Video duration: 50:38
THOMAS PEARSON: Hello. My name is Tom Pearson. And I'm an Assessment Specialist in the Higher Ed division at Educational Testing Service. And I'll be helping you prepare for the Quantitative Reasoning measure of the GRE®.
During this session, I'll be providing you with a brief introduction to the Quantitative Reasoning measure. I'll be looking at different question types and strategies for answering each of those types. I'll spend some time talking about the online calculator that is available to you as you work your way through the measure.
I'll look at some general problem-solving steps and strategies as they apply to the Quantitative Reasoning measure. And finally, we'll look at some GRE® resources that are available to you in your preparation for the test.
Let's begin with an introduction to the Quantitative Reasoning measure.
So what is the measure trying to assess exactly? Basically, we're testing basic mathematical skills, the understanding of elementary mathematical concepts, and the ability to reason quantitatively and to model and solve problems with quantitative methods.
Now, this is important. The test is more than simply — can you do the math, can you solve for X? It has to do with reasoning within a mathematical context — can you understand, interpret, and analyze quantitative information?
Now, the basic mathematical knowledge that is expected of test takers includes basic concepts of arithmetic, algebra, geometry, and data analysis. So those are the four basic content areas — arithmetic, algebra, geometry, and data analysis. The test also includes high school-level mathematics and statistics. It's generally no higher than algebra II. And the test excludes trigonometry, calculus, and higher-level college mathematics.
Two very important free tools to help you in your preparation, the first being the GRE Math Review. This is a 100-page math refresher in each of the content areas. It includes definitions, properties, examples and exercises with answers at the end of each section.
So this is a great tool for kind of establishing a baseline of your mathematical skills as you start your preparation for the test.
Maybe it's been a few years since you've had algebra. Maybe it's been a while since you've thought about geometry. So this math review can give you a good sense of where you stand, where your strengths are, where your weaknesses are as you begin your preparation for the test. And it also includes links to additional help at the Khan Academy®.
So that's one tool, the Math Review. The other important tool is the GRE Math Conventions. This tool includes mathematical notation, symbols, terminology and guidelines that are used in the test. It's sort of the basic — the rules of the game as it were. Now, the most important math conventions we give to you in the directions. So you don't have to worry too much about this if it's not something you're interested in pursuing in great depth.
The most important math conventions that you need to take the test will be provided to you in the directions to the test. But you can look at and study the GRE Math Conventions if you wish. And both the Math Review and the Math Conventions are available at www.ets.org. And you can see the link there at the bottom of the slide.
All right, let's look in a little more depth at each of the four content areas. Arithmetic includes the following topics — elementary operations, number line, estimation, percent ratio and rates, absolute value, and properties of integers, such as divisibility, odd and even, and prime numbers. Again, all of these will be discussed with examples in the Math Review.
Similarly, algebra includes algebraic expressions and manipulations, functions and their graphs, coordinate geometry. It includes solving equations and inequalities as well as modeling and solving word problems with algebra.
Our third content area, geometry, includes elementary geometric figures, concepts such as lines, circles, triangles, quadrilaterals and other polygons. It includes angle measure, area and perimeter, volume and the Pythagorean theorem. And it also will include intuitive geometric concepts. For example, the sum of any two sides of a triangle is greater than the third side.
And note that the ability to construct proofs is not something that is measured on the test. And again, all these topics are discussed in the Math Review.
Data analysis will include basic descriptive statistics, such as mean, median, mode, range, interquartile range, percentile and standard deviation. It will include frequency distributions. There will be interpretations of data presentations. And we'll see an example of that in one of the problems we look at later in the slides. It touches on elementary probability as well as counting methods.
Let's now turn to the content of the Quantitative Reasoning measure. The computer-delivered Quantitative Reasoning test contains two 35-minute sections. Each section has 20 questions. And the test contains the following question types — multiple choice — select one answer; multiple choice — select one or more answer choices; quantitative comparison questions in which you're comparing two quantities; and numeric entry questions where you formulate the correct response and enter it into a box.
There are questions that are part of data interpretation sets, several questions about the same data presentation. And we'll look at examples of all these as we go along. This is just a quick overview. There'll be some questions that involve real-life scenarios, word problems in a real-life context of some kind. There are also questions that are more pure math, solving for x in other words. And throughout the test, an on-screen calculator is provided if you need it.
