when you sit down for the official act exam, act math is the second section you’ll see on the test. so just what does this section test? the act math test assesses the mathematical skills students are expected to obtain before grade 12 (meaning through advanced algebra and basic trigonometry, but not calculus). here’s a quick breakdown of what you need to know:

• you have a 60-minute time limit
• you will face 60 multiple choice math questions
• you get to use a calculator for the whole test (but it must be an approved one)

still curious? read on for everything you need to know about the act mathematics test!

what kind of math is on the act?

the act is a standardized test–and that’s great news for test-takers! it means that the test makers aren’t just pulling questions out of thin air: there’s a standardized list of topics that questions can cover. in the mathematics section on the act, an act math question could come from any category below:

pre-algebra

• decimals, fractions, integers, number properties, positive and negative numbers, square roots, scientific notation, factors, ratios, proportions, percents, averages, charts and graphs of statistics

algebra

• elementary algebra
  • exponents, evaluating algebraic expressions through substitution, using variables to express relationships, algebraic operations, and factoring quadratic equations
• intermediate algebra
  • quadratic formula, rational and radical expressions (including some logarithms), absolute value equations and inequalities, sequences and series, systems of equations, quadratic inequalities, functions, matrices, roots of polynomials, probability, and complex numbers

geometry

• coordinate geometry
  • evaluating points and lines on a graph, polynomials, circles and other curves, graphing inequalities, slope, parallel and perpendicular lines, distance, midpoints, and conics
• plane geometry
  • angles and relationships among perpendicular and parallel lines, properties of circles, ellipses, rectangles, triangles, parallelograms, trapezoids, transformations, area, volume, three-dimensional geometry

trigonometry

• trigonometric relations in right angles, values and properties of trig functions, graphing trig functions, using trig identities, solving trig equations

is sat or act math harder?

at the risk of sounding wishy-washy, it depends. it really does. the two exams test math differently (take a look at this act vs sat comparison to see exactly how!). however, you’ll generally find act math easier than sat math if:

you’re good at mental math, including finding shortcuts.
you have some familiarity with algebra ii/trig, or are willing to learn
you prefer using a calculator (there’s a no-calculator math section on the sat–but not on the act!)
you like multiple-choice questions, rather than grid-ins

still not sure? magoosh’s test prep expert, kristin, breaks down the differences in the math sections even more here!

how do you get a 36 on the act math?

have you set your sights high when it comes to your act math score? awesome! we love it. here are a few essential skills to get your math score into that stratospheric range. then, for more advice, check out our guide to getting a perfect 36 on the math test!

1. pacing is key. check out our tips for improving your timing on the act!
    
2. focus on high-frequency topics. familiarize yourself with the most frequently tested math concepts as well as other key concepts for act math to make sure you have them down pat by test day!
    
3. balance hard and easy problems. practice with hard math problems–but don’t forget to practice easy and medium problems, too! after all, they’re just as common on the exam.
    
4. practice using formulas. ensure you don’t just memorize math formulas–work on using them in context by applying them to act math practice questions!

the post how to ace act math: topics and tips appeared first on magoosh blog | high school.

what do you need to know about the act format? the act test is comprised of four core test sections: english, math, science, and reading. there is also an optional writing test. this page covers everything you need to know about the act sections with links to additional useful resources.

act sections in a nutshell

act english test

the act english section (or, as the act refers to it, the act english test), primarily looks at your understanding of standard english grammar, mechanics, and usage, as well as the elements that make up strong writing: language use as it relates to style, tone, and voice.

here’s what you need to know about the act english section:

• section on act: 1
• time: 45 minutes
• number of questions: 75
• question types: 3
  • production of writing (development and organization): 22-24 questions
  • knowledge of language (style): 10-14 questions
  • conventions of standard english (grammar and punctuation, sentence structure): 38-42 questions
• score scale for act english: 1-36

all of these questions are passage-based, though some may have to do with underlined portions of the passage (words or phrases) and others may have to do with paragraphs, sections, or passages as a whole.

act math test

the act math section tests math most high school students have seen by junior year. in other words, there’s no calculus here: the test’s “easiest” problems start with pre-algebra and max out with basic trig. the hard questions tend to be hard because of the thought processes and logic involved in them, not because they contain particularly complex concepts.

here’s what you need to know about the act math section:

• section on act: 2
• time: 60 minutes
• number of questions: 60
• content areas: 6
  • pre-algebra
  • elementary algebra
  • intermediate algebra
  • coordinate geometry
  • plane geometry
  • basic trigonometry
• score scale for act math: 1-36

you can use a calculator on act math–but you’ll be going so fast (1 question a minute) that you’ll want to make sure you use it smartly. practice is crucial!

also, unlike sat math, act math will not give you a list of formulas. so practice putting those math formulas to use well before test day! simple memorization won’t cut it here–put them to practice.

act reading test

act reading is a passage-based section. you’ll read passages and answer multiple-choice questions on each one. while this isn’t exactly like what you do in your english classes (you’ll likely never have seen these questions before), it’s absolutely possible to develop the skills in your act prep to master an entirely new passage with gusto!

here’s what you need to know about the act reading section:

• section on act: 3
• time: 35 minutes
• number of passages: 4 (fiction, social science, humanities, and natural science)
• number of questions: 40 (10 per passage)
• question types: 3
  • key ideas and details (information directly stated in the passage): 22-24 questions
  • craft and structure (purpose and author’s choices): 10-12 questions
  • integration of knowledge and ideas (connections between passages/parts of passages): 5-7 questions
• score scale for act reading: 1-36

the absolute best way to prepare for act reading is to read–a lot. read actively, read a variety of materials, read outside your comfort zone. and know that in 2021, the reading section of the act will start adding a graphic to one passage (this will then be tested with an integration of knowledge and idea question).

for act reading, a few skills are key:

• fast, accurate reading
• understanding potentially confusing vocabulary in context
• figuring out why authors have made certain choices in each passage

these sound like challenging skills to learn–but they can be learned! with practice, you’ll learn how to think like the test-makers think.

act science test

act science is a test mostly of scientific reasoning. sure, it will likely be helpful to know a handful of basic (very basic) scientific facts for some of the questions. but for the majority, understanding graphics, working through logical reasoning, and evaluating conclusions will serve you much better than knowing the atomic number of argon (it’s 18, by the way).

here’s what you need to know about the act science section:

• section on act: 4
• time: 35 minutes
• number of questions: 40
• question types: 3
  • interpretation of data (analyzing data from the passage and/or graphic): 16-20 questions
  • scientific investigation (scientific/experimental process questions): 8-12 questions
  • evaluation of models, inferences, and experimental results (validity of conclusions/inferences): 10-14 questions
• score scale for act science: 1-36

act science sounds like another section that would be challenging to prep for. and while it is unlike anything you’ve likely seen before, act science can absolutely be prepared for! in this case, working closely with high-quality science materials specific to the act will do you the most good. upping your graphic literacy by working with charts, graphs, and tables is also important.

act writing section

act writing is the optional essay test on the act. in the essay prompt, you’ll read a variety of different perspectives on a topic, then present your own opinion.

here’s what you need to know about the optional writing section on the act:

• section on act: 5 (optional)
• time: 40 minutes
• number of questions: 1
• question types: 1
• score scale for act writing: 2-12

beware! even though you don’t need to take the essay section to take the exam, some colleges may require it…and you really don’t want to go back to the test center on a weekend morning just to write an essay. so do your research about what the colleges you’re applying to require!

keep in mind that the scoring on act writing is different than act scoring more generally. not only is the score scale different (2-12, rather than 1-36, like the other sections—scored by two graders on a 1-6 scale), but it’s also not counted in your act composite score. admissions committees will see test scores for your essay in the college admissions process, but just as a separate section score, not as part of the composite.

which act section is the hardest?

let’s face it: the act wouldn’t include all of these sections if they weren’t in some way challenging for at least some test-takers. with that said, the “hardest” act section is going to depend on you and your background, skills, and study habits (you can take an 2022世界杯入门名单 to figure out what those are for you). in the meantime, here are a few challenges we commonly see our students face in different act sections:

act english: answering so many questions (75!) in such a short time period (45 minutes!). practicing pacing will be key so that you can have more time to tackle the more challenging english questions

act math: combining concepts from multiple subject areas (think: number properties and similar triangles) into the same problem. you want to ensure that you’re familiar with the most frequently-tested topics before tackling the harder, more obscure ones.

