SAT study guide
SAT Study Guide
Introduction (1)
What is the SAT? (1)
How is the SAT scored? (2)
How do I take the SAT? (2)
How do I get my scores? (3)
Strategies for Test-takers (3)
What is the guessing penalty? (3)
When should I guess? (3)
The Importance of Time (4)
What should I remember on exam day? (5)
The Math Section (5)
The Critical Reading Section (6)
The Writing Section (7)
The Essay (7)
WELL-SUPPORTED (9)
But Spelling, Punctuation and Grammar Count! (18)
Improving Sentences (22)
Introduction
What is the SAT?
The SAT Reasoning Test, formerly the SAT I: Reasoning Test, is a college entrance examination administered several times a year by the non-profit College Board. The exam is used by many colleges and universities in the United States, among other factors, to determine acceptance into undergraduate school. The SAT is not the only test you can take for college admission, but it is the most widely used. The other most common test is the ACT. Interestingly, SAT does not stand for anything (neither does ACT). If you plan to attend college after graduating from high school, the SAT is one of the most important tests you will ever take. You should be sure to take it seriously and study hard, using books such as this one.
The SAT is a timed test of approximately 3 hours and 45 minutes, divided into three sections: the Math section, the Critical Reading section, and the Writing section, which includes a written essay. Each section is graded on a scale of 200-800. Scaling allows the College Board to correct minor variations in difficulty between versions of the tests. On the test itself, there are ten sets of questions: three from each section. Each section has a different time limit; each question has five answers. One of the Writing section sets is the essay, which is also always Section 1. One of the sections is known as an equating
section and does not count in anyway towards your score; it is used to test new questions and compare the difficulty of different exams.
Until recently, the SAT had only two sections. The Writing section was added in 2005 and the other two sections were overhauled. As a result, you now have to use concepts from Algebra 2 in the Math section, but analogies have been removed from Critical Reading. Collectively, the new test is referred to as the New SAT. The maximum possible score on the new test is 2400. According to the College Board, only 107 students out of 330,000 scored a perfect score on the date of its first administration on March 12, 2005.
How is the SAT scored?
The SAT is scored in a different manner than most test you are probably used to. On the SAT, a grading theory is used where the student begins with a raw score of zero. You earn 1 point for each correct answer, lose 1/4 of a point for each wrong answer, and receive no credit (but also lose no points) for leaving a question blank. However, note also that there are 10 questions in the math section which require you to provide your own numerical answer; fortunately, no points are deducted for wrong answers on these questions. The raw score is then converted using a chart developed by the College Board into a scaled score. Note that although scores under 200 are possible, the College Board cancels all scores which fall into this category.
For the essay, a special method of scoring is used. The essay is scanned into a computer and transmitted over the internet to approved graders. Two graders read your essay and give you a score of 0-6. A zero score is only awarded if your answer is completely off topic. If the two scores are more than one point apart, a third grader plays tie-breaker. The two scores are added together (or the third grader's score is doubled) to produce a score in the range of 0-12. This score is then used in conjunction with your writing multiple-choice raw score to produce the scaled score.
The reason for this method of grading is that the College Board's exam writers understand you are not likely to be able to answer every question on the test. They also know that guessing on these tests would ruin the accuracy of the scores. The loss of points for wrong answers is known as the guessing penalty.
How do I take the SAT?
You can register for the SAT in one of two ways, either via the College Board's website or by obtaining a mail-in registration form from your high school's counseling center.
Detailed instructions for registering and other related information are available from either location. Make sure to save a copy of your Admission Ticket and to bring it with you to the testing center, along with a Photo ID. It is recommended that you register online, as this will automatically tie your registration to an online account. About 4 weeks after you take the test, your score will then become available online, and you will be able to easily view it.
There are several options available: You can choose to order the Question and Answer Service, which provides you with your answer to each question, the correct answer, and the question itself, as well as why each answer is the correct one. You may also order the College Board's SAT Online Course. Other official study guides are available, and their purchase is up to you.
How do I get my scores?
Scores first become available online about 2 weeks after the date on which you took the test. After about 3 weeks, you will receive a Score Report in the mail, detailing your performance. If you registered online, you will simply need to log in to your College Board website account and click View my Scores in the My Organizer pane. If not, you will need to set up an account. You will need the SAT registration identification number found on your Admission Ticket, and all other related information. The website will provide with complete instructions on exactly what to do.
Strategies for Test-takers
What is the guessing penalty?
The guessing penalty, as explained in the Introduction, is the subtraction of 1/4 of a point for each question which you answer incorrectly. However, the term guessing penalty is misleading. Consider the following situation: assume that Fred takes the SAT and randomly guesses on five questions. Because each question has five answers (excluding the ten student response math questions), probability says that he has a 1 in 5 chance of getting each question right. This means that in the 5 questions, he is likely to get one correct and 4 incorrect. He will receive 1 point for each correct answer and lose 1/4 points for each incorrect answer. Quick SAT study question: What will his raw score be? The answer is zero. This means that there really is no guessing penalty. Unless you are very unlucky (or lucky), your net score is unlikely to change noticeably if you guess randomly.
The point of all of this is that guessing is not necessarily harmful to you. Below, you will learn exactly when to guess. For now, understand that though the SAT may be graded differently than the way in which you are used to having your tests graded, it does not mean that it is a harder system. Learn to accept it now and look on the SAT as a challenge, not a necessary evil.
When should I guess?
Contrary to popular belief, the idea that you should never guess, is, like the guessing penalty, completely false. There is a time to guess on the SAT. On some questions you will
not have any idea what the answer is; you are likely on these questions to even be confused as to what it is asking. You should never guess on such a question. You are much more likely to get it wrong in this situation.
However, should you be able to eliminate answers, especially in the Writing and Critical Reading sections, it may be in your best interest to guess. Before guessing, ask your self a few questions:
1. What are the chances of getting this question correct? If you can eliminate one or
more answers, it is probably a good idea to guess.
2. How sure am I of the answer? If you cannot make an educated decision about your
guess, even if you can eliminate one or more answers, it may be unwise to answer the question. Guessing in the blind is never a good idea.
3. How hard is the question? Remember, the SAT (with the exception of passage-based
reading) is a powered test. Balance what you know is the probable difficulty level of a question with how sure you are of your guess.
A basic rule of thumb is if you have eliminated at least two answers and are reasonably sure of your guess, then choose an answer and move on. There's no point in worrying about it. Make your best possible choice, take a breath, and get ready for a whole new battle (that is, question).