Let's talk a little bit about how the Quantitative Reasoning measure is scored. The Quantitative Measure of the computer-delivered test is section-level adaptive. So what does that mean? Well, the computer will select your second section. There are two sections of the test. And your second section is based on your performance in the first section. So we sometimes call the first section the routing section.
It's the same for everybody in any given administration of the test. So depending on how well you do on that first section, you're routed to either a more difficult or a less difficult second section.
Now, within each section, all questions contribute equally to the final score. So it's important that you do your best on all questions throughout the test in both sections of the measure. Both sections are important since the final score on the measure is based on the total number of correct answers and the overall difficulty level of the questions that you encountered.
Your score on the measure is based on both sections. And it considers the number of right answers as well as the difficulty level of the questions. A raw score is computed. Your raw score is simply the number of questions that you answered correctly. And then this raw score is converted to a scaled score through the equating process.
This is a process which accounts for variations in difficulty among different test editions as well as differences in difficulty among individual tests introduced by the section-level adaptation. So any given scaled score reflects the same level of performance regardless of which section was selected in your test administration and when the test was taken.
All right, let's now turn to the different question types and some strategies for answering each of those types.
Here are the question types in the Quantitative Reasoning measure once again for you.
There's the multiple choice — select one answer choice, in which you're presented with five answer choices, you choose one and only one correct answer; multiple choice in which you can select one or more of the answer choices, so one or more choices from a list of choices; quantitative comparison questions in which you're comparing two quantities; numeric entry questions where you formulate an answer and you enter that answer in an answer box, or sometimes boxes, plural. And then there are questions that are a part of data interpretation sets.
All right, so here are some strategies for the multiple choice — select one answer choice. First off, very basic but very important, you should use the fact that the correct answer is there. The correct answer is one of the choices.
If you do your calculation to answer the question and the answer you come up with is not one of the answers presented to you, you probably have done something wrong. So you may need to rethink — or you do need to rethink your approach to the problem. So bear in mind, the correct answer is one of the choices.
You should examine the answer choices to get a better sense of what is being asked. And for questions that require approximations, by all means, take a look at the answer choices and see how close an approximation is needed.
In other words, if all the answer choices are rounded to the nearest ten or to the nearest hundred, well, that gives you a sense of the level of accuracy that is expected in order to answer the question correctly. And it might give you a sense of how much work you need to do to formulate a correct response.
On the other hand, if all the answer choices are very detailed and lots of very fine distinctions between the answer choices, well, that tells you something. I mean, there's a higher level of accuracy required for that particular question. So take a look at the answer choices. See if they provide you any guidance as to the best approach to any particular question.
All right, so here is a sample "multiple choice — select one answer choice" question. It's a word problem set in a real-life context. A car got 33 miles per gallon using gasoline that costs $2.95 per gallon. Approximately, what was the cost in dollars of the gasoline used in driving the car 350 miles? Select one answer choice it tells you at the bottom of the screen.
Remember, I said look at the answer choices. See if they give you any indication as to how to approach the problem. They're all nice round numbers — $10, $20, $30, $40, and $50. So that tells you the degree of accuracy that you need for this question. Also, note that the question itself tells you, approximately, what was the cost? So these are clues that can help you in answering this question both correctly and efficiently.
So here is the question with the correct answer choice selected. It's $30 that was the approximate cost of the gasoline used in driving the car 350 miles. Now, you could if you wanted fire up the calculator for this question. And divide 350 by 33 to get 10.606. And multiply that by the cost of gasoline, $2.95. And come up with a very detailed figure.
Or you could look at the answer choices. See they're all rounded to the nearest ten. Notice that the question itself is telling you to determine approximately what was the cost in dollars. And use that as your guide. And do the math in your head.
And come up with $30 as the best answer choice and be confident in that choice as the correct response. So in other words, use what's there to guide you. Don't do more work than you have to do to come up with the correct response.
Now, we're going to turn to multiple-choice questions with one or more answer choices. And let's look at some of the strategies for this question type. First off, you need to note whether you're being asked to indicate a specific number of answer choices — the question may say select three choices that fit a certain description — or whether you're being asked to indicate all the choices that apply. In other words, you're not told how many of the possible answer choices are correct. It could be one or more. It could be all of them in that case.
