Well, after the high drama of writing, the math section is pretty tame. Except the whole oh, my god, are they serious? part. Caveat: I’m assuming that the SAT is still a harder version of the PSAT, and that this is a representative test.

Metric
| Old SAT
| Old PSAT
| ACT
| New PSAT |

Questions | 54 44 MC, 10 grid | 38 28 MC, 10 grid | 60 MC | 48 40 MC, 8 grid |

Sections
| 1: 20 q, 25 m 2: 18 q, 25 m 3: 16 q, 20 m | 1: 20 q, 25 m 2: 18 q, 25 m | 1: 60 q, 60 m | NC: 17 q, 25 m Calc: 31 q, 45 m |

MPQ
| 1: 1.25 mpq 2: 1.38 mpq 3: 1.25 mpq | 1: 1.25 mpq 2: 1.38 mpq | 1 mpq | NC: 1.47 mpq Calc: 1.45 mpq |

Category
| Number Operations Algebra & Functions Geometry & Measurement Data & Statistics | Same | Pre-algebra Algebraelem & intermed. Geometry coord & plane Trigonometry |
1) Heart of Algebra 2) Passport to Advanced Math 3) Probability & 4) Data Analysis Additional Topics in math |

It’s going to take me a while to fully process the math section. For my first go-round, I thought I’d point out the instant takeaways, and then discuss the math questions that are going to make any SAT expert sit up and take notice.

**Format**

The SAT and PSAT always gave an average of 1.25 minutes for multiple choice question sections. On the 18 question section that has 10 grid-ins, giving 1.25 minutes for the 8 multiple choice questions leaves 1.5 minutes for each grid in.

That same conversion doesn’t work on the new PSAT. However, both sections have exactly 4 grid-ins, which makes a nifty linear system. Here you go, boys and girls, check my work.

The math section that doesn’t allow a calculator has 13 multiple choice questions and 4 grid-ins, and a time limit of 25 minutes. The calculator math section has 27 multiple choice questions and 4 grid-ins, and a time limit of 45 minutes.

13x + 4y = 1500

27x + 4y = 2700

Flip them around and subtract for

14x = 1200

x = 85.714 seconds, or 1.42857 minutes. Let’s round it up to 14.3

y = 96.428 seconds, or 1.607 minutes, which I shall round down to 1.6 minutes.

If–and this is a big if–the test is using a fixed average time for multiple choice and another for grid-ins, then each multiple choice question is getting a 14.4% boost in time, and each grid-in a 7% boost. But the test may be using an entirely different parameter.

**Question Organization**

In the old SAT and ACT, the questions move from easier to more difficult. The SAT and PSAT difficulty level resets for the grid-in questions. The new PSAT does not organize the problems by difficulty. Easy problems (there are only 4) are more likely to be at the beginning, but they are interlaced with medium difficulty problems. I saw only two Hard problems in the non-calculator section, both near but not at the end. The Hard problems in the calculator section are tossed throughout the second half, with the first one showing up at 15. However, the coding is inexplicable, as I’ll discuss later.

As nearly everyone has mentioned, any evaluation of the questions in the new test doesn’t lead to an easy distinction between “no calc” and “calc”. I didn’t use a calculator more than two or three times at any point in the test. However, the College Board may have knowledge about what questions kids can game with a good calculator. I know that the SAT Math 2c test is a fifteen minute endeavor if you get a series of TI-84 programs. (Note: Not a 15 minute endeavor to get the programs, but a 15 minute endeavor to take the test. And get an 800. Which is my theory as to why the results are so skewed towards 800.) So there may be a good organizing principle behind this breakdown.

That said, I’m doubtful. The only trig question on the test is categorized as “hard”. But the question is simplicity itself if the student knows any right triangle trigonometry, which is taught in geometry. But for students who *don’t* know any trigonometry, will a calculator help? If the answer is “no”, then why is it in this section? Worse, what if the answer is “yes”? Do not underestimate the ability of people who turned the Math 2c into a 15 minute plug and play to come up with programs to automate checks for this sort of thing.

**Categories**

Geometry has disappeared. Not just from the categories, either. The geometry formula box has been expanded considerably.

