Posts filed under: new SAT

The five questions in this short quiz would all, I’m fairly sure, fit into the “Heart of Algebra” category that will be so heavily emphasized on the new SAT. Obviously, without even a full practice test released yet, it’s tough to know for sure if I’ve got the style right, but based on what I’ve seen, you’d better be able to cruise through these if you’re planning to take the SAT after January 2016.

(UPDATE: Try another set of questions in the style of the new SAT here.)

You need to be registered and logged in to take this quiz. Log in or Register

How’s everyone else doing on this quiz?

36% got 5 right
29% got 4 right
14% got 3 right
12% got 2 right
7% got 1 right
2% got 0 right

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 13 through 18 in the “calculator not permitted” section. For the “calculator permitted” section, see questions 1 through 5 here, 6 through 11 here, 12 through 15 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here. You can see questions 1 through 6 from the “calculator not permitted” section here, and 7 through 12 here.

Question 13 (link)

Difficulty: Hard
Is this new? Kinda. (This basic concept is tested on the current SAT, but the algebra in this question is a bit more involved.)

The thing you need to know here is that the only way two linear equations will have no solution is when they create parallel lines (and parallel lines have equal slopes). So put both equations into y=mx+b form, and then see what a needs to be to make the slopes the same:

\dfrac{1}{2}x-\dfrac{1}{4}y=5

-\dfrac{1}{4}y=-\dfrac{1}{2}x+5

y=2x-20

So the slope we’re looking for is 2. Proceeding to the next equation:

ax-3y=20

-3y=-ax+20

y=\dfrac{a}{3}x-\dfrac{20}{3}

So you know \dfrac{a}{3}=2. That means a must equal 6.

Question 14 (link)

Difficulty: Hard
Is this new? Yes. The current SAT generally stays away from work/rate problems. 

This is another one of the “new” questions that appeared in the first document we got about the new SAT months ago. I hated it then, and I still hate it now.

The basic idea is that you need to see, from the right-hand side of the equation, that the equation is telling you how much the printers will print in 1 hour, which is \dfrac{1}{5} of the job.

From there, you need to recognize that each fraction on the left represents one of the two printers. From there, hopefully it’s intuitive that the \dfrac{1}{x} represents the slow printer, and the \dfrac{2}{x} represents the fast one.

Question 15 (link)

Difficulty: Hard
Is this new? Not really—It’s basically a right triangle question. The only thing new is that the current SAT doesn’t have chord as part of its vocabulary. 

This question doesn’t immediately look like a right triangle question, but it’s totally a right triangle question—and one where plugging in will work, at that! Say r = 3, so that AB = 6 and CD = 4. Now draw some lines:

test_specifications_for_the_redesigned_sat_na3_pdf 8

Of course, since PD is a radius, it has a length of 3 just like AP and PB.

2^2+QP^2=3^2

4+QP^2 = 9

QP=\sqrt{5}

Only one answer choice has \sqrt{5} in it, so we can feel pretty good about choice D without even doing the last step of the plugging in process, which is to put 3 in for r in each answer choice. Of course, when you do, D is the only answer to give you \sqrt{5}.

Question 16 (link)

Difficulty: Hard
Is this new? Yes. There is no trigonometry on the current SAT.

A calculator would be nice here—plug in, graph, and you’re done. Without a calculator, you need to know the relationship between angles with opposite sines.

Since the sine function has a period of 2π, subtracting π from x inside the function will result in a negated sine. You might recognize that this really ends up being a graph translation problem. The red graph shifts the blue graph π units to the right (if we had added π and shifter π units to the left we would have had the same result).

Question 17 (link)

Difficulty: Hard
Is this new? Yes. There are no circle equation questions on the current SAT. This could appear on the current SAT Math Subject Tests, though.

 To solve this one, you need to know the general equation of a circle, which is this:

A circle with radius r and a center of (ab) has the equation (x-a)^2+(y-b)^2=r^2.

You also need to know how to complete the square, because otherwise you’re not going to be able to wrangle what you’re given into that form.

x^2+y^2-6x+8y=144

Do a little rearranging:

(x^2-6x)+(y^2+8y)=144

Now, what binomial square begins with x^2-6x? What binomial square begins with y^2+8y?

  • x^2-6x is the beginning of the (x-3)^2=x^2-6x+9 binomial square
  • y^2+8y is the beginning of the (y+4)^2=y^2+8y+16 binomial square.

So, to complete those squares, we need to add 9 and 16 to both sides of the circle equation!

(x^2-6x+9)+(y^2+8y+16)=144+9+16

(x-3)^2+(y+4)^2=169

(x-3)^2+(y+4)^2=13^2

So we have a circle centered at (3, –4) with a radius of 13. The question asked for the diameter, though, so the answer is 26.

Question 18 (link)

Difficulty: Hard
Is this new? Not really. The algebra is trickier than you’d see on most current SAT questions, but I would not be surprised to see this as a #18 in the current SAT grid-in section.

\dfrac{24}{x+1}-\dfrac{12}{x-1}=1

This is just algebra. Let’s do it. First, get a common denominator:

\dfrac{24(x-1)}{(x+1)(x-1)}-\dfrac{12(x+1)}{(x+1)(x-1)}=1

Hopefully you recognize that the denominator is a difference of two squares:

\dfrac{24x-24}{x^2-1}-\dfrac{12x+12}{x^2-1}=1

\dfrac{24x-24-(12x+12)}{x^2-1}=1

\dfrac{12x-36}{x^2-1}=1

Now you can move that denominator over:

12x-36=x^2-1

And set everything equal to 0:

0=x^2-12x+35

And finally, factor:

0=(x-7)(x-5)

x could equal 7 or 5.

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 7 through 12 in the “calculator not permitted” section. For the “calculator permitted” section, see questions 1 through 5 here, 6 through 11 here, 12 through 15 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here. You can see questions 1 through 6 from the “calculator not permitted” section here, and questions 13 through 18 here

Question 7 (link)

Difficulty: Medium
Is this new? Kinda. The vertex form of a parabola has shown up (rarely) on the current SAT.

And here we see that the new SAT will expect you to know the vertex form of a parabola. Only one of the answer choices is in vertex form, so if you know what that looks like, you don’t even need to confirm that it actually has the right numbers in it—you just need to recognize the structure. Not sure how I feel about that.

Anyway, the vertex form of a parabola is this:

For constants ah, and k, the parabola y=a(x-h)^2+k will have its vertex at (h, k). If you’re nervous about whether y=(2x-4)(x-4) and y=2(x-3)^2+(-2) are equivalent, you can do some algebra (CB walks you through that in the answer explanation) but I’m sticking to my guns—if they’re going to make it so that only one choice is actually in the right form, then you don’t need to bother checking.

Question 8 (link)

Difficulty: Medium
Is this new? Very much so.

Imaginary. Numbers. On the SAT. My whole world is turned upside-down.

No, seriously though, this stuff is easy if you know how to FOIL. All you need to do is remember, at the end, that i^2=-1.

(14-2i)(7+12i)
(14)(7) + (14)(12i) + (-2i)(7) +(-2i)(12i)
98 + 168i -14i -24i^2
98 + 154i -24i^2

Now remember that i^2=-1

98 + 154i +24
122 + 154i

See? Easy.

Question 9 (link)

Difficulty: Medium
Is this new? Not really. The current SAT doesn’t usually ask questions that require this kind of equation solving, but it’s not unheard of.

Good ol’ fashioned algebra, here. Finally!

\dfrac{5(k+2)-7}{6}=\dfrac{13-(4-k)}{9}

First, simplify the numerators…

\dfrac{5k+3}{6}=\dfrac{9+k}{9}

Now cross multiply!