act reading: reading all of the passages/answering all of the questions in the given time. pacing and familiarizing yourself with act reading passages will go a long way in this section.

act science: getting frustrated by the complex and/or unfamiliar graphic information, which is especially pronounced for hard act science questions. once you’re more comfortable with all the graphs and tables, this section is not so bad.

act writing: taking a strong stance, while leaving time for outlining, writing, and proofreading. if you consistently practice with our guide to scoring a perfect 12 on the act essay, you’ll be able to write a strong essay in no time!

the post your complete guide to act test sections appeared first on magoosh blog | high school.

the sat is ⅓ writing, and a pretty huge chunk of that third relies on knowing sat grammar and punctuation rules. having a solid grasp on the rules of written english, along with plenty of sat grammar practice, is pretty important to acing sat english. keep reading to get our top 50+ sat grammar and punctuation rules, along with a practice quiz and free, downloadable english grammar cheat sheet!

table of contents

• how the sat tests english grammar and writing rules
• sat grammar tips for success
• 50+ sat grammar and punctuation rules
  • sentence structure rules (with video!)
  • punctuation rules
  • effective language use
  • style and tone
  • voice
  • subordination and coordination rules
  • parallel structure rules
  • modifier placement rules
  • verb rules
  • agreement rules
  • pronoun rules
  • possessive determinant rules
  • diction rules (frequently confused words)
  • logical comparisons
  • conventional expression rules (including prepositions)
  • rules about items in a series
• learn and test your knowledge with the sat grammar rules pdf
• sat grammar practice quiz
• sat grammar and punctuation rules wrap-up

how the sat tests english grammar and writing rules

generally, students are more willing to study math topics than grammar topics…but that’s a mistake! because there are fewer grammar topics than there are math topics, they carry more weight on average.

and here’s the kicker: if you’re a native english speaker, a lot of that grammar (though not all) is already pretty natural to you. topics like transition words, subject-verb agreement, and tenses may take a keen eye at times, but if you train yourself to watch for them on the test, you’ll be using your innate english knowledge, which you exercise every time you speak. spend time on sat grammar practice, study our downloadable english grammar cheat sheet on your phone or computer, and get ready to seriously bump up your sat writing score.

when it comes to identifying grammar questions on the sat, you’ll want to focus on these types of questions:

of course, this is an excerpt from a longer passage with many other questions—both about sat grammar and vocabulary (although a strong knowledge of sat grammar rules can definitely help you figure out the best vocabulary words in context, too). which brings us to the big question…should you read the whole passage or try to answer the questions without reading it? after all, a lot of the questions are on the sentence level…

but what you’ll find is that even the sentence-level questions rely on a lot of context. the best thing to do is to practice sat grammar and time yourself as much as possible until you can at least skim the passage and then answer the questions. you’ll be glad you did when you see your score!

examples of sat grammar in the writing section

on test day, you’ll open your booklet to the writing section and see passages with numbered, underlined portions. the underlined sections could be as short as a word or as long as several sentences. you’re then asked to pick the best answer from among four.

you’ve already seen one example, but here are a couple more.

a handful of questions on test day will use the “delete” option—don’t rule it out without good reason! it’s right just as often as the other answer choices are.

what do these sat grammar questions have in common? their format, first of all. they all have the same first answer choice. they all refer to a passage. in addition to grammar, sat writing will also test your understanding of organization and flow, style (like when to use the passive voice), as well as some vocabulary, so you can expect to see a hodgepodge of question types on the official exam.

sat grammar tips for success

here are our top sat grammar tips to help you prepare for test day and choose the right answer.

• you won’t need to know sat grammar rules verbatim on test day…but you will definitely need to know them in practice. that’s why we’ve given you correct and incorrect sentences below to get you off on the right foot!
• grammar is important in not just one but two sat sections: writing and the essay. mastering it can thus boost your scores in two sections!
• keep reading to improve your grammar, but make sure what you’re reading is professionally proofread. major national newspapers, magazines, and books are all good sources…though be careful about dialogue in books, as authors often have characters use ungrammatical language to sound more natural.

tips for how to choose the correct answer

• the first answer choice in sat writing (when the question has no stem) is always no change—and this is correct as often as the others.
• you may sometimes see a choice to delete (or in sat terms, delete) the underlined portion—this can absolutely be the right choice.
• for the vast majority of questions, don’t go with what “sounds right.” this is a common trap because we often speak ungrammatically and what sounds right isn’t always right. (for example, “could of” sounds just fine…but it’s not actually a correct expression!)
• be careful that when you correct one mistake, you’re not introducing another. wordiness in particular often sneaks in there. read through all the answer choices to make sure you have the best one.
• identify the part of speech that’s underlined for a clue to what the question is testing. preposition? you might be looking at an idiom. adverb? check your modifiers. more on this below!

50+ sat grammar and punctuation rules

now that we’ve gone over how the sat english tests grammar and writing rules, along with our top sat grammar tips, let’s dive into the top 50+ sat grammar and punctuation rules you should know for test day.

sentence structure rules

fragments

sat grammar rule #1: each sentence needs a noun, a verb, and to express a full thought.

which of the following two examples is not a sentence?

he was tired.
because he was tired.

only the first one is a sentence. the second is a fragment.

both of these, however, are clauses. a clause is a phrase that contains a subject and a verb. it may or may not be a sentence.

so what’s the point of a clause?

breaking up long sentences into clauses helps us to better understand what each part of the sentence is doing. this will also help you be able to tell if you are dealing with an actual sentence, or if you are dealing with a fragment or a run-on. (hint: this is what the sat is really testing you on when it mentions clauses.)

subordinate clauses
now let’s take a look at a sentence that is made up of two clauses:

because he was tired, charlie decided not to run the race.

the first clause, “because he was tired,” is not a sentence. we call this a subordinate clause because it is not the most important part of the sentence. hence, it is subordinate or secondary to the main part or main clause of the sentence, “charlie decided not to run the race.” in other words, the big idea is that charlie didn’t run the race.

a subordinate clause is also called a dependent clause, because it “depends” on another clause. otherwise, it’s just a sentence fragment. similarly, a clause that is a complete sentence is called an independent clause. it’s independent; it doesn’t need to rely on any other clauses to be a sentence. the sat is not going to test you on the exact terminology, but it will test you on the ability to determine whether a clause is a fragment.

subordinating conjunctions
if i take a simple sentence like “he studied” and add a subordinating conjunction in front of it, what was a sentence is no longer a sentence; it is now a subordinate clause. subordinating conjunctions include because, although, even though, since, nevertheless, whereas, while. there are more, but a good shortcut to identifying whether a word is a subordinating conjunction is to ask yourself the following: does it provide a reason, contrast, or condition for the main clause?

unless you finish all of your broccoli, you can’t have any ice cream.

• • = condition

sarah was a star athlete, whereas her sister, maggie, would rather not get up off the couch. = contrast

sat grammar practice exercise

run-on sentences

sat grammar rule #2: don’t combine whole sentences with a comma.

if you can put a period between two pieces of a sentence without making a fragment, then they can’t be put together without some kind of grammatical glue. using a comma in place of a period is called a “comma splice,” and it’s one of the most common writing errors that high schoolers make. (so be sure to keep these sat grammar rules in mind!)

• • ✓

i can’t hear you. can you please use the megaphone?

• • ✗

i can’t hear you, can you please use the megaphone?

the sat loves to give you complex sentences to pick apart. a well-constructed sentence is made of one independent clause and one dependent clause. a run-on sentence is when there are too many independent clauses. there are four main ways to remedy a run-on.

• • • transform one of the excess independent clauses into a dependent one.
    • split the two independent clauses into two separate sentences.
    • use a semicolon between the two independent clauses.
    • use a comma and then a conjunction to make a compound sentence.

sat grammar rule #3:”and” alone is not enough
although using a comma and “and” together to join sentences is correct, “and” alone doesn’t finish the job. instead, it leaves you with the same problem that using a comma alone created: a run-on sentence. the same is true for “but,” “or,” and “so”—all of them need commas to combine sentences.

• • ✓

i can’t hear you, so can you please use the megaphone?

• • ✗

i can’t hear you and can you please use the megaphone?

of course, all of those words also have other jobs. “and,” “but,” and “or” can just combine lists of nouns, verbs, or adjectives, and then they don’t need commas.

• • ✓

i bought glow-in-the-dark sunscreen and went to the beach last night.