There are only three plane geometry questions on the test. One was actually an algebra question using the perimeter formula Another is a variation question using a trapezoid’s area. Interestingly, neither rectangle perimeter nor trapezoid formula were provided. (To reinforce an earlier point, both of these questions were in the calculator section. I don’t know why; they’re both pure algebra.)

The last geometry question really involves ratios; I simply picked the multiple choice answer that had 7 as a factor.

I could only find one coordinate geometry question, barely. Most of the other xy plane questions were analytic geometry, rather than the basic skills that you usually see regarding midpoint and distance–both of which were completely absent. Nothing on the Pythagorean Theorem, either. Freaky deaky weird.

When I wrote about the Common Core math standards, I mentioned that most of geometry had been pushed down into seventh and eighth grade. In theory, anyway. Apparently the College Board thinks that testing geometry will be too basic for a test on college-level math? Don’t know.

Don’t you love the categories? You can see which ones the makers cared about. Heart of Algebra. Passport to Advanced Math! Meanwhile, geometry and the one trig question are stuck under “Additional Topic in Math”. As opposed to the “Additional Topic in History”, I guess.

**Degree of Difficulty**;

I worked the new PSAT test while sitting at a Starbucks. Missed three on the no-calculator section, but two of them were careless errors due to clatter and haste. In one case I flipped a negative in a problem I didn’t even bother to write down, in the other I missed a unit conversion (have I mentioned before how measurement issues are the obsessions of petty little minds?)

The one I actually missed was a function notation problem. I’m not fully versed in function algebra and I hadn’t really thought this one through. I think I’ve seen it before on the SAT Math 2c test, which I haven’t looked at in years. Takeaway— if I’m weak on that, so are a lot of kids. I didn’t miss any on the calculator section, and I rarely used a calculator.

But oh, my lord, the problems. They aren’t just difficult. The original, pre-2005 SAT had a lot of tough questions. But those questions relied on logic and intelligence—that is, they sought out aptitude. So a classic “diamond in the rough” who hadn’t had access to advanced math could still score quite well. Meanwhile, on both the pre and post 2005 tests, kids who weren’t terribly advanced in either ability or transcript faced a test that had plenty of familiar material, with or without coaching, because the bulk of the test is arithmetic, algebra I, and geometry.

The new PSAT and, presumably, the SAT, is impossible to do unless the student has taken and understood two years of algebra. Some will push back and say oh, don’t be silly, all the linear systems work is covered in algebra I. Yeah, but kids don’t really *get it* then. Not even many of the top students. You need two years of algebra even as a strong student, to be able to work these problems with the speed and confidence needed to get most of these answers in the time required.

And this is the PSAT, a test that students take at the beginning of their junior year (or sophomore, in many schools), so the College Board has created a test with material that most students won’t have covered by the time they are expected to take the test. As I mentioned earlier, California alone has nearly a quarter of a million sophomores and juniors in algebra and geometry. Will the new PSAT or the SAT be able to accurately assess their actual math knowledge?

**Key point:** The SAT and the ACT’s ability to reflect a full range of abilities is an unacknowledged attribute of these tests. Many colleges use these tests as placement proxies, including many, if not most or all, of the public university systems.

The difficulty level I see in this new PSAT makes me wonder what the hell the organization is up to. How can the test will reveal anything meaningful about kids who a) haven’t yet taken algebra 2 or b) have taken algebra 2 but didn’t really understand it? And if David Coleman’s answer is “Those testers aren’t ready for college so they shouldn’t be taking the test” then I have deep doubts that David Coleman understands the market for college admissions tests.

Of course, it’s also possible that the SAT will yield the same range of scores and abilities despite being considerably harder. I don’t do psychometrics.

Examples:

Here’s the function question I missed. I think I get it now. I don’t generally cover this degree of complexity in Precalc, much less algebra 2. I suspect this type of question will be the sort covered in new SAT test prep courses.

These two are fairly complicated quadratic questions. The question on the left reveals that the SAT is moving into new territory; previously, SAT never expected testers to factor a quadratic unless a=1. Notice too how it uses the term “divisible by x” rather than the more common term, “x is a factor”. While all students know that “2 is a factor of 6” is the same as “6 is divisible by 2”, it’s not a completely intuitive leap to think of variable factors in the same way. That’s why we cover the concept–usually in late algebra 2, but much more likely in pre-calc. That’s when synthetic division/substitution is covered–as I write in that piece, I’m considered unusual for introducing “division” of this form so early in the math cycle.