Elimination When it's neccesary to eliminate, there are two strategies: You could eliminate answers that are bias. Remember, the test makers are most likely write the test in a neutral way; You could eliminate off-the-wall options. For example, when the problem is asked for the area of a square, there's no way any signs of a pi could appear. Usually, it happens to be the numbers that have big difference with other numbers.
The Importance of Time
As in life, the second biggest enemy on the SATs is the clock; the first biggest is yourself. The test requires you to answer a certain amount of questions in a certain amount of time, and thus it is vastly advantageous for you to use your time wisely. Some of these tips are fairly obvious, but it is surprising how many people neglect to do these under the pressure of the moment.
1. Familiarize yourself with the directions ahead of time. You can easily waste more time
figuring out how to answer questions than actually answering them. Before you take the test, make sure you know the types of questions you'll face, especially since the information is readily available before you even set foot in the test room.
2. DON'T SPEND TOO MUCH TIME ON ONE QUESTION! That is - DON'T SPEND
TOO MUCH TIME ON ONE QUESTION! This has been the downfall of even the cleverest, smartest people. For every challenging, thought-provoking question there are four or five easy, quickly-answered questions following it. The SATs aren't about impressing anyone--they're all about getting points. Get as many as you can.
3. Budget your time, especially in the first few minutes. The more questions you answer
in the beginning, the more time you have to chew over more complex problems in the
end.
A good strategy (especially on the Math sections, which are much quicker to digest) is to blow through the exam, burning through the easy questions and immediately skipping any question that'll make you think too hard. Then, just keep going through the questions in the next passes. This way, you'll be less likely to dwell on hard problems and you'll at least be guaranteed the easy ones.
What should I remember on exam day?
Above all, remember to bring the following list of items:
1. Your Admission Ticket
2. Valid photo ID
3. Two or more No. 2 sharpened pencils
4. A calculator, preferably a graphing one (although you should make sure it is approved
for use on the SAT, some graphing calculators (especially ones with a QWERTY keyboard) are not allowed)
5. A pencil sharpener
6. A snack (to be consumed during breaks, not during test time)
Barring these, a good testing day begins long before it actually arrives. Studying, using guides such as this one, is an essential step. Being prepared will boost your self-confidence and improve your score. Start a week before the test and get a lot of sleep. You'll find you do much better. On testing day, after that good night's sleep, make sure you eat a reasonably-sized breakfast. These tips will help give you an edge; remember what one of the first people who took the New SAT said: "It was a test of mental and physical endurance". At 3 hours and 45 minutes (officially), you will find your brain clogged down. Better to delay the clogging by following these steps than wishing you had when it happens.
The Math Section
Introduction to the Math section
The Math section of the SAT covers several areas: first, it covers general concepts concerning numbers which you are expected to know. Second, it encompasses geometry, especially the properties of angles. Often, parts of the first two are used in coordination with the third area, Algebra. Also, some concepts from Algebra 2, particularly second degree equations, are tested as well.
The Math section contains, in general, around 50 questions. Each question has five answers. However, the Math section is different in that ten of these questions do not have multiple-choice answers. You are required to grid in your answers on these questions.
Don't worry- they are similar to the other types of questions; make sure and study the probability portions of Chapter 1. Several different types of answers are allowed on this section.
Although the New SAT contains more straightforward math questions, you must still become used to solving unusual problems involving logic. This logic will build on general rules you can learn in the chapters listed above. The reason is linked to the College Board's philosophy about the test; they see it as more of an aptitude test than an assessment test. In the end, no one can teach you the logic behind the test, but you can study hard, and that will improve your score a lot.
Degrees and Angles
1. A circle contains 360 degrees. Every circle contains 360 degrees. Each degree is
1/360 of the total distance around the outside of the circle.
2. A line is a 180-degree angle. A line is a flat angle. Half a circle, or a semicircle, is also
a 180 degree angle.
3. When two lines intersect, four angles are formed.
4. If two lines are perpendicular, their slopes are negative reciprocals. If line A has a
slope of 3, and line B is perpendicular to A, it has a slope of -1/3.
5. When two lines intersect, the angles opposite each other will have the same
measures. These are known as vertical angles. The total of all four angles created is 360 degrees.
6. The sum of the lengths of any two sides of a triangle must be larger than the length of
the remaining side.
7. A rectangle + two triangles on opposite sides = a trapezoid.
The Critical Reading Section
Sentence Completion
The object of the Sentence Completion section of the SAT is for the test-taker to complete the given sentence using context clues.
A sentence completion question may have one or more blanks, and five answers to choose from. For example:
Scotland is a nation in the north of the Island of Great Britain; it has a(n) _________ economy that distinguishes it from the rest of the United Kingdom.
A. dignified
B. self-sufficient
C. fancy
D. unique
E. complex
We can use context to help us select the correct answer. The blank is clearly describing Scotland's economy. From the rest of the sentence, we can determine that we know two things about Scotland - it's in the north of Great Britain and its economy distinguishes it somehow. So, the blank must relate to one of those things. The geographic location of Scotland is probably irrelevant when describing its economy, so the blank must mean something relating to 'distinguishing'.
While distinguished can sometimes relate to being (A) dignified, that is not how the word is being used in this context, and while (B), (C), and (E) may be true, only (D) unique means the same thing as being distinguished, i.e. standing out.
Therefore, the answer is D. unique.
The Writing Section
The Essay
The First WELL of the Excellent Essay:
WELL-STRUCTURED
The Importance of Structure
1. Just starting to write as soon as you read the essay prompt is a BAD STRATEGY!
Unfortunately, this is how many students approach the test. Why do they do this?
Because most students do not realize that THE TEST GRADERS ARE EVALUATING YOUR EFFORTS BASED IN LARGE PART ON THE ORGANIZATION OF YOUR IDEAS.
2. Taking 2 minutes to OUTLINE YOUR ESSAY enables you to CREATE A ROADMAP
for your essay.
1) Writing your essay without first creating a general outline is like taking a car trip
without first having any sense of the directions you’ll need to follow in order to
arrive at your destination.
2) Without a map travelers get lost. Without an outline, student essays ramble.
3) Take the two minutes to create a map of your essay – it is time very well spent.
(And NOT, contrary to popular misconception, time wasted.)
Success Secret for the Test
ESSAY GRADERS ARE LOOKING FOR SOLID STRUCTURE
QUESTIONS ESSAY GRADERS WILL ASK THEMSELVES:
z Is there a sensible progression of ideas?
z Does the student logically move from Point A to Point B?
z Does the essay have a Beginning, a Middle and an End?
