The GMAT doesn’t allow you to use a calculator—which means you need to be quick and accurate with basic calculations. Be able to multiply and divide decimals. Know common higher powers and roots. Have fractions down to a science: Knowing right away whether 3/8 is less than 5/12 will mean you have more time later to work on more complicated calculations.

Even questions that don’t contain variables can still be tackled by choosing numbers wisely. For example, if a question asks you about “a multiple of 6,” it’s probably quicker to work with a particular multiple of 6 (say, 12) than the abstract “multiple of 6.” While studying, identify the kind of problems where this strategy can be applied.

Estimation is an important part of the PS section. Often, a question will test your ability not to compute, but rather to make reasonable approximations. For example, the fact that 11 goes into 56 a little more than 5 times means that 11/56 must be slightly less than 1/5, or 0.2.

Know what problems this strategy is useful for—and how best to apply it. Remember to always choose your numbers wisely. If a problem asks you about “1/7 of n garbage trucks,” plug in a multiple of 7 for n. Try to avoid plugging in 0 or 1 in almost every case.

Whenever possible, give a label to the exact quantity that you’re trying to find, rather than, say, its square root. In other words, always be sure that the solution to the problem is also the solution to the equation you’ve set up. This will help avoid careless – and all too common – “last step” errors.

If you know that the correct answer must be less than the value in choice C, you can immediately eliminate choices D and E (or choices A and B, if the answers are in decreasing order). Don’t guess randomly!

If you’re stumped but you notice that three of the choices have a factor of ab, try to figure out where that factor comes from. If you know that the right answer choice should not have a factor of ab, eliminate all those choices in one fell swoop.

Exponent rules are frequently tested on the Problem Solving section. Make sure to know what fractional exponents and negative exponents mean like the back of your hand. Also, be ready to answer answers about quantities with absolute values less than 1 being raised to odd and even powers.

Stuck on a question? Check to be sure you’ve used every fact supplied to you. Conversely, if you’ve solved a problem without using every fact, double-check your work; you probably haven’t solved the problem correctly.

Be aware of how your approach would change for these two question types. Are there certain numbers you might want to plug in for one type but not for the other? Why or why not?

]]>The good news is that with some strategic practice, you too can train yourself to think like a DS whiz. To master this section, start by becoming familiar with the structure of DS questions and the concepts they most commonly test.

Here are some concrete tips to get you on track:

No excuses: On Data Sufficiency, they’re always the same! Know in the blink of an eye what choice C is. On test day, if you find that Statement 1 is insufficient, be able to cross out choices A and D without hesitation.

Each statement alone will be sufficient if both of the statements **on their own** contain all the information necessary to answer the question. The statements will be sufficient together if they contain every piece of necessary information **between them**. Take the area of a parallelogram: Do you need to know every side length to determine the area? If you have every side length, can you find the area?

Statement 2 may tell you that x is negative, but that fact has no bearing on Statement 1 when viewed by itself. Explore all the possibilities offered by each statement individually. If you’ve scrutinized Statement 1 and found it sufficient, be equally merciless when it comes to Statement 2.

Don’t pay so much attention to the statements that you forget the rest of the question. Often, half the information that you need is in the set-up.

If the question asks for the value of x and you whittle the problem down to an equation like 305x = 2(500) – 10205, don’t waste your time solving for x! It’s only important to know that you COULD solve if you wanted to. Remember, all linear one-variable equations have a unique solution, but quadratic equations—equations with an x^2 term—can have zero, one, or two solutions.

Again, you never need to solve a DS problem—you only need to know that you** could**. A system of n independent linear equations with n variables can be solved for ALL of the n variables. The key word here is “independent”: Equations are independent if they’re not multiples of one another. For example, y = 2x and 3y = 6x are NOT independent equations because the second equation is just three times the first. If on test day you don’t feel comfortable declaring that a system of equations is solvable, get the system down to one single-variable equation and then reassess.

Although any GMAT math concept is fair game on the DS section, prime factorization shows up frequently and reliably. If x is divisible by 15, will x^2 be divisible by 27? What about x^3?

Be comfortable representing these overlapping sets with Venn diagrams. This topic is a DS favorite. A statement like, “The number of widgets that were not made in Factory A or Factory B is three times greater than the number of widgets that were made in Factory B” can be difficult to unpack in the heat of the moment. Train yourself to answer questions about sets methodically and quickly.

This means that there’s a 60% chance that the correct answer will treat the statements on an individual footing. It can be tempting to use all the information the problem provides, but keep these odds in mind. Choices C and E, as a group, are 20% less likely to be correct than choices A, B, and D, as a group.

The area of a square, for instance, contains just as much information as the side length of the square. If you know the area, you can find the side length; conversely, if you know the side length, you can find the area. Often on the DS section, Statement 2 will just be a repackaging of the same information provided by Statement 1.