Some questions will ask for possible values of a quantity in a given scenario. And it may be efficient to determine the least and/or greatest possible value before considering the answer choices. Determine a range, in other words, highest to lowest that are possible. And then examine the answer choices to see which ones fall into that range.
And the third strategy listed here — avoid lengthy calculations by recognizing and continuing numerical patterns. So again, if you can save yourself some work, save yourself some lengthy calculations and the time and effort that that would require, then you should by all means do so.
Here is a sample "multiple choice — select one or more answer choices" item type. Which of the following integers are multiples of both two and three? Indicate all such integers. All right, so we're not telling you how many. We just say indicate all such integers. And then you're presented with six possible answer choices. It could be one. It could be two. It could be three. It could be four out of six, five out of six. It could possibly be all six.
There are some visual clues that indicate to you that you are dealing with the "select one or more answer choice" item type. For instance, note at the bottom of the screen, in the gray box, we tell you to select one or more answer choices. Also note that the answers are marked by square boxes rather than ovals.
We do this in the verbal measure as well. When you're dealing with multiple-choice, multiple-selection item types, we indicate that visually by the use of these square boxes. The answer choices are marked with a square box rather than an oval. So we try to provide test takers with a lot of clues to indicate to them that they're dealing with this particular item format.
Here is the same question with the correct responses indicated with an x in the square box. So all such integers here would be 12, 18, and 36. Now, you should note there's no partial credit for these question types. In other words, you have to get all three and only those three correct to have this scored as a correct response.
All right, now, let's turn to the quantitative comparison question. These are not necessarily something you've encountered before. They're kind of a specialized thing. So it's good to become familiar with these before you take the test to see what these are all about. Basically, they ask you to compare two quantities, quantity A and quantity B, and then determine which of the following statements describes the comparison.
Quantity A is greater. Quantity B is greater. The two quantities are equal. Or the relationship cannot be determined from the information given. Those four choices are always the answer choices presented to you in this question type. They never change. So you want to be aware of that and know what you're getting into.
Here is a sample quantitative comparison question. Looks like we're dealing with a geometric figure, a triangle PQR. We also have point S on the line PR. We are told that PQ equals PR. So that's what we're dealing with here. And then we're being asked to compare quantity A, which is line segment PS, with quantity B, which is line segment SR.
Then we have our four option choices, four possible answers. Quantity A is greater. Quantity B is greater. Two quantities are equal. The relationship cannot be determined from the information given. And notice again down at the bottom in the gray box, we tell you select one answer choice.
So here's the question with the correct response indicated, that being the final answer choice. The relationship cannot be determined from the information given. So the relationship just can't be determined. Remember, geometric figures are not necessarily drawn to scale. That's one of the basic math conventions. That's one of the conventions we give you in the directions to the Quantitative Reasoning measure. So geometric figures are not necessarily drawn to scale.
That being the case, we don't know from the information given about the relative sizes of PS and SR. It looks like S could maybe sort of be the midpoint in this line segment PR. But we don't know that for certain. So there's not enough information to determine whether A is greater, B is greater, or the two quantities are equal. So in this case, the relationship cannot be determined from what we're provided.
Some strategies for the quantitative comparison question type. You should certainly become familiar with the answer choices. Remember, they don't change. They never vary. They're always the same. Avoid unnecessary computations if you can. Save yourself the time and the effort. Geometric figures not necessarily drawn to scale, one of the important math conventions that we provide to you in the directions to the test.
You can try plugging in numbers. If you're dealing with an algebraic expression of some kind, you can substitute easy numbers for the variables and compare the resulting quantities. That can be useful. But you have to be careful. The numbers you choose may not necessarily be conclusive. There may be other possibilities that you haven't thought of.
So if you try this strategy, try positive 1 as well as negative 1, a positive number and a negative number. Try 0, plugging that in. Try a really big number as well as a really small number. So try to cover as many possible categories as you can if you're going to plug in numbers.
Another useful strategy, depending on the math that's involved, is simplifying the comparison and seeing if that helps you make a determination.
All right, turning to another question type, that being the numeric entry question. With these, you formulate an answer yourself and enter it into an answer box. So there are no choices presented to you. And now, with these, you should enter your answer as an integer or a decimal if there's a single answer box. Enter your answer as a fraction if there are two separate boxes, one for the numerator and one for the denominator.