KEY POINTS TO REMEMBER ABOUT OUTLINING:
z It creates Focus.
z It saves time. (Really, it does. When you know where you are going, it’s a lot easier to get there.)
z It provides an intelligent Road Map that prevents rambling.
How to Create an Outline:
The Four Paragraph SAT Essay of Excellence
NOTE: Being that this is a timed test, students WILL NOT HAVE TIME to elaborate upon every possible aspect the question touches upon. Test Makers know this. Test Graders know this. Students who score well know this, too. By coherently and intelligently responding to the essay prompt in four well-written paragraphs students will enable themselves to earn an excellent score on the Essay Writing Section of the SAT.
There are Four Main Sections a student wants to outline before they begin to write their essay.
Paragraph 1 – The Main Idea (including a Thesis Statement)
Paragraph 2 – Supporting Paragraph #1 (Point A)
Paragraph 3 – Supporting Paragraph #2 (Point B)
Paragraph 4 – The Concrete Conclusion (re-connecting to the Thesis)
Outline these four Big Points in two minutes BEFORE YOU BEGIN!!
NOTE: Yes, this is a formula - a formula for success on the SAT. With only 25 minutes to complete a well-written essay, strategy is a HUGE factor for success on this test.
Put yourself in the shoes of a Test Grader.
If a Test Grader sees that you can write four well-structured paragraphs that progress from Point to Point, including a Thesis Statement and a Conclusion, you are on you way to an excellent score. Remember, all you need to do is write four paragraphs – there is no time for a 33 page doctoral thesis on the essay prompt - so do NOT try to write one.
PITFALLS:Watch out!
z Avoid the temptation to skip the outline process.
z Avoid the temptation to abandon the outline after taking the time to create it.
Samples Outlining Activities:
Sample #1
DIRECTIONS: Please explain the following quote and whether or not you agree or disagree with the statement.
Nothing great was ever achieved without enthusiasm. (Ralph Waldo Emerson)
Two-Minute General Outline:
Paragraph 1 - You must care about something in order to really create greatness. Paragraph 2 - I agree, being negative will never result in producing amazing results. Paragraph 3 – Examples exist everywhere proving this point.
Paragraph 4 – Without genuine passion, excellence is unattainable.
Sample #2
DIRECTIONS: Please explain the following quote and whether or not you agree or disagree with the statement.
The person who lies for you will lie against you. (Harry Truman)
Two-Minute General Outline:
Paragraph 1 – A liar is a liar.
Paragraph 2 – Betrayal will eventually happen.
Paragraph 3 – Honesty is a principle without exception.
Paragraph 4 – People who lie for you reveal their true character so beware.
Though these are only rough statements, one can see that these essays now have a clear sense of direction – and as a result they will be MUCH EASIER to write because a road map is now in place telling us where to go and what to accomplish.
WELL-SUPPORTED
The Importance of Support
z Test Makers and Test Graders are looking for the strong, solid support of ideas in student essays.
z An idea without strong, solid support is like a roof without a strong, solid foundation – it is going to collapse.
z Many students will offer a strong, solid idea but not follow it up -- as a result, they do not earn excellent scores for their efforts.
How to Support: Know (and Use) The Umbrella Theory
Think of the four main points of your outline as if each of them were an umbrella.
z Items properly placed under the umbrella are shielded from a storm of point subtraction.
z Items not placed properly under the umbrella are at risk of being rained on by a storm of point subtraction.
z Supporting ideas that are sensible and properly placed will fit nicely underneath the umbrellas of your outline.
z Rambling statements that shoot off in all sorts of nutty directions will not fit under your umbrella.
What to support: Paragraph 1 - THE MAIN IDEA/THESIS STATEMENT
Each Well-Written Essay has a Thesis Statement that needs to be supported.
z What is a Thesis Statement?
z A Thesis statement expresses the MAIN IDEA OF THE ENTIRE ESSAY.
z Why do you need a Thesis Statement?
z You need a thesis statement because it controls the direction, focus and purpose of the essay.
z How do you create a Thesis Statement?
z A great way to create a thesis statement that will ensure you address the question you have been asked is to CONVERT THE QUESTION PROMPT INTO A THESIS STATEMENT.
How do you support a Thesis Statement?
z The well-written thesis statement will be like an umbrella for your entire essay – everything will fit underneath its scope.
z The TOPIC SENTENCES of Paragraphs 2, 3 and 4 will be the specific tent poles of support for your thesis by the time you are finished. (More on that in a bit.)
z Remember, support for the thesis statement will be found throughout the entire essay.
In one sense, the purpose of the entire essay is to support the thesis.
THE THESIS STATEMENT GUIDES YOUR ESSAY
HOW TO CONVERT A QUESTION PROMPT INTO A THESIS STATEMENT
1. Read the Question Prompt.
2. Change the Prompt from a question into a firm statement.
Example:
Question Prompt: Why do you like vanilla ice cream?
Thesis: Many reasons exist for me to like vanilla ice cream.
Question Prompt: Do you agree that the United States should avoid raising taxes? Thesis: I completely disagree with the idea that the United States should avoid raising taxes.
３.Use this converted statement as the basis for your thesis.
More examples:
Question: Considering that most teenage driving fatalities occur after dark, do you believe that teenage drivers should be banned from driving at night?
Converted to Thesis Statement: Because most teenage driving fatalities occur after dark, I believe teenagers should not be allowed to drive their cars after the sun sets. Question: If at the age of eighteen a person can join the military and die for their country, do you feel that they should then also be allowed to go into a bar and be served an alcoholic beverage?
Converted to Thesis Statement: If a person can join the military and die for their country, they should definitely be able to enter a bar and be served an alcoholic beverage.
Tips to Remember about Thesis Statements:
z They need to be direct and focused.
z They need to serve as an umbrella which can be used for the entire essay.
z They must address a specific topic and put forth a clear main idea.
What to support: Paragraph 2 - THE TOPIC SENTENCE
z Paragraph 2 will begin with a TOPIC SENTENCE.
z This topic sentence will have been generated from your outline.
z This topic sentence will need to be supported by the paragraph that follows.
z What is a Topic Sentence?
The topic sentence will directly state the focus, direction and purpose of the paragraph.
z Why do you need a Topic Sentence?
A Topic Sentence is needed for two distinct reasons:
1. So that the point of the paragraph is clear and precise.
2. So that the thesis statement is provide with solid support.
z How do you create a Topic Sentence?