• • ✗

i bought glow-in-the-dark sunscreen and i went to the beach last night.

because there’s no “i” in the second half of that corrected sentence, the “and” is just combining “bought” and “went” into a short list, and that’s okay.

punctuation rules

without punctuation, written language would become much more difficult to understand. those tiny marks between and sometimes inside of words give us cues for reading emphasis, breaks, tone, and more. that’s why the sat will likely test you on punctuation rules.

commas

of all punctuation marks, commas may be the most versatile and the most commonly misused. there are so many ways to use commas that deciding whether to use one or not can be overwhelming. follow the sat punctuation rules below to know whether to add a comma.

sat grammar rule #4: every comma should serve a purpose, so never place one without a reason.

sat grammar rule #5: commas may go between independent clauses, but only when paired with a coordinating conjunction (fanboys).

• the dog raced after the car, but he was left in the dust.

 
strong style=”color: #c5168c”>sat grammar rule #6: commas may go after dependent clauses, prepositional phrases, or other descriptive phrases at the beginning of sentences.

• on the other hand, the fortune teller had been correct.

 
sat grammar rule #7: commas may be surrounding non-essential phrases, such as appositives.

• betsy, a lovely woman, offered to knit us socks.

 
sat grammar rule #8: commas may go between elements in a list.

• the child ate cotton candy, fried cookies, and ice cream at the fair.

 
sat grammar rule #9: commas may go between two adjectives that describe the same noun and are interchangeable.

• my nephew made me a funny, cute card.

 
sat grammar rule #10: commas may separate transitional words/phrases from the rest of the sentence.

• however, the judge disagreed.

 
sat grammar rule #11: commas go before or after quotes.

• he asked, “why?” “because it just makes sense,” she replied.

 
sat grammar rule #12: commas are used when addressing a person by name.

• peter, would you pass the pepper?

 
sat grammar rule #13: if you see a comma or an “and” underlined in a writing multiple choice question (but not used together), ask yourself whether they can be replaced by a period. if they can, then there’s the problem.

semicolons

this comma-colon hybrid can be an excellent alternative to a period between independent clauses.

sat grammar rule #14: a semicolon is meant to separate two clauses that, by themselves, could function as sentences.

sat grammar rule #15: if two sentences are talking about the same topic, a semicolon placed between them may improve the flow of ideas from one to the other. also, if you are writing a list and the items within the list include commas, separating the items with semicolons prevents confusion between all the commas.

• that supermarket is my favorite; it carries all my favorite foods for reasonable prices.
• on our trip, we visited washington, d.c.; chicago, illinois; and new york, new york.

colons

sat grammar rule #16: colons can also appear between independent clauses, but should really only be used when the second clause clarifies, or paraphrases the first.

the platypus is a strange animal: it lays eggs and has a bill, but is still considered a mammal.

the other way you can use a colon is after an independent clause that is followed by a list.

we will need the following for our camping trip: a tent, sleeping bags, and hiking boots.

hyphens

sat grammar rule #17: hyphens appear not just within sentences, but within words themselves.

adjectives can be formed of two or more words connected by hyphens, but these adjectives may only appear before the noun they describe.

the hand-made necklace was made of local turquoise.

compound numbers written as words must also use hyphens.

he has answered sixty-three math questions so far.

some of the most common places to see hyphens are after certain prefixes, between prefixes and proper nouns, after single-letter prefixes, and with the suffix -elect.

the all-cotton t-shirt depicted the president-elect.

apostrophes

sat grammar rule #18: contractions use apostrophes to stand in for letters that have been removed.

you shouldn’t poke the cat, so don’t do it.

apostrophes are also essential for indicating possession.

sat grammar rule #19: if a noun is singular or is plural but doesn’t end with an ​s​, add ​‘s.​ if it is plural and ends with an ​s​,​ just add an apostrophe.

our architect used the pta members’ ideas in the design for the children’s playground.

sat grammar rule #20: be careful! several contractions are homophones for possessive pronouns, which do not use apostrophes. for example, ​it’s​ means ​it is​, while ​its​ is a possessive pronoun.

parentheses

sat grammar rule #21: parentheses surround non-essential elements within sentences.

commas do the same, but parentheses are a bit more eye-catching, so they provide some extra emphasis for the phrases they surround. remember that parentheses always come in pairs.

the road trip (which was all my brother’s idea) was ruined when the car broke down.

dashes

sat grammar rule #22: like commas and parentheses, dashes set off non-essential elements in sentences.

they can be used in pairs, to set off phrases in the middle of sentences, or alone, to set off phrases at the end of sentences. dashes provide even more emphasis than parentheses, and they do a great job of catching readers’ eyes and drawing attention to certain parts of a sentence.

my friends — megan, john, and ricky — offered to help me move.
his face revealed a mix of emotions — confused, yet happy.

quotation marks

sat grammar rule #23: quotation marks always appear in pairs and closing quotation marks must be placed outside a sentence’s end-punctuation.

as their name suggests, quotation marks surround direct quotes. they can be used around an entire sentence or within sentences.

“i hope to see you again.”
“four score and seven years ago,” began lincoln.

sat grammar practice exercise

effective language use

redundancy

sat grammar rule #24: watch out for synonyms in the same sentence.

redundancy is simply repeating the same information over and over again, repeatedly, many times. (see what i did there?) in true sat style, redundancies will sometimes be hidden in a sentence like, “annually, my family goes to the beach every year.” we only need to include either annually or every year, since they mean the same thing.

sat grammar practice exercise

style and tone

sat grammar rule #25: hard to understand = bad! look out for over-formality, wordiness, and passive voice in sat writing and essays.

if you think a sentence sounds academic (and therefore good) because it’s hard to follow, then think twice. why is it tough? is it because of the structure or the meaning? unnecessarily complicated sentence structures are not a good thing. so make it clear.

every sentence in english includes a verb to describe an action. but there are basically two different kinds of verbs: connecting verbs and acting verbs.

connecting verbs don’t carry a whole lot of meaning. instead, they just join the pieces of the sentence. “be” is by far the most important connecting verb. there’s no action, really. it’s just a kind of glue.

some examples of sentences with connecting verbs:

my hobby is collecting other people’s receipts. i have a stash of thousands of them under my bed. not many people do as much digging through trash as i do.

these kinds of verbs are important—we couldn’t form sentences without them—but they’re inherently weak. they’re just not very descriptive.

if you use the action for the verb, you’ll usually have a stronger sentence. it’ll generally be more concise, too. notice how each of the acting verbs below were already included in the first set of sentences, but in different forms.

i collect other people’s receipts. i’ve stashed thousands of them under my bed. not many people dig through trash as much as i do.

yes, you should write formally for your sat essay. but you have to be natural. students often go too far, and while trying to sound more academic, they throw in crazy constructions. and more often than not, those clunky sentences have a connecting verb in the middle and an action verb transformed into some other part of the sentence.

fix them by finding the action and making it the main verb.

notice in that last one that there’s still a passive structure—”was convicted of”—but that doesn’t mean it’s wrong. it’s more important that the action is the main verb.

voice

sat grammar rule #26: you don’t want to say anything in a casual, lax way—or too formally, either.

the sat wants to make sure you know when to use the appropriate level of voice. that is, you don’t want to say something in a casual, lax way when writing an essay, or, in this case, editing an essay that the sat has provided.

in the sentence pairs below, one maintains a consistent tone whereas the other lapses into casual speak.

1a) the sat places harsh demands on a student’s attention, forcing them to be focused for almost four hours straight.

1b) the sat is really hard because you have to keep your mind on the task for four straight hours.

2a) working in groups can lead to high employee productivity while making people feel good about working with others in the office.

2b) working in groups can lead to high employee productivity while fostering good will between colleagues.

language can also be too stilted, or formal. the sat—and any manual of style!—will tell you to avoid overly formal language. can you spot the offender below?

3a) the sat is highly onerous on minds exhibiting a propensity for divagation.

3b) students who are serious about sat prep are more likely to seek outside help, and are thus more likely to see a score improvement.

if you are not quite sure what 3a) is saying, you are not alone. it uses over-inflated, pompous language. remember, there is a major difference between choosing a particularly apt word and using words like “divagation” (don’t worry, that word will not be on the test!) for the sake of sounding more intelligent. the latter will confuse your reader, and ultimately turn them off from what you have to say.

sat grammar practice exercise

subordination and coordination rules

conjunctions

sat grammar rule #27: when we have independent clauses (sentences that can stand on their own), and want to join them, we can use a comma and either a coordinating or a subordinating conjunction.

first, let’s deal with coordinating conjunctions:

• f: for
• a: and
• n: nor
• b: but
• o: or
• y: yet
• s: so

 
known as fanboys, these seven conjunctions will help you remember the coordinating conjunctions. it is important that you know the difference between these conjunctions and that you use the conjunctions appropriately.

kelly wanted to get straight a’s, so she studied every night and weekend.
 