The question on the right is a harder version of an SAT classic misdirection. The test question doesn’t appear to give enough information, until you realize it’s not asking you to identify the equation and solve for a, b, and c–just plug in the point and yield a new relationship between the variables. But these questions always used to show up in linear equations, not quadratics.

That’s the big news: the new PSAT is pushing quadratic fluency in a *big* way.

Here, the student is expected to find the factors of 1890:

This is a quadratic system. I don’t usually teach these until Pre-Calc, but then my algebra 2 classes are basically algebra one on steroids. I’m not alone in this.

No doubt there’s a way to game this problem with the answer choices that I’m missing, but to solve this in the forward fashion you either have to use the quadratic formula or, as I said, find all the factors of 1890, which is exactly what the answer document suggests. I know of no standardized test that requires knowledge of the quadratic formula. The old school GRE never did; the new one might (I don’t coach it anymore). The GMAT does not require knowledge of the quadratic formula. It’s possible that the CATs push a quadratic formula question to differentiate at the 800 level, but I’ve never heard of it. The ACT has not ever required knowledge of the quadratic formula. I’ve taught for Kaplan and other test prep companies, and the quadratic formula is not covered in most test prep curricula.

Here’s one of the inexplicable difficulty codings I mentioned–this is coded as of Medium difficulty.

As big a deal as that is, this one’s even more of a shock: a quadratic and linear *system*.

The answer document suggests putting the quadratic into vertex form, then plugging in the point and solving for a. I solved it with a linear system. Either way, after solving the quadratic you find the equation of the line and set them equal to each other to solve. I am….stunned. Notice it’s not a multiple choice question, so no plug and play.

Then, a negative 16 problem–except it uses meters, not feet. That’s just plain mean.

Notice that the problem gives three complicated equations. However, those who know the basic algorithm (*h(t)=-4.9t ^{2} + v_{0} + s_{0}*) can completely ignore the equations and solve a fairly easy problem. Those who don’t know the basic algorithm will have to figure out how to coordinate the equations to solve the problem, which is much more difficult. So this problem represents dramatically different levels of difficulty based on whether or not the student has been taught the algorithm. And in that case, the problem is quite straightforward, so should be coded as of Medium difficulty. But no, it’s tagged as Hard. As is this extremely simple graph interpretation problem. I’m confused.

Recall: if the College Board keeps the traditional practice, the SAT will be *more* difficult.

So this piece is long enough. I have some thoughts–rather, questions–on what on earth the College Board’s intentions are, but that’s for another test.

**tl;dr** Testers will get a little more time to work much harder problems. Geometry has disappeared almost entirely. Quadratics beefed up to the point of requiring a steroids test. Inexplicable “calc/no calc” categorization. College Board didn’t rip off the ACT math section. If the new PSAT is any indication, I do not see how the SAT can be used by the same population for the same purpose unless the CB does very clever things with the grading scale.

April 16th, 2015 at 5:01 pm

[…] Here’s my evaluation of the math section. […]

April 16th, 2015 at 6:23 pm

5670/3 = 1890, not 1870. 1890 is pretty trivial to factor in comparison (add the digits; the rest is transparent).

April 16th, 2015 at 7:05 pm

Oops, that’s a typo. I’ll fix. But you’ve got 2×3^3x5x7, which is a tremendous amount of factoring to do under pressure,

April 16th, 2015 at 8:53 pm

“…you’ve got 2×3^3x5x7, which is a tremendous amount of factoring to do under pressure”

Is it? I’m not calibrated on what makes factoring difficult, but I’d do this:

*) 1890 is a multiple of ten, so pull out the 2 and the 5, (189 left)

*) 189 isn’t a multiple of 2, 5 or 10, but the digits to add to a multiple of three, so pull out a 3 (63 left)

*) 63 is a multiple of three, but is also 7×9, so …

2x5x3x(3×3)x7

I see detail work here, but if the victim knows even/odd for 2s, ends in 5 or 0 for fives, ends in 0 for 10s, the “add the digits” trick for 3 and 9 and their times-tables what is the tremendous amount of factoring? Is the division that difficult (honest question … I know I’m not calibrated)? Or is this something else?