]]>David Ingber is the Faculty Manager at Knewton

If you watch the World Cup over the next few weeks, you are bound to hear the commentators utter sentences like this: “England, which have not brought the World Cup trophy back to their homeland in over 50 years, face a difficult road ahead of them.”

For whatever reason, such a sentence is perfectly acceptable in the soccer world. However, such sentences are terrible on the SAT and GMAT. If you want to succeed on either test, you must tune out such egregious grammatical errors! “England” is a singular noun. Therefore, the sentence should read like this: “England, which has not brought the World Cup trophy back to its homeland in over 50 years, faces a difficult road ahead of it.”

Just as the players will angrily disagree with every yellow card they are given, so the commentators’ subjects, verbs, pronouns, and antecedents will wantonly disagree. We only hope that you will watch the games on mute.

Please celebrate in a responsible, grammatically sound manner. Go USA!!!

]]>In an earlier post, I discussed Dan Meyer’s visionary talk about the future of math education. Halfway through the talk, Meyer mentions a classic problem in which students must determine how long it takes to fill a tank with water.

A typical textbook would give students all (or most of) the necessary pieces and then ask them to construct the puzzle — that is, plug the numbers into a formula. Meyer, however, advocates doing away with all the information and simply posing the question: “How long will it take to fill the tank with water?” Students then have to figure out what they need to answer the question. This approach forces them to think patiently and creatively.

By no small coincidence, we at Knewton teach the exact same question in the Data Sufficiency portion of our GMAT course:

An empty rectangular tank has uniform depth. How long will it take to fill the tank with water?

- Water will be pumped at the rate of 480 gallons per hour (1 cubic foot = 7.5 gallons).
- The tank is 100 feet deep and 30 feet wide.

- (A) Statement (1) ALONE is sufficient, but statement (2) alone isÂ not sufficient.
- (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- (D) EACH statement ALONE is sufficient.
- (E) Statements (1) and (2) TOGETHER are NOT sufficient.

Okay, so the question isn’t *exactly* the same. Meyer’s tank is an octagonal prism; ours is a rectangular solid. Details aside, both questions require the same level of critical reasoning.

Imagine that the prompt was pared down to simply, “How long will it take to fill the tank with water?” Give it a moment’s thought. What do you need to know to answer this question? This is, at its core, what the Data Sufficiency section tests. You never have to compute the final answer on a DS question; you just have to know what it would take to find the answer.

Here, you need to know how quickly the water is flowing into the tank — that is, you need a rate. Next, you need to know how big the tank is — you need a volume. The prompt tells you that the tank is rectangular, with uniform depth, so its volume will be a product of its length, its width, and its depth. In total, then, you need four quantities to answer the question: (1) rate, (2) length, (3) width, (4) depth.

Statement 1 gives you quantity (1), the rate, but nothing more; it can’t be sufficient. Statement 2 gives you quantities (3) and (4), the width and depth, but since it doesn’t give you the length or the rate, it can’t be sufficient, either. When you put the statements together, you’ve got quantities (1), (3), and (4), but you still don’t have quantity (2), the length of the tank. Even together, the statements are not sufficient.

If you approach this question critically, you can polish it off in a matter of seconds. “I need four quantities. I see only three quantities. Answer **choice E** is correct.” If you approach it passively, though, you get lost in the numbers and are far more likely to get it wrong. Nearly 25% of students do.

To be sure, questions like this are rare. In general, standardized math tests and creative problem solving do not go well together. As we change the way we teach students by drawing them further into the conversation, we will also need to change the way we assess them. The sooner, the better.

]]>High school sometimes feels like it hinges on two tests: the driving test and the SAT. And though you’re probably more excited to start driving than you are to start applying to college, there’s at least one respect in which the SAT is nicer than the driving test: It doesn’t penalize you for guessing!

“Now waaaaaaait a minute,” you say. “I know a thing or two about this SAT. If I guess on a question and get it wrong, I lose a quarter of a point. So how does it make sense to say there’s no penalty for guessing?”

Let’s work through it. For every multiple-choice question on the test, there are five answers choices. So if you take a random guess, you have a 1 in 5 chance of guessing correctly.

What if you guess on five questions in a row? Well, if you have a 1 in 5 chance of getting each question right, then odds are you’ll get lucky once in those five questions. For that one correct guess, the SAT will reward you with 1 point. For each of the four incorrect guesses, you’ll be docked a quarter-point, which means you’ll lose (1/4) + (1/4) + (1/4) + (1/4) = 1 point.

End result: You gain 1 point, and you lose 1 point, so you’re back to zero. No penalty!

Now imagine that you’re *not* guessing randomly. Maybe you’ve got a math problem that you can’t quite solve, but you know the answer has to be positive. If one of the answer choices is negative, your guessing odds go up. Or maybe you’ve got a sentence completion question, and you know the correct answer has to mean something like “angry.” If you’re sure that three of the words in the answer choices actually mean “friendly,” your odds go way up. Given that guessing randomly didn’t lose you any points, you can see how much you stand to benefit from guessing wisely!

