. (e.g. diligence, leadership, organizational skills, etc.)

These other traits aren't measured by the SAT, and the result is that the students whose minds work differently (but are just as capable and may very well have the same IQ as the kid who does well) are less likely to have the same opportunities as a kid whose mind works this particular way.

For instance, my son and I have roughly the same IQ (though strengths and weakness in completely different areas). This math comes much more easily for him than it does for me. Same IQ; different SAT math abilities. Good for him, too bad for me. I am definitely a harder worker though (at least at this point in his life), and way more organized than he is, not to mention more likely to go out of my way to make a good impression on a teacher and organize work with other students.

He's teaching me math; I'm teaching him the other parts of how to be a "good student."

If there was a group project in school, trust me, you would want me on your team ;)

I've always been the over-doer, the leader, the ringleader, the enthusiast, the organizer, the cheerleader, the one with the artistic sensibility to make something look beautiful. These qualities served me well in college as well as in the working world.

And it's not that I can't do well on the SAT -- it's that I need a few more minutes than someone whose mind works differently than mine (I'm not saying forever.....just A FEW MORE MINUTES).....

We all have strengths and weaknesses -- even two people with the same IQ. This test measures a certain type of thinking; I would say, having taken the SAT 6 times this year as well as having had a successful book publishing career for over two decades where I had to read multiple books per week, and yet still find reading this fast for the SAT very challenging, that if your brain isn't wired to work a particular way for this test, you're out of luck on the doors that are opened by doing well on the test.

Similarly, people who are able to afford tutors have an advantage. Those who can't afford a tutor will most likely have to work longer if they want the same score as the kid who's got the personal tutor, and can teach this specific material from years of experience, point out the traps and pitfalls, give test taking tips, etc.

Should the kid who doesn't have the money to be tutored be less entitled to the same opportunities because it might take him or her a little longer to figure out how to do well on this test? I don't think so. I wouldn't want to penalize the kid who takes a little more time to study diligently for this test because the material wasn't served up on a silver platter like it was to the more affluent kid.

Believe me, I love this test, and when I'm done with my project after next Saturday's SAT I'm going to miss it. BUT, having gone deeply down the SAT rabbit hole, and speaking as someone for whom this type of speed does not come naturally, I personally believe it's capturing a certain type of intelligence.

Allison said...

Debbie,

The problem with your argument is that WHATEVER the test measures, someone will always be better and some will always be worse. Why should we pick YOUR way of being better? Just because you value your own defn of brains more?

It seems that what you want is just to have whatever the test measures be something that you do well and others do not.

Why should we privilege your type of intelligence/assets/traits/skills/knowledge rather than someone else's?

Let's say we give you that 10 more minutes--and you did better. What then about the kid who would have done better STILL with just 2 more minutes past that? or 3? or 4? Why should we bias it for your benefit not theirs? (This doesn't even cover that the test does have accommodations for special needs/handicaps already.) How unfair! Should that kid who didn't have the time to take as many practice tests as you be less entitled to the same opportunities?

Heck, you don't even seem to see that EVEN IF you got your 10 more minutes, that doesn't solve anything--because surely it would help all sorts of other people too. Let's pretend 50% of the people did better with ten more minutes (and 50% did worse due to fatigue). Your raw score might improve, but your normed score wouldn't necessarily. It could even fall. And now we've ceilinged out the test a bit more so the high kids can't be differentiated--should the bright kid be less entitled to the same opportunity because he could only distinguish himself by speed?

No one is entitled to anything.

The rich we will always have with us. All tests can be gamed with enough effort, time and money, and they will always be able to game any test with huge sums spent on tutors. Nothing can stop that, no matter how unfair you view it.

There's no end to how nice things are nicer than non-nice things. Being rich is better than being poor. Being happy is better than being unhappy. Being born to good parents is better than being born to bad ones. How would you like to even the scales? Harrison Bergeron style? Or not that extremely, just enough that the scales put you on top?