April 16th, 2015 at 9:34 pm

The normal factoring question on the SAT, which would always be given not as part of a word problem, is something like x^2 -8x + 15. There has never been a quadratic word problem, period, on the SAT–much less one so difficult.

April 16th, 2015 at 7:29 pm

What makes #10 a quadratic question? After doing the multiplication, you get a constant term of 15 + 5k, and all the other terms include x. For the polynomial to be divisible by x, the constant term must be zero, so k = -3. But at no point does it matter what’s going on in the rest of the function except that all the terms include a factor of x. There could be a term like (x arctan 3^(-x) ) and it still wouldn’t matter.

For #25, where does 1870 come into it? I get the equation w^2 – 3w – 1890 = 0, where 1890 is 5670/3. I hope you’ll agree that 1890 is pretty easy to factor, but that didn’t help me — it’s not obvious to me how to divide the factors into two numbers that differ by 3 (with knowledge of the eventual answer, this works out to be 3*3*5 vs 2*3*7… but with six prime factors, there are 64 possibilities to exhaust), and using the quadratic equation means having to find the square root of 7569. What have I missed?

I agree that #28 is a really nasty problem to encounter unprepared. (And I didn’t like #25 either.) It was always my understanding that it was a design goal of the SAT that preparation was unnecessary, hence the limitation to very basic subject matter. I share your bafflement at overtly reversing that.

I don’t get at all why it would be better to use meters in a trajectory problem instead of feet — my only exposure to those came in a physics class, where meters are obligatory, and a calculus class, where choice of units is kind of beside the point. For the calculus class you don’t need to justify acceleration values by reference to earth gravity; some problems do and other identical-in-form problems don’t.

April 16th, 2015 at 7:33 pm

well, really there are 32 possibilities to exhaust if I want to split 1890 into two numbers; the symmetric options are different, but I think anyone who gets to the point where they’ve determined that (w+45)(w-42) = w^2 + 3w – 1890 will immediately intuit how to factor w^2 – 3w – 1890.

April 16th, 2015 at 7:56 pm

This is a standardized test. As a rule, standardized tests don’t require the testers to go exhaustively through a list. 42 is the 16th factor, working up, unless I missed one.

April 17th, 2015 at 6:01 am

I get 42 as the 23rd factor “working up”, although some factors will be duplicates since 3 and 3 are the same. 13th factor using a slightly more complicated enumeration system that doesn’t list duplicates.

If you have to qualify yourself with “unless I missed one”, you’re not really enumerating properly. The whole point of enumerating a list exhaustively is to prevent yourself from overlooking part of it.

April 17th, 2015 at 4:11 pm

Students are not taught to enumerate factors. I teach my students, but am considered odd. But I’m not a detail person, so “unless I missed one” referred not to my having missed a factor pair, but to my having missed counting one of them on the way.

April 17th, 2015 at 5:41 pm

“Students are not taught to enumerate factors. I teach my students, but am considered odd.”

I learn something new every day!

Some of the Algebra problems that my son works are either directly of the form “list all the factors” or have a part that works out this way. I’m still struggling with him to be systematic so that he knows he hasn’t skipped any. I didn’t realize that this focus on “be sure you have them all and haven’t skipped any” could be considered odd. I thought (well, still think…) that this was simply a gap that needed to be closed.

My current plan is to get some (probably a lot …) PSAT/SAT/ACT test prep books to nail down any gaps. Any suggestions?

April 17th, 2015 at 6:04 pm

Here’s what most teachers mean by “list all the factors: 2,3,4,5,10,12,60 Then oops, you forgot the 1 and 6.

here’s what I mean:

1 60

2 30

3 20

4 15

5 12

6 10

That way you are less likely to forget any. Most teachers don’t do that, and kids end up forgetting a lot.

Kaplan and the Official book are the best practice tests, in my opinion. Practice by section, not whole tests. And good luck!

April 18th, 2015 at 8:03 am

That solves the problem of overlooking one half of a factor pair, but it doesn’t even address the problem of overlooking both halves. Here’s an enumeration of the factors of 60, which is 2*2*3*5, “working upward”:

April 18th, 2015 at 4:28 pm

That’s not how kids think at all. I’m not saying it’s wrong, but I’d never teach it.

April 16th, 2015 at 7:59 pm

On 10–would anyone know what you described without an understanding of quadratics?