(9)(5k+3)=(6)(9+k)
45k+27=54+6k
39k=27

k=\dfrac{27}{39}=\dfrac{9}{13}

Question 10 (link)

Difficulty: Medium
Is this new? Kinda. Systems of equations on the current SAT can almost always be solved easily in one or two steps. This is not that kind of systems question.

More algebra! I’m so happy! Buckle up…

First, get the first equation into a useful form:

4x-y=3y+7
4x-4y=7

Now multiply it by 2:

8x-8y=14

Now add that to the second equation to eliminate the y terms and find x!

8x-8y=14
+(x+8y=4)
9x=18

So you know x = 2. From there, it’s easy to find y:

2+8y=4
8y=2
y=\dfrac{1}{4}

So what’s xy? Well, it’s \left(2\right)\left(\dfrac{1}{4}\right)=\dfrac{1}{2}.

Question 11 (link)

Difficulty: Medium
Is this new? No.

After the last two questions, I kinda feel like they’re joking with this one.

\dfrac{1}{2}x + \dfrac{1}{3}y = 4

Multiply that all by 6 and, well, you’re done.

6\left(\dfrac{1}{2}x + \dfrac{1}{3}y\right) = 6(4)

3x + 2y = 24

Question 12 (link)

Difficulty: Hard
Is this new? Yes. Trig doesn’t appear at all on the current SAT.

And we get to finish this post up with some trig! Wahoo!

There are a few ways to go here. One is to know your unit circle well enough to know that choices A, B, and D don’t make sense. Honestly, that’s how I thought through this when I first looked at it.

There’s a better way, though, which I’m ashamed not to have recognized at first. It’s what makes this an SAT question, and not just a math test question. The graphs of sine and cosine are in phase, such that \sin x =\cos \left(\dfrac{\pi}{2}-x\right)! Therefore:

\sin\dfrac{\pi}{5}=\cos\left(\dfrac{\pi}{2}-\dfrac{\pi}{5}\right)

\sin\dfrac{\pi}{5}=\cos\left(\dfrac{5\pi}{10}-\dfrac{2\pi}{10}\right)

\sin\dfrac{\pi}{5}=\cos\dfrac{3\pi}{10}

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 1 through 6 in the “calculator not permitted” section. For the “calculator permitted” section, see questions 1 through 5 here, 6 through 11 here, 12 through 15 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here. You can see questions 7 through 12 from the “calculator not permitted” section here, and questions 13 through 18 here.

Math_Sample_Question__1___SAT_Suite_of_AssessmentsQuestion 1 (link)

Difficulty: Easy
Is this new? No.

This is an interesting question, because it’s really just testing whether you know that when a line is translated, its slope doesn’t change at all! The slope of the line in the figure is -\dfrac{3}{2}, which is easy to see from the two darkened points—the line moves down 3, and right 2. Therefore, the translated line will also have a slope of -\dfrac{3}{2}.

Question 2 (link)

Difficulty: Easy
Is this new? Kinda. The content is familiar, but the way the question is asked is new.

Heart of Algebra questions (the easy ones, anyway) will usually look like this, apparently. The thing to pay attention to in this equation is where the variable, x, is. If x represents the number of years since 2004, we can conclude that the average must have been 27.227 in 2004, and has increased slightly every year since then, at a roughly constant rate. The 0.8636 number, which is multiplied by x, must be the number that’s added to the average every year—the estimated annual increase in the average number of students per classroom.

Question 3 (link)

Difficulty: Easy
Is this new? Not really, although they’re calling it easy here and this would not be an “easy” question on the current test.

In this “Passport to Advanced Math” question (I promise that I’ll get over these crazy category names soon, but I’m not quite there yet) we’re being tested on whether we recognize the opportunity to complete a binomial square. That a^2+14a bit is most of what we get when we square the quantity a+7. What’s missing?

(a+7)^2=a^2+14a+49

Do you see the opportunity to substitute? We already know thata^2+14a=51!

(a+7)^2=51+49

(a+7)^2=100

Because the question tells us that a is positive, we can ignore the negative square root and just say:

a+7=10

(If this weren’t the “calculator not permitted” section, we also could have solved this by graphing very quickly.)

Question 4 (link)

Difficulty: Medium
Is this new? No.

Just do a little cross-multiplying here:

\dfrac{2}{a-1}=\dfrac{4}{y}

2y=4a-4

y=2a-2

Question 5 (link)

Difficulty: Medium
Is this new? No.

They’re calling this one a medium difficulty question, but I just don’t buy it, unless they think the simple presence of a cubic term is going to send kids running for the hills.*

If y=x^3+2x+5 and z=x^2+7x+1, then:

2y+z=2\left(x^3+2x+5\right)+x^2+7x+1  =2x^3+4x+10+x^2+7x+1  =2x^3+x^2+11x+11

*With the elimination of the guessing penalty on the new SAT, there’s less reason than ever before to go running for the hills.

Question 6 (link)

Difficulty: Medium
Is this new? No.

I have to say that so far I’m liking the “calculator not permitted” questions a lot better. I don’t really know why, since most of the “calculator permitted” questions weren’t really calculator questions either, but it seems like we’ll get to do our fun algebraic manipulation work mostly in the “calculator not permitted” section.

A few ways to go here, but what I usually do with a question like this is just take it one step at a time until I’ve isolated the thing I want to isolate. In this case, that means I’m going to first raise both sides to the –1 power, and then square both sides.

a^{-\frac{1}{2}}=x

\left(a^{-\frac{1}{2}}\right)^{-1}=x^{-1}

a^{\frac{1}{2}}=x^{-1}

\left(a^{\frac{1}{2}}\right)^{2}=\left(x^{-1}\right)^2

a=x^{-2}

That’s not an answer choice, so let’s apply what we know about negative exponents:

a=\dfrac{1}{x^{2}}

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 27 through 30 in the “calculator permitted” section. See questions 1 through 5 here, 6 through 11 here, 12 through 15 here, 16 through 20 here, and 21 through 26 here.

Question 27 (link)

Awesome question. You could spend some time wrestling this into ymxb form, but you don’t need to if you’re paying attention. Look at the choices! One of them is a parabola. There’s no x^2 term in y-x=k(x+y), so you know you’re not going to get a parabola. Cross off D. The other thing you should notice is that A and C have non-zero y-intercepts. Since the given equation is doesn’t have any constant terms, it’s going to go through the origin. You can cross off A and C, and you’re left with B. This is one of the short-cuttiest questions yet!

Question 28 (link)

OK, don’t be intimidated here. All you need to do is take some of the information you’re given, and use it correctly. Put the point (–4, 0) into the given equation, and you’ll be able to solve for c. Simple as that!

f(x)=2x^3+3x^2+cx+8

f(-4)=2(-4)^3+3(-4)^2+c(-4)+8

0=-128+48-4c+8

72=-4c

-18=c

Is it weird that they tell you that c is a constant but don’t feel the need to tell you that p is a constant? Yes, I suppose. But do you need p at all to solve for c? Nope. Not at all. Is this question slightly harder because you had to puzzle with that for a minute before getting to work? You bet.

Question 29 (link)

This question is kinda fun, although I bet a lot of students who encounter it will disagree with me. When the question says that those two expressions are equivalent forms, what that means is that you can set them equal to each other.

\dfrac{4x^2}{2x-1}=\dfrac{1}{2x-1}+A

So let’s do some algebra to get A in terms of x!

\dfrac{4x^2}{2x-1}-\dfrac{1}{2x-1}=A

\dfrac{4x^2-1}{2x-1}=A

Do you see the difference of two squares up top there?

