the ranges of “good scores” vary a lot depending on both your goals and your age. the more competitive your dream schools are, the higher your target score should be. on the other hand, you can expect your scores to increase as you go through high school. for that reason, if you’re a sophomore, 1300 is a good score, while a freshman should be very pleased with scores of 1200 or higher.

want to find out exactly what a good score would be for you and your goals? read on for more information about sat scores—from average sat scores to sat score charts, we have all the data you need.

 

table of contents

• what is a good sat score range for colleges?
• sat score scale: the basics
• what are sat percentiles?
• what is a good score for scholarships?
• how does the adversity index impact my score?
• how can i improve my scores?

what is a good sat score range for colleges?

let’s face it: at the end of the day, there is no “sat passing score.” what is considered a good score depends almost entirely on the colleges that you’re applied to.

keeping in mind the general sat score range to aim for, let’s take a closer look at good scores for your dream school. just to make things a little easier on you, we’ve put together this table of score ranges for the top universities in the united states. the numbers are from the middle 50% score range (meaning 25% of admitted students had lower scores and 25% had higher scores).

type the name of your chosen school in the search box to find its middle 50% sat score range!

sat score scale: the basics

before you send your score report to admissions officers, make sure you understand the sat scoring scale. what is the sat out of? what’s the top sat score? knowing these things can help you figure out what a good score is for you and what your goals should be.

basically, sat test scores are given both by section and overall. here’s a quick breakdown of the sectional sat score scale and how it contributes to the composite score. note that this “new” sat scoring applies to tests from 2016 onwards—scores were different on previous versions of the sat, with a top sat score of 2400!

• you’ll receive two sectional scores, one math and one verbal (combined from the reading and writing sections).
  • math score range: 200-800 points
  • evidence-based reading & writing score range: 200-800 points
  • the average sat score on each section is 500 points, give or take 30 points (in 2020, the average ebrw score was 528 and the average math score was 529).
     
• your math and reading/writing sectional scores add up to a composite (combined) score. the highest composite score you can earn on the sat is 1600 points.
  • composite score range: 400-1600 points
  • the average composite score is about 1000 points (the average overall composite score in 2020 was 1051).
     

what are sat percentiles?

another way of evaluating your scores is to look at percentiles. your sat percentile measures the percentage of test-takers who got a lower score than you did. this means that percentile numbers, much like score numbers, indicate better test performance when they’re higher.

here’s how to find your score percentiles on your online score report.
 

basically, sat percentiles compare your scores to average scores. because the sat is a standardized test, it means that these scores are easily comparable. this information falls along a bell curve:

if you’re in the 91st percentile, for example, it means that only 9% of all other scores are higher than yours. in contrast, in the 30th percentile, 30% of all other test takers would have lower scores than you… and roughly 70% of the scores were higher than yours. this would place you in the bottom half of the year’s test-takers, in terms of performance.

sat percentiles are calculated annually. click the arrows below to see the college board’s most recently released sat user percentiles (meaning only juniors and seniors).

what is a good score for scholarships?

many colleges around the country have what are called guaranteed scholarships. these scholarships are automatically awarded to accepted students who have earned a certain sat score.

a larger number of colleges also have general merit scholarships. these scholarships have the same sat requirements, but you are in competition with other accepted students for a limited number of awards. these scholarships may require a separate application, along with a personal or themed essay.

scholarships based on academic merit often have minimum scores provided in their descriptions. take note of any score requirements you find during your research, then average all those scores. the result is your minimum score goal for scholarships. to see the types of scholarships out there, check out our article what’s a good sat score for scholarships?


 

how does the adversity index impact my score?

if you’ve been paying attention to sat news lately, you may have heard about college board’s new adversity index. this is a measurement that they will give to colleges to contextualize your scores in terms of relative advantage/disadvantage.

by creating this new measure, the college board hopes to show how students from low-income and minority populations perform compared to other students from similar backgrounds. while this has been controversial, the adversity index will not affect scores themselves.

instead, what it will give schools is an understanding of your percentiles based on both your “environmental context” and your “high school context.” again, this won’t change your score at all, but instead give universities one more measure with which to interpret your scores.


 

how can i improve my scores?

check out this video for tips to boost your score, and read on for resources to help you before and during test day! you can also use these tips if you’re planning on retaking the sat.

• make use of a sat study schedule—knowing how to study for the sat is a huge factor in improving your score!
• take a free, full-length sat practice test to establish a baseline score.
• review our tips for pacing yourself during an exam.
• find ways to remain calm when a question flusters you.
• learn how to avoid careless mistakes in the future. (magoosh’s sat study guide has great tips to help with this!)

a final note

“how do my sat scores stack up?” it’s a question that almost every test taker has asked at some point! by taking a look at the score ranges for the colleges you’re applying to, comparing your scores to the national average scores with percentile rankings, and working your test prep to keep improving your score when necessary, you’ll have everything you need to understand how your score will impact your college admissions!

still unsure whether you want to take the sat or act? wondering how your scores stack up to potential act scores? check out magoosh’s sat to act conversion!

all magoosh sat prep plans are digital sat ready! start studying today.

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free 2022世界杯入门名单 with answers and explanations

magoosh’s full-length act sample test has been perfected over the years by magoosh’s act experts, who have combined decades of experience teaching the act exam. practice tests are an essential part of act prep, so this is a great place to start.

to access our practice test pdf (featuring all act sections, including the optional essay, as well as the answers and explanations at the end), just enter your email:

examples by section from the free 2022世界杯入门名单

sample act english practice question:

this question comes from page 3 of the magoosh pdf.

sample act math questions:

the two questions above come from page 15 of the magoosh pdf.

sample act math questions:

the two questions above come from page 15 of the magoosh pdf.

sample act reading question:

for the full passage associated with this question, see pages 24 and 25 of the magoosh pdf.

sample act science question:

the table and question above come from pages 38 and 39 of the magoosh pdf.

where to find additional act practice exams and practice content

so you should be taking a practice test a week—but what happens if you’re studying for more than seven weeks? there are tons of other places you can look to find great sample tests and other content! for example:

free official act practice exams

once you’ve done the magoosh test, i strongly recommend doing some additional practice acts, starting with the official 2022世界杯入门名单s. these tests are the real deal because they were used in previous act administrations. there is one currently available on the act website: the 2022-2023 practice test (pdf).

every year, the act features a different test on their website, but they cycle through the same tests: the above test and additional practice tests that were also used in a real act administration.

within the pdf, you’ll find the answer key for that test at the very end. that will help you figure out your act scores: the composite score (overall score), as well as your sectional scores.

additional resources from magoosh

magoosh act is another great way to get test-like practice with full-length practice tests before the official exam. the 1300+ questions mean that you can take 4 full-length practice tests—and still have questions for quizzes and practice. you can also choose between a live cohorted class with an instructor (which includes all our lessons and practice questions) or access to the self-study option by itself.

you can also get an extra test and lots of other practice by purchasing the magoosh act book.

official prep guides from the act test maker

the official act prep guide 2021-2022 offers six practice tests. you can find other tests in the official act prep guide 2016-2017, or real act prep guide, 3rd edition.

be warned—many of the practice tests in those books are exactly the same as the online ones linked above. the official act materials also do not contain the in-depth text and video explanations that magoosh offers for its own questions, but they are still excellent sources of practice.

how to use this practice test

once you’re ready to take an 2022世界杯入门名单, where do you start? here are our best act tips for the method you should use for the best results!

• begin with a diagnostic exam. this is so crucial to understanding the exam you’ll be taking, getting perspective on your strengths and weaknesses, and figuring out where your score is compared to where you want it to be (though don’t try to use it to make any score predictions—yet!).
• set your act score goal.
• spread the remaining tests out over your study schedule and work hard to reach that score goal. depending on how long you’re studying, you should be taking at least an exam every month—but every week or every two weeks is better.
  • if possible, take your tests on saturday mornings, because that’s when act test dates are.
  • try to emulate test day conditions as much as possible. take the test respecting time limits, without interruptions.
  • find a quiet spot where you can practice.
  • use an authorized calculator, no. 2 pencils, and bubble sheet for answers
• use an error log to review your answers. this can be as simple as tracking your mistakes in a notebook. by understanding and tracking your errors, you’ll be better prepared to face those question types and concepts the next time you see them!

a note about finding high-quality free resources

if you’ve finished this 2022世界杯入门名单 with answers and explanations and you’re looking for more free act practice resources, you can check out our post on the best online act prep. as you continue your search for as many practice tests as you need, the most important thing to do before spending hours taking a practice test you find on the internet is to make sure they come from a reputable source.

there are a lot of terrible act prep resources out there—most of them are free, but a good number also cost money. read reviews from external resources and for the resources that cost money, take free trials if you can to see if they match the quality of the resources featured in this post. you may be tempted to use the free ones, thinking all prep must be good prep, but it’ll help you so much more on test day if you practice with resources that match the act as closely as possible.

if money is an issue, there are also ways to get the above resources for free—you can check your library for the act prep books or see if your school can purchase magoosh accounts on behalf of their students. finally, you can also find additional resources on the magoosh blog. for example, this act math practice!

we also have a free sat practice test you can try as well. you can also do an act to sat score conversion if you want to take both and see which you do better on!

however you decide to get your practice in, we hope this blog has been helpful for you!

the post free 2022世界杯入门名单s (pdf) with bonus answers and explanations appeared first on magoosh blog | high school.

in this post, we’ll give you up-to-date information on both international and u.s. sat test dates and score release dates for the 2022-2023 school year.

2022-2023 sat test dates* (u.s.)

*if you mail in your late registration (rather than register online or by phone), the deadline is a week earlier.

2022-2023 sat test dates (international)

*college board announced that digital sat scores will be released in a matter of days, not weeks, so these scores will likely come out sooner.