16 is a much friendlier number. You are treating the test as someone who’s good at math. Remember, I did the test very quickly. I didn’t find it incredibly difficult for me. I’m thinking about the kids who will be taking it–only 10-20% of whom are as competent as we are.

It’s 1890–a typo. And that’s a lot of work. See my notes on another comment.

It was always my understanding that it was a design goal of the SAT that preparation was unnecessary, hence the limitation to very basic subject matter. I share your bafflement at overtly reversing that.Precisely.

April 17th, 2015 at 3:54 am

What’s “an understanding of quadratics”? To me, the only thing distinguishing quadratics from other polynomials is the quadratic equation, and yes, a lot of people could know what I described without knowing the quadratic equation.

If the question is “would anyone be familiar with what it means for a polynomial to be divisible by x, without first having had a lot of experience working with quadratics in school?”, then I’d be comfortable answering “probably not”, but that criterion also makes the trig question that asks for the definition of tangent a “quadratic question”.

If I’m breaking #10 down into concepts, here are what I see as the more basic elements of the problem:

– can the student factor the 5 out of “5x^3 – 15x + 40”?

– ok, if they can factor out numbers, can they factor the x out of “5x^3 – 16x”?

And that’s it.

April 17th, 2015 at 7:41 pm

I’d be comfortable answering “probably not”, but that criterion also makes the trig question that asks for the definition of tangent a “quadratic question”.No, it doesn’t. And if you aren’t familiar with quadratics, you aren’t familiar with higher degree polynomials.

April 16th, 2015 at 9:53 pm

>it’s not obvious to me how to divide the factors into two numbers that differ by 3

Here was my exact thought process:

3 is really small. The factors have to be very close in size. Try taking the largest and smallest prime factors together (to balance the middling other factors). 7*2 is 14, but that’s too small—1870 is really big; you’d have to multiply 14 by a much bigger number to get something with four digits. Slap another small prime factor on: 7*2*3 is 42; that leaves 3*3*5, which is 45. Done.

This takes maybe five or ten seconds—if you’re very fluent with numbers, multiplication, and magnitudes. There’s value in testing for fluency, but it certainly doesn’t seem to be the SAT’s goal (cf. cutting the vocabulary section), so this is a curious choice. I agree that the problem is certainly not “medium” for the typical test-taker.

April 16th, 2015 at 10:05 pm

Yeah, I would find it a really interesting test problem for algebra 2, when the kids have plenty of time to mull. It’s not a question for a standardized test, though. And you’d have to know how to factor.

April 16th, 2015 at 8:59 pm

Problem #26 is just testing if the victim knows/remembers the Indian Princess “SOHCAHTOA,” right?

This is easily taught/learned in a day (you don’t have to *USE* it, just understand how the names go …), but also isn’t generally taught before the kids take this PSAT test. So … we’re testing here for *either* already in Trig by sophomore year *or* have taken a test-prep course? WTF?

April 16th, 2015 at 10:06 pm

SOHCAHTOA is covered in geometry, but at the end of the year. So as I said, you either know it and it’s easy, or you have absolutely no idea what it means.

April 17th, 2015 at 4:52 am

But maybe that’s the point. That you *have* to have taken a class and covered it, rather than just treating it like an MCSE exam where you can just memorize all the possible question-and-answer pairs (plus a few simple algorithms).

April 17th, 2015 at 7:40 pm

That’s the definition of achievement test. But then they can’t set the knowledge level too high.

April 19th, 2015 at 6:40 pm

My first year of teaching Honors (11th grade) Physics, I was telling the class how to add vectors, talking about sines and cosines and getting very blank looks. “We never did this,” they said. So I stopped, improvised a lesson on basic trig, and went home that night to make up some practice problems.

The next day after school, the geometry teacher who most of them had had the previous year came up to my room. He was mad. “Anyone who tells you they didn’t do trig last year, tell them to come to my room and I’ll take 5 points off last year’s grade!” He showed me 8 pages of trig worksheets they had done (and kindly left me a copy of each). These were Honors students.

Needless to say (?), I now begin vector addition with a fairly substantial “review” of sines, cosines, and tangents (“Everyone from 9th graders to Harvard professors uses SOH-CAH-TOA”).