Since you will know what the focus and purpose of Paragraph 2 needs to be (because you did an outline that sketched out the main idea of this paragraph before you started writing this essay… remember section 1, Well-Structured?) you will know what you are going to be writing about and why. Take your topic sentence from your outline.
z How do you support a Topic Sentence?
There are 3 Major Types of Support in the Excellent Essay:
1. Logical reasoning.
EX: If THIS happens, then THAT will be the result.
2. Personal Examples.
EX: Once, when I was younger, I learned THIS the hard way.
3. Specific, vivid details.
EX: Green slime oozed from the nostril of the dead gazelle.
Remember, topic sentences are also like an umbrella.
z Use logical reasoning, personal examples and specific, vivid details to support your topic sentences.
Students will want to include all three types of support for their topic sentences in order to score well on the SAT Essay Writing Section.
What to support: Paragraph 3 - THE DIFFERENT/OPPOSITE PERSPECTIVE
z Paragraph 3 will begin with a TOPIC SENTENCE.
z This topic sentence will have been generated from your outline.
z This topic sentence will need to be supported by the paragraph that follows.
NOTE: For an effective, excellent essay the topic sentence of Paragraph 3 will address A DIFFERENT/OPPOSITE PERSPECTIVE from the one addressed by Paragraph 2.
ESSAY GRADERS ARE LOOKING FOR A MULTIPLE OF PERSPECTIVES ON THE
SAME ISSUE IN THE HIGH-SCORING SAT ESSAY.
MAKE SURE THE PERSPECTIVE OF PARAGRAPH 3 IS DIFFERENT/OPPOSITE FROM THE PERSPECTIVE OF PARAGRAPH 2.
Students encounter problems when they do not address a different/opposite perspective in Paragraph 3:
z Students who only look at the question from one perspective do not demonstrate the sophisticated thought process test graders like to see.
z Students who only look at the question from one perspective tend to be repetitive in their thoughts and ideas.
z Students who only look at the question from only one angle tend to not score as high as those who view the question from a different/opposite perspective.
How to Create a Different/Opposite Perspective for Paragraph 3:
z Think in terms of, “The Other Side of the Coin.”
z Take the other side of the argument.
z Play “Devil’s Advocate.”
z Change sides for a minute to consider all angles.
For example…
z If paragraph 2 is discussing the need for teenage drivers to stay off the road at night, paragraph 3 can address why teenage drivers might argue that they deserve to be able to drive on the road at night.
z If paragraph 2 is arguing that being able to fight and die as a soldier in the military has nothing to do with being able to responsibly handle being served an alcoholic beverage in a bar, then paragraph 3 can talk about how silly it is that being served alcohol requires more maturity than being asked to handle a weapon in the army.
ADDRESSING A DIFFERENT/OPPOSITE PERSPECTIVE IN PARAGRAPH 3 ADDS DEPTH, SOPHISTICATION AND COMPLEXITY TO YOUR ESSAY IN AN EASY-TO-IMPLEMENT MANNER.
Good phrases to incorporate in Paragraph 3:
z However…
z On the other hand…
z Another way of looking at this is…
z Opponents might say…
z While most may agree, there are others who feel…
What to support: THE CONCLUSION
Each Well-Written SAT Essay has a Concrete Conclusion!
Paragraph 1 + Paragraph 2 + Paragraph 3 add up to Paragraph 4 (The Conclusion) What is a Conclusion?
z A conclusion is a definite ending whereby the reader of the essay will feel that the ideas are tied together and that the essay has been completed.
Why do you need a Conclusion?
z Essays need endings. Including a conclusion shows the reader you understand the proper structure for the SAT essay and have implemented it successfully.
How do you create a conclusion?
z Think of your essay as if it were a circle. You started at one point and now (in the conclusion) you need to bring everything back around again to complete the loop.
Techniques a student can use to craft a strong conclusion.
z Restate and re-affirm your position.
z Offer a solution to a problem.
z Make a recommendation for a course of action.
z Summarize your major points.
z Restate your thesis.
Conclusion examples:
1) In conclusion, teenage drivers are dangerous after dark and should be prohibited
from taking the wheel at night. Too many pieces of evidence demonstrate that
their general recklessness presents real peril after the sun has set. Stopping
them from taking the road doesn’t just protect them, it protects all of us.
2) Of all the crazy laws in our country, the one that says an eighteen year old can
die for the American flag but not have a beer in a bar before doing so is the
dumbest. Sure, people can argue about how the drinking age being set at twenty-one has some advantages but all in all, if I were thinking about serving my
country, I’d find it ridiculous that Uncle Sam will permit me to shoot a man before
sharing a glass of wine with him.
Keep in mind…
z The conclusion is the final say.
z The purpose of the conclusion is to hammer home an idea and make a point.
z This is no time for wishy-washy language nor unclear positions. Take a side and assert your belief. Strength counts.
DO’s for the conclusion…
z Tie up the major points of the essay.
z Use strong, forceful language that MAKES A POINT.
z Give the reader a sense that the essay is completely finished.
z Hammer home an idea and let the reader know precisely the position the author of the essay has taken on the subject matter being discussed.
z Draw a direct line to the thesis statement.
DON’T’s for the conclusion…
z Start talking about new points and side issues that haven’t been mentioned in the essay prior to this paragraph.
z Give the reader a sense that the essay might go on and may not be finished with yet. z Use weak, soft language that doesn’t really make a point about anything in particular. z Doesn’t leave an impression that the author of the essay has any particular belief about the subject matter being discussed.
z Draws no connection to the thesis statement.
Remember…
z Conclusions can be short (after all, this is a timed test and it might just come down to the last few nitty-gritty moments) but short conclusions can be effective.
z Not writing a conclusion, however, is ineffective and will lower your score.
ALWAYS WRITE A CONCLUSION!!!
Two more Principles of the Well-Supported Essay you MUST use to Earn Higher Scores:
1. Use an example from your Personal Experience.
Test Graders are looking for you to support your essay with personal experience.
z Your own life.
z A friend’s life.
z A family member’s life.
Test Graders want to see the generalities of the ideas you write about exemplified through concrete, real-life examples taken from your own personal experience.
If I were writing an essay about the need to keep teenagers off the road at night, I might want to tell the story about the neighborhood teenager who crashed his mother’s new car and ended up in a wheelchair three days after he first got his license. Why?
z It’s got relevance.
z It’s got juicy details.
z It is unique. (Out of all the people taking the SAT in the entire country, I am probably the only one who can tell such a story because it really happened in my life.) ALWAYS USE A PERSONAL EXAMPLE IN YOUR ESSAY!!