✗
 
kelly wanted to get straight a’s, and she studied every night and weekend.

kelly wants to achieve something, so she does such-and-such.
 
✓
 
kelly was able to score at the top of the class, and she was even considered for the position of valedictorian later on.
 
✗
 
she was able to score at the top of the class, but she was even considered for the position of valedictorian later on.

“but” and “yet” are conjunctions that show contrast. “and” is a conjunction that shows that two independent clauses are related. in the examples directly above, we want a conjunction to connect, not to contrast, the idea that kelly did two important things.

parallel structure rules

creating parallel structure

sat grammar rule #28: whether it is with nouns, adjectives, verbs, or adverbs you need to make sure the form is consistent.

• • ✓

the girls at the sleepover said they wanted ice cream for dessert.

• • ✗

the girls at the sleepover said she wanted ice cream for dessert.

✓ the dog is hyper, large, and smelly.
✗ the dog is hyper, large, and doesn’t smell good.

sat grammar rule #29: remember that if you cannot physically count a noun, use the words “less,” “much,” and “amount.”

i had less strength after running the marathon.

sat grammar rule #30: remember that if you can physically count a noun, use the words “fewer,” “many,” and “number.”

i had fewer apples than i thought because my dad ate some.

modifier placement rules

misplaced modifiers

sat grammar rule #31: misplaced modifiers don’t modify the correct part of the sentence.

a modifier is an optional adverb, adjective, or phrase that changes—or modifies—the meaning of a sentence. for example:

i play the flute is an independent clause that totally makes sense by itself.

now let’s look at the sentence with a modifier (bolded):

i play the flute, a woodwind instrument.

the thing about modifiers, though, is that they have to be placed in the right part of the sentence to make sense. let’s see what happens if we switch up phrases:

a woodwind instrument, i play the flute.

don’t see what’s wrong with this arrangement? well, the first example makes it pretty clear that i play a woodwind instrument. the second example, however, makes it seem like i am the flute.

having played composers such as beethoven and mozart at a very young age, alicia keys’ songs have many elements of classical music.

do you spot the error here? the sentence seems to be implying that alicia keys’ songs played beethoven and mozart at a very young age rather than alicia keys, which is silly. a better way to correct the non-modifier part of the sentence is:

having played composers such as beethoven and mozart at a very young age, alicia keys has many elements of classical music in her songs.

sat grammar practice exercise

 
sat grammar rule #32: dangling modifiers don’t actually modify anything in the sentence.

hence, it is “dangling,” or not attached to anything. usually, the modified clause is in passive voice.

• • ✓

typing furiously, she finished her essay.

• • ✗

typing furiously, the essay was finished.

sat grammar practice exercise

sat grammar rule #33: squinting modifiers logically make sense as modifiers for two subjects in the same sentence.

therefore, it makes the sentence ambiguous in meaning. squinting modifiers are the trickiest to identify because the sentence doesn’t have any obvious errors. furthermore, squinting modifiers don’t come up as often on the sat, probably because their ambiguity makes it hard to come up with one correct answer choice. just in case, though, it’s still a good idea to know what squinting modifiers look like and the sat grammar rules needed to correct them.

• • ✓

students who don’t practice often for the sat score lower than they would like.

• • ✗

students who don’t practice for the sat often score lower than they would like.
 
sat grammar rule #34: when encountering a modifier problem, first identify the incorrectly-placed modifier.

let’s say the following is an sat question:

sonia was waiting for her book to come in the mail eagerly.

the underlined portion is what you’re supposed to change. first identify the incorrectly-placed modifier: “eagerly.”

sat grammar rule #35: change a modifier problem in one of two ways: figure out either where to place the modifier, or how to correct the subject.

in the same example sentence above, figure out where to place the modifier “eagerly,” or how to correct the subject (if it should go before or after “waiting”). scan for the answer choice that matches your prediction.

a lot of times–not always, but a lot of times–the corrected modifier or subject should be at the beginning of the correct answer! if you find that to be the case and you only see one answer choice that matches, you already know the right answer without reading through the choices. pick it and move on! (and don’t be afraid to pick “no change” if you think that’s the case!)

sat grammar practice exercise

adverb placement

sat grammar rule #36: determine if there are any words modifying a verb or adverb, and verify that they are also adverbs.

an adverb is a word that modifies a verb or adjective. it usually answers any of the following questions: how much, how often, and where?

he frequently studied.

• • (how often did he do the verb

study

• • ?)

she was extremely displeased. (to what extent was she the adjective displeased?)

he was known to run through the mountains. (where did he do the verb run?)

few people use adverbs properly in everyday speech. and when there is a disconnect between the way people speak and the proper way to speak, you can bet the college board is waiting for a grammar question to snare you.

to see how we can fail to spot the proper use of the adverb, consider the following examples. if you rely on your ear, you will probably think that both sentences are fine as is.

mary yelled to tina, “drive safe – it’s wet out there.”

the new pc was the most clever designed computer the market had seen in years.

in the first case, the verb is “drive.” the word that modifies “drive” is “safe.” “safe” is an adjective, and you cannot modify a verb with an adjective—you need an adverb. to make “safe” an adverb simply add -ly: drive safely.

for the second sentence, we need an adverb to describe how a computer is designed. “clever” is an adjective. so by adding -ly, we now have an adverb: most cleverly designed computer.

verb rules

verb tenses

sat grammar rule #37: make sure the times given in the sentence are consistent and logical.

any time you see a verb underlined, you should check that the tense given feels natural with the times that the rest of the sentence presents. do you smell anything fishy in this sat grammar example?

ernest hemingway’s short stories, including the favorite “indian camp,” continue to be highly influential pieces of fiction despite the fact that they have been written over fifty years ago.

if you do, then you might be onto something. it might be rotten. the time “over fifty years ago” sounds pretty strange when put next to “have been written.” you don’t need to know why; you just need to know it’s wrong.

the good news is that for native english speakers, the different times that different tenses signify are already hard-wired into your thought patterns. all you have to do on the sat is make sure the times given in the sentence are consistent and logical.

sat grammar rule #38: if there’s a sequence of events, make sure their tenses put them in the right logical order.

this one is about making sure words in the sentence all match up, by watching out for when events happened. in the past, or in the present?

• • ✓

having just been swimming, maria smelled like chlorine.

• • ✗

having just been swimming, maria had smelled like chlorine.

each english verb has a few basic forms. kids who learn english in other countries can often rattle off lists like do/did/done and eat/ate/eaten faster than native speakers can. and there are some verbs, like swim, which even native speakers get a little confused about at times. take a look and make sure you know these sets.

• • • swim/swam/swum
    • ring/rang/rung
    • forget/forgot/forgotten
    • forgive/forgave/forgiven
    • lay/laid/laid
    • rise/rose/risen
    • swing/swung/swung

there are countless others, most of which you wouldn’t think twice about. but some of them might have you making things up in conversation (e.g., “i would’ve swang if i’d known he was going to keep throwing strikes.”) that just don’t fit sat grammar rules.

present perfect

sat grammar rule #39: has/have + participle = describes an action/event that happened in the past and continues in the present.
to illustrate, let’s take a look at the following sentences:

1) last night, i walked my dog.