April 27th, 2015 at 2:38 pm

Ed has a post on this subject: The Myth of They Weren’t Ever Taught.

My brother taught part of a semester/quarter at a non-exclusive state university. The department chair told him not to assume that the kids had actually learned/remembered anything from any of the pre-req classes. Which is a challenge when one is teaching engineering.

April 17th, 2015 at 6:59 pm

I wonder if the reduction in geometry problems may not be due to the extra difficulty girls may have with geometry, as opposed to algebra.

At least I would expect that there is a significant gap between girls and boys on geometry vs. algebra — geometry engages visual spatial abilities and logical abilities, and algebra engages calculation abilities, and one would think that the two draw on the different strengths of the two sexes.

April 17th, 2015 at 7:37 pm

There hasn’t been much complaining about the SAT gender gap in quite some time. I don’t think that’s why. Besides, this test is much harder, which favors boys over girls.

April 17th, 2015 at 7:57 pm

Well, I think that the underlying embarrassment, for the College Board, of girls not doing nearly as well as boys at the very upper end never goes away.

And it may be that one way at least to keep the boys at the upper end from starting to do still better, which making the test harder might indeed cause, is to compensate by making the problems more in the wheelhouse of the girls.

April 18th, 2015 at 4:29 pm

But the problems aren’t more in the girls wheelhouse. they’re more in the boys.

April 18th, 2015 at 5:17 pm

Don’t get your claim here.

Is it not the case that algebraic problems go more to the strengths of girls, and geometric problems go more to those of boys?

If you’re claiming the opposite, what makes you think so?

April 18th, 2015 at 5:21 pm

Is there such evidence? I don’t know of it. I was thinking that it was harder math, which plays to boys’ strengths. In any event, I stand by my original point, that I don’t think gender played a part in this. I could be wrong, but there’s been no issues with gender since the CB changed AP scoring back in the 80s.

April 17th, 2015 at 9:33 pm

Any thoughts on permission slips to eat an oreo? “But note the wording where the parent signs at the bottom: “Without a signed permission slip, my child understands that he/she will not be able to sample the Oreo.”

The actuality is that the child will not be allowed to “sample” — or, as some people say, eat — the Oreo but to these Proggies allowed to = able to

http://www.sondrakistan.com/2015/03/28/%e2%80%a6first-seize-the-language/

April 18th, 2015 at 1:53 am

As someone else mentioned, the key to factoring 1890 is to notice that 3 is really small. This is a big clue for someone with good “number sense”.

If I had a calculator I would take the square root to get 43.74. It would then be “obvious” that the factors must be one of the following: 41 and 44 (wrong), or 42 and 45 (correct), or 43 and 46 (wrong). [Why? One factor must be lower than the square root, and one must be greater than the square root — and they differ by 3.]

Without a calculator some estimation is required: first figure out that 1890 is between 40^2 and 45^2. So the square root is somewhere in between those two numbers (meaning that the smaller number must be less than 45 and the larger number must be greater than 40). Try 38 and 41, 39 and 42, etc. up to 44 and 47. It’s pretty easy to tell that the answer must be 42 and 45 without actually doing the calculations (e.g., the other pairs of numbers won’t end in a zero when you multiply except for 40 * 43 which is wrong). If you realize that 1890 has a lot of factors (because it is divisible by a fairly high power of 3) you won’t even consider any choice besides 42 and 45 (both of which have a lot of factors themselves).

April 18th, 2015 at 4:27 pm

You are missing the point.

April 20th, 2015 at 9:28 pm

Since I’m a white guy who sent his kid to an all white SAT prep class I am not up on the cutting edge of cheating. But to be able to hot wire the test with a calculator? WTF? Or am I Rumplestilskin? That’s how my ABC nephew got a 2400?

No wonder you are so ticked about the cheating.

April 26th, 2015 at 7:52 pm

Back when I took these standardized tests as a kid, I don’t recall time ever being an issue, but going through these two new PSAT math sections I felt extremely rushed. A few of the problems are just hard enough that I had to experiment with different approaches and getting too bogged down in an approach that turns out to be wrong could be deadly.

It seems like in addition to math reasoning you are now being tested on time management skills as well. If you have to practice these ahead of time, then so much for the SAT/PSAT being tests you should be able to take without preparation.