2. Be Specific, Use Details
Why be specific and use details?
Test graders read lots and lots of papers. Lots of them are dull papers, too. Exciting language will stand out and capture their attention.
z Bland language lays flat on the page.
z Descriptive language leaps off the page.
z General thoughts put readers to sleep.
z Specific details make readers pay attention.
*In other words, VIVID LANGUAGE BRINGS WRITING TO LIFE.
Students MUST use descriptive language in order to achieve high scores!!
Never use the word GOOD!
Use fantastic, amazing, incredible, tremendous, spectacular, extraordinary, unbelievable, phenomenal, stupendous, remarkable, astonishing, etc…
Never use the word BAD!
Use horrible, disgusting, terrible, appalling, dreadful, horrifying, detestable, hideous, abominable, offensive, despicable, etc…
Enhance Your Descriptions by Tapping the Five Senses
z Write about how the sunshine’s sizzle sounded like a Sunday morning omelet.
z Write about how the old car smelled like a bad tuna fish sandwich.
z Write about how the baby’s skin felt like the petal of an orchid.
z Write about how the fresh rain tasted like an angel kissing your tongue.
z Write about how the broken computer looked like a fallen soldier.
ALWAYS USE SPECIFIC DETAIL AND VIVID LANGUAGE IN YOUR ESSAY!!
PITFALLS:Watch out!
z Not following your outline.
z Making generalized statements that lack support.
z Not including personal experience.
z Using dull, non-descriptive language.
The Third WELL of the Excellent Essay:
Well-Reasoned
z The Importance of Well-Reasoned: Simple, Strong and Straightforward.
With only 25 minutes to write an excellent SAT essay a student must know what they can and cannot accomplish. Almost all of the topics could lend themselves to 5 page typed papers if a student were given the time to write one.
This is NOT what the Test Graders are looking for.
Test Graders are looking for students to demonstrate solid, strong proficiency in the area of essay writing when they only have a limited amount of time. The secret to successfully accomplishing this is found in the three S’s: SIMPLE, STRONG AND STRAIGHTFORWARD.
z Make SIMPLE Arguments
Being Simple does not mean being stupid, unsophisticated or unintelligent. Being simple means being; clear, precise and achievable.
Simple equals Clear.
z On a timed test, the clarity of an argument adds to a student’s score. Some students
try to tackle too much and get embroiled in convoluted thought processes that tangle them up more than they help them out. BEING CLEAR allows you to score higher.
Simple equals Precise.
z On a timed test, the preciseness of an argument adds to a student’s score.
Communicating a specific idea in a clear-cut manner allows a Test Grader to see evidence that the student can fashion a distinct, precise perspective in a limited amount of time, which allows you to score higher.
Simple equals Achievable.
z On a timed test, completing one’s thoughts - and not leaving ideas or essays incomplete - adds to a student’s score. Many students begin elaborate, complicated patterns of logic that require 26 sentences to explain. This makes it very tough for them to finish the essay and opens the door up for them to make avoidable mistakes like constructing run-on sentences, engaging in convoluted logic or simply not finishing the essay because they have run out of time.
*You do not want to run out of time. As discussed earlier, you need a strong conclusion to score well. If you try to tackle an overly complicated thought process, you might not finish in the allotted time frame. ACHIEVING COMPLETION OF YOUR ESSAY IN THE TIME ALLOWED IS A BIG PART OF SCORING WELL. Simplicity allows you to do this.
z Make STRONG Arguments
Break out the big guns right away! Find your biggest, most solid, most forceful, most dominant, #1 point and ADDRESS IT FIRST!!!
Some students have 2 small points and then one really big one and they figure that they will save the biggest one for last and go with the smaller ones first. This is a bad strategy for the SAT essay!
Remember, on a holistically graded test such as this, strong impressions count and the best way to make a really strong impression is by going with your number 1, biggest, strongest, most influential point first. Besides, there might not even be enough time to get to your #1 argument if you wait. (And wouldn’t that be a shame if you had saved your best for last but never got a chance to use your best due to time constraints?)
Remember…
z Each paragraph should make a specific, solid point.
z Each sentence should make a specific, solid point that relates to the overall, specific, solid point of the paragraph.
z Use your best first.
z Make STRAIGHTFORWARD Arguments
Avoid over-complicated, intricate, sophisticated patterns of logic and reasoning. They suck up tons of time, open up the door for too many simple grammatical and punctuation mistakes to happen and they tend to cause students to ramble and go on straying tangents. Remember the Umbrella Theory? Arguments that stray from underneath the umbrella of the main point (either of the topic sentence or of the thesis statement) open up
a student’s test to a rainstorm of point subtraction.
Remember, a straightforward argument…
z Creates an achievable goal because time is limited.
z Prevents straying from the essential point.
z Avoid Over-Extended logic patterns that create opportunities for missteps.
Something Else your Essay MUST HAVE to Earn Higher Scores:
A POINT OF VIEW (P.O.V.)
Test Graders want to see you take a clear position in your answer.
z Test questions do not have right or wrong answers – they are subjective, interpretive, open-ended questions. Therefore, ASSERT YOUR OPINION.
z You will not be penalized for the stance you take on a matter, nor on your beliefs. Your ideas are your ideas, however… YOU WILL NOT SCORE HIGH IF YOU DO NOT TAKE A STANCE OR PUT FORTH YOUR BELIEFS!
Remember…
HAVING A CLEAR POINT OF VIEW INCREASES YOUR ESSAY SCORE
NOT HAVING AN OBVIOUS P.O.V. DETRACTS POINTS FROM YOUR SCORE.
z Having an obvious P.O.V., countering with an opposing P.O.V. and then re-asserting an additional reason to support your own P.O.V. raises your scores into the upper echelons of the grading rubric.
Also remember…
You will not be marked down by a scorer because they disagree with your argument. You will be marked down by a scorer if you do not take a position and make an argument.
SPECIAL SECTION: Do NOT B.S.
z Trying to impress scorers by using big vocabulary words you are not sure how to use properly detracts from – not adds to – your score.
z Trying to impress scorers by sounding artificially sophisticated when you do not know what you are talking about detracts from – not adds to –your score.
z Writing simply for the sake of writing (i.e. aimless, wandering, & imprecise sentences written just to fill space on the page) detracts from – not adds to - your overall score.