"if your brain isn't wired to work a particular way for this test, you're out of luck on the doors that are opened by doing well on the test."

Yes, you are. And I'm out of luck at being a football player, a ballerina, a cellist, a movie actor, or a million other things, but somehow, you don't seem to view sports, arts, or entertainment the way. Different skills open different doors. To have such a narrow view as to place undue value on the ones that this test opens is perhaps a problem.

Text Savvy said...

I don't know that setting a time limit is necessarily all about assessing speed. That's why I played around as long as I did with this problem: to see what universals, if any, one could apply to make this problem a practical no-brainer.

And there is. It wouldn't matter if it was x + 6 and x + 12 or x + 24 and x + 48. Because sqrt(2)*p = 2p/sqrt(2), increasing the diagonal length by double the increase in side length will always give you a value of p*sqrt(2).

If you know that, you should be rewarded for it.

Jen said...

Wow, Allison, that's quite a diatribe!

Much of what you write is true, the problem is that the gatekeeping for careers in football or other sports or the arts is based on repeated performances (and practice) over years. There is rarely one performance or test that determines much of the rest of your career in the way that the SAT may (or would like to do).

That combined with the fact that some of the things that the SAT tests (willingness to game the system by practicing, ability to sit in the same seat for hours on end with only short breaks, ability to "see" the problem quickly, under pressure) may well not be all that useful for what it purports to do.

That's the point. Are there people who aren't ending up where they may be best educated because a standardized test isn't the best way to determine their future? Would it be better to do what some colleges do now and look at grades, and even in some cases syllabi and final exams? I don't know. My family scores high on standardized tests, so this current system works well for us. However, I'm honest to point out that it may not be the best system, just because I like it!

Perhaps, a better answer is in multiple performances analyzed than in one high-stakes test. (Or that test taken 2-3 times.)

Or in separate tests at different levels or in untimed tests or...

Jen said...

"I'm honest to point out "

Heh. I'm really just honest *enough* not quite so thoroughly honest as I suggested above.

Amanda said...

For me it is quicker and easier to write and solve the quadratic than it is to use the 1:1:root 2 triangle 'shortcut' - this way I don't have to deal with surds in a denominator or multiply the expression through by (root 2 + 1)

Catherine Johnson said...

This doesn't even address the fact that the goal of a test is to discriminate the population of test takers.

wait!

what happened to standards-based scoring is a good thing and can work?

Catherine Johnson said...

As a teacher, I don't give tests to separate the men from the mice.

I give tests to find out whether **everyone** in my class has learned the material I've been trying to teach. I would be thrilled if every student in my class got every answer right.

That's my goal.

Catherine Johnson said...

On the midterm, I asked everyone to diagram the sentence "Jesus wept."

Everyone did!

They all thought that was pretty funny.

Catherine Johnson said...

I should have timed myself ... I got 4 square-root 2 using the Pythagorean theorem in under 75 seconds.

Where I would lose time on this one (and in fact where I did lose time) would be in trying to suss out whether there's a shortcut I'm not seeing.

Specifically, the fact that the figure is an isosceles right triangle caused me to think there had to be some way x+4 and x+8 translated easily into 'x' and 'xsquareroot2'.

The reason a test-taker who's been prepped 'spends' time looking for shortcuts is that there are problems on the SAT where the standard way of solving the problem is the wrong way because it takes too much time.

Often you can't just solve a problem you know how to solve.

You have to see the shortcut.

Catherine Johnson said...

I got my test back and found that I MISSED A NUMBER TWO: I missed one of the gimmes.

A classic, classic error.

The only way I can finish the test is by racing through the early problems, and racing through the early problems astronomically increases your chance of tripping and falling splat on your face.

PWN has an email from a student who scored a 740, using only PWN's book. The student said that he had explicitly set a goal of scoring 700; he planned from the get-go not to finish any math section.