\dfrac{(2x+1)(2x-1)}{2x-1}=A

2x+1=A

Question 30 (link)

Ah, finally some trig!

You’re going to want to drop an altitude segment down from the top vertex. That’s going to end up being a perpendicular bisector and make two right triangles. (You knew that, because you know those congruent base angles mean this is an isosceles triangle, right? RIGHT???)

Remember your SOH-CAH-TOA. We’re dealing with cosine here, which is the adjacent leg over the hypotenuse.

The cosine of x, then, is just \dfrac{16}{24}=\dfrac{2}{3}.

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 21 through 26 in the “calculator permitted” section. See questions 1 through 5 here, 6 through 11 here, 12 through 15 here, 16 through 20 here, and 27 through 30 here.

Question 21 (link)

This one’s just about as easy as a medium question can come, but it requires careful reading of both the question and the graph. The thing you need not to miss in the question is that the dishes are 10 square centimeters in area. So when bacteria cover 7 square centimeters of the dish, they also cover 70% of the 10 square centimeter dish.

Let’s just look at each choice one at a time:

  • A: Nope. At time t = 0, neither dish is even close to being 100% covered.
  • B: Yes, this is the answer. Dish 2 has 2 square centimeters covered, and Dish 1 has 1 square centimeter covered. Because the dishes are 10 square centimeters in area, those percentages are correct.
  • C: Nope. Dish 2 covers twice as much as Dish 1 at t = 0. That’s not 50% more, that’s 100% more.
  • D: Nope. The opposite is true—Dish 1 grows much more quickly in the first hour than does Dish 2.

Question 22 (link)

Pay SUPER close attention to units here. They’re your lifeblood. First, figure out what 11.2 gigabits is in megabits:

11.2 gigabits × 1024 megabits per gigabit = 11,468.8 megabits

Now figure out how many seconds are in 11 hours:

11 hours × 60 minutes per hour × 60 seconds per minute = 39,600 seconds

Now figure out how many megabits can be transmitted in that period based on the 3 megabits per second rate:

3 megabits per second × 39,600 seconds  = 118,800 megabits

Finally, figure out how many images that’ll let the tracking station receive:

118,800 megabits / 11,468.8 megabits per image ≈ 10.4

That’s 10 full images.

Question 23 (link)

This is one of the most old-school SAT questions we’ve seen–all you need to do is make one quick substitution and solve, but make sure you don’t go one step too far! The question asks for x^2, not x!

x^2+y^2=153

y = -4x

x^2 + (-4x)^2 = 153

17x^2 = 153

x^2 = 9

Question 24 (link)

I hope you don’t mind me plagiarizing myself here—the next three question were floated by College Board way back when they announced the new SAT, and I solved them in a couple posts then. This section and the following two are reprints from here and here.

To get this, first find the area of the hexagon, and then subtract the area of the circle. Once you’ve done that, you can multiply by 1 centimeter to get the volume. Then you can use the given density to find the mass. Sound like fun???

A regular hexagon can be divided into 6 equilateral triangles, like so:

The area of an equilateral triangle is \left(\dfrac{1}{2}\right)\left(b\right)\left(\dfrac{b\sqrt{3}}{2}\right). (Derive that for yourself—it’s good practice! Or see here.) For each of those equilateral triangles, then, the area is \left(\dfrac{1}{2}\right)\left(2\right)\left(\dfrac{2\sqrt{3}}{2}\right)=\sqrt{3}. There are 6 of those, so the area of the hexagon is 6\sqrt{3} square centimeters.

The hole drilled through the middle has a diameter of 2, so it has a radius of 1, and therefore an area of π square centimeters. So the area of the hexagonal face of the nut, minus the hole drilled through it, is 6\sqrt{3}-\pi square centimeters.

Multiply that area by the height of the nut, 1 centimeter, and you get the volume:6\sqrt{3}-\pi cubic centimeters.

Now, you’re told that density = mass/volume, you’re given the density 7.9 grams per cubic centimeter), and we just found the volume. We’re asked for the mass to the nearest gram.

7.9=\dfrac{m}{6\sqrt{3}-\pi}

7.9\left(6\sqrt{3}-\pi\right)=m

57.2706… = m

Since we’re asked for the answer to the nearest gram, we write 57.

Question 25 (link)

Yikes, right?

If the bank converts Sara’s purchase to dollars, and adds a 4% charge, then the $9.88 she’s charged doesn’t convert directly to rupees. First we need to get that 4% fee out of there. If x = the dollar cost of her purchase without the fee, then 1.04x = 9.88.

x = \dfrac{9.88}{1.04} = 9.5

So her purchase was worth $9.50, which means the exchange rate that day, in rupees per dollar, is:

\dfrac{602\text{ rupees}}{9.5\text{ dollars}}\approx 63.36\text{ rupees per dollar}

Round that to the nearest whole number and the answer is 63.

Question 26 (link)

If Sara buys the prepaid card, she doesn’t have to pay the 4% fee her credit card charges her, but she could lose money if she doesn’t spend the whole card. The exchange rate doesn’t actually matter anymore—we can actually forget about dollars entirely at this point.

The question is really how much Sara has to spend on her credit card, with the 4% fee, before she’d save money by buying the prepaid card, even if she doesn’t spend it all. Say y = the amount of rupees Sara spends:

7,500 = 1.04y
7,211.54 = y

If Sara spends 7,211.54 rupees on her credit card, it costs her the same as buying a 7,500 rupee prepaid card. Therefore, if Sara spends 7212 rupees or more on her prepaid card, she gets a better deal than if she spends the same amount on her credit card.

 

 

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 16 through 20 in the “calculator permitted” section. See questions 1 through 5 here, 6 through 11 here, 12 through 15 here, 21 through 26 here, and 27 through 30 here.

Question 16 (link)

Math_Sample_Question__16___SAT_Suite_of_AssessmentsWell, this is embarrassing for CB, but now there have been 2 questions in a row where an answer choice (and the correct answer choice, no less) is printed incorrectly. (See Question 15 in my previous post.) In this case, choices A and C, screengrabbed at right, should have || symbols, not blank squares. Someone didn’t check his or her font library carefully enough before choosing a math font!

Typo aside, this is a good question. The thing you need to know about similar triangles is that their corresponding angles are congruent. Since you have vertical angles at point C, which must be congruent to each other, you know how the corresponding angles are laid out. I’ve color coded them below.

 

Do you see what’s going on at points B and D, and points A and E? Those are alternate interior angles of lines AB and DE, so if they’re congruent, then those lines must be parallel!

Question 17 (link)

Another Heart of Algebra question, another scenario described that we need to identify an equation for. For this one, it’ll be helpful to use units. You know you want to end up with gallons, so make sure all the miles and hours units in your equation cancel out!

Hopefully it’s obvious that you need to start with “17 –” something. That right there eliminates choices C and D, amazingly.

Each remaining choice has a 50t in it. The 50, we know, is a speed, in miles per hour. The t is time, in hours. So the units of 50t is miles per hour × hours = miles. Remember that we want our final units to be gallons.

The units of 21 is miles per gallon. So between choices A and B, we’re choosing between \dfrac{\text{miles per gallon}}{\text{miles}} and \dfrac{\text{miles}}{\text{miles per gallon}}. Which of those simplifies to gallons? B does.

Question 18 (link)

They’re not even asking you to solve systems of equations anymore—they’re just asking you to be able to write them. This is just the first step on medium and hard questions on the current test, but on this medium question it’s the only step. There are x cars and y trucks, so you xy should equal the total number of vehicles, 187. You also know how much each kind of vehicle has to pay in tolls. Cars pay $6.50 and trucks pay $10. So 6.5x + 10y should tell you the total amount collected in tolls, $1338.