You can use the computer mouse and keyboard to enter your answer. For a single answer box, you can transfer the number to the box from on-screen calculator. You have to be a little careful, though, using the calculator. And we'll talk about that later.
Enter the exact answer unless the question requires you to round your answer. So pay attention to whether the question actually tells you to round your answer. And if so, then you need to do that to get the correct response.
OK, here is a sample numeric entry question. A merchant made a profit of $5 on the sale of a sweater that cost the merchant $15. What is the profit expressed as a percent of the merchant's cost? Give your answer to the nearest whole percent. So we tell you very clearly, very explicitly give your answer to the nearest whole percent. So we don't want to see any decimals in the answer box.
In dealing with numeric entry questions, do exactly what the question is telling you to do first off. Second off, you don't have the kind of security blanket that you do with the single selection multiple choice of knowing that the answer is presented to you as one of the answer choices. You just have an empty box. You've got to come up with the right response and enter it. So you have to be very confident you're solving the problem correctly.
Another little thing to watch for down here at the bottom — again, useful information in the gray box. Enter your answer as an integer or a decimal in the answer box. Backspace to erase. So those are sort of the generic instructions that you'll see at the bottom of all the numeric entry questions.
But notice for this particular question, you're being told to give your answer to the nearest whole percent. So get everything clear in your mind as to what exactly you're being asked before you do the calculations.
All right, here is that same question with the correct response entered into the answer box. The answer to the nearest whole percent is 33%. Basically, the calculation involves figuring out what percent of 15 is 5. $5 is the profit. And the cost was $15. So the profit expressed as a percent of cost, 33%.
Now, this screen also shows you the calculator that's available to you in the Quantitative Reasoning measure. If you want to, you can fire up the calculator. Divide 5 by 15.
But note that it solves it as 0.33333, et cetera. So if you transfer that into the answer box, that is not the correct response. That's a wrong answer.
Because we're telling you to answer to the nearest whole percent, not as a decimal. So you have to be careful. You have to be careful. You want to make sure you're doing the task as it's being spelled out for you.
Some strategies for numeric entry questions. As I was saying at the end of last slide, make sure you answer the question that is being asked. If you're being asked to round your answer, make sure that you round to the required degree of accuracy. You have to get that right if you want to get the question scored as a correct response.
And by all means, examine your answer to see if it is reasonable with respect to the information given. In the previous question about the sweater, if you had come up with an answer like 110% or something like that or 130%, that would not seem especially reasonable with respect to the information given.
So give your answer a quick reality check to see if it makes sense. Maybe you got off track slightly. Even though you have the math skills to answer the question, maybe you just got off track somehow. So make sure that your answer is reasonable given what you're told.
Now, here is a numeric entry question with two answer boxes. So you'll be giving your answer as a fraction. Rectangle R has length 30 and width 10. And square S has length 5. What is the ratio of the perimeter of S to the perimeter of R? More specific instructions here tell you to give your answer as a fraction. There's one box for the numerator and another for the denominator.
Again, important information at the bottom of the screen in the gray box telling you to enter your answer as a fraction with the numerator and denominator in their respective answer boxes. Backspace to erase. So if you need to erase — you change your mind — you simply use the backspace key.
And here is the question with the correct answer indicated, 1 over 4. Notice this test taker has reduced the fraction down to 1 over 4. You don't have to do that unless the question tells you to do that. But here, the instructions are to give your answer as a fraction. So it's completely acceptable to enter 20 over 80, 20 being the length of the square.
Square S has length of 5. 5 times 4 is 20. 20 over 80 — 80 would be the perimeter of rectangle R. So 20 over 80 is a correct response. 2 over 8 would be a correct response. 1 over 4 is a correct response. So unless we tell you to reduce, you don't have to. So any of those would be considered a correct answer as long as you're giving your answer as a fraction.
I want to turn now to another question type, that being the data interpretation question. Data interpretation questions are grouped together. They come as a set — two, three, four questions all relating to the same table or graph or some kind of a data presentation. So you get a table or a graph and then two, three, maybe four questions based upon that data presentation.
These questions will ask you to interpret or to somehow analyze the given data. And the types of questions can be multiple choice, either single selection or more than one selection. They can also be numeric entry.