That strategy netted him a 740.

That's the strategy I should have used -- it's the one all the tutors tell you to use -- but I just couldn't bring myself to do it!

Catherine Johnson said...

If you don't want to think, just plug/chug w/Pythagoras. It works out in about 45 sec if you write down every step since the numbers are so easy

If you've had test prep, you know that this is a "back solve" problem (something I never quite mastered because I didn't have test prep).

Tutors teach students exactly which problems to backsolve, and students learn to recognize them instantly.

(Another area of no common sense-y in my case. A couple of weeks before I took the test, I asked Chris to explain to me how you know which questions to back solve...which meant I never got over my ingrained habit of NOT backsolving...)

For the longest time, I thought: I'll just solve the problems!

Wrong!

It's not a math test; it's a video game.

Some problems you solve, some problems you back solve, some problems you plug in, etc.

Catherine Johnson said...

lgm wrote: If you don't want to think, just plug/chug w/Pythagoras. It works out in about 45 sec if you write down every step since the numbers are so easy.

My steps:

First I thought.

Then I plugged and chugged with Pythagoras.

Catherine Johnson said...

Do the problem in 50-70 seconds without stopping to wonder whether this is what they wanted you to do?

I think there's probably time to do both -- and I also think that most test takers **are** going to do both---

Catherine Johnson said...

A study was done that showed that the best students were the ones most helped by a little extra time.

Steve's right.

Extra time helps the better students.

Catherine Johnson said...

Jen wrote: That combined with the fact that some of the things that the SAT tests (willingness to game the system by practicing, ability to sit in the same seat for hours on end with only short breaks, ability to "see" the problem quickly, under pressure

Add to that the willingness of your parents to spend \$84K for one year of SAT preparation, which includes 10 or more real, full-length, timed, proctored tests.

SteveH said...

"And I'm out of luck at being a football player, a ballerina, a cellist, a movie actor, or a million other things, but somehow, you don't seem to view sports, arts, or entertainment the way."

That's because they ARE different. Those are all performance-based careers, and I assume that the million other things fall into the same category. If my son decides to apply to a music conservatory, they won't care much about his SAT scores. It's all about the audition. If my son decides to major in physics, the special case knowledge and performance-level speed skills required to compete with his well-prepared peers on the SAT is only meaningful in the sense that he would be willing to do the preparation work and play the game. I would rather he prove that doing something more meaningful - even if it's the SAT II or AP tests.

Debbie Stier said...

@Catherine (aka "Cathy)

re: A study was done that showed that the best students were the ones most helped by a little extra time.

Steve's right.

Extra time helps the better students.

THANK YOU. I can feel that is the truth so I'm glad to know I'm backed up by a study.

Any idea how I can find that study?

Glen said...

There's a lot of wisdom in what Allison is saying, that no matter what test they use, 7% of applicants to Stanford will be admitted and 93% rejected--they'll just be different people depending on the test design.

But Steve's complaints resonate with me. Steve, you're SUCH an engineer. Engineers learn to resist the urge to dash through familiar problems, because dashing through real-world engineering problems gets people hurt. In engineering, you do better work and make faster progress on a scale of months by being very deliberate and methodical on a scale of minutes.

It's frustrating that we methodical engineering types are penalized for traits that are a valued asset in certain fields.

And that's the problem with the SAT, in my opinion: what would be a weakness in one field is an asset in another in the real world, but we're reaching the point where almost all fields go through college, and all colleges are entered through a single test with a single set of criteria.

Allison is right. The problem isn't which criteria are used. It's that that every year a wider variety of kids have to squeeze through the same filter to reach a wider variety of jobs. We don't need a better filter. We need more ways to go around the filter.

SteveH said...

"Steve, you're SUCH an engineer."