Your equations, then, are xy = 187, and 6.5x + 10y = 1338. That’s choice C.

Question 19 (link)

OK. There a few ways we might go here. One is to just plug the (d, p) coordinates we know (9, 18.7) and (14, 20.9) into each answer choice and see which one gives us a true equation. Since calculators are allowed on this question, that might be the fastest way. You might also enter each equation into your graphing calculator and use the table function to see which equation gives you the points you want, like so:

See how Y2 has the points you want? That means the equation entered into Y2 is the right equation!

Of course, you can also use the points you’re given to find the equations algebraically. First you find the slope:

\dfrac{20.9-18.7}{14-9}=0.44

…then you find the intercept using one of the points:

18.7 = 0.44(9) + b
14.74 = b

That’ll tell you that the equation you want is p = 0.44d + 14.74. If you ask me, though, using the calculator is the way to go here.

Question 20 (link)

This question is essentially asking for the slope of the line of best fit, so our job is to find 2 points on the graph, and then use the slope formula: slope = \dfrac{y_2-y_1}{x_2-x_1}.

Math_Sample_Question__20___SAT_Suite_of_Assessments 2Remember that it’s fine to approximate here! The answer choices are really far apart, so even if our approximations aren’t very good, we’ll know the right answer. Using my numbers, here’s the slope calculation:

\dfrac{3000-2000}{2003-1996}=\dfrac{1000}{7}\approx 143

143 isn’t an answer choice, but it’s awfully close to 150, and not very close at all to any other answer choice. 150, therefore, is the right answer.

 

 

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 12 through 15 in the “calculator permitted” section. See questions 1 through 5 here, 6 through 11 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here.

Question 12 (link)

Now we’re finally getting into some fun stuff! We know the company will bring in 12n dollars for n items sold, and we know it costs the company 7n + 350 to make n items. To figure out how many items need to be sold to make a profit, solve the following inequality:

12n > 7n + 350
5n > 350
n > 70

Again, the focus here is not on the algebra itself—it’s on whether you can figure out how to set the algebra up. That, I suppose, is what the College Board means by “Heart of Algebra.”

Question 13 (link)

I’m flummoxed here because they’re calling this a medium difficulty question, and it seems far easier than most of the other easy questions we’ve seen together so far, as long as you remember that the number of males doesn’t necessarily equal the number of females. (Again, this is a reading question as much as it is a math question.) If the average age of the males is 15, and the average age of the females is 19. If the number of males equalled the number of females, then the average age would be exactly 17, but we don’t know that the preserve has the same number of males and females.

Question 14 (link)

In spite of myself, I kinda like this question. It’s all about being cautious and modest with data. The only statement that data about 2000 16-year-olds in the US can support is a statement specifically about 16-year-olds in the US. If you want to draw conclusions about people in the world, then you have to sample the whole world. So cross off B and D.

Then, you have to remember something that will be drilled into your head in any social science class (and hopefully has already been drilled into your head a bit): correlation does not imply causation! Just because exercise and sleep are positively correlated (this question uses the word associated, which is a more general term, but association does not imply causation lacks that alliterative flair) we cannot conclude that one causes the other.

Question 15 (link)

Before we start discussing this, note that the CB, at the time of my writing this post, has F’d up a bit. Answer choice D, which should be the right answer, is misprinted. It looks like that over there on the right, but it should look like this: P=50(2)^{\frac{n}{12}}. That’s a big boo-boo to have a typo in the right answer—hopefully they’ll fix it soon.

Anyhoo, you can plug in here if you like. You know the population is supposed to double every 12 years, and you know the starting population is 50. So if you say n = 12, the right answer should resolve to 100. Let’s check:

A: 12 + 50(12) \neq 100
B: 50 + 12(12) \neq 100
C: 50(2)^{12(12)}\neq 100
D: 50(2)^{\frac{12}{12}}= 100

Only D works!

If you want to understand why D works, you need to first understand exponential growth in general. The basic setup of exponential growth (compound interest in a savings account is one classic application of this) is that you have a starting value (a), a growth rate over a certain period (r), and a number of periods (p). Where you’ll be after p periods is ar^p. Why?

Well, look at just the first 3 periods, one period at a time. You start with a. After 1 period, a grows by a factor of r, so you have ar. After a second period, ar grows by a factor of r again, so you have ar^2. After a third period, ar^2 grows by a factor of r again, and you get ar^3. Et cetera. I guess if this is going to start appearing on the new SAT, I should make a whole post about it. 🙂

Anyway, in this case, the growth period is 12 years, but we’re given n in years. That’s why we need our exponent to be \dfrac{n}{12}. That way, in 24 years, when there have only been 2 growth periods, the exponent \dfrac{24}{12} will equal 2, as it should.

 

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 6 through 11 in the “calculator permitted” section. See questions 1 through 5 here12 through 15 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here.

Question 6 (link)

More of this business where a scenario is described and you need to select the equation that accurately captures it. I actually think this is really important and something worth testing, but the fact that we’ve seen two of these in only 6 questions makes me think I’m going to be pretty tired of them pretty quickly once the new test arrives.

The trick here is that the $5 fee is untaxed, so you need to calculate the room fee, at $99.95 per night, apply tax to that, and then add the $5. You probably work with percents often enough to know that the 1.08 in each answer choice* represents 108%, which is what we need to multiply the room fee by to add 8% tax.

If you stay at the hotel for x nights, then the per-night fee will be 99.95x. Add 8% tax to that and you get 1.08(99.95x). Then add the $5 untaxed fee, and you get answer choice B: 1.08(99.95x) + 5.

* Where are the trap answers? If you put 1.08 in each answer choice, you’re making something a lot of students struggle with, percents, too easy! This ends up being a question that’s more about testing whether students caught the word “untaxed” than whether they really know how to work with percents.

Question 7 (link)

Jeez. The figure here might intimidate you for a second, but look at the question! All it’s asking is how many solutions the system has. Remember this well, as I imagine it’ll appear on this new test fairly often: When the graphs of each function in a system converge in one intersection point, that’s a solution to the system. That happens twice in this figure, so there are two solutions.

Question 8 (link)

Ooh—a grid-in! It’s all about reading a table, though. Nothing too exciting.

There are 7 metalloids in the solids and liquids columns, and there are 92 total elements that are solids or liquids. The fraction you want here is 7/92.

Question 9 (link)

Now this question should feel pretty reminiscent of the current SAT to anyone who’s been prepping with me. It’s a solving for expressions question, through and through! How do you go from what you’re given to what you want? You’re given info about -3t+1, and you want to know about 9t-3? Multiply everything by –3!

Oh, and don’t forget that when you multiply or divide a negative through an inequality, you need to flip the direction!

-\dfrac{9}{5} < -3t+1 < -\dfrac{7}{4}

\left(-\dfrac{9}{5}\right)\left(-3\right)>\left(-3t+1\right)\left(-3\right) >\left(-\dfrac{7}{4}\right)\left(-3\right)

\dfrac{27}{5}> 9t-1 > \dfrac{21}{4}

27/5 = 5.4 and 21/4 = 5.25, so any number between those works just fine. Note that while this question is probably tougher than anything else we’ve seen yet, it’s still classified by CB as easy.

Question 10 (link)

More table reading! Here, we’re asked which age group had the highest percentage of voters. Easy enough—just divide the number of voters by the number of survey respondents in each row! You’ll see that the 55- to 74-year-olds were the best voters: 43,075/59,998 ≈72%.

Note that you might be able to eyeball the first two age groups and see that they’re not in competition, but you’re going to need to calculate the latter two. The people aged 75 and over had a 70% voting rate—too close to 72% to call by eyeball.