2022 psat test dates

these are the test dates for the psat/nmsqt and psat 10 test in 2022:

psat event	date
regular test day	wednesday, october 12, 2022
saturday test day	saturday, october 15, 2022
alternate test day	tuesday, october 25, 2022
online scores available	dec 5-6 (date varies by state)

faqs

 
whew! great job for sticking with us! if you have more questions about sat test dates, let us know in the comments. and if you’re all fired up and ready to start prepping for your best sat test date, well, we can help with our self-paced, online magoosh sat prep.

all magoosh sat prep plans are digital sat ready! start studying today.

the post sat test dates: your best test date (2022-2023) appeared first on magoosh blog | high school.

 
so what will you see in the pdf?

well, 154 questions, to start. but beyond that, the test has…

• sat reading test (65 minutes, 52 questions)
• sat writing and language test (35 minutes, 44 questions)
• sat math section – no calculator test (25 minutes, 20 questions)
• sat math section – calculator test (55 minutes, 38 questions)
• an answer key
• information on grading your test
• links to text and video explanations for every single question on ech section of the test

where to find full-length sat practice tests (even official ones)

there are a few great places to look for full-length sat practice tests! here are our top recommendations:

• khan academy: khan academy has paired up with the college board (they’re the test creators) to offer free online practice sat tests. there are four full exams on the khan academy site, which can also be found on the college board site. after you’ve taken an exam or two, you can then sharpen your skills with practice in different areas with resources on khan academy’s site and elsewhere.
• prep books: but beware! while prep books can be great for lessons, they tend to be better for learning than for full-length practice tests. yes, the college board’s book is awesome, but guess what? those eight tests are the same eight tests you can find on their website, just printed and bound (they’re transparent about this). there are a few great books out there and a few to avoid—you can check them out in magoosh’s post on the best sat books and our review of the princeton review’s practice tests!.
• magoosh prep: you can choose between a live cohorted class with an instructor (which includes all our lessons and practice questions) or access to the self-study option by itself.

why do i need to take a full-length practice test?

to maximize your score on the sat, you’ll need three things:

• lessons
• practice questions
• practice tests

and despite what a lot of students believe, those last two aren’t interchangeable! lessons are super valuable for reviewing content that you might not have seen for a while—or ever. practice questions are great for making sure you’ve mastered (and continue to remember) the lesson content.

learn it, practice it…why the third step? well, first of all, the official sat won’t have an “algebra” problem set or a “geometry” problem set. it’ll have all kinds of question types mixed together within the three sections. that means that studying different content areas is a very different experience from test day, when you’ll be in front of the exam for at least three hours (more if you’re taking the essay). so you need to get ready with multiple practice tests!

how to take an sat practice test

ready to mimic the official experience? fantastic! all students can benefit from taking an sat practice test–unless your test is the next morning, in which case, get your rest! for the rest of you, here’s what to do:

1. set aside approximately four hours of uninterrupted time to take the practice test.
2. take the test in a quiet place where you won’t be distracted.
3. mimic test day conditions by turning off your phone and leaving it in another room.
4. take the test in a quiet place where you won’t be distracted.
5. try to take the entire practice test in one sitting.
6. eat a healthy, energizing snack before taking the practice test.
7. give yourself a brief, 10-minute break after the reading test.
8. give yourself a brief, five-minute break after the math (no calculator) test.
9. use a countdown timer and remember to reset it for each test.

lucky for you, you don’t have to remember half of these steps if you follow our simulated sat practice test below, proctored by magoosh curriculum manager kat.

all you’d need to do is print your test, find a quiet place, set aside a few hours, and press play to simulate a test-day experience in the comfort of your own home!

what to do after practice tests

after the test, it is critical that you check your answers and make note of any questions you missed. in fact, it’s a good idea to spend at least as much time examining your results as you did taking the test. why? well, did you get a question right because you knew the answer, or because you were guessing? did you get a question wrong because you filled in the wrong bubble? (practice tests help a lot with this latter problem, by the way!)

make sure to check the explanations for every question you get wrong, so that you don’t repeat the same mistakes on test day. after you’ve examined your results, do some practice in your weak areas.

signing up for magoosh sat prep is a great way to learn or relearn concepts you need help with and get access to additional practice problems.

after working on your weak areas and brushing up on your strong areas, take another test and do the same process all over again! in the next section, we highlight some resources that will help you structure your sat prep after taking your first practice test.

free practice materials

while free practice resources for the sat may not be as easy to find once you’ve finished the work available here and on the college board site, don’t worry! there are plenty of other online resources–both free and low-cost (including this blog!)–that will allow you to target your weaker areas and keep your strong areas strong.

• magoosh sat ebook
• complete guide to sat scoring
• sat diagnostic test
• sat question of the day
• sat flashcards (prep and vocab)
• magoosh sat 7-day free trial

a final word

you’ve made it this far. congratulations! the sat can have many implications for the college admissions process, which can intimidate students. getting this far shows commitment—the same kind of commitment you need to master the sat.

so what are you waiting for? dig out those #2 pencils, find yourself a quiet corner, and get on it! it’s time to start boosting your test score.

the post free sat practice test with answers and explanations (pdf) appeared first on magoosh blog | high school.

*list prices given at time of publication and subject to change. click on the book cover found in each review to see the current price on amazon.

best sat books of 2021-2022: reviews

not all sat prep books are created equal: some you’ll want to use in their entirety (though only a few), others you’ll want to use only parts of, and others you’ll want to steer clear of. this is particularly true now that prices are falling and kindle versions are available—in some ways, that’s awesome, but it can also encourage students to buy a ton of books without seriously evaluating their quality.

to help you maximize your study time (and save your hard-earned cash) i’ve reviewed the options for the best sat books on the market and graded them from most to least helpful. keep in mind that some publishers do a good job on some things (like practice tests or a specific section of the sat) and a sub-par job on other things. this doesn’t necessarily mean that you shouldn’t buy the book, but it might mean that you won’t use the book in its entirety.

magoosh’s complete, free sat study guide

i want to start by pointing out that magoosh has a great, free sat study guide! we’ve spent years and years reviewing all of the sat study guides out there. we knew that our experts could cut through all of the unnecessary stuff and give you the best info out there…so that’s what we did! even better? it doesn’t cost you a cent. we even have a free sat practice test you can take after studying!

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sat prep black book: the most effective sat strategies ever published

how the current edition compares to the previous one
this book hasn’t been updated for a while (the most recent version was published in 2017), so steer clear of the first edition: it’s written for a previous edition of the test and won’t help you prepare for the current sat!

things to love

• this book provides in-depth explanations (and we mean really in-depth explanations) of the four official sat practice tests from the college board. the college board’s practice is great—but if you’re struggling to understand why you got questions wrong even after reading their explanations, this is a great place to turn.
• it’s also a great place to get your strategy set for test day. not only does mike barrett give you multiple workable methods to approach questions, but he also goes through possible issues and solutions to every type of everything: passages, questions, answers, even test-taking vocabulary.
• the book even covers what non-native english speakers should focus on (albeit in a slightly perfunctory way).

things to consider
is this the be-all and end-all of sat prep? well, no.

• in the first place, we definitely have some pedagogical (teaching) differences. barrett encourages you not to think about the average time you spend per question, and while we don’t think you should obsess over this, you should definitely be aware of it.
• he also doesn’t encourage taking notes while you read or taking a diagnostic test. those strategies can be helpful for many test-takers.
• in addition, barrett vastly underplays the importance of the sat essay, noting that a lot of test-takers won’t need to take it. while that may be true of some test-takers, more and more schools are requiring the sat essay—particularly the most competitive schools that students aiming for a 1600 are probably targeting.
• finally, you won’t find any practice questions here. barrett has his own arguments for why you should only use college board resources. we definitely disagree with that; for students aiming for top scores, a much deeper pool of questions is important (as long as they’re up to snuff)!

takeaway
at the end of the day, this is a pretty great addition to your sat library, particularly for those students just starting their test prep journey.

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kallis’ sat pattern strategy

how the current edition compares to the previous one
the 2018 kallis book and the previous 2016 edition are similar in many respects, but there have been a few distinct changes. most of these changes are good. the sat reading and sat math practice questions have been updated to better reflect the content of the real sat, with more recently written passages and a more even mix of math topics, respectively. there are a few changes that seem like downgrades though, with less tutorial support for certain topics in sat writing & language and sat math.

things to love

• the content review for each section is thorough, clear, engaging, and most importantly, helpful. of course, many publishers do content review fairly well.
• where most books flounder is in writing the practice tests, but kallis is relatively strong on that front. kallis isn’t perfect—it’s exceedingly difficult to write questions that mimic the real test—but they do a respectable job. and in the newest edition, the quality is more respectable than ever. the sat math practice is much improved–there no more long strings of problems that just focus on one topic, and kallis has added more complex word problems that resemble the college board ones more closely. the sat reading practice also now includes more passages that were written in the last 20 years, compared to the 2016 volume’s disproportionate use of late 1800s/early 1900s readings.
• the amount of practice content is great too: 6 full practice tests!

things to consider

• the writing questions tend to be more difficult than those on the actual test and don’t quite have the feel of those questions.
• the reading passages are sometimes spot-on. other times, they are too stylistic and more reminiscent of the old sat. there’s also slightly more of a focus on trap answers than on the actual test.
• the book does a great job in presenting the content you need to know for the new test. however, it doesn’t do the greatest job of discussing strategies. for instance, it shows you the tedious and not necessarily intuitive way of setting up complex equations rather than the tried-and-true way of plugging in values. likewise, the writing section recommends always reading the question first instead of reading the passage first, or at least a paragraph at a time. i advocate for a nuanced approach, one that takes into account both tactics.
• even with the addition of some more sat-like word problems, most of the word problems in the math section still tend to contain far fewer words than those on the actual exam. the contexts used in the examples are often a little too relatable. gone is the more esoteric fare of the actual test—bacteria in a petri dish, strength of a satellite signal, etc.
• frustratingly, support and advice for certain key writing & language and math topics that were in the 2016 version of kallis’ sat pattern strategy have been removed from the 2018 book. there is noticeably less support and advice for stats and functions, and advice on certain writing features such as passive voice or reporting verbs have been removed altogether.

takeaway
an all-around excellent prep book for the sat, one that is great for the motivated self-studier, especially one who aims to score in the top 20%. i intend to use it to tutor, which is a lofty compliment, given that i say this about very few prep books outside of the official materials.

(quick update: i ended up using this book for my class, and it turns out that quite a few of the writing questions have debatable answer choices or at least wrong answers that aren’t quite wrong enough. for that reason, i’ve lowered the grade from an a- to a b+.)