2) i have walked bucky every night for the last two years.

in the first sentence, i am doing the action, “walk,” only once. this is the simple past. in the second sentence, i am describing something that has taken place on a number of occasions in the past and continues on till today (meaning tonight i will most likely walk bucky). this is the present perfect.

sat grammar practice exercise

past perfect

sat grammar rule #40: had + participle = describes an action/event in the past that happened before another action in the past.

whenever you are dealing with two events in the past, one of which started or happened before the other, you must use the past perfect tense to describe the event that started first.

before i moved to california, i had walked bucky in the mornings, not at nights.

sat grammar rule #41: whenever we use the past perfect, we must also have another verb in the sentence that is in the simple past.

here’s an example using the sentence above:

first event: i had walked bucky in the morning = past perfect construction

second event: i moved to california = simple past

another way to think of the past perfect is with specific dates. let’s say i moved to california in 1984. i walked bucky every morning from 1981 to 1984. the sentence implies that once i moved to california i no longer walked bucky in the morning. that is, an event that happened repeatedly in the past stopped when another event happened. that interrupting event uses the simple past.

sat grammar practice exercise

imperative mood

sat grammar rule #42: “command words,” or words such as “request that,” “order that,” “require that” are followed by “be + verb participle.”

the teacher demanded that the hyperactive be seated the entire class.

he requested that she be present at the meeting.

subjunctive mood

sat grammar rule #43: the subjunctive mood implies conditionality, in other words a hypothetical, i.e., something that isn’t and can’t be reality. to show this, we change the verb “was” to “were.”

were we space aliens, we could travel the cosmos.
were she responsible for the break-in, she would not have an airtight alibi.
if i were president of the united states, i would make “taco tuesdays” a national holiday.

if the verb is already “were,” then we just leave it. when something is possible, then you don’t need to change the verb to “were.”

notice how i didn’t mention the “interrogative mood.” that’s because nothing changes with the verb; therefore, there’s no easy way for the sat to test that the interrogative mood is being used.

in general, “mood” is so rare on the test that i think it showed up in one question of the four practice tests in the college board book. so if you have to skip one section in your sat grammar practice, this might be the one.

agreement rules

subject-verb agreement

sat grammar rule #44: the sat likes to put extra information between the subject and verb. if you cross out everything between the subject and the verb, the correct choice will be much more obvious.

subject-verb agreement is a basic building block of english language sentence structure. because it is so normal in your speech and writing patterns, your eyes can glide right past well-hidden subject-verb agreement errors. be sure to watch for underlined verbs, and clauses set off by commas or dashes.

• • ✓

the teachers, who loaded up their trays with pizza, cookies, and chocolate milk, stand at the back of the cafeteria.

✗ the teachers, who loaded up their trays with pizza, cookies, and chocolate milk, stands at the back of the cafeteria.

when the sat adds all that info, they will often make the word closest to the verb the opposite of the subject. if the subject is singular, the word in front of the verb might be plural, just to throw you off. sneaky, eh?

pronoun rules

sat grammar rule #45: a pronoun refers to a noun in the sentence.

sat grammar rule #46: a pronoun can either take the subjective or objective form depending on its role in the sentence.

he called me.

• • • subject: he (thing or person committing the action)
    • object: me

i called him.

• • • subject: i
    • object: him

mary lent her book to tom. she told him to return it to her soon.

• • • for mary, she is the subject pronoun and her is the object pronoun. notice how in the second sentence, the recipient of the action is mary, which means the pronoun must be in the object case. so we use her.
    • the second sentence does not use he but rather him. if we are talking about the subject of the sentence and we are referring to tom, then we want to use he. if tom is the object of the sentence, then we want to substitute him.

sat grammar rule #47: when you’re listing other people as well as yourself, i or me should come last.

this is an sat english favorite. it’s shown up on a number of tests, and it’s a perfect trap for anybody who trusted their 8^th grade english teacher just a little too much. there is, admittedly, plenty of truth in these sat grammar rules: when you’re listing other people as well as yourself, i or me should come last. think of it as being polite and holding the door for the other names.

when the names are the subject, use i.

• • ✓

my pinkie toe and i have been through some hard times.

• • ✗

the world is against my pinkie toe and i.

• • ✗

a polar bear ate my pinkie toe and i.

by the same token, don’t start a sentence with “…and me.”

• • ✓

my pinkie toe and i fought off a polar bear.

• • ✗

my pinkie toe and me fought off a polar bear.

if the names are not the subject—a notable example being after prepositions like against—then use me, the object form.

• • ✓

the world is against my pinkie toe and me.

• • ✓

a polar bear ate my pinkie toe and me.

if these sat grammar rules are a bit unclear, then take out the other player (e.g. “my pinkie toe”) and see how it sounds. saying “a polar bear ate i” sounds good to approximately nobody, so don’t write it—even if there’s a pinkie toe that comes first.

also, you can roughly assume that i is more likely correct near the beginning of the sentence, while me is more common in the middle or end.

possessive determinant rules

sat grammar rule #48: its = possessive. it’s = it is.

the sat is going to try to trip you up on a distinction that many—even seasoned writers—flub. many assume that an apostrophe always means possessive. it doesn’t. it can also be a contraction, or a shortening. in this case, the shortening is of “it is.”

a marathon is a challenging race. it’s a combination of endurance, grit, and focus.

sat grammar rule #49: ‘s = singular possessive. s’ – plural possessive.

next, we have agreement in number. by number i mean either singular or plural. for instance, in the previous sentence, the subject is marathon, so the pronoun needs to be singular. we want to use “it’s,” not “they are” or “they’re.” had the subject been marathons, the pronoun would be “they are” or “they’re.”

on this subject, ross always made me remember the distinction between “you’re” and “your” better than even my english-teacher mother could.

sat grammar practice exercise

diction rules (frequently confused words)

sat grammar rule #50: know that diction questions will appear, and always be on guard if you can’t seem to find a mistake in the question.

one of the least common—but no less diabolical—errors you can encounter is called the diction error. they pop up about once every other test, if that often. you may even be familiar with this error type—it’s the old switcheroo, where one word that sounds an awful lot like another word is incorrectly used in a sentence.

the scientists, once they discovered that each had tackled a similar line of stem-cell research, realized that they would both benefit if they corroborated in their research.

the word should be “collaborated,” and not “corroborated,” which means to confirm or give support to evidence.

the first two years are the most formidable—it is during this time that many of our enduring personality traits form.

the word should be “formative,” not “formidable,” which describes something that is so big or amazing that it commands respect (think of lebron james’s skills on the basketball court).

so what to do? carefully, look to see if the correct word is being used. below is a list of some of the most common diction errors.

• • • affect vs. effect
    • conscientious vs. conscious vs. conscience
    • could of vs. could have vs. could’ve (hint: “could of” is always wrong!)
    • crutch vs. crux
    • elicit vs. illicit
    • empathetic vs. emphatic
    • flaunt vs. flout
    • imitated vs. intimated
    • ingenuous vs. ingenious
    • irregardless – no such word

logical comparisons

sat grammar rule #51: note the word “than.” it almost always signals that we’re making a comparison, so you know to check for a problem there.

after noticing the “than,” we should look for the adjective or verb that it’s being paired with, e.g. “enjoy more than” or “is easier than.” in the sentence below, our adjective is tougher.

oranges’ skins are tougher than lemons’.

sat grammar rule #52: when making comparisons, find the nouns—where the problem might happen.

some of the most difficult sat grammar questions deal with correct comparisons between nouns. there are two major reasons why they cause such a problem. the first is that we make these kinds of “mistakes” pretty often in conversation. to the untrained proofreader, they seem pretty natural.

the second is that they can often be fixed with a possessive. if you want to compare the color of sam’s and brian’s tongues, then the sentence “sam’s tongue is redder than brian” is clearly weird, whereas “sam’s tongue is redder than brian‘s” would be just fine. that’s all well and good, but the sat might not make it so easy for you. instead, the two nouns may already be plural.

• • ✗

oranges’ skins are tougher than lemons.

• • ✓

oranges’ skins are tougher than lemons’.

conventional expression rules

sat grammar rule #53: when it’s a preposition that’s underlined, it’s all about your ear.

ask yourself, “does that sound right?” of all our sat grammar tips, this might be the only time that i say to go with your gut about an error. most times, you should be trying to find why something is wrong as it’s written.

the test-makers want to know if you can choose the right preposition to go with words which actually do carry the meaning they’re expected to. for instance, you might disagree with an idea, but you can’t object with it. instead, you object to it. both “disagree” and “object” mean exactly what we expect them to in those phrases. it’s only the preposition that’s weird.

sat grammar rule #54: if you see an underlined preposition, find the words before and/or after that should get paired with it.

pretty often, you’ll find the verb or adjective that came before the preposition, like in the “disagree” vs. “object” example above, but sometimes it’s about the words after the preposition, instead.

in the sentence “i’m on my classmate’s phone,” you’d note the word “phone,” and ask whether “on” and “phone” link well together, which they do. similarly, an actor might appear in a movie, on tv, or at a restaurant.

whether it’s the word before or after the preposition, you have to find that related word. then, it’s all about your ear.

rules about items in a series

parallelism

sat grammar rule #55: words in a sentence, or sentences in a paragraph, should all be alike in structure.

parallelism involves making sure words in a sentence, or sentences in a paragraph, are all alike in structure. you’ll see this most frequently with a list of verbs.

learn and test your knowledge with the sat grammar rules pdf

whew! great job, guys. now it’s time to take what you’ve learned in this post and start self-studying! just enter your email to download the sat grammar rules pdf for free! think of this as your english grammar cheat sheet for when you’re studying and reviewing grammar on the go!

sat grammar practice quiz

ready to test your knowledge with an sat grammar practice quiz? let’s get started!

unlike on test day, we’ve only underlined the word the question tests to make it easier for you to read on-screen. however, just like on test day, the instructions for these questions are as follows.