Seven and a half years of engineering in college and I never needed special knowledge of 45-45-90 triangles. Rarely, if ever, is a problem a special case. The only reason I'm learning about these things after 30+ years of engineering and programming is to help my son prepare for the SAT. Remember, I never prepared (even via math competitions) for the SAT. You can't do that these days.

This whole issue revolves around whether the SAT is a reasonable filter, not whether a filter should be used. Colleges will always filter students. Top colleges do accept students with lower SAT scores than other applicants. They are trying to not make the SAT the ultimate factor, but it is still a very critical factor. Colleges filter one way or another - demand exceeds supply. There is no way around that. However, I wouldn't like it if colleges used only subjective filters. There is no way around some sort of filter.

The issue I raise in this thread is not whether a filter should be used or not, but to point out how the SAT is just a filter based on preparation for taking a math test on a very limited domain of material. What response do I get? I get defense of the questions and of the timing. And, I get the position that, in effect, all filters have the same problems.

They don't.

The only commonality is that high demand drives enormous preparation effort and the spending of lots of money. It would be nice if all of this money and effort were directed towards more meaningful learning activities. You could start by allowing kids to replace SAT grades with SAT II or AP scores. I would rather be working with my son on AP Calc than trying to optimize his recognition of shortcuts of special cases in a very limited domain of material.

With the enormous pressure (effort and \$) being applied to the SAT, they can't tell the difference between preparation and aptitude.

One cannot start with an argument that tries to justify the SAT and then claim that all filters are variations of the same thing.

The goal is not to eliminate all objective measures. The goal is to spend the effort and dollars on more meaningful work. If they focused on AP tests, that would drive real changes into school systems rather than just into external SAT test prep organizations and tutors. There would at least be a chance that more students without external resources won't be left behind and that real learning will be improved.

SteveH said...

"Any idea how I can find that study?"

Debbie - I'll see if I can dig it up, but the general idea is that the lower level students don't know the material and more time won't help much. However, at the top level, speed and tricks are used to separate students, not general knowledge of the material. Since ETS has to separate students using a limited domain of material, timing becomes critical.

Catherine Johnson said...

Allison wrote: It's just a test.

It's not just a test.

SAT II American History is just a test.

AP Bio is just a test.

The SAT is not just a test.

US News uses the SAT to rank colleges and universities, and for colleges and universities the US News ranking is make or break.

C. gets loads and loads of marketing material in the mail every day, some of it promising merit aid, entirely on the basis of his SAT scores.

He has been recruited by 3 elite schools, entirely on the basis of SAT scores.

My home value is based on the SAT scores of kids attending our local high school.

Speaking of which, last year our high school principal sent out a lengthy memo on the SAT, all of it pulled from the Frontline report on the test.

He told the community that SAT scores are in no wise a measure of the quality of a high school but, rather, measured "test taking strategies" or some such.

This year comes word that our high school principal has just returned from an 8-day, all expenses paid trip to China paid for by the Chinese government and College Board.

So now I've got College Board working with the Chinese government to raise my property taxes.

aaaarrrrggghh

Catherine Johnson said...

Our world is filled with tests. You want to be a lawyer, doctor, or engineer? You'll take exams in college, in post-bac, and for your licenses. You'll take exams if you want to be a plumber or electrician or beautician. That's just how our society is set up. Railing that it's unfair or an arms race isn't adding anything.

Our world is filled with fuzzy math and constructivism!

I never let a lost cause keep me from railing.

Catherine Johnson said...

Glen wrote: Steve, you're SUCH an engineer. Engineers learn to resist the urge to dash through familiar problems, because dashing through real-world engineering problems gets people hurt.

oh my gosh!

you guys always nail it -- you've just articulated something that has been nagging at me but that I haven't been able to put into words.

The math test is all about speed; that's the main method it uses to produce error, I would say.

(I mentioned earlier that I got the second question wrong on one of the math sections. That happened because I was working too fast, trying to bank time for the "hard" questions.)