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ivy global: ultimate sat guide and sat 6 practice tests

ivy global is a lesser-known publisher that produces high-quality practice materials. they have two main books, the ultimate sat guide and sat 6 practice tests, and offer two free practice tests on their website (available to anyone whether you bought a book or not). the ultimate sat guide offers a thorough concept review for each section and three full practice tests. the 6 practice tests book gives a quick run-down of the sections and question types and offers 6 practice tests. each practice test is different from the others from ivy global, so you don’t have to worry about paying for the same material twice. the ultimate sat guide is a more refined version of ivy global’s new sat guide 2.0 but the test material is nearly the same. the only difference is some of the concept reviews and errors from the previous edition were fixed and improved slightly.

things to love

• in both books, the text and layout design are clear and understated. they don’t go for the whimsical or frenetic styles that other test prep books have, which can definitely be a plus if you want something a little more no-nonsense
• in the main book, the new sat guide, the introduction to the sat and test-taking strategies sections are clear and well-formatted.
• the section-specific explanations cover the necessary material and use lots of examples. not enough to teach you a concept if you’ve never seen it before, but definitely enough to help you refresh your memory and fill in any gaps.
• their practice tests are strong. the questions match the difficulty of the real thing, and there’s a good range of difficulty in each section.

things to consider

• the one small weakness is that there are no answer explanations in the back of the book, just the correct answer choices. answer explanations are available online, which isn’t too big of a deal, but it does add an extra step.
• this book might have jumped into the a range if it had more extensive review of strategies and made it clear how they are most advantageous for different topics and situations. similar to the kallis book, this is a great resource, but probably shouldn’t be your only test prep resource.

takeaway
great general-purpose practice source that deserves more attention!

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princeton review: sat premium prep, 2022 (formerly cracking the sat, 2020)

how the current edition compares to the previous one
the 2022 sat premium prep is the new evolution of the cracking the sat series. although it is presented as an update, the 2022 edition matches the 2020 edition aside from a few minor edits. previous reviewers have noted that the 2020 edition is also a near reprint of older editions from 2018/2019 so you may want to consider buying any version since they’re essentially the same over the last 3 years. the benefit of the premium version is that it gives you eight practice tests (four in the book and four online) whereas the conventional version gives you five practice tests (four in the book and one online). the rest of the content is the same.

things to love

• hey, want to learn a lot about sat math? this book has almost 300 pages of material to review, strategies to learn, and practice sets to, well, practice. if you want a total math review with lots of practice, this is a great place to start.
• the verbal sections are really strong in how they introduce the test format, then walk you through progressively more difficult (and sat-like) examples.
• unlike a lot of other books, the verbal here is pretty solid as well. in particular, the princeton review seems to actually understand the college board’s incorporation of graphs into the verbal section on the new sat—a nuanced concept that few publishers get right.
• also, the practice tests here are different than those included in the princeton review’s 10 practice tests for the sat, so if you’ve bought both books, you’ll really get your money’s worth.

things to consider

• what’s that you say? seems overwhelming? it is, a little.
• if you’re looking for practice on reading or writing and language, the book does offer good advice and practice sets—just not very much advice and not many practice sets (think 50-75 pages).
• yes, students do struggle with sat math, but there’s no need to push it on them to the exclusion of the other sections. after all, you need to look at that composite score, too!
• meanwhile, other parts are unnecessarily confusing, mainly because they use a ton of acronyms (including the unfortunate “pood”).

takeaway
a great place to overhaul your math scores if you have a lot of time before the exam. also worth looking into for the use of graphs in the verbal sections.

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pwn the sat: math guide

how the current edition compares to the previous one
the most recent edition of this book is from 2016, when the sat changed—so definitely don’t get earlier editions! they won’t prep you for the current test.

things to love

• many test prep books make the mistake of being dry—really dry. and that’s a problem when your subject matter is already dry to begin with. mike mcclenathan steers clear of any such aridity, injecting a “hey-we’re-in-this-together” voice as he takes you through all the important parts of the sat.
• you never feel like he’s writing this book to meet some deadline. it’s clearly a labor of love. his affection for the material and the test comes across in nearly every page. but it isn’t that he just wants to geek out on the material—he genuinely seems to care that students improve their score.
• luckily, the book gives us a clean breakdown of every question type in the official guide. that way, you can practice a concept on real sat questions and have a rough idea of how difficult they are (the difficulty of these questions depends on where the question shows up in the section—easier questions are at the beginning; harder questions at the end).
• overall, a strong book for the self-studier who needs an accessible refresher of the math tested on the sat.

things to consider

• this book alone isn’t enough to improve your score—though it is a good start. you’ll want to make sure to complement it with the official guide (as the author encourages). even then, i recommend a book like barron’s 1600 math book so you can get lots of extra practice questions.
• the questions in the pwn book, while okay, don’t quite impart the flavor of the current test. it seems that some of them are still steeped in the old-sat-style of asking questions.
• another issue is there is no indication of whether a question in a practice set is ‘easy,’ ‘medium,’ or ‘difficult.’ this is problematic because, often, the section that introduces the topic uses clear, easy-to-follow examples. then there’s a jump in difficulty in the problem sets, often because what you learned at the beginning of the chapter isn’t enough to answer the harder questions. for students who are just starting out, they might get easily frustrated and think, as they are wont to do, that they aren’t good at math. the problem sets would have been improved had they included easier questions and broken up the questions into discrete difficulty levels.

takeaway
on its own, this book isn’t enough to prepare you for test day. in combination with a few other resources, though (namely, the official guide), it’s a great resource for students aiming for top math scores.

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500+ practice questions for the new sat

how the current edition compares to the previous one
this book hasn’t been updated since 2015, but it does focus on the “new” (i.e. 2016-onwards) sat. so don’t get earlier editions, but this one is geared towards the current test.

things to love

• the princeton review has created questions that, while perfectly legitimate, aren’t quite as complex and nuanced as those found on the actual test. and you know what? that’s not necessarily a bad thing—if you are just starting off.
• unlike kaplan’s books, for instance, which—at least for verbal—are much easier than the real test in an inaccurate way, the princeton review mostly stays true to the underlying subtleties of the questions and answer choices. it just doesn’t have the hard-level questions that make up 15-20% of the actual test.

things to consider

• you’ll have to pick up another book to help you with strategies and to review the fundamentals.
• you’ll need a book that also has practice tests, since this book is made up of just one drill after another.

takeaway
as a companion guide to a book of strategies and fundamentals, this book is a great place to start.

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10 practice tests for the sat, 2022 edition

note: the practice tests in this book are not the same ones that appear in the princeton review’s cracking the sat (reviewed above). all commentary here applies to the 10 practice tests book alone.

how the current edition compares to the previous one
there are no changes from the 2020 edition (which had four new practice tests added and some older passages updated) to the 2022 edition. tests 1-6 are the exact same six tests seen in the 2017 book. the two new passages were written in 2008 and 2014, while the older essay prompt readings were from the early-to-mid 20th century.

things to love

• the verbal sections here are pretty strong, and there are also great medium- and low-level math items. the science passages in the reading section are outstanding, and so is the princeton review’s use of graphic stimuli in these sections. these types of problems are going to be a new kind of challenge for many students on test day, and here we have difficult, test-like problems! woohoo!
• answer explanations are also pretty good. the explanations of wrong answers are brief or even missing, but the clear explanations of why the right choice is right make this a relatively small issue.
• the upgrades from the previous edition are nice. 10 practice questions are literally greater than 6! and the two modernized essay prompts a minor but nice improvement.

things to consider

• the math here verges on the easy side, but not so much so that it’s unfaithful to the test. it’s just missing those tough problems that you’ll need to answer correctly for a 700+ sectional score. if you’re looking to get your score up in the top percentiles, you’ll want to supplement the tests with practice problems from books like the college board’s official guide.

takeaway
a good book for most students. if you’re hoping to score in the stratosphere (90th percentile or above), you might begin here, depending on where you are now, and move on to more challenging materials as you master these problems.

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barron’s sat, 31st edition

how the current edition compares to the previous one
the 2021 edition has been updated with few new changes compared to the 2020 edition. the most major change is the removal of all guides related to the sat essay that has been recently removed from the test. the second update that might prove more meaningful is the addition of an index at the end of the book. the index allows you to go straight to the page for nearly every concept on the test which is helpful if you’re looking for targeted practice. aside from the essay removal and index addition, the book is nearly identical to the previous edition.

things to love

• the math is great! the strategies are solid and helpful, the concepts are broken down well, and it hits that goldilocks sweet spot of around 200 pages—enough to help you enormously without being overwhelming.
• answer explanations are great on the practice sets (though less so on the practice tests…see below).
• there’s a flashcard app! this is exciting.
• the index makes it easier to find the content you want to review.

things to consider

• how long do you think it would take you to memorize 74 pages of word definitions in teensy-tiny type? is there enough time before test day? if not, and you’re looking for verbal help, you’d better look elsewhere. other than those 74 pages, there are only about 60 pages of reading test help here, and ditto for writing and language. furthermore, writing and language is full of lists that made even my eyes glaze over. how helpful is a list of conjugated irregular verbs when you’re studying for the sat? well…sorry, i just fell asleep there for a moment. those lists are not helpful at all.
• practice test answer explanations are really weak (“a is the right answer because xyz. b is not the right answer because it is not xyz”).
• yes, there’s a flashcard app…but is your time going to be best spent studying the 3,000 words the book claims are on that app? definitely not.

takeaway
this is great for math practice and really, really not student-friendly for the verbal.

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the official sat study guide 2020

how the current edition compares to the previous one
there are a few small, inconsequential changes to the math support in the official sat study guide (2020), compared to the previous 2018 version. a few example problems have been changed, with one algebra equation replaced by another, a geometry question replaced with a different geometry question on a similar topic, and so on. again, very minor changes, likely due to feedback or analysis that the college board received regarding the prior version of the book. the truly big change here is the eight practice tests: two of the practice exams are brand new, while the other 6 also appeared in the 2018 sat og.

things to love

• the questions in this book are mt. sinai level. the sat gods part the skies and give us a taste of what to expect when the test debuts in a couple of months. all the nuances, all the traps, all the idiosyncrasies are there for us to behold, and the more you understand them, the better prepared you’ll be come test day

things to consider

• the practice questions in this book are available for free online, and, since the questions are by far the best thing about this book, you might rightly start to wonder whether you need to actually purchase this book. what follows might indeed make you think you don’t need the book at all.
• the explanations are really weak. how weak? well….hi, i’m going to play pretend today. i’m going to pretend i’m the person or people who wrote the explanations for the questions.choice (a) is the best answer because the information in the passage best supports (a). lines 11-13, <insert quote here> show this.(b), (c), and (d) do not specifically answer the question.to be fair, the explanation of the right answer is often more thorough. as for why the wrong answer is wrong, forget about it. that is about as specific as it gets. in the end, you are likely to find these explanations laconic to the point of infuriation. this is problematic, since understanding your mistakes is one of the best ways to improve.as a result, i’ve put together video explanations to the official sat study guide questions. you can check them out for free in our magoosh sat youtube channel.
• the strategy tutorials leave a lot to be desired. admittedly, i like the way the book breaks down the test so and covers all its components and aspects. however, there is so much terminology that i imagine students getting bogged down thinking they have to know what nonrestrictive and parenthetical elements are, or that these needlessly complicated terms are known as “conventions of punctuation.”as indicated in my example above, the strategy support for sat writing & language is particularly frustrating. what we don’t get is a solid explanation of grammar concepts and how they relate to the test. nor do we get strategies on how to approach these questions. in fact, i feel like the first few hundred pages are more for people like me—people who want to understand how the test is constructed—and less for students, who need help understanding how to solve the actual questions.

takeaway
in sum, you don’t need to buy this book. for review, you are much better off going with any of the major publishers on the market. for practice questions, there is no better source than the college board, but the fact that they have made this content available for free online makes this book unnecessary, if not unhelpful. now let’s hope the clouds don’t part and the college board strikes me down with lightning. (my alternative grade for this book? ‘r’ for redundant.)