So here is a sample data interpretation question. The question itself is on the right-hand side of the screen. The data is on the left-hand side. So the data has to do with annual percentage change in dollar amount of sales at five retail stores from 2006 to 2008.
And you have stores P, Q, R, S, and T. And then you have a column showing the percentage change from 2006 to 2007 for each store and then from 2007 to 2008 in each store. So that's your data.
Again, in a test-taking experience, you'll probably encounter maybe two, three, four questions based on this data presentation. So that's not going to change from question to question. That remains the same.
Now, the question is over here on the right-hand side. And it has to do with, if the dollar amount of sales at store P was $800,000 for 2006, what was the dollar amount of sales at the store for 2008?
So a couple of things just to think about as you begin figuring this question out and arriving at a right answer, notice this question has to do with only one of the stores, store P. So there's no need to study the data presentation in great, great detail to answer this question. It has to do strictly with store P.
So when you start a new set of data interpretations, it's a good idea to glance at the data, get a sense of what it's about, what is going on, what's involved. But you don't have to study every particular detail down to the most minute level.
So if to answer this first question you only need to look at store P, then that's where the focus of your attention will be.
And here is the question with the correct answer indicated for you. So the dollar amount of sales at store P for 2008 would be $792,000.
All right, so how do we arrive at that answer? Well, we know that we're told that the amount of sales at store P was $800,000 in 2006. We know from the data from the table that there was a 10% increase from '06 to '07.
So we take $800,000 times 1.1 to account for that 10% increase. And that gets us to $880,000. Then we know there was a 10% decrease from 2007 to 2008. So we multiply $880,000 times 0.9 to account for that 10% decrease. And that gives us $792,000.
So that's the correct response. And notice, again, for this question, all we have done is grapple with the data regarding store P. Haven't paid much attention at all to stores Q, R, S, and T. Now, that doesn't mean there won't be questions that follow this one that have to do with those stores. There probably will be. But for any particular question, just deal with the material — with the data that you have to deal with to arrive at the correct response.
Some strategies for these data interpretation questions. Scan the data briefly as I've been saying. Scan it briefly to see what it's all about. But don't spend time studying all the information in great detail. Bar graphs and circle graphs as well as other graphical displays of data are drawn to scale. So you can read or estimate data visually from such graphs, if you have a bar graph or a circle graph for instance.
The question should be answered on the basis of the data presented and based on everyday facts, such as the number of days in a year and your basic knowledge of mathematics.
All right, now, I want to spend just a little bit of time talking about the on-screen calculator.
So there is an on-screen calculator in the computer-based GRE Quantitative Measure. It can be operated with a keyboard or a mouse. It has four arithmetic functions as well as square root, memory, and parentheses.
It has a Transfer Display button, which allows you to transfer a number into a numeric entry question box. But you have to be careful with that as in the question we looked at earlier in the presentation. The calculator respects order of operations. If you're a little fuzzy as to what order of operations is all about, you can read all about it in the Math Review. It has a nice discussion of order of operations.
Bear in mind that, most questions, you're not going to need the calculator. They don't require difficult computations. So don't spend precious time firing up the calculator if you really don't need it. Use the calculator only when you're dealing with very large numbers, with long, complicated divisions or multiplications, square roots, things like that, tasks where you really need the calculator. Otherwise, it's probably best not to bother.
OK, now, let's turn to some general strategies as they apply to the Quantitative Reasoning measure.
OK, some general strategies for the quantitative measure. By all means, you should become familiar with the different item formats and with the item directions and the directions to the measure as a whole before you take the test.
Especially looking through the directions that appear at the beginning of the test is very useful. There's a lot of important and very useful information in the directions that begin the Quantitative Reasoning measure. So become familiar with those so you're not surprised or caught off guard when you take the test.
You also want to read carefully each test question so that you don't overlook information or misread the question. You may have the math skills. But you just get off track because you misread something or overlook something about approximation or about rounding.
You don't want to answer something that is not being asked in other words. Be careful not to make unwarranted assumptions. For example, not all numbers are integers, nor are all numbers positive. And finally, you want to search for general mathematical relationships among the quantities in a question.
Continuing with general strategies, remember, the geometric figures are not necessarily drawn to scale. So avoid estimating sizes by sight or by measurement when you're dealing with geometric figures, like the triangle we looked at earlier on. If applicable, draw your own diagram or figure. Make a list to help sort out what the question is asking.