(bonus: now you know the instructions, you won’t have to waste time reading them on test day!)

ready? here you go!

 10%

question 1 of 10

1. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
 

no change

all the same

therefore

so

question 1 of 10

question 2 of 10

2. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
 

no change

i and my sisters

my sisters with me

my sisters and i

question 2 of 10

question 3 of 10

3. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
  

no change

house; just like

house as though

house similar to

question 3 of 10

question 4 of 10

4. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
  

no change

number

weight

size

question 4 of 10

question 5 of 10

5. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
 

no change

white plate, like at home; they

white plate, like at home, but they

white plate, like at home, instead they

question 5 of 10

question 6 of 10

6. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
 

no change

was

might be

might have been

question 6 of 10

question 7 of 10

7. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
 

no change

invisible generations, who were vividly illustrated and turning to dust

invisible generations, whose books had been vividly illustrated and turning to dust

invisible generations, the tomes vividly illustrated and turning to dust

question 7 of 10

question 8 of 10

8. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
 

no change

embarked

entered

enraged

question 8 of 10

question 9 of 10

9. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
 

no change

although

nevertheless

despite that,

question 9 of 10

question 10 of 10

10. my grandmother wasn’t a rich woman, but somehow my sisters and me always viewed a visit to her house; as though it were a visit to a fairy-tale castle. in retrospect, i think this probably had to do with the sheer amount of mysterious objects she’d accumulated. when she made cookies for us, they weren’t served on a plain white plate, like at home, they were served in a series of cookie jars with figures of ballerinas, tiny dogs, or even quotes painted at the bottom. her “grown-up” book collection wasn’t comprised of just the dry, academic tomes my parents had around the house, but instead were made up of books from the seventeenth, eighteenth, nineteenth centuries, inherited from invisible generations, vividly illustrated and turning to dust. other trinkets around the house attested to the unknowability of the past, from the watercolor portraits fading in the sunlight to the tiny dish that showed our long-distant ancestors engaged in that medical “cure,” blood-letting. even granny’s dog fascinated us: unlike maggie, our placid and patient mutt, her purebred cocker spaniel bairn watched us adroitly, with thinly disguised disdain that quickly produced a low growl if we got too close.
  

no change

aloofly

arcanely

absurdly

question 10 of 10

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sat grammar and pronounciation rules wrap-up

if you’ve read this all in one go, your eyes are probably crossed by now. don’t worry! that’s why we made this english grammar cheat sheet for you (it’s a lot shorter!): magoosh 50+ sat grammar rules pdf. read through it, quiz yourself, come back and brush up on areas where you might be rusty.

in the meantime, don’t forget to prepare for the sat writing test format as well as its content, and keep quizzing yourself (here and elsewhere) until you’re happy with your progress.

need more help?

with magoosh sat, 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 50+ sat grammar and punctuation rules to know (with video) 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.

retaking the act? you’re not alone—lots of test-takers find that after their first official exam, they want different results. but the reality is most students sign up for an act retake and do exactly the same thing over again. it’s time to learn from the experience! let’s do some soul-searching about how to study for an act retake.

are you wondering how to retake the act? you need to focus on all aspects of the testing experience. this means evaluating your previous preparation–academically, strategically, mentally, and physically–and knowing which areas you need to improve to get a better score. by assessing all of the key areas, you’ll have a much better chance of giving the act the one-two (three-four) punch on your retake and avoid the frustration of the same old result.

how to prepare for an act retake

1. brush up on your weaknesses…

as soon as you can, write down everything you remember about the question types you struggled with on the act.  did you want to kick yourself when you couldn’t remember your trig identities? did you spend too much time reading (and rereading) a passage and then ran out of time to answer the questions?

as you continue to practice for your retake, keep a running list of your weak areas so you can practice them more. writing them down is crucial. it keeps you accountable and ensures you have a game plan for your studying.

if you haven’t done so already, i suggest going through these lists of what to study for the act english, math, reading, science, and writing sections. don’t fall prey to unfamiliar questions again. it’s okay if you are still weak in certain areas, but knowing your strengths and weaknesses is half the battle: this is how you know when to tackle problems and when to take a guess and move on.

2. …but capitalize on your strengths

each question counts exactly the same. so if you are really good at a certain type of question, make sure you practice these too so you can nab as many points on them as possible. particularly if you’re retaking the entire exam, you should be very careful to not brush up only on your weaknesses when you study for a retake.

i’ve seen countless students be disappointed by their overall score when they see their weak section scores go up, but their strong section scores go down. but the act has recognized this is a problem. and with that in mind, they’ve come out with some exciting news!

as of september 2020, the act will allow you to retake individual sections, rather than the test as a whole! this is great news if you’re struggling with an individual subject but don’t want to risk lowering your score in other sections.

3. use the incredibly helpful test information release service

if you took a december, april, or june test, i highly suggest you order the test information release service if you have the time to wait for the report.

this service provides you with a booklet of all the questions and your answers on the act, so you can see exactly which questions you missed that you can use as you figure out how to study for an act retake, providing you with extremely useful data about why you ran into trouble.

4. define your goals for the retake

after you receive your act scores, make a plan. do you want to increase your score by 2 points? 10 points? do you want to remember to relax and breathe so you can think of a good response on the essay section? write down a concrete list of your goals for your retake and a timeline and plan of attack for accomplishing them.

as you make this list, consider what your goals are for each section. remember, sectional retakes will be a possibility after september 2020!

finally, if you are targeting certain colleges, make sure you know the act score ranges of admitted students, so you know what you are aiming for and make sure you have the time and resources you need to accomplish this gain.

5. determine your retake study method

it’s time to be honest with yourself about how you study best to figure out how to study for an act retake successfully. there are a few main categories of test preparation:

1. self study (or guided self study like our one month act study plan or act streaks schedule!)
2. classes
3. tutoring

everyone learns differently. evaluate how you studied the first time and ask yourself if it was best suited to your needs. do you need a class that will hold you accountable for your homework? would self study fit better with your busy schedule? do you need a tutor who can help you with stress management? could you benefit from a combination of all of these methods? now’s the time to make a change and give yourself a fighting chance on your act retake.

6. use score choice

being strategic in your preparation also means keeping in mind the bigger picture. many colleges allow you to choose which composite score you send, which is great because it puts you in control of what test report you send, taking the risk out of retaking the test. even better is what is popularly known as superscoring.

although it is more common with the sat than the act, some colleges will allow you to combine your highest section scores across test administrations. that means you can combine your best individual section scores into a super-awesome overall score. as you determine how to study for an act retake, considering using the superscore may provide you with an important tool to cut your study time down.

keep in mind that there are a handful of competitive schools out there that require you to send all of your scores. so if you are eyeballing these schools, make an effort to identify how many times you should retake the act.