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using the best sat books in your studies

there you have it, the best sat books to use in your prep. for more information on how to put the best sat books to use in a complete study plan, check out our study schedules for the sat!

questions, comments, concerns? leave me a comment, and let’s get this discussion on the best sat books started!

the post best sat books 2021-2022 | book reviews appeared first on magoosh blog | high school.

the sat essay—which will be discontinued for most students taking the sat starting in june 2021—requires you to read a complex, opinion-driven essay and write an essay that discusses how the writer goes about trying to persuade his or her audience. while the passages are different, the directions are always the same:

“consider how the [insert author’s name here] uses

• evidence, such as facts or examples, to support claims.
• reasoning to develop ideas and to connect claims and evidence.
• stylistic or persuasive elements, such as word choice or appeals to emotion, to add power to the ideas expressed.”

if you live in a state that requires the essay as a part of their sat school day administration, read on for the top tips on making sure that you ace your sat essay.

sat essay tips

1. know the fundamentals of rhetoric
2. understand how the essay is scored
3. practice with sample prompts and essays
4. get skilled in annotating and outlining
5. vary your sentence structure and vocabulary
6. work on understanding the analysis
7. read with a critical eye

1. know the fundamentals of rhetoric

it’s a good skill to know how writers go about persuading us. indeed, the ancient greeks developed a glossary of terms to describe the way a speaker or writer aims to persuade his or her audience:

rhetor

this is a fancy way of referring to the speaker/writer, the person trying to argue a point. those he or she aims to persuade is the audience. for the sat, the writer of the article is the rhetor; the audience is made up of those who originally read the work. you, the sat reader, however, are not the audience. instead, you should think of yourself as a referee or judge. your job is to describe how the rhetor is trying to persuade his or her audience.

to understand this, the next few terms are essential and tie back to the directions listed above:

pathos

compare the following two sentences:

closing the school down will exert a negative effect on the community at large.

by closing down the school, administrators will displace hundreds of young children who have only just begun to forge friendships; additionally many local residents employed by the school might be forced to move from the area.

both sentences are saying the same thing. but the first sentence likely leaves you feeling cold; the language is vague and technical. the second, by contrast, tugs at your heartstrings (the poor children!). were the second sentence written on a petition to save the school, you’d be far more likely to sign it than the first sentence, i’m guessing. and that’s the point of pathos: it hopes to persuade us by appealing to our emotions.

ethos

it is possible to make the sentence with the school even more persuasive without appealing even more to our emotions. how? well, compare the following:

according to the united states department of education, closing down the school will displace hundreds of young children who have only just begun to forge friendships; additionally, many local residents employed by the school might be forced to move from the area.

all i did was attribute—or credit—the idea to an entity. but not just any entity. i appealed to the highest educational authority in the land. after all, if i put “i think”, you might wonder, who the heck i am. but by putting the united states department of education, i’ve invoked the highest authority in the land in matters of education. ethos refers to the credibility of the speaker.

on the sat essay, ethos will often take the form of “a study released by harvard medical school”. that is, the writer will quote where he or she is getting the information from. and it will never be their neighbor or that one lady they talked to on the bus. writers will always quote leading authorities to give their claims greater authority. that way, their audience is more likely to be persuaded.

logos

you might be thinking that the kids should just be able to go to another school. and surely there are more jobs in the area. those are valid objections and that’s why writing doesn’t just aim to persuade us at an emotional level (pathos) but also at an intellectual or logical level (logos). how does the following use logos to build on the pathos?

happy hills private school is a one-of-a-kind institution for gifted children recruited from all over the country. for many decades it has grown to such a degree that a large community has sprung up consisting of many who depend on the school for their livelihood. if the school shuts down, these educators, administrators, and custodians will have to move elsewhere and many local businesses, which depend on their patronage, will be forced to close. additionally, by closing down the school, administrators will displace hundreds of young children who have forged deep friendships

we now have the necessary context to understand the logic behind the idea that a closure of a school means a serious disruption in the lives of students and for the community that depends on the school.

logos, or logical statements, can often be identified by “if…then” statements. notice the bolded part above. the second bolded part (“by closing…friendships”) also has a similar structure: if you close the school, this will happen (“by closing down the school, etc.”)

all writing that you’ll see will use a combination of ethos, pathos, and logos. sometimes in the same sentence:

according to the education department’s report, if the school is closed down, hundreds of students will be torn from a nurturing environment and cast into alien—and possibly hostile—environments.

(okay, maybe i got a little carried away with the pathos there!)

2. understand how the essay is scored

knowing how the sat essay is scored is fundamental for understanding what the graders are looking for. the sat essay contains three scores, one for reading, one for analysis, and one for writing. two graders will score the essay and these scores will be added up. in the end, we get a range of 2-8 for each of the three areas. a score report will look something like this: 7 reading/6 analysis/6 writing. while that adds up to 19, the sat will deliver the score split three ways.

it’s also a good idea to know what these three different categories are. the reading score reflects your ability to understand the passage that you have to read. for instance, if you misinterpret what the author is trying to say this is going to hurt your score.

analysis, which i will go over in-depth in tip #6, is your ability to analyze how the author goes about persuading his or her audience. remembering the fundamentals of rhetoric is a great first step.

finally, writing is just what it sounds like: how do you use words to create sentences and convey your thoughts? do you so in a way that is grammatically sound and your meaning is clear?

3. practice with sample prompts and essays

i can pontificate all day on what the test writers are looking for, but unless you actually look at how specific essays are graded, along with copious feedback from the graders, you won’t really get a sense of what the test writers are looking for.

you also won’t get a sense of what separates a writing score of 4 from a writing score of 3, an analysis score of 2 from an analysis score of 1, and so on. but you get all of that by studying the prompt, essay responses, essay scores, and feedback.

use the prompts linked below to write your essay first and compare your essay to the corresponding sample essays to see where you’d likely score. in looking at the samples, you’ll likely see things that the writers do well, and things they don’t do so well, which will give you a sense of the issues you need to work on. maybe you can write wonderful flowery sentences, full of phrasal twists and turns. but when you read the passage, you are not exactly sure what to analyze or exactly what the essay graders are looking at when they grade for analysis. or you might get the gist of the analysis, but you feel that you can’t get your thoughts down on paper.

click here to check out official college board sample sat essay prompts and responses from real students

4. get skilled in annotating and outlining

whenever you are faced with a timed essay, it is a natural response to want to begin writing as soon as the teacher/proctor says “time.” if you don’t plan on how you’ll attack the essay, however, your essay will lack the organization the test graders are looking for. most likely, you’ll describe the main points of the essay and just list out what you think are the rhetorical devices the author uses. the essay will lack any overarching point.

instead, first write down a few main points the author is making. then, quickly write down three distinct areas in which the author is using rhetoric. this second bit will help you focus your analysis. often, it is a good idea to break up paragraphs either by the different areas of analysis used in the essay or by the specific points the author is trying to make and how he or she is specifically going about persuading the reader. by outlining you’ll have a clear idea of what you are going to write about, versus frantically grasping onto unrelated ideas just to keep the writing afloat.

another way of outlining is annotation, which is when you’re underlining and taking notes in the margins. for some students, annotating while outlining allows them to engage more actively with the reading and therefore, improve reading comprehension. click here for an example of how the writer who scored an 8 in magoosh’s sample essay above annotated their reading (note the points in the margin have been written out to facilitate understanding. on the test, this writer would have annotated in a much more rushed and abbreviated manner!).

5. vary your sentence structure and vocabulary

the sat is an important test. the essay is very important for some. you need to understand what the test writers are looking for. this post will help you that.

what do you notice about these four short sentences? do they put you off from reading more? the reason is that the sentence structure is almost exactly the same: subject + verb + object. additionally, these four sentences lack any transitions, such as the word “additionally”.

changing up your sentence structure makes your writing far more compelling. and using transitions will help tie ideas together both between and within sentences.

finally, you’ll want to avoid using vague words such as “good”, “big”, especially if you repeat them. notice the first two sentences use the word “important”. i’m not saying you should avoid this word altogether. but repeating it so closely together smacks of monotony, much as the sentence structure does.

now, let’s take the intro sentence again:

the sat is an important test. the essay is very important for some. you need to understand what the test writers are looking for. this post will help you that.

…and vary up the sentence structure and vocabulary, while offering some helpful transition words.

many know the sat might be the most important test for college admissions. yet, for some, the essay can also play a significant role. for this group, understanding how the essay has changed and what the test graders expect is paramount. hopefully, this post will help you with that.

6. work on understanding the analysis

in tip #1, i talked about rhetoric, or the tools an author uses to persuade us. understanding these tools is the first step to analyzing the essay. but you’ll want to go a step further. since each essay is very specific, it’ll be doing things that can loosely be categorized as falling under pathos, logos, or ethos. make sure you describe these specific things. for instance, let’s take an official sat essay from college board.

it is not enough to say, “the author uses pathos because he reminds the author of a childhood experience and such experiences appeal to our emotions.” this is pretty obvious and superficial. digging deeper means looking at the very specific choices the author makes to really get into our psyche. here is one possible way to describe this:

to illustrate just how much darkness has become a scarce resource, paul bogard draws upon memories of the night sky from when he was a child. the author, though, is not merely content to describe the night sky but dramatizes the darkness: “i knew woods so dark…eyes.” furthermore, he uses metaphorical descriptions to capture the intensity of the sky (“sugary trails”). as readers, we are readily transported to the vista unfolding above him. this description also allows the author to set up the dramatic contrast with tonight’s sky when he describes many children today who will “never know” such a sky. this last bit creates an effect of urgency: something must be done.

notice i didn’t say “pathos” anywhere. instead, i described—in meticulous detail—how the author constructs the paragraph to elicit a strong emotional response from the reader. i also analyzed how he constructed the passage, an example of logos; yet i didn’t call logos out by name. instead, i describe the logic of the transitions and how this affected the emotional effect of the paragraph.