The reading test isn't quite as insane in terms of the speed requirements, but it's bad enough.

I'm a fast reader, and I always finish with time to spare -- so I didn't quite grok how much the timed nature of the reading test bothers me.

But it does.

The speed-reading nature of the Critical Reading sections bothers me for the same reason the speed-math nature of the math sections bothers an engineer: good readers don't whip through passages at breakneck speed.

Good readers CAN whip through passages at breakneck speed, but if they're good readers** that's exactly what they don't do.**

Same thing with writing.

A good writer can spend days on one sentence.

Catherine Johnson said...

Should the kid who doesn't have the money to be tutored be less entitled to the same opportunities because it might take him or her a little longer to figure out how to do well on this test?

\$84K for one year of SAT tutoring.

That's what my kid is competing with when he applies to elite schools.

I believe that if C. had had one year of SAT math tutoring from Advantage he would have scored 100 points higher on SAT math.

Ed thinks colleges should ask candidates to disclose how much tutoring they had and from whom.

Catherine Johnson said...

That's the point. Are there people who aren't ending up where they may be best educated because a standardized test isn't the best way to determine their future?

I don't know the answer to this, but I would bet a modest sum of money there are students who end up without merit aid because of the test.

I'm trying to remember whether I posted the link about colleges that are SAT-optional for admissions BUT give merit aid on the basis of SAT scores.

SAT scores are money.

Catherine Johnson said...

Steve wrote: Seven and a half years of engineering in college and I never needed special knowledge of 45-45-90 triangles.

NO!!!!!!!

NOT SERIOUSLY!!!!!!

(I'm joking --- )

SteveH said...

One of the reactions to the SAT is to call for the elimination of all nationally-normed objective measures. I don't agree with that. I just want my son to spend his time on more productive work. If he gets a 5 on AP calculus, what more do you need to know about math?

What about those kids who don't get to math at that level? How about SAT II Math, level I or II? I think there is still a left-over history of somehow testing aptitude. This indicator is completely lost with all of the preparation effort. As I have mentioned in many other threads, a little preparation can make you look soooo smart.

lgm said...

>>Ed thinks colleges should ask candidates to disclose how much tutoring they had and from whom.

The need for tutoring just indicates that the teacher is not teaching the whole enchilda, ime. I have to tutor son #1 in Alg 2 this year. I figured out the real easy way is to have him either read son #2's notes or the appropriate section in Dolciani or send him to Son #2's teacher. Son #2 has an excellent teacher who is writing her own notes with coherent examples. Son #1 has a person who is copying out of Amsco's review book and isn't copying the visual or abstract explanations, just the verbal. She doesn't start at the beginning of the concept, so that makes it harder for those who didn't have Alg I the year prior. Makes it more time consuming for Son #2, but I can't say that the need for tutoring is his fault. He's visual, not verbal in his math learning style. What his teacher cannot accomplish in 40 minutes can be accomplished at home in 10. You don't know how much I wish for choice in scheduling, but we all know certain teachers would have no one sign up if that were an option in public school.

Back in the Jurassic, I had to know visual, verbal, and symbolic explanations of the concepts. The deliberate restriction of presenting concepts in solely the verbal mode harms visual-spatial kids - many of whom are the future engineers.

And for an alternate viewpoint..6 yrs of engineering school (B.S., M.S.) and I did use prealg and alg in my jobs.

lgm said...

Make that 'more time consuming for son #1' not 2 please.

Catherine Johnson said...

The need for tutoring just indicates that the teacher is not teaching the whole enchilda, ime.

Right - normally that's exactly my view, but my year of SAT prep has changed my view of SAT scores & coaching specifically.

It's just a very strange test. It's not really a 'content' test, and yet it's not really a 'normal' IQ test although it correlates with IQ tests...

The items on the SAT aren't exactly being taught anywhere, and I'm not sure they should be.