Remember…
Test Graders are masters at detecting B.S. so don’t even try it.
z Make real points.
z Take a definite P.O.V.
z Implement the proper, 4 paragraph structure.
z Use specific details and vivid language.
z Include a personal example from your own, unique life.
*If you do all of this, there is no need to B.S. – you will be in good shape.
The Fourth WELL of the Excellent Essay:
Well-Executed
The Importance of Execution
Time is limited. MAKING A STRONG IMPRESSION in this limited amount of time is a key ingredient to success.The best way to make a strong impression is to prove that you can write well.The way you prove that you write well is through the use of proper grammar, spelling and punctuation.No matter how brilliant your ideas are, if you do not know how to write a PROPERLY PUNCTUATED essay that USES CORRECT GRAMMAR and is ALMOST ENTIRELY FREE OF SPELLING ERRORS, you will not earn an excellent score.And the truth is, you do not deserve one, either. These are the basic tools of the writer’s trade… KNOW THEM!!!
Remember…
A student does NOT have to be perfect to earn an excellent score.
z Small mistakes such as misplaced commas will not ruin you.
z Too many simple errors will definitely hurt your score.
But Spelling, Punctuation and Grammar Count!
Simple Tools to Ensure a Higher Score
Proofread Your Work.
z It always amazes me to watch students who finishes their essay with 2 minutes to spare and fold their paper over and stare into space as if they have nothing to do.
z What they NEED to do is PROOFREAD their work!
z On timed tests when students are working quickly and efficiently, test takers often forget a word in a sentence, omit a period or question mark and misspell easy words they know how to spell. Finding even one of these small mistakes might be the last little thing you need to inch your score up a notch and move you into a higher echelon in the Test Grader’s eyes.
z PROOFREAD!!! If you have the time and you really care about earning the best score you can, there will not be a wasted moment of time on the SAT Essay writing section.
Students who have the time but do not bother to check over their work are not students who traditionally perform excellently well on the test. Proofread! It matters.
Avoid Carelessness.
z Pay attention to details.
z Make it a point to do away with sloppiness.
z If you practice NOT BEING SLOPPY during your practice essays, you will find that your performance will tend not to be sloppy as well.
Be legible.
z Students will NOT be marked down for their handwriting HOWEVER, without being legible, scores might suffer. (If I can’t read it, I can’t evaluate it.)
Remember, the Essay is Graded Holistically
Students get one score (on a scale from 1-6).
z Simple, proper execution can raise scores.
z A lack of simple execution can lower them.
z Excellently Execute your Essay:Use Proper Grammar
I’ll say it again, these are the basic tools of the writer’s trade and this is the foundation for what your score will be based upon. KNOW THIS MATERIAL!!!
Subject/Verb Agreement
All subjects and verbs need to agree.
z Writing things like, “They is going to the store” is the same as begging for a low score. z Basic subject/verb agreement is a fifth grade Language Arts skill strong essay writers need to have mastered.
Commas
Make sure you properly use commas where they are needed.
z One or two misplaced commas will not kill you.
z An entire essay riddled with misplaced commas will definitely hurt.
z If you are not sure how to properly use a comma, seek help.
Apostrophes
Possessives, contractions, plurals – hey, they need apostrophes (sometimes).
z Proper apostrophe use is a fairly simple skill to know.
z Students who forget apostrophes are sending a secret signal to the Test Grader that they are really not a solid, competent writer.
z Forgotten apostrophes are usually due to sloppiness, laziness or a lack of attention to detail. REMEMBER THEM!
Parallel Sentence Structure
Smooth, clear writing has parallel sentence structure.
z Use the same grammatical form to express equal, or parallel ideas.
z Pair a noun with a noun.
z Pair a phrase with a phrase.
z Pair a clause with a clause.
z Pair an infinitive with an infinitive.
Pronouns
z A Pronoun by itself has no definite meaning.
z A Pronoun is only clear when the reader knows what the pronoun refers to.
z Be sure that the references you utilize pronouns for are clear.
Colons and Semicolons
In general, I recommend students AVOID USING COLONS AND SEMICOLONS.
z Students do not gain extra points for using colons and semicolons properly.
z Students can lose points for using colons and semicolons improperly.
z If there is really nothing to be gained but something to be lost, why bother getting involved in the first place?
NOTE: Many students love to try and impress the Test Graders by showing off their vast use of sophisticated punctuation when they really do not have an excellent grasp of how to use the punctuation properly.
This is NOT necessary - avoid doing it yourself. You’re only creating potential potholes on the road you’ll be travelling.
Capitalize!
z All proper nouns must begin with a capital letter.
z All sentences must begin with a capital letter.
z Capitalization is easy – try not to shoot yourself in the foot.
Indent your paragraphs
z You will be writing 4 paragraphs.
z Indent all 4 Paragraphs.
z Again, this is very simple stuff that low scoring students do not do.
Use Proper Spelling
z If you are not sure how to spell a word, use a different word. The English language is very flexible.
z Remember, synonyms you do know how to spell are better than enriched vocabulary words you do not know how to spell.
z Test score do not rely on any one word so do not feel the need to force things.
NOTE: Do NOT use vocabulary words with which you are not familiar to impress test graders. It’s a BAD IDEA.
z Trying to impress Test Graders by using 14 letter vocabulary words that you are not sure about how to use properly is a sure sign of an amateur test taker. Remember, using words incorrectly can do more harm than good.
Use Proper Punctuation
z All sentences require punctuation. Use it properly. Keep it simple.
z The period.
z It’s a silly mistake to forget these at the end of sentences.
z The question mark.
z It’s a silly mistake to forget these at the end of questions.
z The exclamation point.
z A great tool – but not if used every other sentence. Be judicial!
z Quotation marks.