7. read with a critical eye

assuming you’ve read all of the above and have a good idea of how the sat essay is constructed, you can start to read a little differently. what do i mean? well, for the reading comprehension section, i recommend that students read articles from the new york times or some other popular online newspaper. while reading the article, put on your grammar hat and analyze the sentences. do you notice the subordinating conjunctions? how about the use—and the correct use, mind you—of em-dashes? (see what i did there?)

but it’s not just about grammar. by analyzing professional writing, you can improve your writing, noticing the transitions and the vocabulary such articles use. of course, it doesn’t hurt with your overall comprehension, something that bleeds into both the writing section and the reading comprehension of the sat.

finally, with more pointed pieces—such as those you’ll find in the new york times op-ed section—you’ll be able to see how authors use the tools of rhetoric. in other words, you’ll be analyzing and comprehending just as you’ll have to do on the actual sat essay.

all magoosh sat prep plans are digital sat ready! start studying today.

the post top 7 sat essay tips appeared first on magoosh blog | high school.

sat math includes the whimsically-named (and slightly intimidating) section “passport to advanced math.” the good news is that it is actually not advanced math. it’s actually the skills that underpin some of the more advanced stuff you’ll be doing in calculus and pre-calculus. in this light, the name passport doesn’t seem as fanciful, since in order to travel in the land of advanced math you’ll need a “passport” showing that you have the fundamentals down.

passport to advanced math includes 16 of the 58 questions spread out over the two math sections. if you’re already comfortable with the other math and are shooting for that perfect math score, you should definitely spend more time in this area. in this post, we’ll look at some critical concepts under the sat passport to advanced math category, along with some practice questions!

table of contents

• exponents and roots in sat passport to advanced math questions
  • sat exponent basics: what you need to know
  • practice with exponents
• quadratic equations
  • what do quadratic equations look like?
  • quadratic equation patterns
  • quadratic formula on the sat
  • practice with quadratic equations
• bigger polynomials
  • practice with polynomials
• coordinate geometry
  • quadratic graphing shortcuts
  • practice with coordinate geometry
• additional topics in sat passport to advanced math

exponents in sat passport to advanced math questions

you are likely to see a few exponents on the sat. exponents often times scare the bejeezus out of students. the word exponent alone conjures up numbers so big that they seemingly dwarf the number of atoms in the known universe. but do not fear: there are no atoms on the test, and the exponents on the sat deal with far smaller numbers. in order to have a chance on the harder questions, you’ll have to first know your basics.

once you are consistently doing well on these, you won’t have to worry about exponents. the thing, though, is it is very easy—especially during the stress of the actual exam—to make simple (and easily avoidable!) mistakes regarding exponents.

first, watch one of magoosh’s sat experts, chris, discuss exponents in sat passport to math questions; then, we’ll dive into the topic in slightly more depth!

https://www.youtube.com/watch?v=_oon2igpbc4&feature=youtu.be

sat exponent basics: what you need to know

#1: the base (it’s the big number)

an exponent all alone would be nothing more than a tiny speck floating in space. every exponent needs a base:

\(3^2 \): base 3
\(2^5\) : base 2

#2: don’t add or subtract the bases

\(3^2+2^2\) does not equal \(5^2\)
\(6^2-3^2\) does not equal \(3^2\)

#3: when multiplying the same bases, add the exponents; when dividing, subtract

\(3^2(3^5) = 3^7\)
\(2^4(2^4) = 2^8\)

\(3^{10}/(3^5) = 3^5\)
\(2^4/(2^2) = 2^2\)

it’s also important to see how this plays out with variables:

\({x^y}*{x^z}=x^{y+z}\)
\({x^y}/{x^z}=x^{y-z}\)

it’s pretty easy to see why rule #3 exists if we use actual numbers and expand the equation:

\( {2^2}*{2^3}=2^{2+3}=2^5\)

\({2^2}=2*2\)
\({2^3}=2*2*2\)
\({2^2}*{2^3}=2*2*2*2*2=2^5\)

be careful that you only do this when the bases are the same!

\(3^2(2^5) ne 6^7\)

#4: when taking both a base and exponent to an exponent, multiply the exponents; when taking the root, divide

when you have a power of a power, you can multiply those exponents. don’t add them!

\((4^2)^3\) does not equal \(4^5\) or \(4^8\)
\((4^2)^3 = 4^6\)

since roots are the opposite operation of powers, just like division is the opposite of multiplication, you can divide an exponent by the radical.

again, if we use real numbers and expand it, the reasons why are pretty clear.

\((2^3)^2=2^(3*2)=2^6\)
\((2^3)^2=(2*2*2)^2=(2*2*2)*(2*2*2)=2^6\)

\(sqrt[3]{2^6}=2^{6/3}=2^2\)
\(sqrt[3]{(2*2)*(2*2)*(2*2)}=(2*2)=2^2\)

when using variables, this is what this rule looks like:

\((x^y)^z=x^{y*z}\)
\(sqrt[z]{x^y}=x^{y/z}\)

#5: sometimes you’ll need to distribute exponents and roots

\((xy)^z={x^z}{y^z}\)
\(sqrt[z]{xy}=sqrt[z]{x}sqrt[z]{y}\)

if you have an exponent outside of parentheses that contain two multiplied numbers, you need to find the power of both factors.

\((2*3)^2=6^2=36\)
\( (2*3)^2={2^2}*{3^2}=4*9=36\)
\((2*3)^2 ne 2*{3^2}\)

and there’s another common exponent mistake made in this kind of situation: don’t distribute exponents to terms inside the parentheses that are added or subtracted.

\((x+y)^z ne {x^z}+{y^z}\)

the foil method makes it pretty clear why that doesn’t work.

radicals follow the same rules. if you have numbers under the radical that are added, then you can’t just find the root of each one. you have to combine them first.

but you can take a number under a radical, break it into factors, and simplify it that way.

\(sqrt[3]{56}=sqrt[3]{8*7}=sqrt[3]{8}sqrt[3]{7}=2sqrt[3]{7}\)

knowing this comes in handy on the sat.

#6: the sat may throw radicals other than square roots at you

numbers over radicals occasionally throw people off. just remember that it’s the opposite process of exponents. so if \(2^3\) is \(2*2*2=8\), then \(sqrt[3]{8}=2\). this is totally fundamental, but it’s a good place to start.

#7: the sat may throw fractions in exponents at you

these are just the same as above. if you see a fraction in an exponent on your sat, go right ahead and convert it into a radical if that helps. so \(8^{1/3}\) equals 2, just the same as \(sqrt[3]{8}=2\).

if you have something other than a 1 in the numerator, like \(8^{2/3}\), then just put the denominator into the radical and keep the numerator as an exponent: \(sqrt[3]{8^2}\). from there, it doesn’t matter which operation you carry out first. \(2^2=4\) just as \(sqrt[3]{64}=4\).

#8: the sat may throw a zero exponent at you

any number to the 0^th power is one. \(2^0=1\) and \(9,999^0=1\). we don’t need to worry about why for the purposes of the sat (but if you enjoy math puzzles and want to figure it out, here’s a hint: it has to do with the next rule).

#9: the sat may throw a negative exponent at you

careful not to get \(x^{-2}\) confused with \(x^{1/2}\). instead, \(x^{-2}=1/x^2\). although fractions in exponents are tested more often, negatives are also liable to show up on your sat, so you should get comfortable with this if you aren’t already.

how you can imagine this concept using gummy bears or whatever, i’m not sure, but that doesn’t make it difficult to use—and that’s true for all of the rules above. as long as you know the facts, you can work pretty easily with exponents on your sat that might seem daunting at first.

now that we’ve gone through everything you’d need to know about exponents and roots, let’s try our hand at a practice problem:

practice with exponents

quadratic equations in sat passport to advanced math

the sat wants you to be able to deal not just with linear equations, as you do in the heart of algebra section, but polynomials. and the most common polynomial you’ll have to deal with are quadratic equations.

what do quadratic equations look like?

the basic form of a quadratic function is this: \(f(x)=a{x}^2+bx+c\)

(also note that this is the same as \(y=ax^2+bx+c\).)

any time you see an equation that looks like that, you can graph it as a parabola.

\(y=5x^2+2x+3\) would be quadratic, for example. \(a=5\), \(b=2\), and \(c=3\).

in your introductory algebra classes, you were probably taught the reverse foil method to solve quadratics. the second-degree polynomials (i.e., quadratics) you end up seeing on passport to advanced math can, unsurprisingly, be solved using reverse foil.

take \(x^2-4x-5 = 0\) as an example.

to use reverse foil for a given quadratic, here are the steps:

• find two numbers in which the product equals \(c\) (\(-5\) in the equation) and the sum equals \(b\) (\(-4\) in the equation). in this case, if you multiply \(-5\) and 1, you’ll get \(-5\), and if you add \(-5\) and 1, you’ll get \(-4\).
• using these numbers, you can now solve the equation: \((x-5)(x+1) = 0\)

quadratic equation patterns

being familiar with foil and reverse foil is foundational to success on passport to advanced math, but there are two patterns that can help you bypass foil/reverse foil entirely: perfect squares and difference of squares.

perfect squares refers to the following expression: \((x+b)^2 = x^2 + 2bx + b^2\).

when you’re familiar with this expression, you would be able to foil an expression like \((x+2)^2\) and reverse foil an expression like \(x^2 + 10x + 25\) in no time.

 
difference of squares refers to the following expression: \((x+b)(x-b) = x^2 – b^2\). you’re probably wondering what happened to the middle coefficient. when you actually use foil on an expression like \((x+b)(x-b)\), you’ll see that the middle coefficient cancels out: \(x^2\) \(- bx + bx \) \(+ b^2\).

let’s see how this pattern works when you replace \(b\) with 4: \((x+4)(x-4) = x^2 – b^2 = x^2 – 4^2 =\) \(x^2 – 16\)

if you’re not already comfortable with perfect squares and difference of squares, i highly recommend trying out different expressions where \(b\) = 1, 2, 3, etc. for example, using these formulas, try to figure out what’s \((x+1)^2\), \((x+1)(x-1)\), \((x+2)^2\), \((x+2)(x-2)\), and so on. these expressions come up a lot on the sat, so recognizing them off the bat will really help you save time.