When appropriate, avoid lengthy calculations by rounding before computing an estimate by looking for comparisons and recognizing and continuing patterns. Don't do more work than you have to do in other words. And finally, some questions are most naturally answered by considering several cases of the situation that is described.
For some questions, one way to a solution is simply by guessing and then checking out your guess and then improving and refining on your guess to come to the correct response. You should also evaluate your progress and switch to a different problem-solving strategy if you're getting stuck or if you're spending an inordinate amount of time on a particular problem.
And finally, after arriving at an answer, you should reread the question and make sure that your answer is reasonable, given what was asked. Make sure it passes the reality test in other words.
Let's continue with some general problem-solving steps and strategies.
So your first step is simply to understand the problem. You need to make sure that you understand the information that is given and that you understand the problem you are being asked to solve.
Once you have that worked out, you can move on to step 2, which is to carry out a strategy for solving the problem. You do need a strategy, whether it's an explicit one or one that you just rely on intuitively to determine what mathematical facts to use and when and how to use those facts to develop a solution to a problem. And we'll talk more about specific strategies in just a second. And finally, step 3 is to check your answer for reasonableness. Make sure that it answers the question that was being asked.
So I just mentioned on the previous slide that you need a strategy for solving problems in the Quantitative Reasoning measure. And with that in mind, ETS test developers have identified 14 different strategies that you can use to solve problems in the Quantitative Reasoning measure.
This list of 14 strategies is not meant to be exhaustive or prescriptive. You don't have to use any particular strategy. And there are no doubt other strategies that could be used. They are very useful ways of thinking about and approaching quantitative reasoning problems to obtain a correct answer. Think of them as a useful set of tools, tools in a toolbox to employ as needed.
I'm not going to read them all word for word but just highlight a few of them. The first four, for instance, are all translation strategies.
They have to do with ways of rethinking the problem where one representation of a math problem is translated into another representation that allows you to solve the problem more efficiently — so translating from words to an arithmetic or algebraic representation, translating from words to a figure or diagram, translating from an algebraic to a graphical representation, translating from a figure to an arithmetic or algebraic representation.
Now, some of these are probably pretty intuitive. Some of these you are using whether you are aware of it or not — for instance, in that very first problem we looked at, the one about the car getting 33 miles per gallon with gas costing $2.95 per gallon. And we had to figure out what was the cost in dollars of gasoline to drive for 350 miles. Well, to solve that, we can use the first strategy, translating from words to an arithmetic representation in order to solve the problem.
I very much encourage you to go and look at the discussion of these strategies at the GRE website where they are discussed in detail. And each strategy is linked to actual test questions so you can see the strategy in action as it were, the strategy being applied to an actual problem.
Some general test preparation pointers. By all means, become aware of your strength and weaknesses. And here, again, I would go to the GRE Math Review, all of the four content areas — arithmetic, algebra, geometry, and data interpretation. See where your strengths are. See where your weaknesses are. And kind of start your preparation from there.
Maybe it's been a few years since you've had algebra. But with a little review, you can probably reawaken some of that knowledge. If you set some realistic goals and allow enough time, you can improve. You can improve. So set realistic goals. Allow enough time for some study, for some improvement to occur.
Other available resources — by all means, all of the freely available ETS material at the GRE website — sample tests available for your use as well as university faculty and study groups can be useful resources in your preparation for the quantitative measure.
Let's look at some specific GRE resources that are available to you.
So I've touched on all these resources earlier in the presentation. But here, they are collected for you all in one place.
GRE resources — first off, the Quantitative Reasoning overview on the GRE website in which we present detailed information on the question types in the measure along with sample questions and explanations, discussions of how the question works and why the right answer is the right answer, as well as tips for answering and general problem-solving strategies. Also, the Math Review, a very important resource, and the Math Conventions if you want to take a look at that.
Here is information and a link to the official GRE website, which I've been discussing. The website has a lot of very useful, very comprehensive information and explanations of the test, what the test is all about. It also includes policy information, test dates and locations, and other important information along those lines. The GRE website is a great resource.
This is the end of the presentation. I hope you found it useful. I wish you the very best of luck in all of your future educational endeavors.