初三(下册)数学知识点详解
初三（下册）数学各章节重要知识点总结 二次函数 1. 二次函数的一般形式：y=ax2+bx+c.(a≠0) 2. 关于二次函数的几个概念：二次函数的图象是抛物线，所以也叫抛物线y=ax2+bx+c；抛物线关于对称轴对称且以对称轴为界，一半图象上坡，另一半图象下坡；其中c叫二次函数在y轴上的截距, 即二次函数图象必过（0，c）点. 3. y=ax2(a≠0)的特性：当y=ax2+bx+c (a≠0)中的b=0且c=0时二次函数为y=ax2(a≠0); 这个二次函数是一个特殊的二次函数，有下列特性： （1）图象关于y轴对称；（2）顶点（0，0）；（3）y=ax2 (a≠0)可以经过补0看做二次函数的一般式，顶点式和双根式，即： y=ax2+0x+0, y=a(x-0)2+0, y=a(x-0)(x-0). 4. 二次函数y=ax2+bx+c (a≠0)的图象及几个重要点的公式： 5. 二次函数y=ax2+bx+c (a≠0)中，a、b、c与Δ的符号与图象的关系： (1) a＞0 <=> 抛物线开口向上； a＜0 <=> 抛物线开口向下； (2) c＞0 <=> 抛物线从原点上方通过； c=0 <=> 抛物线从原点通过； c＜0 <=> 抛物线从原点下方通过； (3) a, b异号 <=> 对称轴在y轴的右侧； a, b同号 <=> 对称轴在y轴的左侧； b=0 <=> 对称轴是y轴； (4) Δ＞0 <=> 抛物线与x轴有两个交点； Δ=0 <=> 抛物线与x轴有一个交点（即相切）； Δ＜0 <=> 抛物线与x轴无交点. 6．求二次函数的解析式：已知二次函数图象上三点的坐标，可设解析式y=ax2+bx+c，并把这三点的坐标代入，解关于a、b、c的三元一次方程组，求出a、b、c的值, 从而求出解析式-------待定系数法. 8．二次函数的顶点式： y=a(x-h)2+k (a≠0)；由顶点式可直接得出二次函数的顶点坐标（h, k），对称轴方程 x=h 和函数的最值 y最值= k. 9．求二次函数的解析式：已知二次函数的顶点坐标（x0,y0）和图象上的另一点的坐标，可设解析式为y=a(x -x0)2+ y0，再代入另一点的坐标求a，从而求出解析式.（注意：习题无特殊说明，最后结果要求化为一般式） 10. 二次函数图象的平行移动：二次函数一般应先化为顶点式，然后才好判断图象的平行移
(完整word版)数学家精彩小故事
八岁的高斯发现了数学定理 德国著名大科学家高斯(1777～1855)出生在一个贫穷的家庭。高斯在还不会讲话就自己学计算，在三岁时有一天晚上他看着父亲在算工钱时，还纠正父亲计算的错误。 长大后他成为当代最杰出的天文学家、数学家。他在物理的电磁学方面有一些贡献，现在电磁学的一个单位就是用他的名字命名。数学家们则称呼他为“数学王子”。 他八岁时进入乡村小学读书。教数学的老师是一个从城里来的人，觉得在一个穷乡僻壤教几个小猢狲读书，真是大材小用。而他又有些偏见：穷人的孩子天生都是笨蛋，教这些蠢笨的孩子念书不必认真，如果有机会还应该处罚他们，使自己在这枯燥的生活里添一些乐趣。 这一天正是数学教师情绪低落的一天。同学们看到老师那抑郁的脸孔，心里畏缩起来，知道老师又会在今天捉这些学生处罚了。 “你们今天替我算从1加2加3一直到100的和。谁算不出来就罚他不能回家吃午饭。”老师讲了这句话后就一言不发的拿起一本小说坐在椅子上看去了。 教室里的小朋友们拿起石板开始计算：“1加2等于3，3加3等于6，6加4等于10……”一些小朋友加到一个数后就擦掉石板上的结果，再加下去，数越来越大，很不好算。有些孩子的小脸孔涨红了，有些手心、额上渗出了汗来。 还不到半个小时，小高斯拿起了他的石板走上前去。“老师，答案是不是这样？” 老师头也不抬，挥着那肥厚的手，说：“去，回去再算！错了。”他想不可能这么快就会有答案了。 可是高斯却站着不动，把石板伸向老师面前：“老师！我想这个答案是对的。” 数学老师本来想怒吼起来，可是一看石板上整整齐齐写了这样的数：5050，他惊奇起来，因为他自己曾经算过，得到的数也是5050，这个8岁的小鬼怎么这样快就得到了这个数值呢？ 高斯解释他发现的一个方法，这个方法就是古时希腊人和中国人用来计算级数1+2+3+…+n的方法。高斯的发现使老师觉得羞愧，觉得自己以前目空一切和轻视穷人家的孩子的观点是不对的。他以后也认真教起书来，并且还常从城里买些数学书自己进修并借给高斯看。在他的鼓励下，高斯以后便在数学上作了一些重要的研究了。 为了中华民族的富强-------苏步青的故事 苏步青1902年9月出生在浙江省平阳县的一个山村里。虽然家境清贫，可他父母省吃俭用，拼死拼活也要供他上学。他在读初中时，对数学并不感兴趣，觉得数学太简单，一学就懂。可量，后来的一堂数学课影响了他一生的道路。 那是苏步青上初三时，他就读浙江省六十中来了一位刚从东京留学归来的教数学课的杨老师。第一堂课杨老师没有讲数学，而是讲故事。他说：“当今世界，弱肉强食，世界列强依仗船坚炮利，都想蚕食瓜分中国。中华亡国灭种的危险迫在眉睫，振兴科学，发展实业，救亡图存，在此一举。‘天下兴亡，匹夫有责’，在座的每一位同学都有责任。”他旁征博引，讲述了数学在现代科学技术发展中的巨大作用。这堂课的最后一句话是：“为了救亡图存，必须振兴科学。数学是科学的开路先锋，为了发展科学，必须学好数学。”苏步青一生不知听过多少堂课，但这一堂课使他终身难忘。 杨老师的课深深地打动了他，给他的思想注入了新的兴奋剂。读书，不仅为了摆脱个人困境，而是要拯救中国广大的苦难民众；读书，不仅是为了个人找出路，而是为中华民族求新生。当天晚上，苏步青辗转反侧，彻夜难眠。在杨老师的影响下，苏步青的兴趣从文学转向了数学，并从此立下了“读书不忘救国，救国不忘读书”的座右铭。一迷上数学，不管是酷暑隆冬，霜晨雪夜，苏步青只知道读书、思考、解题、演算，
人教版九年级数学上下册培优讲义机构辅导资料(共30讲)
九年级讲义目录
专题01 二次根式的化简与求值 阅读与思考 二次根式的化简与求值问题常涉及最简根式、同类根式，分母有理化等概念，常用到分解、分拆、换元等技巧. 有条件的二次根式的化简与求值问题是代数变形的重点，也是难点，这类问题包含了整式、分式、二次根式等众多知识，又联系着分解变形、整体代换、一般化等重要的思想方法，解题的基本思路是： 1、直接代入 直接将已知条件代入待化简求值的式子. 2、变形代入 适当地变条件、适当地变结论，同时变条件与结论，再代入求值. 数学思想： 数学中充满了矛盾，如正与负，加与减，乘与除，数与形，有理数与无理数，常量与变量、有理式与无理式，相等与不等，正面与反面、有限与无限，分解与合并，特殊与一般，存在与不存在等，数学就是在矛盾中产生，又在矛盾中发展. =x , y , n 都是正整数） 例题与求解 【例1】 当x = 时，代数式32003 (420052001)x x --的值是（ ） A 、0 B 、－1 C 、1 D 、2003 2- （绍兴市竞赛试题） 【例2】 化简 （1(b a b ab b -÷-- （黄冈市中考试题） （2 （五城市联赛试题）
（3 （北京市竞赛试题） （4 （陕西省竞赛试题） 解题思路：若一开始把分母有理化，则计算必定繁难，仔细观察每题中分子与分母的数字特点，通过分解、分析等方法寻找它们的联系，问题便迎刃而解. 思想精髓：因式分解是针对多项式而言的，在整式，分母中应用非常广泛，但是因式分解的思想也广泛应用于解二次根式的问题中，恰当地作类似于因式分解的变形，可降低一些二次根式问题的难度. 【例3】比6大的最小整数是多少？ （西安交大少年班入学试题） 解题思路：直接展开，计算较繁，可引入有理化因式辅助解题，即设x y == 想一想：设x=求 432 32 621823 7515 x x x x x x x --++ -++ 的值. （“祖冲之杯”邀请赛试题） 的根式为复合二次根式，常用配方，引入参数等方法来化简复合二次根式.