7. prepare both mentally and physically

many students fall victim to test pressure and anxiety on the act, particularly the first time because it is a new experience. the second (or third or fourth) time you take the test, you will have a better idea of what to expect.

so now is the time to do some really great mental and physical preparation.  if you got tired or hungry or overwhelmed, your most important preparation may not be studying questions. instead, learning how to be healthy and manage stress may be the best thing you can do as you determine how to study for an act retake. snacks and sleep can make a world of difference.

and don’t forget that there is such a thing as too much act prep. some students are simply burned out on test prep. if this sounds like you, ask yourself if what you really need for your retake is a temporary break.

how to retake the act: a student’s perspective

for a firsthand account of how to study for an act retake, watch the embedded video below!

in “how to raise your act score: my retake tips” magoosh student, nikolai, explains how he took his act score from 24 to 33 over the course of several retakes.

the post how to study for an act retake appeared first on magoosh blog | high school.

 
are you interested in pushing the boundaries of science and technology, designing video games, or innovating in the business world? if so, then rensselaer polytechnic institute might be a good fit for you!

rpi is known worldwide for a focus on science, engineering, and technology, and is consistently ranked as one of the top fifty universities in america. rensselaer’s campus boasts a business incubator, one of the most powerful university supercomputers, and an annual student video game showcase.

rensselaer polytechnic institute is also considered one of the top six universities in america in terms of median graduate income, meaning their alumni are rewarded not only with state-of-the-art facilities and rigorous academics but the tools they need to build a career straight out of college.

with a student population of 7,500 and only ten miles between campus and albany, new york, you can count on a vibrant student life in addition to a solid education.

if their motto “why not change the world?” resonates with you, rpi could be the place for you!

rensselaer polytechnic institute at a glance

some of those are pretty high numbers…but they are achievable!

let’s take a deeper dive into rensselaer polytechnic institute admissions, and find out what you need to do to make yourself a competitive candidate for rpi!

rensselaer polytechnic institute sat scores

while the composite rensselaer polytechnic institute sat scores are in the above table, here’s a breakdown by section:
 

rensselaer doesn’t have a minimum sat score required for admission, but the average composite sat score of admitted students in 2019 was a 1415 out of 1600.

on the writing test the middle 50% of students scored between a 640 and a 720, while in math the middle 50% scored between a 690 and a 780. this means 25% of admitted students scored lower than a 640 in writing or a 690 in math, while another 25% scored about a 720 in writing and a 780 in math.

if you can get scores above the middle 50% of admitted students, you can be fairly confident about your chances–if the rest of your application stands up. if not, rpi is a selective school, so you should be prepared to support your scores with strong grades and diverse extracurriculars.

rensselaer polytechnic institute act scores

the average act of students admitted in 2019 was a 31 out of 36.

the middle 50% of students, meaning those scoring between the 25% and 75% percentiles, got between a 29 and a 33. that means scoring below a 29 will put you at a disadvantage, but one quarter of students admitted that year scored lower than a 29 and still got in.

if you have a 33 or above, you can be confident about your chances: only 25% of admitted students had scores that high!

rensselaer polytechnic institute gpa average

students admitted to rpi in 2018 had an average gpa of 3.87–which means the rensselaer polytechnic institute admissions office expects you to be near the top of your class.

two-thirds of admitted freshman had a gpa of 3.75 or higher, 21% had between a 3.50 and a 3.74, 8% had between a 3.25 and a 3.49, and only 4% had gpas between a 2.00 and a 2.99.

not sure how to calculate your gpa? you can figure it out with our gpa calculator!

rensselaer polytechnic institute acceptance rate

rensselaer’s acceptance rate is 43%. this year a record total of 20,402 high school students applied to rpi, with 8,773 admitted and 1,755 enrolled.

although a 43% acceptance rate is not too selective (about 2 out of every 5 applicants are admitted!) increasing numbers of applicants may drive rpi’s admissions to become more competitive in coming years.

rensselaer polytechnic institute freshman profile

so who is the typical rensselaer freshman? let’s take a look at the class of 2022:

• international students: 13.6% from 39 countries
• students from u.s: 86.4%
• american indian/alaskan native: 0.1%
• asian: 13.7%
• black/african-american: 4.6%
• hispanic/latino: 10.4%
• multi-race (not hispanic/latino): 7.8%
• white: 61.8%
• unknown: 1.6%

rensselaer polytechnic institute other admissions requirements and info

let’s dig a little deeper, and see what else you need to get into rensselaer polytechnic institute, besides your gpa and test scores:

rpi has the following admissions recommendations:

• four years of mathematics through pre-calculus (calculus recommended)
• three years of science (including chemistry and lab-based physics)
• four years of english
• three years of social studies and/or history

additionally, the rensselaer polytechnic institute admissions committee will be looking for students who demonstrate leadership qualities and talents that will add to the rpi community!

expert advice on how to get into rensselaer polytechnic institute

we talked to ashley c. from the college prep organization transizion, to get her expert advice on how to get into rensselaer polytechnic institute university.

let’s see what she has to say!

rensselaer polytechnic institute faq

does rensselaer accept transfer students?

yes! any student with 12 or more transferable college credits completed after high school is eligible to apply to rpi as a transfer student.

transfer applications are reviewed on a rolling basis, but there are final deadlines to apply to each semester:

• fall: apply by june 1
• spring: apply by november 1
• summer (architecture only): apply by march 1

after your application is submitted, you may expect a decision in approximately 3-4 weeks!

does rensselaer accept the common app?
yes! rensselaer accepts the common app, the coalition application, or the candidate’s choice application.

the rpi admissions office weighs all applications equally, so feel free to use whichever of these application platforms is the most convenient for you!

can non-native english speakers apply to rensselaer?
yes! however, if english is not your first language, you will be required to submit your test of english as a foreign language (toefl) scores as part of your rpi applications!

how to get into rensselaer polytechnic institute

how to get into rensselaer polytechnic institute?

let’s do a quick summary!

here’s your “how to get into rensselaer polytechnic institute” checklist…

you need:

check those boxes, and you’ll be all set for your new life in new york–just don’t forget a jacket!

the post rensselaer polytechnic institute admissions: the sat, act scores and gpa you need to get in appeared first on magoosh blog | high school.

interval notation: the types of intervals

there are a few different types of intervals that commonly arise when studying math, called the open interval and the closed interval, notated respectively as (a, b) and [a, b].

interval notation for open intervals

the open interval uses parentheses, and they signify the fact that the interval contains all the real numbers x that are strictly between the numbers a and b, i.e. the interval does not actually contain the numbers a and b. another way of notating an open interval is the set of all x such that a < x < b.

interval notation for closed intervals

in the case of the closed interval, the square brackets are used to indicate that the endpoints are contained in the interval. therefore we can notate a closed interval as the set of x so that

interval notation for half-open intervals

there are slightly fancier intervals, called half-open intervals, notated as (a, b] and [a, b), which are the respective sets of all x so that , and .

an interval is called bounded when there is a real positive number m with the property that for any point x inside of the interval, we have that |x| < m.

observations on intervals

supposing as in the setup that a < b, then how many numbers are actually in the interval (a, b)? it turns out that there are uncountably infinite numbers in any interval (a, b) where a < b, no matter how close a and b are together.

it is a fact that actually, there are the same quantity of real numbers in the interval (0, 1) as there are in the entire real numbers, also represented by the interval . this seems counterintuitive, because one interval seems so much more vast than the other, but it is not a contradiction, but rather a beautiful subtly of set theory.

calculus and intervals

intervals arise regularly in calculus, and it will be important for you to know the difference between a closed interval and an open interval, since there are some theorems, like the intermediate value theorem, which requires that the interval upon which the function is defined is a closed and bounded interval.

closed and bounded intervals touch on one of the most important concepts in the broader study of calculus, that of compactness. many central theories in calculus revolve around compact sets, which in the setting of the real numbers are exactly the closed bounded intervals.

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for a quick review of integration (or, antidifferentiation), you might want to check out the following articles first.

• ap calculus exam review: integrals
• ap calculus exam review: antidifferentiation

and now, without further ado, here are some of the most common integrals found on the ap calculus exams!

common integrals

the following seven integrals (or their close cousins) seem to pop up all the time on the ap calculus ab and bc exams.

1. remember your trig integrals!

trigonometric functions are popular on the exam!

2. simple substitutions

you need to recognize when to use the substitution u = kx, for constant k. this substitution generates a factor of 1/k because du = k dx.

for example,

3. common integration by parts

integrands of the form x f(x) often lend themselves to integration by parts (ibp).

in the following integral, let u = x and dv = sin x dx, and use ibp.

4. linear denominators

integrands of the form a/(bx + c) pop up as a result of partial fractions decomposition. (see ap calculus bc review: partial fractions). while partial fractions is a bc test topic, it’s not rare to see an integral with linear denominator showing up in the ab test as well.

the key is that substituting u = bx + c (and du = b dx) turns the integrand into a constant times 1/u. let’s see how this works in general. keep in mind that a, b, and c must be constants in order to use this rule.

5. integral of ln x

the antiderivative of f(x) = ln x is interesting. you have to use a tricky integration by parts.

let u = ln x, and dv = dx.

by the way, this trick works for other inverse functions too, such as the inverse trig functions, arcsin x, arccos x, and arctan x. for example,

6. using trig identities

for some trigonometric integrals, you have to rewrite the integrand in an equivalent way. in other words, use a trig identity before integrating. one of the most popular (and useful) techniques is the half-angle identity.