I can't suss that out. I've just begun really thinking about the Critical Reading passages --- I really don't know whether the mode of reading one does on the test is the mode of reading one would teach---?

Catherine Johnson said...

lgm wrote: The deliberate restriction of presenting concepts in solely the verbal mode harms visual-spatial kids

Interesting.

I was thinking this afternoon about a study I read identifying three forms of intelligence -- or, at least, 3 separable elements of IQ. I think it was a factor analysis.

As I recall, the researchers found that verbal, math, and spatial (something like that - I'll find it) were distinguishable in the data, and didn't necessarily track together.

I've recently begun to think that students need to be taught the technical vocabulary of various fields, including the vocabulary of grammar and linguistics.

I'm also 'feeling' (this is a personal experience) that verbal types REALLY need those words.

TerriW said...

Even though my oldest child is 7, I read these SAT threads with interest and an eye to the future.

I often think, as a homeschooler, we have to gun hard for the SAT. I mean, seriously, we can put together a portfolio or transcript or whathaveyou, but since none of that is third party, my gut tells me we're going to live or die on the SAT.

I guess this is somewhat ironic since my own path to higher education was a bit non-mainstream, itself. In a class of about 550 students, I was ranked 4th in 7th grade, 15th in 8th, something even lower in 9th, and lower in 10th -- regular school was a baaaad social fit for me, post-puberty.

So, come age 16/junior year, I started attending my local community college full time with a GPA of about 3.96 that first quarter, and kept it about that high throughout. So, once out of the public high school situation, my grades came right back.

I completed my AA and then transferred as a Junior to a state college on the west coast where I finished my BS. Never took the SAT. Well, took some sort of thing (PSAT?) in the 7th or 8th grade to qualify for honors classes, but that was it. But, I guess, what's the point of needing an SAT score when I can show you how I performed in these 90 college credits here that I've already taken? Assuming they haven't dismantled PSEO (Post Secondary Enrollment Options) by the time my kids are "of age," maybe we won't have to pin so many of our hopes on a single high pressure test.

On the other hand, I was never interested in a prestigious college track, so maybe my experience isn't particularly useful.

SteveH said...

Keep in mind that SAT also means money. Kids can move down a tier in the prestigious college track in exchange for money.

Although there are other tracks through college that are quite reasonable, it's always nicer to have options. One can choose not to play the game, but it's never all or nothing. Do well in school, take the PSAT, take the SAT prep class offered by the high school, and then take the SAT once or twice. This is normal and doesn't require a lot of effort.

It may seem from my posts that I'm torturing my son with preparation, but I really haven't done anything yet. He has only taken the school PSAT and I just barely got him to do the timed PSAT practice test that the school handed out. I'm waiting for those results to figure out what practice he needs.

What I'm trying to do is to figure out what work he can do that will maximize his scores with the least amount of work or angst possible. I'm not going to try to turn him into something he isn't. Hopefully, he will quickly get to a point of diminishing returns and the SAT prep will be minimal. Unfortunately, the SAT game sucks you in when you get near the upper end. Each new question you get correct has a bigger effect on your score and the opportunities and money you receive.

This is annoying to me because SAT prep takes away time from more meaningful things my son could be doing. I'm not going to take a "screw it" attitude, and I'm not going to have my son get weird about it. I haven't gotten to that trade-off point yet and he is a sophomore. SAT test prep has taken up almost 0% of his time.

I have always wondered whether there are gentle (?) long-term SAT prep routes. One, I've figured out too late, is middle school math competitions. It's moot because our school didn't offer such a thing. I don't like general admonishments about reading more because that doesn't deal with some specific skills required by the SAT. Perhaps it would have been good to get him to do the SAT question a day starting years ago. I thought of it, but that seemed to be too weird. However, how weird is it to pack all of the specialized SAT skills into high school? How can one be willing to play the game, but then keep it in perspective?

I tend to believe that in many cases, the means justifies the end. It's not what you do, but how you do it.