Question 11 (link)

More with the voters, and again, whether you get this question right will really hinge on how carefully you read it, rather than how well you can solve an equation. 287/500 voters in the 18- to 34-year-old group voted for Candidate A. Assuming that the random sample of 500 of those folks is perfectly representative of the whole group, that means we can set up and solve a proportion:

\dfrac{287}{500}=\dfrac{\text{Candidate A voters}}{30,329,000}

17,408,846=\text{Candidate A voters}

Of course, the sample won’t really be perfectly representative, which is why we just say about 17 million people in that age group voted for Candidate A.

College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 1 through 5 in the “calculator permitted” section. See questions 6 through 11 here, 12 through 15 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here.

Question 1 (link)

On the one hand, this is as easy as it gets. On the other hand, it already feels like a departure from what you see on the current SAT. This kind of question, where a scenario is explained in English and then the tester has to identify the right equation (or, in this case, inequality) in the answer choices, feels much more ACT-ish. But whatever. Stuff like this is what you can expect from “Heart of Algebra” questions.

The main thing this question is testing is whether you, the tester, can read that the “recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg),” and recognize that you want your answer to say, “≥ 1000,” not just “> 1000.” That’s because if a 20-year-old has exactly 1,000 mg of calcium in a day, he meets the recommended daily intake. He doesn’t need any more.

I doubt many folks reading this would be fooled by C or D, but it’s not hard to see why they’re in there as traps.

Question 2 (link)

Oh for crying out loud. There are 107 words in that question. 107! Ugh.

This is a question that doesn’t require you to be able to calculate anything. It just requires you to know, on a very superficial level, how a margin of error works.

Basically, you need to know that if you want to figure out the mean hours spent reading for the population of psych students at this university, you can ask a small sample of that population and arrive at a pretty good estimate of the actual mean of the whole population, within some margin of error. If you want to be more sure, you ask more people from the same population. The more people you ask, the smaller your margin of error. This makes sense if you think about it: say I wanted to know the average height of 100 people. I could feel pretty OK about my estimate if I only measure, say, 20 of them. I’d feel even better if I measured 60 of them. If I measured 99 of them, I’d be super confident that my mean is close tot he actual mean—even if the 100th person is a 7-foot giant, he can’t bring the average up very much! The larger the sample, the more sure you can be that the sample mean is close to the actual population mean.

So C is the answer because it increases the number of people in the experiment without changing the population from psych students to students from all across the university.

Question 3 (link)

More superficial statistics knowledge being tested here. We’re three questions into the “calculator allowed” section and we haven’t had to calculate anything yet. Given that this graph will be the basis of the next three questions, that means we’ll be through 5 out of 30 questions in this exercise without so much as needing to turn our calculators on. Make a mental note of that.

What the question is saying is that there’s a correlation between the length of a person’s metacarpal bone and that person’s height. It’s not a perfect correlation, which is why all the dots aren’t right on the line of best fit, but it’s a correlation. All the question’s asking you to do is count the dots that are more than 3 inches from the line in either direction. There are 4 dots that are at least 3 cm off the line of best fit, so the answer is B.

Things to watch out for on a question like this:

  • Axis scale. Every horizontal line represents 1 cm on this graph, like you’d expect it would, and that’s the axis you need to pay attention to for this question since it asks about outlier heights. The vertical lines represent 0.1 cm, though, and I bet a future question will test that somehow…
  • Dots they hope you miss. In this case, there’s that one all the way on the bottom left. Did you see that one? Not everyone will. The answer choice you’d expect to be there to catch the people who miss that dot, 3, isn’t there, but I don’t think we can take that as a sign that the new test won’t do that kind of thing with regularity.

Question 4 (link)

More on the same graph. Here, they’re asking about slope. Ah, slope! Our old friend! I hope we’ll be seeing more of you!

Slope, as you know, is a measure of rise over run. Here, given the axes, it’s a measure of height increase over metacarpal length increase. That’s precisely what choice A describes. If you’re having trouble distinguishing between choices A and B (choices C and D aren’t very tempting) pay attention to units. The metacarpal length axis only increases by 1 cm, total. From this graph, it’s easy enough to say that height would be expected to increase about 18cm with a 1cm increase in metacarpal bone length. It would not be easy to predict from this data how much longer someone’s metacarpal bone would be if they were 1 cm taller.

Question 5 (link)

Still on this silly graph. I was expecting these questions to get harder as they go, but this might be the easiest of the bunch! All you need to do is trace the graph. 4.45 cm metacarpal is going to be in the middle of 4.4 and 4.5—trace that up to the line of best fit, then look left to the height axis and see where you are. You’re right on 170!

The danger here, of course, is that you misread the scale on the bottom axis, and there is the 169 there in the answer choices to make you feel good about your mistake if you trace the 4.4 line up by mistake. Still, if you can read a graph, you’ll have no problem nailing this question.

 

I have a lot to say about  the new SAT’s essay; I might not end up being able to squeeze it all into one post, but I’m going to try. In order to give this post a bit of structure—both to help me organize my thoughts and to help you find what you’re looking for—I’m going to break this post into three sections: what we know (including the basic structure of the task), what we don’t, and what I think. If you’re interested in this stuff and you haven’t already, you might want to download the College Board’s Test Specifications for the Redesigned SAT. I’ll be referring to page numbers from that document once in a while.

What we know

The new SAT’s essay will look very different from the current SAT’s essay. Here’s what the new prompt will be for every test administration (from page 76):

test_specifications_for_the_redesigned_sat_na3_pdf

This is a major departure. Although the prompts on the current test do contain some of the same instructional text every time (“Plan and write an essay in which you develop your point of view on this issue. Support your position with reasoning and examples taken from your reading, studies, experience, or observations.”) the task varies significantly from test to test.

The actual passage students will write on a 650-750 word passage that will be generally characterizable as an “argument written for a broad audience.” In the Test Specifications document, two example passages are given—one concerning a decline in reading among America’s young people (page 77), and one concerning light pollution (page 122). Both strike me as passages that could very easily appear on the current SAT in a Critical Reading section.

The most interesting thing here, to me, is that the author’s main claim is given to students in the question, and students are specifically instructed to set their own opinions aside.

The other big headlines about the new SAT’s essay are that it will be optional, and that students will get 50 minutes to complete the task. I’m sure I’m not the first person to have told you either of those things. 

What we don’t know

So, we know a fair amount of things! But we don’t know everything. As I was reading through the Test Specifications for the Essay, I jotted down some as-yet-unanswered questions.

Scoring

test_specifications_for_the_redesigned_sat_na3_pdf 2

It’s not clear yet how the new essay will be scored. The College Board candidly admits that it’s still trying to figure out how to evaluate the thing. As it stands right now, each essay will be evaluated by two readers who will each rate the essays on a 1–4 scale for each of three broad essay traits: Reading, Analysis, and Writing (see screencap above from page 82). There’s no rubric yet, and honestly I don’t feel like it’d be productive for me to speculate much about how this will evolve. I’m in wait-and-see mode on this one. (For my take on the general scoring scheme of the new SAT, which might be summed up as o_O, click here.)

Length

The College Board has been very specific about how long students will have to write the essay, and even a the word count of the passage prompt, but has not specified whether students will be limited in the amount of writing they can produce. The current SAT essay allows students two hand-written pages for 25 minutes. Will the new one double that?

Computer vs. paper

It’s also worth pointing out here that although David Coleman announced that students would be able to take the new SAT on a computer in 2016 in his big announcement back in March, there’s not a single mention of that anywhere in the 211 pages of Test Specifications released last week. I know a lot of students who are pretty fast typists. How will the ability to type impact essay length restrictions? Will there even be length restrictions for computer testers?