quadratic formula

sometimes, reverse foil is not enough to solve quadratics. in that case, you’ll need the quadratic formula or

before you freak out—the quadratic formula has that effect on students—let’s apply it to a real quadratic equation:

in this case, there are no integers whose product is \(-5\) and that sum to \(-5\). for instance, the only factors of \(-5\) are 1/\(-1\) and 5/\(-5\). none of those will sum to \(-5\). so in order to find out which two numbers, when multiplied equal \(-5\), but when added equal \(-5\), we have to use the quadratic formula.

comparing \(x^2-5x-5 = 0\) to the expression \(ax^2-bx-c = 0\), you can see that a = 1, b = -5, and c = -5.

plugging those values into the quadratic formula we get:

when should i expect the quadratic formula on the sat?

practice with quadratic equations

bigger polynomials

on the sat, sometimes you have to deal with polynomials in which the powers can be very high, as in the following:

the key to dealing with polynomials of powers higher than three on the sat is usually simplifying them. this is typically a matter of factoring an \(x\) or an \(x^2\) out of each term until you’re left with a simple quadratic in the parentheses. however, if you’re presented with something more complicated, you may want to use the answer choices as clues for how to break the quadratic down.

practice with polynomials

coordinate geometry

other topics include the graphs of complex polynomials. the point of the test, though, isn’t to throw ridiculously complex equations at you; it wants to test your ability to recognize simple patterns in all that complexity. in other words, can you navigate around all the noise to find a relatively straightforward approach to solving the question?

when testing your knowledge about quadratics, functions, and polynomials, college board is essentially interested in your ability to understand “relationships between algebraic and graphical representations of functions.” however, we’re going to use some less grand, simpler terms.

coordinate geometry, or more colloquially, graphing, is another area which is filed under sat passport to advanced math, as long as (and here’s the important part) the equation is a polynomial. most of the time, this will mean a parabola. sometimes, though, you’ll get a graph of some monstrous polynomial like \(y^5+3y^4-2y^2+1\).

the good news is you’ll probably only have to decipher the graph to figure out how many times it crosses through the x-axis or something else relatively straightforward.

it’s really the parabola that is going to show up more often. what you need to know is that parabolas are symmetrical, meaning that each side occupies the same area on both sides of either y-axis or x-axis.

the equation of a parabola can be defined as \(f(x)= ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants (meaning they are some fixed number). since \(a\) will often equal 1, it helps knowing good-old-fashioned foil, as you’ll see in the first practice question below.

quadratic graphing shortcuts

since the sat doesn’t ask you to actually draw any graphs, we won’t focus on that. you may be able to find the answer to the question without even needing to know how to do this. instead, let’s look at a few shortcuts.

shortcut #1: a shows the direction of the parabola

shortcut #2: c shows the y-intercept

shortcut #3: use reverse foil to find the x-intercepts

shortcut #4: know the parabola formula to find the vertex

shortcut #5: use a graphing calculator, but only if you need to!

practice with coordinate geometry

practice question 1

practice question 2

additional topics in sat passport to advanced math

the one thing to know about sat passport to advanced math is that it loves to combine topics. as such, these question types could easily get you flustered on test day. knowing what to expect is half the battle.

this post covers the most common content for this question type, but you may also see:

• complex numbers
• trigonometry
• systems of equations
• function notation
• interpreting and analyzing more complex equations in context

to get a sense of all the different types of concepts that pop up in the sat passport to advanced math section, check out the official study guide. take the practice tests to see the 16 questions per test. if you miss a question because of conceptual misunderstanding, you’ll want to go back and review the content in this section of the study guide—or bookmark this page to come back to!

the post sat passport to advanced math: tips and practice (video) appeared first on magoosh blog | high school.

covering 29% of the concepts on the sat, the problem solving and data analysis section is the second most common on sat math, after heart of algebra. keeping in line with real-world scenarios, these sat problems will ask you to infer information based on a study with any number of participants or interpret data from a graph. this is actually beneficial for sat students, given that you’ll be learning lots about cause and effect and inferential statistics in college.

now, let’s talk about how to approach the various types of problem solving and data analysis questions on the sat, plus some practice questions to get you started! feel free to use the table of contents to navigate directly to the topics you want to learn.

table of contents

• quick facts about sat problem solving and data analysis
• ratio, proportion, units, and percentage
• subjects and treatments
• sat statistics (mean, median, and mode)
• what’s important to know about averages on the sat?
• sat graphs: tips and tricks

quick facts about sat problem solving and data analysis

you can expect to see about 17 problem solving and data analysis questions on the sat math section, although they will not be outright labeled by question type. some answers will be single-step questions, while others will be multistep problems. if you fear mental math, there’s no need to sweat it in this section—you’ll be allowed to use a calculator. like the other sat math sections, you’ll receive a problem solving and data analysis subscore on a scale of 1 to 15.

ratio, proportion, units, and percentage

this problem solving and data analysis question type shouldn’t come as a surprise since it has probably been part of your math courses for the last five years (yep, you most likely went over this stuff, in some form, all the way back in early middle school). i’ll start with ratios.

understanding ratios

a good way to think of ratios is apples and oranges. say i have two oranges and three apples, the ratio of oranges to apples is 2:3. seems straightforward. what if i have four apples and 6 oranges? if you answered 4:6, that is not quite correct. you have to think of the ratio the way you would a fraction, in lowest terms. both 4 and 6 can be divided by ‘2’, giving you 2:3. notice how that is the same ratio as 2:3.

this highlights an important conceptual idea: ratio is not about total number. it is about the number of one thing, to the number of another thing, reduced, so that the ratio is expressed as two prime numbers.

one last thing about ratios. let’s say you have a ratio of 1:2. this is not the same thing as ½. the bottom number in a fraction is always the total. the total of a ratio is always the parts of a ratio added together. in this case, 1:2 is 1 + 2 = 3. so if i have 1 apple to two oranges, 1/3 of the fruit are apples and 2/3 are oranges.

if you have more than two ratios, make sure to add up all of the ratios. for instance, if the ratio of blue marbles to red marbles to green marbles is 2 : 5: 7, red marbles account for 5/14 of the total (2 + 5 + 7 = 14).

do you think you got that? well, here are some practice questions to test your knowledge of problem solving and data analysis.

understanding percentages

percentages can be surprisingly complicated on the sat. part of that is because we can’t always translate them into fractions, which are easier to work with algebraically. while it’s easy enough to think of \(50%\) as \(frac{1}{2}\), it’s rarely so easy to make the conversion on the sat, especially when the percentages given are, say, 35% or 15%.

finding a percent is pretty easy, as long as you have a calculator. just divide the part by the whole and multiply the decimal that comes out by 100. so if you ate 10 out of a serving of 12 buffalo wings, then you ate (10/12)100=83.33%. remembering that formula can save you some grief when you have to use it algebraically.

however, the sat won’t just test you on the simple process of finding the percentage of a number (like calculating a tip). instead, it’ll ask you to calculate in reverse (finding the whole from the part), find a combination of percentages, find a percent change, or give some other scenario-specific piece of information.

being prepared for percent change questions, in particular, will take you far on problem solving and data analysis.

the equation for percent increase \( = frac{text{new number – original number}}{text{original number}}*100 \).

the equation for percent decrease is \( = frac{text{original number – new number}}{text{original number}}*100 \).

practice questions

subjects and treatments

this is not an official title but the name i’m giving to questions that deal with studies trying to determine cause and effect.

in order to understand how to approach subjects and treatments questions, let’s talk about randomization. the idea of randomization is the essence, the beating heart, of determining cause and effect. it helps us more reliably answer the question of whether a certain form of treatment causes a predictable outcome in subjects.

randomization can happen at two levels. first off, when researchers select from the population in general, they have to make sure that they are not unknowingly selecting a certain type of person. say, for instance, that i want to know what percent of americans use instagram. if i walk on a college campus and ask students there, i’m not taking a randomized sample of americans (think how different my response rate would be if i decide to poll the audience at a rolling stones concert).

on the other hand, if i went to a city phone directory and dropped a quarter on the page, choosing the name that the center of the quarter was closest to, i would be doing a much better job of randomizing (though, one would rightly argue, i’d still be skewing to an older age-group, assuming that most young people have only cell phones, which aren’t listed in city directories). for the sake of argument, let’s say our phone directory method is able to randomly choose for all ages.

after throwing the quarter a total of a hundred times on randomly selected pages (we wouldn’t want only people whose names begin with ‘c’, because they might share some common trait), our sample size consists of 100 subjects. if we were to ask them about their instagram use, our findings would far more likely skew with the general population. therefore, this method would allow us to make generalizations about the population at large.

subjects and treatments practice question

subjects and treatments: a summary

here are the key points regarding subjects and treatments (aka cause and effect questions) on sat problem solving and data analysis:

1. results from a study can only be generalized to the population at large if the group of subjects was randomly selected.
2. once subjects have been selected, whether or not they were randomly selected, cause and effect can only be determined if the subjects were randomly assigned to the groups within the experiment/study/treatment.
3. there are three basic types of averages on the sat that you should be pretty comfortable with at this point, and all of them start with the letter “m.” those are the mean, the median, and the mode. in case those aren’t second nature, let’s define them, quickly.

sat statistics (mean, median, and mode)

the sat math test often asks you to do some statistics problems involving averages. finding the mean is the most commonly used average and, as it so happens, the most commonly tested when it comes to sat statistics. the formula is pretty simple:

{a+b+c+….}/n where n is the number of terms added in the numerator.  in the set of numbers {2,3,4,5}, 3.5 would be the mean, because 2+3+4+5=14, and \(14/4=3.5\)

if the numbers in a set are listed in order, the median is the middle number. in the set {1,5,130}, 5 is the median. in the set above, {2,3,4,5}, the median is 3.5, which is the mean of the middle two terms since there’s an odd number of them.

the mode is just the number that shows up the most often. it’s perfectly possible that there is no mode or that there are several modes. in the set {5,7,7,9,18,18}, both 7 and 18 are modes.

what’s important to know about averages on the sat?

averages come up in algebra or word problems. you’ll usually have to find some value using the formula for a mean, but it may not be as simple as finding the average of a few numbers.  instead, you’ll have to plug some numbers into the formula and then use a bit of algebra or logic to get at what’s missing.

for example, you might see a question like this:

if the arithmetic mean of x, 2x, and 6x is 126, what is the value of x?

to solve the question, you’ll need to plug it all into the formula and then do some variable manipulation.