Basically, if you can eliminate any answer choices, it’s in your interest to bubble something in. Educated guessing is rewarded on the SAT.

To be clear: this isn’t a trick. The test is designed with this aim in mind. A student who can eliminate lots of answer choices possesses more knowledge than a student who can’t eliminate any answer choices, and that knowledge deserves to be reflected in a higher score. If you never guess, you’ll never be rewarded for that knowledge.

Here’s another way to convince yourself: The SAT can’t tell colleges what you “really” know — it can only tell them what your score is. You may feel uncomfortable guessing on a question that you’re not 100% sure about.

You think you’ll be “caught.” However, the only thing that matters on the test is your final score. Since guessing tends to increase that score, be brave, and go for it!

]]>The academics team at Knewton has been buzzing about a video that one of our expert teachers, Chris Wu, sent around this morning. It’s a TEDx talk by Dan Meyer, a high school math instructor in Santa Cruz. The talk focuses on the virtues of what Meyer calls “patient problem solving” — where fewer formulas and inputs are fed to students and more active problem formulation is required of students.

For example, rather than giving students a train’s average speed and the distance the train needs to travel and asking them what time the train will arrive, why not ask them for the train’s arrival time and let them, in a group conversation, determine what information is needed to solve the problem? When they realize they need some kind of distance measurement, make them consult maps to find the distance in question; when they need a rate, let them research a train’s average speed. Students will learn to manipulate equations in the process, but more importantly, they’ll learn to think creatively about the real world. The result, as Meyer says, is that “the math serves the conversation; the conversation doesn’t serve the math.”

Math and science education is a hot topic in the US these days, and as the Obama administration prepares to institute new school standards that will set the curriculum in 48 states, we must improve not only what we teach but also how we teach. Part of the “how” fix boils down to making the educational experience less monolithic. In an earlier post on this site, Kalyan Dudala bemoaned the lecture format that has hounded him since his undergraduate days. A three-hour lecture on organic chemistry may work for some students, but it certainly doesn’t work for most. People learn at different rates and in different ways.

Motivation is also an issue. The lecture format is designed, in part, to allow the teacher to verify that you came to class. But fear of punishment doesn’t make somebody a good learner. Once we’ve done away with one-size-fits-all lectures and customized the student experience, we will still need to make sure that students are engaged. In other words, it’s not enough to make sure that everyone can successfully plug values into formulas. Teachers must also communicate why these particular math and science concepts are being studied. Sean Carroll put it well in a recent New York Times interview: Students need to see that the quantitative aspect of science “can be in the service of interesting rather than boring problems.” If they do, the subject becomes inherently worthwhile. On the other hand, if math and science are presented as a series of rote equations, students are unlikely to learn the concepts. Or — just as bad — they’ll learn them, but they won’t grasp the fact that in life, the things you need in order to solve a problem are seldom handed to you. Nor, for that matter, is the problem.

Of course, some people do approach reality best through the medium of abstract equations. That type of intelligence — which spawns the most profound scientific breakthroughs — can and will be ministered to as well. In fact, that’s the beauty of where math and science education appears to be headed. If, as Dan Meyer says, “math is the vocabulary for your own intuition,” then the curriculum of the future will essentially amount to letting you speak your mind — and clearly.

]]>The adaptive learning engine that we’re building at Knewton is part of a larger trend of adaptive approaches to some of the world’s biggest problems. Rather than rely on rules that are etched in stone — rules that frequently have no bearing on the situation at hand — an adaptive approach responds to the environment. It reacts dynamically to what’s actually going on. It, well, adapts.

In New York, the grueling morning commute takes the average person 39 minutes. In Mumbai, it’s 47 minutes. In Tokyo, it’s 67 minutes! Think of all the energy, both personal and environmental, that’s wasted. At least part of the problem lies in how traffic signals are coordinated. You’ve probably experienced it yourself. You’re parked at a red light with a dozen other honking cars, wondering why the empty street before you has a green light. Sure, the traffic lights change according to a schedule, but why does it need to be that way?

Enter adaptive technology. These days, proposals abound on how to better control the flow of traffic. One team of Hungarian researchers has suggested putting wireless communication devices on cars and using the data culled from them to adjust traffic signals. Another proposal by a Belgian team involves making lights self-adaptive. By counting the number of cars on the road, traffic lights can determine whether they need to be green more (or less) often.

Both these ideas underscore the potential benefits of adaptive approaches. If a system moves from being strict and centralized to being fluid and atomized, its individual parts will function all the more smoothly.

At Knewton, we’re interested in applying the principles of adaptivity to LSAT prep and GMAT prep, and eventually to all learning experiences: By accounting for unique performance on individual concepts, we’ll be able to identify a study path that is tailored to each student. The usefulness of adaptivity in education is clear, and many real-world systems would see similar benefits.

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David Yourdon is a Math Content Developer at Knewton*.