Maybe more importantly, how can handwritten essays be evaluated on the same criteria as typed essays? When you think about who will be more likely to take the test on a computer—students at schools that can afford to provide computers for every student taking the test— it’s easy to see how this might work counter to the College Board’s goal of “delivering opportunity.”

What a good essay looks like

test_specifications_for_the_redesigned_sat_na3_pdf 3The College Board has quite notably not released an example of what would be considered a strong essay. It does provide annotations of the two sample prompt passages (example on right, from page 81), but those read much more like the notes of someone about to write a set of Critical Reading questions about the passage than they do like an outline.

The prompt specifically instructs students not to write about whether they agree or disagree with the prompt, but the scoring guidelines as they stand do require students to “provide a precise central claim.” For now, I’m assuming that means that good essays will begin with sentences like, “Dana Gioia combines emotional appeals with concrete and authoritative data to argue that the decline of reading in America will have a negative effect on society.”

When we finally see some exemplary essays (Ideally in two sets—one of good handwritten essays and one of good typed essays) we’ll know a lot more.

Who’s going to care

Huge question mark here. At least at first, there will probably be a lot of colleges who don’t care, which will mean many students will be able to take a 3-hour multiple choice test, and then go home without slogging through the 50-minute essay. This might compel students to a bit more proactive in compiling their college list before they do much testing.

Competitive students who don’t know which schools they’re going to apply to will probably have to take the SAT essay just in case they choose to apply to schools that require it later. I’m betting that at least a few of the most competitive schools will be on board right away. The Dean of Admissions at Harvard seems to like the new SAT, for example, so students with Harvard aspirations should probably plan to take the essay.

What I think

The College Board set out to address the weaknesses of the current SAT essay, and has successfully neutralized a couple of them. First, students will no longer be able to fabricate examples about their uncle George to support their argument—truth will matter in the new SAT’s essay. I’m super psyched about that. One of the only truly tiresome parts of the otherwise super-fun job of being an SAT tutor is when a student says “my friend says he just made all his examples up and he got a 12 so why aren’t you telling me to do that?” I’ll be very glad to stop having that conversation.

Second, a major, and valid, complaint about the current SAT’s essay is that it doesn’t really tell colleges anything worth knowing about a student. There is certainly some skill involved in writing a coherent essay on a prompt you’ve just been given that you might not have ever thought about before, but that skill isn’t really required for serious college work—written assignments in college will afford students plenty of time to research and form well-reasoned positions. In forcing students to leave their opinions at the door and analyze how an author makes his or her argument, the new SAT’s essay task serves as a better proxy for college preparedness. I do, however, pity whoever gets hired to read these things. They’re going to be super boring.

The other major complaint about the current SAT’s essay, that length is so highly correlated with scores, will probably still hold. Frankly, that’s fine by me. Good essays aren’t better because they’re longer, they’re longer because they’re better. For the most part, students who have more to say write longer essays.

I obviously think the whole new SAT will be susceptible to prep just like any other standardized test, but I think the new SAT’s essay might be the most “preppable” thing on there. Further, I’m pretty sure that’s intentional. Look at this quote from page 76:

test_specifications_for_the_redesigned_sat_na3_pdf 4

What do you make of that?

As for what prep’s going to look like, here’s my best guess: students will go through the prompt and annotate it with a few shorthand codes: “AS” for evidence from authoritative source, “F” for well-known fact, “E” for emotional appeal, “W” for stylistic word choice, etc. That’ll give them a picture of which common devices the author uses to make his point. Then students will write a quick intro (“Dana Gioia combines emotional appeals with concrete and authoritative data to argue that the decline of reading in America will have a negative effect on society.”), and devote a paragraph to each important device and how it furthered the author’s argument.

It’s going to be a boring 50 minutes, writing this essay. But students who learn to do it will will very possibly also learn to better, more careful readers.

tumblr_m16jbnRs3q1ro1guto1_400As you know, I’ve been poking around in the 200-page document recently released by the College Board about the new SAT (coming March 2016). When the new exam was announced a month ago, one of the sound bites that got traction in the press was that the SAT would be returning to the old, familiar 1600 point scale. That struck me as odd: the 1600 scale was certainly more familiar to reporters writing stories about the SAT, who probably took the test before 2005, but anyone who works with the test, and certainly anyone prepping for it, acclimated to 2400 a long time ago. Anyway, it turns out that the return to 1600 will be a lot more complicated than College Board originally let on.

As it stands now, students who take the new SAT will receive not only scores of 200–800 for Math and Evidence-Based Reading and Writing (the sum of which will comprise the 1600 score) and an essay score, but also a whole bunch of other scores. I’ll try to make sense of it all for you below.

The scores students and parents will care about

Students will receive two area (domain) scores: one for Math, and one for Evidence-Based Reading and Writing (EBRW). Each of these will be on a 200–800 scale. They will combine to form the composite score on a 400–1600 scale. The essay will not factor into theses scores.

The scores colleges might care about

Students will also receive three test scores, which will be on a 10–40 scale. These will be the Reading Test score, the Writing and Language (WL) Test score, and the Math Test score. The essay will still not factor into these scores. The Essay Test, which will be the fourth (and optional*) test, will get its own score. More on that later.

* My guess is that enough selective colleges will ask students to submit essay scores that the essay won’t be optional for ambitious students.

The scores no one will care about

There are nine of these. That’s right—nine scores you’re probably not going to care about.

Subscores

Students will receive seven subscores, on a 1–15 scale. These might actually provide some value to students hoping to raise their area scores, but which will probably not matter much (or at all) in the admissions process. Below is a list of the planned subscores:

  • Command of Evidence (R, WL)
  • Relevant Words in Context (R, WL)
  • Expression of Ideas (WL)
  • Standard English Conventions (WL)
  • Heart of Algebra (M)
  • Problem Solving and Data Analysis (M)
  • Passport to Advanced Math (M)

The letters in parentheses next to those subscores are the tests in which the content comprising the subscores will be. So the first two subscores, Command of Evidence and Relevant Words in Context, will be the result of student performance on items in both the Reading and the Writing and Language Tests.

Not every item on the SAT will contribute to a subscore. For example, as I point out in my post about the new SAT’s math section, questions categorized “Additional Topics in Math,” including geometry, trigonometry, complex numbers, etc. will contribute to overall math test and area scores, but not subscores. Simple, right? Who could be confused by this?

Cross-test scores

Rounding out the scores nobody will care about are the cross-test scores, which will be on a 10–40 scale, just like the test scores. Students will receive two of these: Analysis in History/Social Studies and Analysis in Science. Cross-test scores will be determined based on student performance throughout the entire SAT. So, for example, a math question requiring students to do statistical analysis on demographic information will contribute to the Analysis in History/Social Studies cross-test score, as will student performance on reading and writing tasks pertaining to social studies.

Essay scores

The College Board is pretty noncommittal about this so far, stating that this decision will be reassessed pending further research, but the current plan is that two readers will rate student essays on a scale of 1–4 for each of three traits. Those ratings will be added together to give students a 2–8 rating for Reading, Analysis, and Writing. That’s right: 2/3 of a student’s essay scores will be determined based on how well he/she communicates an understanding of the content and structure of the prompt passage. This makes more sense than it initially seems, given the nature of the new SAT’s essay assignment. (I’ll post about the new essay soon enough, but in the meantime you can check out a sample prompt on page 122 of the College Board document linked at the top of this post.)

I’m betting whether College Board likes it or not, they’re going to have to provide some sort of single-number essay score or people are going to go nuts. My guess is that’ll be either an average of those three scores, or a sum.