April 27th, 2015 at 1:44 am

Question 25 (Janice’s fence) can be quickly solved using guess and check. The length is 9 feet less than three times its width. The area is 5,670 square feet. What is the perimeter?

Approximate the garden as three adjacent squares, each width * width. The area of each square is a bit more than 1/3 of 5,670 ft², so round off the area of each square to 2,000 ft². Guess that the width is about 45 ft², because (45 ft) squared is 2,025 ft².

Check the guess:

The guessed length is 3 * 45 ft – 9 ft = 135 ft – 9 ft = 126 ft.

The guessed area is (45 ft) * (126 ft) = (50 ft)(126 ft) – (5 ft)(126 ft) = 6300 ft² – 630 ft² = 5,670 ft².

The guess is correct. The perimeter = 2 * length + 2 * width = 2 * (126 ft + 45 ft) = 2 * 171 feet = 342 feet = Answer A.

Some essential features of this method:

* Correctly identify the given facts.

* Correctly identify the desired output (the perimeter).

* Use check-by-substitution.

* Keep track of units.

* Keep track of known quantities (as opposed to guessed quantities).

Note that:

* No calculator is required.

* No prime factorization is required.

* It helps to know that 4.5 * 4.5 = 20.25.

* It helps to know that 45 = 50 – 5.

* It helps to know that 50x = 100(x/2).

* It helps to be able to use the distributive property.

* If the guess had not been correct, it would have been close, and could be adjusted.

April 27th, 2015 at 3:16 am

Oh, good lord. That’s not guess and check. That’s estimation. If you have enough math knowledge to do that, you don’t need to estimate. Really, some of you don’t seem to understand how much math you know and think you’re being clever for entirely the wrong reasons.

May 6th, 2015 at 8:37 am

“Of course, it’s also possible that the SAT will yield the same range of scores and abilities despite being considerably harder. I don’t do psychometrics.”

What will also be interesting to note is whether individual scores still correlate with general intelligence (g). Several different research teams have concluded that the SAT is primarily a test of g on the basis of something like a 81% correlation with IQ test scores.

This development is of utmost interest to teachers, because it will directly impact how and to what degree the SAT can be “taught.” Obviously no test meets the criteria for theoretical perfection (it would have to be totally unpredictable) and any test can be “gamed,” in theory, but my observation has been that in general, the better the test, the greater the intellectual level necessary to “game” it. In other words, the resulting “gamed” score would often end up reflecting the taker’s actual intellectual prowess anyway. (As a psychometrics test coach and a computer programmer this is more or less what I think happens in a lot of the TI-84 cases you cite. Speed factor aside, simply knowing what algorithm to use and how to set it up takes the bulk of mental effort on many such problems.)

However, if the SAT is de-emphasizing “puzzle” logic of the sort psychometric test writers love in favor of straight-line “concept-themed” mathematics, then the game changes for SAT preparation, as you imply. The role of math teachers becomes more important in permanently ingraining relevant math *concepts*, and the SAT coach becomes more like a subject tutor.

Such a test might in some ways be more difficult to “game” in traditional psychological ways, but would probably say less about the student’s intellectual ability than about the quality of his scholarization (the two are certainly correlated, but definitely not on a straight line). Whether you think this is a good thing depends, I suppose, on what criteria you think colleges should be using for admission.

May 12th, 2015 at 10:26 pm

Has your school pushed geometry down per common core? Some curriculum spreads geometry out, inserting it throughout other subjects. Do you think it is best to have a stand alone high school geometry class for math skills? How about for standardized tests? In your ideal non-politicized math world, what would your high school curriculum look like?

May 13th, 2015 at 4:48 am

We aren’t doing integrated math (which is where it is spread out). These days, teaching a pure geometry class to anything but the top tier kids is a criminal waste of time. A lot of kids won’t pass, and a good chunk could use the time to learn more algebra. It’s quite possible to cover geometry with a huge dose of algebra. Not if you follow CC, of course, but I don’t intend to.

There wouldn’t be one high school math curriculum for everyone.

June 10th, 2016 at 6:44 pm

We didn’t have calcs in 70s, 80s on the SAT. You can test all the concepts and look for aptitude fine without calculators. This whole “tech push” is insane. They want calcs allowed and then they actually go back in and add questions to require them. And it’s a cheating hazard. Just cut it out. KISS!