7. trigonometric substitution

it’s no secret that the ap calculus exams consist of challenging problems. perhaps the most challenging integrals are those that require a trigonometric substitution.

the table below summarizes the trigonometric substitutions.

for example, find the integral:

here, the best substitution would be x = (3/2) sin θ.

now we’re not out of the woods yet. use the half-angle identity (see point 6 above). we also get to use the double-angle identity for sine in the second line.

note, the third line may seem like it comes out of nowhere. but it’s based on the substitution and a right triangle.

if x = (3/2) sin θ, then sin θ = (2x) / 3. draw a right triangle with angle θ, opposite side 2x, and hypotenuse 3.

by the pythagorean theorem, we find the adjacent side is equal to:

that allows us to identify cos θ in the expression (adjacent over hypotenuse).

finally, θ by itself is equal to arcsin(2x/3).

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a slope field shows the direction of flow for solutions to a differential equation.

what is a slope field?

a slope field is a visual representation of a differential equation of the form dy/dx = f(x, y). at each sample point (x, y), there is a small line segment whose slope equals the value of f(x, y).

that is, each segment on the graph is a representation of the value of dy/dx. (check out ap calculus review: differential equations for more about differential equations on the ap calculus exams.)

because each segment has slope equal to the derivative value, you can think of the segments as small pieces of tangent lines. any curve that follows the flow suggested by the directions of the segments is a solution to the differential equation.

each curve represents a particular solution to a differential equation.

example — building a slope field

consider the differential equation dy/dx = x – y. let’s sketch a slope field for this equation. although it takes some time to do it, the best way to understand what a slope field does is to construct one from scratch.

first of all, we need to decide on our sample points. for our purposes, i’m going to choose points within the window [-2, 2] × [-2, 2], and we’ll sample points in increments of 1. just keep in mind that the window could be anything, and increments are generally smaller than 1 in practice.

now plug in each sample point (x, y) into the (multivariable) function x – y. we will keep track of the work in a table.

ok, now let’s draw the slope field. remember, the values in the table above represent slopes — positive slopes mean go up; negative ones mean go down; and zero slopes are horizontal.

spend some time matching each slope value from the table with its respective segment on the graph.

it’s important to realize that this is just a sketch. a more accurate picture would result from sampling many more points. for example, here is a slope field for dy/dx = x – y generated by a computer algebra system. the viewing window is the same, but now there are 400 sample points (rather than the paltry 25 samples in the first graph).

slope field for dy/dx = x – y

analyzing slope fields

now let’s get down to the heart of the matter. what do i need to know about slope fields on the ap calculus ab or bc exams?

you’ll need to master these basic skills:

• given a slope field, select the differential equation that best matches it.
• given a slope field, estimate values of a solution with given initial condition.
• sketch a slope field on indicated sample points, from a given differential equation.

we’ve already seen above how to sketch a slope field, so let’s get some practice with the first two skills in the list instead.

sample problem 1

the slope field shown above corresponds to which of the following differential equations.

  a. dy/dx = y²

  b. dy/dx = sin y

  c. dy/dx = -sin y

  d. dy/dx = sin x

solution

look for the clues. the segments have the same slopes in any given row (left to right across the graph). therefore, since the slopes do not change with respect to x, we can assume that dy/dx is a function of y alone. that eliminates choice d.

the horizontal segments occur when y = 0, π, and -π. however, the only point at which y² equals 0 is y = 0 (not π or -π). that narrows it down to a choice between b and c.

finally, notice that slopes are positive when 0 < y < π and negative when -π < y < 0. this pattern corresponds to the values of sin y. (the signs are opposite for -sin y, ruling out choice c).

the correct choice is b.

sample problem 2

suppose y = f(x) is a particular solution to the differential equation dy/dx = x – y such that f(0) = 0. use the slope field shown earlier to estimate the value of f(2).

  a. -2.5

  b. -1.3

  c. 0.2

  d. 1.1

solution

because f(0) = 0, the solution curve must begin at (0, 0). then sketch the curve carefully following the directions of the segments. it helps to imagine that the segments are showing the currents in a river. your solution should be like a raft carried along by the currents.

then find the approximate value of f(2) on your solution curve.

the best choice from among the answers is: d. 1.1.

slope fields and euler’s method

often if you see a slope field problem in the free response section of the exam, one part of the problem might be to use euler’s method to estimate a value of a solution curve.

while the slope field itself can be used to estimate solutions, euler’s method is much more precise and does not rely on the visual representation. check out this review article for practice using the method: ap calculus bc review: euler’s method.

summary

• slope fields are visual representations of differential equations of the form dy/dx = f(x, y).
• at each sample point of a slope field, there is a segment having slope equal to the value of dy/dx.
• any curve that follows the flow suggested by the directions of the segments is a solution to the differential equation.

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properties of logarithms

first of all, let’s review what a logarithm is. more specifically, we need to understand how the logarithm function can be used to break down complicated expressions.

you might want to check out the following article before getting started: ap calculus review: properties of exponents and logartithms. however, the most important properties for us will be the product, quotient, and power properties for logarithms. here, we focus on a particular logarithm: the natural logarithm, ln x, though the properties remain true in any base.

in other words, logarithms change…

1. products into sums,
2. quotients into differences, and
3. exponents into multiplication.

using the properties

i like to think of the logarithm as a powerful acid that can dissolve a complicated algebraic expression.

let me illustrate the point with an example.

notice how the original expression involves a huge fraction with roots and powers all over the place. after applying the properties of logarithms, the resulting expression mostly has only plus and minus. (of course, there is a trade-off — there are now three natural logs in the simplified expression.)

logarithmic differentiation

now let’s get down to business! how can we exploit these logarithmic simplification rules to help find derivatives?

the most straightforward case is when the function already has a logarithm involved.

example 1

find the derivative of .

solution

first simplify using the properties of logarithms (see work above). then you can take the derivative of each term. but be careful — the final term requires a product rule!

the general method

in the above example, there was already a logarithm in the function. but what if we want to use logarithmic differentiation when our function has no logarithm?

suppose f(x) is a function with a lot of products, quotients, and/or powers. then you might use the method of logarithmic differentiation to find f ‘(x).

1. 1. first write y = f(x)
   2. next, apply the (natural) logarithm function to both sides of the equation.
   3. then use the product, quotient, and/or power properties to break down the expression on the right. in other words, simplify ln(f(x)) algebraically.
   4. now apply the derivative operator to both sides.
   5. use your rules of differentiation to find the derivatives. note that the left-hand side requires implicit differentiation. (check out: ap calculus review: implicit variation for details.)
      
   6. solve for dy/dx by multiplying both sides by y. in your answer, don’t forget to replace y by f(x).

example 2

use logarithmic differentiation to find the derivative of:  

solution

let’s follow the steps outlined above. the first two steps are routine.

on the other hand, step 3 requires us to break down the logarithmic expression using the properties. the work in this step depends on the function. in our case, there is a product of two factors, so we’ll start with the product property. the power property helps to break down the radical. finally, don’t forget the cancellation rule: ln(e^x) = x.

next, in steps 4 and 5 apply the derivative and work out the right-hand side.

finally, in step 6, we solve for the unknown derivative by multiplying both sides by y. don’t forget to substitute back the original function f(x) in place of y.

functions raised to a function power

a very famous question in calculus class is: what is the derivative of x^x ?

• the answer is not x x^{x – 1}, because the power rule for derivatives cannot be used when the power includes a variable.
• the answer is not x^x ln x, because the exponential rule for derivatives cannot be used when the base includes a variable.

so what is the derivative of x^x ?

well, it turns out that only logarithmic differentiation can decide this one for us!

in fact any time there is a function raised to a function power (that is, neither the exponent nor the base is constant), then you will have to use logarithms to break it down before you can take a derivative.

let’s see how it works in the simplest case: x^x.

first, write y = x^x.

then, apply the logarithm to both sides:   ln y = ln x^x.

break down the right-hand side of the equation using the algebraic properties of logarithms. in this case, only the power property plays a role.

ln y = x ln x

now you can take derivatives of the functions on both sides. but be careful… the function on the right requires a product rule.

(1/y)(dy/dx) = (1) ln x + x(1/x) = ln x + 1

finally, multiply both sides by the original function (y = x^x) to isolate dy/dx.

dy/dx = x^x(ln x + 1)

and there you have it! the derivative of x^x turns out to be trickier than you might have thought at first, but it’s not impossible.

summary

• logarithmic differentiation is a method for finding derivatives of complicated functions involving products, quotients, and/or powers.
• you can use the algebraic properties of logarithms to break down functions into simpler pieces before taking the derivative.

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