Seriously, WTF

jackiechanBelow is the College Board’s own summary of this scoring scheme, from page 23 of the document linked at the top of this post. The College Board is careful throughout the document to point out that these are only preliminary blueprints and that things might change before March 2016; these caveats certainly apply to the scoring scheme as well. I suspect that we’ll see some evolution of this system.

test_specifications_for_the_redesigned_sat_na3_pdf

Before I wrap this up, I want to say a few more things.

First, I’m no education expert. I’m a good teacher, but my Master’s degree is in environmental policy, not education. I’m in this business because I like helping kids; I happily leave the architecting to the architects. That said, there’s a lot here not to like from the perspective of someone who likes helping kids. I’m dreading the first time someone calls me looking for tutoring because their Passport to Advanced Math subscore is keeping their area score down. I don’t know how vague euphemisms help anyone whose mission is “delivering opportunity” (an impressive euphemism in its own right) to disadvantaged students.

Second, I want to quote this bit of text from the College Board’s discussion of test scores (the 10–40 scores for Reading, Writing and Language, and Math). This probably also applies to cross-test scores, which are on the same 10–40 scale. From page 22:

The SAT will be the anchor of a vertically aligned, longitudinal assessment system that is designed to monitor student growth across grades in each of these areas annually.

As others have pointed out before me, this is about the Common Core. David Coleman, the President of the College Board who presided over the construction of this new test, was on the team that designed the Common Core. As I’ve said, I’m no education expert and I don’t feel comfortable getting into a debate on the merits of the Common Core here or anywhere else. I do think, however, that the Common Core, and how the new exam complements it, should feature more prominently in the mainstream press’s coverage of the SAT overhaul.

The College Board released a big document today that fills in some of the blanks about the new SAT, which will make its debut in March 2016. I woke up early to dig though it like a little kid on Christmas morning.

Rather than trying to jam all my reactions into a single monolithic post, I’m going to be making frequent, short updates as I read more, and my reactions evolve. In this post, I’m reading and giving my initial reactions to the part of the document that pertains to the math section. The math content begins on page 133 of the document if you want to follow along.

Test structure

As we’ve known since March, the new SAT will test students on three somewhat vaguely defined areas: Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math. We now know that in addition to the main Math scores on the 800 scale, students will receive subscores in each of those topics. College Board also reveals that the test will contain questions on Additional Topics in Math that won’t contribute to the subscores of the three main sections, but will contribute to the overall math score. This, apparently, is where geometry will live.

Page 135 of the College Board document contains a breakdown of content specifications. The math section will consist of a calculator section (55 minutes, 37 questions) and a no-calculator section (25 minutes, 20 questions). The calculator section will consist of 30 multiple choice questions, 6 short grid ins, and one extended thinking grid-in, which will be in the Problem Solving and Data Analysis category, and which will be weighted more heavily—4 points instead of 1. The 20 question, no-calculator section will be all multiple choice.

All told, the new SAT’s math section will consist of 80 minutes of testing on 57 questions—10 more minutes and 3 more questions than the current SAT.

Heart of Algebra

This is all about linear equations. While it’s clear that there’ll be more linear equations on the new SAT than there were on the old one, there doesn’t appear to be anything new. Students will need to solve linear equations and systems of equations, and recognize properties of graphs (e.g. slope, x– and y-intercepts). Here’s a screenshot of an example question (p147):

test_specifications_for_the_redesigned_sat_na3_pdf

This is a straight-up solving for expressions question. Students who’ve been prepping assiduously for the current SAT should have no problem recognizing the one-step solution: multiply the whole equation by 6:

6\left(\dfrac{1}{2}x+\dfrac{1}{3}y\right)=6(4)

3x+2y=24

Solving for expressions will also come into play in questions like this grid-on (p162):

test_specifications_for_the_redesigned_sat_na3_pdf 3

How do you go from –3t + 1 to 9t – 3? Multiply everything by –3, of course! Here, presumably, students are being tested on whether they remember that inequalities reverse direction when multiplied or divided by negative numbers. Multiply everything by –3 and you get:

\dfrac{27}{5}>9t-3>\dfrac{21}{4}

5.4>9t-3>5.25

Any value in that range works.

Unlike the current SAT, though, it looks like systems of equations on the new SAT will not always be solving for expressions questions. Check out this multiple choice question (p148):

test_specifications_for_the_redesigned_sat_na3_pdf 2

No simple way to get xy there—you actually have to solve for each variable individually. Also, four answer choices instead of five? WHAT IS HAPPENING!?

To solve, first get x by itself in the second equation:

x = 4 – 8y

Now substitute for x in the first equation and solve for y:

4(4 – 8y) – y = 3y + 7
16 – 32yy = 3y + 7
16 – 33y = 3y + 7
9 = 36y
0.25 = y

Now use y = 0.25 to solve for x:

x = 4 – 8(0.25)
x = 4 – 2
x = 2

Finally, multiply to get the answer:

xy = (2)(0.25)
xy = 0.5 = \dfrac{1}{2}

So…fun, right? No. Not fun. Soul-suckingly boring. But I’m not really complaining; the ability to solve linear equations is important, and in my experience a lot of students need to be reminded how to do it by the time they reach SAT-taking age. This is a good change.

Here’s one more Heart of Algebra question I love (p167):

test_specifications_for_the_redesigned_sat_na3_pdf 4

You can do a lot of algebra to shoehorn this into y=mx + b form, but nimble students should be able to look at the original equation and see two things: 1) there’s no x^2 term, so (D) is out, and 2) there won’t be any constant terms—the y-intercept is 0, so (A) and (C) are out. Awesome question.

Problem Solving and Data Analysis

It seems like the Data Analysis part of this category won’t be much different than current SAT questions that require students to read tables and graphs. Again, while this skill is tested on the current SAT, it looks to be a much more prominent feature in the new one. Students probably won’t have to calculate best-fit lines, but they’ll need to know what a best-fit line is, and why it’s important. Check this out (p172):

test_specifications_for_the_redesigned_sat_na3_pdf 5

Super-rough estimates are good enough here—the manatee population was about 1,000 in 1990 and about 4,000 in 2010. That’s a 3,000 manatee increase in 20 years. \dfrac{3,000\text{ manatees}}{20\text{ years}} = 150 manatees per year.

The real interesting thing in Problem Solving and Data Analysis, though, is the Extended Thinking question. Students taking the new SAT will have to wrestle with stuff like this (p180):

test_specifications_for_the_redesigned_sat_na3_pdf 6

Yikes, right? Questions like this will be worth 4 points, and should serve to silence everyone who says the new SAT will be too easy compared to the old one.

If the bank converts Sara’s purchase to dollars, and adds a 4% charge, then the $9.88 she’s charged doesn’t convert directly to rupees. First we need to get that 4% fee out of there. If x = the dollar cost of her purchase without the fee, then 1.04x = 9.88.

x = \dfrac{9.88}{1.04} = 9.5

So her purchase was worth $9.50, which means the exchange rate that day, in rupees per dollar, is:

\dfrac{602\text{ rupees}}{9.5\text{ dollars}}\approx 63.36\text{ rupees per dollar}

Round that to the nearest whole number and the answer to part 1 is 63.

Now on to part 2. Take a deep breath. We can do this.

If Sara buys the prepaid card, she doesn’t have to pay the 4% fee her credit card charges her, but she could lose money if she doesn’t spend the whole card. The exchange rate doesn’t actually matter anymore—we can actually forget about dollars entirely at this point.