\(frac {x+2x+6x}{3}=126\)
 
\({x+2x+6x}=378\)
 
\(9x = 378\)
 
\(x=42\)

medians and modes, on the other hand, don’t show up all that often in problem solving and data analysis. definitely be sure that you can remember which is which, but expect questions on means, most of the time. as for other types of statistical analysis, you may also be asked to solve some problems involving standard deviation.

what is a “weighted average”?

basically, weighted means uneven, here; the numbers that you’re looking at don’t carry the same importance. for example, if i’m trying to find the average number of fleas that my pets have, and each cat has 150 while each dog has 200, then those two numbers have equal “weight” only if i have the same number of cats as dogs. let’s say i have 1 of each.

\(frac {150+200}{2}=175\)

that’s just a normal mean, so that’s no problem. well, the fleas are a problem, i guess. and the fact that i’m counting fleas might have my family a little worried…anyway, the math is easy. but that’s a non-weighted average.

for a weighted average, i would have a different number of cats than dogs. let’s say i had 3 cats and 2 dogs. (and they all have fleas…things are starting to get kinda gross. sorry.)

in order to give them the appropriate weight, we’d have to multiply each piece appropriately and change the total (denominator) to reflect it.

\(frac {3(150)+2(200)}{5}=170\)

but if you expand that, you’ll see that it’s the same as the standard mean formula.

\(frac {150+150+150+200+200}{5}=170\)

just make sure you divide by five (because i have five pets) not two (for two types of pets).

finding average rates

average rates are a type of weighted average. your sat problem solving and data analysis section will include a problem or two about these, and you need to be sure not to fall for the common trap.

maria’s drive to the supermarket takes her 20 minutes, during which she averages a speed of 21 miles per hour. she takes the same route home, but it only takes 15 minutes to cover the equal distance. what was maria’s average speed while driving?

1. 15.5 mph
2. 21 mph
3. 24 mph
4. 24.5 mph
5. 28 mph

this is a tricky, multi-step problem, and you can’t plug in the answer choices to solve it, sadly.

let’s first find all of our information, because the question has only given you part of it. you need to know the formula r=d/t (rate = distance/time), also expressed as d=rt (easily remembered as the “dirt” formula). we’re going to use it both ways.

using that formula, let’s look at the first leg of her trip. she traveled for 1/3 of an hour at 21 mph, so she must have traveled 7 miles.

that’s \(21*0.333=7\)

using that info, we can figure out the rate of her trip back home. going 7 miles in 1/4 of an hour on the way home, she went an average of 28 mph.

that’s \(7/0.25=28\)

so now we need to find the total average. that’s not the average of the two numbers we have! because each mile she traveled on the way there took more time than each mile on the way home, they have different weights!

✗ \(frac{21+28}{2}=24.5\)

instead, you need to take the total of each piece—total time and total distance—to find the total, average rate.

✓ \(frac{14 text{ miles}}{.333 text{ hours} + .25 text{ hours}}=frac{14 text{ miles}}{.5833 text{ hours}}={24 text{ mph}}\)

average rate practice question

weighted averages that you won’t see on your sat

i’ve never seen an sat problem solving and data analysis question that asks you to find an average based on percent weights (e.g. finding a final grade in a class where quizzes count for 70%, attendance for 20%, and participation for 10%). finding that average is a little more complicated, so it’s nice that we don’t have to worry about it.

 
simply put… if you’re finding the average of two sets of information that already are averages in their own right, as the number of fleas per cat and the number fleas per dog, you can’t just take the mean of those averages. you have to find the totals and then plug them into the formula. you should be excited about these kinds of problems, if for nothing more than having the opportunity to bust out your handy-dandy, brand-spanking’ new sat calculator.

sat graphs: tips and tricks

among the math skills that sat problem solving and data analysis tests, reading data from tables or graphs is one of the more straightforward tasks. but there are a number of simple mistakes that might make you miss out on points if you’re not careful. the best way to avoid those totally avoidable slip-ups is to train yourself to follow a pattern.

1. scanning before reading the question
   it’s tempting to jump right into the question, especially if you’re feeling clock pressure. but don’t do it! usually, the question will make pretty much no sense if you don’t have the context that the figure gives you. you’ll end up reading the question, looking over the table/graph, rereading the question, and then finding the information you need. why waste the time doing it twice? scan the figure first.

   you’ll want to read the headings, the axes, and the units of measurement, then make note of any missing information or obvious patterns.

   after you’ve done that, go ahead to the question.

2. add in any information from the question
   as is often true for other types of sat math problems, the written question might have some info in it that the figure doesn’t include. just like you would write in angle measurements, fill in any extra info; there’s no reason to try to keep it in your head.
3. find the areas of the table that the question asks about
   your sat might try to make the question more confusing by adding in a whole lot of superfluous information into the visual. check what the question asks for, then circle the area in the table or graph that answers the question (or gives you info that will lead to the answer).

   for the easiest sat data questions, you’re already finished—they just want you to locate the info. but it might ask you to go another step.

4. write the math out
   if you’re asked about relationships between two things, look carefully at the relationships between 4-6 pieces of information (two xs and two ys), and write out the pattern. if you’re looking for some variable, write out the equation. if it’s not clear how to go about that, maybe you should try plugging answer choices in to see if they work.
5. double-check your units
   after you’ve done whatever math you need to, check your units. it’s easy to make a mistake by using minutes instead of hours, and the sat takes advantage of that in the incorrect answer choices.

   a lot of sat prep has to do with training yourself against hasty mistakes, and tables and graphs are classic places to make those slips. follow the rules and you’ll be much safer.

that’s all for the sat problem solving and data analysis section! we hope this breakdown was helpful for you. to read up on the other two sections of sat math, check out our posts on heart of algebra and passport to advanced math.

the post sat math: problem solving and data analysis appeared first on magoosh blog | high school.

heart of algebra is one of 3 sections on the sat math test, and will include 19 of the total 58 questions. in short, “heart of algebra” refers to any problem that has an algebraic equation in which none of the powers next to a variable is higher than one. for example:

pretty straightforward, right? the computation underpinning heart of algebra is straightforward, but the sat typically won’t be giving you problems this simple. while you will encounter several problems that ask you to solve for variables, much of the heart of algebra content are real-world word problems that you have to unwrap before solving. for example:

how to prepare for the sat heart of algebra section

there are two ways to make sure you’re prepared for heart of algebra questions:

1. understand how to solve linear equations and systems of equations, along with common variations, like inequalities.
2. practice translating real-world scenarios and information into a mathematical equation.

to help you get the swing of things, we’ve broken down and demonstrated the key concepts and skills that will serve you in the heart of algebra section. feel free to explore these concepts through the clickable table of contents below:

algebra basics: balancing the equation

first thing’s first: in order to interpret real-world scenarios as equations, you’ll need to be super clear on how to balance an equation!

so, in an algebraic equation with an equal sign, think of the equation in balance. if you do anything to one side you do the exact same thing to the other side. if you multiply one side by 4, you have to multiply the other side by 4. if you subtract 3 from one side, you subtract 3 from the other side.

the goal in balancing the equation is isolation. that’s right — you want to isolate x, or get x (or whatever variable happens to be in the equation) by itself. think of the variable as the unpopular person in the room. everyone – that is, the other numbers – want to get away!

\(4x + 3 = 15.\)

to get x by itself, first subtract 3 from both sides of the equation.

\(4x + 3 – 3 = 15 – 3\)

\(4x = 12\)

now divide both sides of the equation by 4:

\(frac {4x} {4} = frac {12} {4}\)

\(x = 3\)

let’s try one more:

\(sqrt{x} = 9\)

notice how x has a square root sign on top of it. we want x to be all by itself. therefore we have to get rid of the square root sign by squaring both sides:

\((sqrt{x})^2 = 9^2 \)

\(x = 81\)

the gist? by knowing how to isolate variables by balancing equations, you’re well on your way to success on the heart of algebra section!

inequalities

although inequalities may seem tricky, if you’re comfortable balancing equations, you shouldn’t have much problem with balancing and solving inequalities. the main way that inequalities differ from equations is that the sign changes if you multiply or divide by a negative number.

inequalities come up a lot on sat heart of algebra, so if you need a refresher, make sure to check out this video:

system of equations

now, let’s apply a more advanced version of balancing equations to a very important concept in heart of algebra: system of equations.

system of equations refer to when you have two equations with two unknowns and by combining these two equations, you can solve for these unknowns. there are two methods for combining equations: substitution and elimination.

for a refresher on these methods, check out the below video and then try your hand at solving the practice problem that comes after:

algebra basics practice: system of equations

now, let’s go back to that original question we started this post off with:

coordinate geometry basics

ok, let’s break down each of the concepts you’ll need to understand, step-by-step, to fully grasp coordinate geometry.

quadrants

there are 4 quadrants in a coordinate plane.

simple, right?

what’s wonky is how those quadrants are arranged on the plane:

a good way to visualize the quadrant layout is to imagine yourself on a cross-country trip in which you start off in new york (the northeast = quadrant i), drive on over to seattle (the northwest = quadrant ii), then drive down to los angeles (southwest = quadrant iii) and finally end up in miami (southeast = quadrant iv).

the x-axis & y-axis

the next things to be familiar with are the x-axis and the y-axis. the x-axis runs from left to right (this is the horizontal line), whereas the y-axis runs up and down (this is the vertical line).

integers and scale of graphs

next, you need to understand integers, or what we like to call “people numbers.”

why people numbers? think of it this way: you can have one more person or one person fewer. however, you can’t have .3 people or ½ a person. in “coordinate geometry land,” you’ll have integers running from left to right, like on a number line, from negative to positive. on the left hand side the negative numbers will decrease until they hit the middle of the graph (the intersection of the x- and y-axes—or what’s known as the origin). at that point, you’ll see the number zero. then the numbers continue up from zero.

for the y-axis, the vertical one, the negative numbers start at the bottom, decreasing until they get to zero. then, from the origin going up, each number will go up as well.

on the sat, the scale of the x- and y-axes will vary from question to question. not all axes increase and decrease on an interval of 1. sometimes, the scale of the x-axis will be totally different from the scale of the y-axis! the test-makers will try to trick you on this point with the answer choices, so definitely make sure to pay attention to the scale of each axis.

coordinates

next up are the coordinates. coordinates are always presented as two numbers: the first number corresponding to the x-axis, and the second one to the y-axis. for instance, in the graph below, point a has the coordinates (2, 4). that is you “go over” two points to the right on the x-axis and up four points on the y-axis. point b has the coordinates (-1, -3).