The question is really how much Sara has to spend on her credit card, with the 4% fee, before she’d save money by buying the prepaid card, even if she doesn’t spend it all. Say y = the amount of rupees Sara spends:

7,500 = 1.04y
7,211.54 = y

If Sara spends 7,211.54 rupees on her credit card, it costs her the same as buying a 7,500 rupee prepaid card. Therefore, if Sara spends 7212 rupees or more on her prepaid card, she gets a better deal than if she spends the same amount on her credit card.

Passport to Advanced Math

Passport to Advanced Math is going to involve a lot of manipulating unwieldy equations, and recognizing what parts of those equations mean. The SAT has tested this kind of thing in the past, to the chagrin of many test takers. (An example that comes to mind is the January 2008 test, which is included on the DVD if you got the Blue Book edition that comes with a DVD. See section 8, question #14, with the stacked pails.)

Let’s have a look at what that might look like on the new SAT (p186):

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Umm…yeah. This is a hateful question. To get it right, you need to recognize that the units of each term are jobs per hour, and that the equation is set up by someone trying to solve a question that’s not actually been asked: How quickly could each printer do the job by itself? Solving gets you x = 15, which means the slow printer works at a rate of 1/15 jobs per hour, and the fast printer works at a rate of 2/15 jobs per hour.

Once you’ve got that figured out, then the rest is easy—the answer must be B).

Passport to Advanced Math will also contain more straightforward questions that will be familiar to any student that’s taken algebra 2 (p188):

test_specifications_for_the_redesigned_sat_na3_pdf 12

To get this one, just substitute (and pay careful attention—the SAT will still try to nail you by asking for x^2 but making x (or at least one value of it) a choice.

x^2 + (-4x)^2 = 153

x^2 + 16x^2=153

17x^2=153

x^2=9

 

Additional Topics in Math

Here’s where all the surprises are. The following concepts will now be tested on the SAT:

  • trigonometry (using degrees and radians)
  • complex numbers
  • far more thorough knowledge of circles than is required on the current SAT (students will have to find chord lengths, for example, or working with circles in a coordinate plane)

The whole of geometry from the current SAT seems to live under this category, too. As noted above, this stuff doesn’t contribute to any of your math subscores (it remains to be seen how subscores will be used, but I’m guessing they’ll have limited value except for diagnostic purposes as students prepare), but will contribute to your overall math score.

Check this bad boy out (p193):

test_specifications_for_the_redesigned_sat_na3_pdf 7

With the exception of the word “chord,” this question would be just as at home on the current SAT—it’s a question that doesn’t immediately look like a right triangle question, but it’s totally a right triangle question. Another nice thing about this question is that it’s proof that plugging in will still be useful on the new SAT. Say r = 3, so that AB = 6 and CD = 4. Now draw some lines:

test_specifications_for_the_redesigned_sat_na3_pdf 8

Of course, since PD is a radius, it has a length of 3 just like AP and PB.

2^2+QP^2=3^2

4+QP^2 = 9

QP=\sqrt{5}

Only one answer choice has \sqrt{5} in it, so we can feel pretty good about (D) without even doing the last step of the plugging in process, which is to put 3 in for r in each answer choice. Of course, when you do, (D) is the only answer to give you \sqrt{5}.

What will trig look like on the new SAT? It’ll look like this no-calculator question (p194):

test_specifications_for_the_redesigned_sat_na3_pdf 9

A calculator would be nice here—plug in, graph, and you’re done. Without a calculator, you need to know the relationship between angles with opposite sines.

sin_x_-_Wolfram_Alpha

Since the sine function has a period of 2π, subtracting π from x will result in a negated sine. :/

More to come…

Above are just a few quick impressions. I’ll be posting more in the coming days as I’ve had more time to dig into the details. And of course, remember that if you’re prepping for the SAT now, you don’t need to worry about this stuff at all. The SAT doesn’t change until March 2016.

I’ve been receiving reports from a bunch of March SAT takers (here are two examples) that they had strange experimental sections, most notably an 18-question reading/writing section that might be a preview of the new Evidence-Based Reading and Writing (EBRW) section that will appear on the new SAT in 2016. There were some reports of strange sections appearing on the January SAT as well.

Of course, the College Board needs to start testing this material out on students before it goes live—that’s one of the important uses of the experimental section. But for students who have been preparing faithfully for the current test, material that’s so far out of line from the norm can be very disorienting. If you’re taking any SAT between now and March 2016, you need to be mentally prepared for the possibility that your experimental section on test day will look nothing like any practice test you’ve taken. Don’t let it throw you off your game.

Note that I am very much not saying that you should intentionally tank a section if you suspect it’s experimental. There’s always a risk that you could be wrong, and that would suck. I’m just saying that you should not panic if you come across a section that looks structurally different from any section you’ve seen in your myriad practice tests. It doesn’t mean that you aren’t well-prepared. In fact, recognizing that something is amiss is an indicator that you are well-prepared. So take a deep breath, go with the flow, and do your best on every section you face.

If you’re preparing for the SAT right now, then you probably don’t need to worry about this—the test won’t change until March 2016. Even if you’re a sophomore right now, if you’re already thinking about the SAT then your focus should be on the current test, not the new one. 
Well, that was certainly interesting! I believe you can still watch the announcement by College Board President David Coleman if you want here, but you’ll get the information much faster by reading one of the thousands of articles about the new SAT that are already online from tons of major news outlets. You can also get it straight from the horse’s mouth here and here. A lot more information will be released on April 16th.
I’m not going to write a super long response to the announcement right now, but here are a few of the most salient changes and my reactions to them.
  • Back to the 1600 scale. Two sections make up that score are Math, and “Evidence-Based Reading and Writing.”
  • There will still be an essay, but it will be scored separately and it will be optional. Also, it will be 50 minutes long and the question (but not the passage it’s about) will be known to all students in advance. Most people in the biz expected major essay changes, and we got them.
  • No more guessing penalty. Right answers help your score. Wrongs and blanks count the same. This is how the ACT does it, and many in the prep world saw this coming, too.
  • Calculators will be allowed on some math sections, prohibited on others. This is so a student’s number sense can be tested more effectively.
  • The test will be available on paper and digitally. Hopefully, that means students will get to choose how they want to take it.
  • No more sentence completions. Vocabulary will still play a role in the test, but the focus will be much more on words with multiple meanings and uses that will be important for students to use every day in college and beyond. In the presentation, David Coleman used the word “synthesis” as an example.
  • The “Evidence-Based Reading and Writing” (you know what, I’m just gonna call that EBRW) will have charts and graphs in addition to pure text for students to grapple with. Seems to me like that’s inspired by the ACT science section.
  • The Math section will narrow its focus to “problem solving and data analysis,” the “heart of algebra,” and “passport to advanced math.”It’s not exactly clear what those mean yet—I can see how each topic currently tested could be recategorized into one of those. If the new test will really be narrower, we’ll have a better idea how much narrower no April 16.
  • The new SAT will focus on the Founding Documents (e.g. the Declaration of Independence) and the Great Global Conversation (e.g. “Letter from a Birmingham Jail”). I’m a bit conflicted on this. On the one hand, what could be more important than engaging students in the documents upon which our society is based. On the other hand, does it trivialize those documents, or make students likely to hate them, to include them as an integral part of the SAT?
So, yeah. The new SAT will be quite different from the current one. Here’s how I think it’ll still be the same:
  • It will still be about 4 hours long (assuming you do the optional essay).
  • It still won’t be easy.
  • It will still be difficult to raise your critical reading (or EBRW) score if you haven’t been engaging in the practice of careful reading for many years.
  • The math will still find ways to test simple concepts in very tricky ways.
So those are my initial reactions. Did you watch the announcement? What did you think?