even though the numbers on the x-axis and y-axis are always broken down via integers, coordinates can have fractions, as point c shows below (1/2, -7/2).

slope

ok here’s the fun part that involves putting together the basics from above.

remember the term “rise over run”? you’ve probably had this hammered in for so long you can’t even remember when you first heard it. sadly though, most people forget exactly what it means.

so here it is: you can figure out the slope of a line by seeing how many squares on the coordinate plane one point is above the other. in the graph below, point d has a y-coordinate of 2 and point e has a y-coordinate of -3. so point d is five squares higher than point e. this is the “rise,” how far up it goes from the lowest point, is 5.

when you are figuring out the slope, which consists of a fraction, always put the “rise” in the numerator.

the “run,” on the other hand, is how far away two points are in a left-to-right sense.

if you count how many squares point d (3, 2) is from point e (-2, -3), in a left to right sense, you’ll get the “run,” which in this case is 5.

maybe you’re wondering why on earth it’s called the “run.” it’s just one of those conventions you have to learn. the key is you always put the “run”, or whatever you want to call it, in the denominator.

now, the good news. there’s actually a faster way of doing this besides just counting squares! it’s all in this formula (which your algebra teacher would hate me for leaving out):

for example, let’s have \(y_2 = 2\) (the y-coordinate in point d) and \(y_1 = -3\) (the y-coordinate in point e). that means \(x_2 = 3\) (the x-coordinate in point d) and \(x_1 = -2\) (the x-coordinate in point e). using the formula, we’ll get:

\( 2 – (-3) \)
_______
\(3 – (-2)\)

which equals:

\( 5/5 \) or \( 1 \), meaning that the slope is 1.

if you switch the order, the equation will look like

\( -3 – 2\)
_______
\(-2 – 3\)

which equals:

\( -5/-5 \), which is also \( 1 \).

coordinate geometry practice

every now and then, the sat gets diabolical and throws a coordinate geometry question without the coordinate plane. to make matters worse, the question is usually difficult to begin with—even if you had the plane.

the key is to translate the information by drawing out a mini-coordinate plane. remember, this is a timed test, so it doesn’t have to be anything fancy (don’t worry, the test isn’t going to require you to carefully graph out parabolas the way your math teacher does). as long as your graph can help you better visualize the problem, you’re halfway to getting the problem right!

of course, the other half is always a bit tricky. case in point: the question below. if you’re able to crack it in less than two minutes, you’re in excellent shape for heart of algebra!

if you have a comment or question about this problem, leave it for us below!

functions

you’re going to find about 2-6 function questions on your sat, and if you haven’t been working with them recently in math class, they might throw you for a loop.

just to be clear, we’re talking about equations that look a bit like this: \(g(x) = frac {x-4} 2\)

you may have a couple of immediate reactions to equations like this.

for one, you may have the instinct to graph it out right off the bat, which is great, but not necessarily what you’ll need to do on the sat.

or, if you’re not so familiar with functions, you might make a lethal error and assume the \(g\) here is a variable. it’s not, by any means, but it’s also important to note that these functions aren’t always a separate skill from solving algebraic equations. they’re just wearing costumes: \(g(x) = frac {x-4} 2\) is the same thing as \(y = frac {x-4} 2\).

although function questions tend to come later on in the section, which is typically where the hardest questions on the test are found, functions are really not that difficult and can be solved pretty quickly with some practice. if it helps, you can change the function notation to look like an algebraic equation that you’re more familiar with.

try that concept out with relatively simple function question, based on the equation above:

here are some things to keep in mind about functions:

the “g” in g(x) is just shorthand

let’s say you have a friend who has a particular way of eating grapes. let’s call her gina. and let’s say gina brings grapes to school every day, eats four, splits the rest into two equal piles, then throws away one of the piles and gives away what’s left.

the thing is that she brings a different number of grapes every day. meaning, what she gives away at the end also varies. does this sound familiar? you might recognize this process from above. it’s just an example of that earlier function \(g(x) = frac {x-4} 2\), where:

• x is how many grapes gina comes to school with originally
• \(4\) represents how many grapes gina eats
• \(frac {x-4} 2\) represents both the pile that gina throws away and the pile she gives away, and
• \(g(x)\) is just shorthand for gina’s process: \(frac {x-4} 2\)

what that \(g\) really means is a process (not the final result) without writing out each step of the way. gina brings \(x\) grapes, and she then “\(g\)s” the grapes. in a way, “\(g\)” a verb, not a noun.

functions inside of functions

so let’s say you’ve got a whole group of friends with unique grape-eating habits. gina does her grape ritual and then hands off what’s left to hailey. hailey goes through a whole other process. she won’t eat gina’s grapes unless she gets the same amount of grapes from somebody else, and she then squashes one of them under her foot.

so hailey’s equation would be \(h(x) = 2x – 1\). what she has, then, is described by \(h(x)\). but first we need to think about how many gina gave away.then, put the two processes together into one: \(h(g(x))\). take it from the inside out, and you’ll have no problem.

if hailey’s grapes are passed along to even more friends with specific grape eating habits, just keep nesting the function, e.g. \(f(h(g(x))\). on the sat, you likely won’t go beyond one function nested in another.

in the end, sat functions like this will end up being no different than the simpler-seeming sat equations. really.

functions practice: nested practice

let’s try out gina and hailey’s equations with a practice question:

heart of algebra: you got this!

the above concepts really do constitute the “heart” of the heart of algebra section, so if you familiarize yourself with them and practice these types of problems, you’ll be in great shape! for more practice problems, make sure to check out sat math: algebra practice questions.

you can also check out the other question types on sat math, like problem solving and data analysis, read more about how to score a perfect 800 on the sat math section, and check out some additional sat math practice!

the post sat math: heart of algebra appeared first on magoosh blog | high school.

you have to perform well on the sat exam in order to have your pick of colleges, but where do you start? we’ll share our ten best study tips to help you in preparing for the sat. best of all, some of the tips are easier than you might think!

how to prepare for the sat: top study tips

1. read lots of nonfiction outside of class

the sat is very heavy on reading—you’ll have five long, dense passages one after another in an hour. it’s not that the questions are difficult. indeed, they are pretty straightforward. but it’s the reading part that is tough, because your brain will likely run out of energy sometime during the section (hopefully not during the first passage!).

to prevent this from happening, do lots of nonfiction reading when preparing for the sat. the reason is four of the five passages you’ll see will be nonfiction. high school reading curriculum tends to lean mostly towards fiction. to counterbalance that—and prepare for sat reading in general—pick up a magazine (time for the newbies, the new yorker for the more ambitious) or a newspaper (the new york times is excellent all around).

since your “reading brain” won’t sprout overnight, this is an area in which you’ll want to get a jumpstart. so hit the books (er, online magazines) now.

2. learn how to do mental math

one of the two math sections on the sat will not allow you to use a calculator. however, you don’t have to get stuck doing equations with only a no. 2 pencil.

prepare for sat math by using your brain as your “mental calculator.” you’ll save a lot of time on test day if you know the answer to, say, 3 × 13 right off the bat. not sure where to start? magoosh’s free, downloadable math ebook has shortcuts and example problems to make mental math easier.

3. brush up on grammar

almost half of the verbal section is made up of grammar questions. while many have to do with “big picture” essay questions, others rely on basic grammar. you might dread the thought of learning grammar, but it is one of the easiest topics to improve on when preparing for the sat.

4. use the best sat prep materials

there are lots of study materials out there. not all are created equal; in fact some will hurt your score by providing you with questions that aren’t representative of the actual test. take a look at some of the best study resources below or check out this comprehensive review of the best sat prep courses.

• best sat books
• sat reading guide
• sat writing and language guide
• sat math guide
• sat question of the day

5. try some mixed sample tests

there are essentially three different parts to the sat test: math, reading, and writing and language. often when students prep they think, hey, i’ll just do a bunch of math tonight. studies have shown that it is much more beneficial to do study sessions in which you do, say, 35 minutes of math and 35 minutes of writing and language. this will mimic what you’ll do on test day,–switching from section to section.

our free, full-length sat practice test is a great way to prepare for sat test day—you’ll be comfortable with the format, as well as switching between sections after a certain time limit.

6. don’t cram!

cramming is a bad idea. while you are doing it, it feels like you are retaining so much information. but within a week much of that information vanishes. (cramming is much like trying to build a skyscraper from a deck of cards.)

instead, you should make sure to prep at least a few times a week and review what you’ve learned. you’ll find it is easier for you to learn when you are re-exposed to information you recently attempted to learn.

and there is no need to do more than three hours total prep on a day; you’ll start to get diminishing returns. you’ll also want to take a break in between all that studying to let the information sink in. so break up studying throughout the week, and break it up throughout the day. this one-month sat study schedule is a good way to structure your study time.

7. figure out, and work on, your weaknesses

you’ll naturally be good at some things on the sat, and it’s good to maintain that edge by practicing those concepts from time to time. but it is better to figure out where you struggle. take a diagnostic test to see which areas you need to work on when preparing for the sat.

8. sneak in sat prep during “dead time”

many of us have certain parts of the day where we are just, well, sort of hanging out. maybe we’re waiting for a friend, the bus, or an annoying commercial break to end. well, don’t be a victim of waiting; use these easy ideas!

• review vocabulary flashcards on your phone
• read a chapter or two of our free sat study guide
• work on your mental math (quick, what’s 5 × 16?)
• discover top tips from students who got a perfect score on the sat

9. find an sat study partner

don’t go at it alone. find a partner (or two!) and keep each other accountable when preparing for the sat. share strategies, resources, and sat study tips. test each other, compete against each other, and, most importantly, commiserate with each other. the sat is a rite of passage, full of its ups and downs. it is best not to go at it alone.

10. prepare for sat test day by reviewing this checklist

don’t be fooled by the title of this video—this is definitely not about cramming! instead, use it to prepare for sat test day, including what to bring to the test center and how much sleep you should get beforehand.

bonus video: how to study for the sat and get a perfect score

finally, if you’re wondering how to study for the sat with a perfect score goal in mind, this video is for you. even if you’re not aiming for a perfect score, chris goes over some great sat study tips for all students!

good luck!

need more help?

with magoosh act, you can choose between a live cohorted class with an instructor (which includes all our lessons and practice questions) or access to the self-study option by itself.

all magoosh sat prep plans are digital sat ready! start studying today.

the post top 10 ways to prepare for the sat appeared first on magoosh blog | high school.