有趣的数学家故事
有趣的数学家故事 蒲丰试验 一天,法国数学家蒲丰请许多朋友到家里,做了一次试验.蒲丰在桌子上铺好一张大白纸,白纸上画满了等距离的平行线,他又拿出很多等长的小针,小针的长度都是平行线的一半.蒲丰说:“请大家把这些小针往这张白纸上随便仍吧！”客人们按他说的做了。 蒲丰的统计结果是：大家共掷2212次，其中小针与纸上平行线相交704次， 2210÷704≈3.142。蒲丰说：“这个数是π的近似值。每次都会得到圆周率的近似值，而且投掷的次数越多，求出的圆周率近似值越精确。”这就是著名的“蒲丰试验”。 数学魔术家 1981年的一个夏日，在印度举行了一场心算比赛。表演者是印度的一位37岁的妇女，她的名字叫沙贡塔娜。当天，她要以惊人的心算能力，与一台先进的电子计算机展开竞赛。工作人员写出一个201位的大数，让求这个数的23次方根。运算结果，沙贡塔娜只用了50秒钟就向观众报出了正确的答案。而计算机为了得出同样的答数，必须输入两万条指令，再进行计算，花费的时间比沙贡塔娜要多得多。 这一奇闻，在国际上引起了轰动，沙贡塔娜被称为“数学魔术家”。 工作到最后一天的华罗庚 华罗庚出生于江苏省，从小喜欢数学，而且非常聪明。1930年，19岁的华罗庚到清华大学读书。华罗庚在清华四年中，在熊庆来教授的指导下，刻苦学习，一连发表了十几篇论文，后来又被派到英国留学，获得博士学位。他对数论有很深的研究，得出了著名的华氏定理。他特别注意理论联系实际，走遍了20多个省、市、自治区，动员群众把优选法用于农业生产。 记者在一次采访时问他：“你最大的愿望是什么？” 他不加思索地回答：“工作到最后一天。”他的确为科学辛劳工作的最后一天，实现了自己的诺言。 21世纪七大数学难题 美国的克雷数学研究所于2000年5月24日在巴黎宣布了众多数学家评选的结果:对七个“千禧年数学难题”的每一个悬赏一百万美元。 “千年大奖问题”公布以来，在世界数学界产生了强烈反响。这些问题都是关于数学基本理论的，但这些问题的解决将对数学理论的发展和应用的深化产生巨大推动。认识和研究“千年大奖问题”已成为世界数学界的热点。不少国家的数学家正在组织联合攻关。可以预期，“千年大奖问题”将会改变新世纪数学发展的历史进程。
初三下册数学知识点
初三下册数学知识点 1 二次函数及其图像 二次函数(quadratic function)是指未知数的次数为二次的多项式函数。二次函数可以表示为f(x)=ax bx c(a不为0)。其图像是一条主轴平行于y轴的抛物线。 一般的，自变量x和因变量y之间存在如下关系： 一般式 y=ax∧2; bx c(a≠0,a、b、c为常数)，顶点坐标为(-b/2a，-(4ac-b∧2)/4a) ; 顶点式 y=a(x m)∧2 k(a≠0,a、m、k为常数)或y=a(x-h)∧2 k(a≠0,a、h、k为常数)，顶点坐标为(-m，k)对称轴为x=-m，顶点的位置特征和图像的开口方向与函数y=ax ∧2的图像相同，有时题目会指出让你用配方法把一般式化成顶点式; 交点式 y=a(x-x1)(x-x2) [仅限于与x轴有交点A(x1，0)和B(x2，0)的抛物线] ; 重要概念：a，b，c为常数，a≠0，且a决定函数的开口方向，a>0时，开口方向向上，a牛顿插值公式(已知三点求函数解析式) y=(y3(x-x1)(x-x2))/((x3-x1)(x3-x2) (y2(x-x1)(x-x3))/((x2-x1)(x2-x3) (y1(x-x2)(x-x3))/((x1-x2)(x1-x3) 。由此可引导出交点式的系数a=y1/(x1*x2) (y1为截距) 求根公式 二次函数表达式的右边通常为二次三项式。 求根公式 x是自变量，y是x的二次函数 x1,x2=[-b±(√(b -4ac))]/2a (即一元二次方程求根公式)
求根的方法还有因式分解法和配方法 在平面直角坐标系中作出二次函数y=2x的平方的图像， 可以看出，二次函数的图像是一条永无止境的抛物线。 不同的二次函数图像 如果所画图形准确无误，那么二次函数将是由一般式平移得到的。 注意：草图要有1本身图像，旁边注明函数。 2画出对称轴，并注明X=什么 3与X轴交点坐标，与Y轴交点坐标，顶点坐标。抛物线的性质 轴对称 1.抛物线是轴对称图形。对称轴为直线x = -b/2a。 对称轴与抛物线的交点为抛物线的顶点P。 特别地，当b=0时，抛物线的对称轴是y轴(即直线x=0) 顶点 2.抛物线有一个顶点P，坐标为P ( -b/2a ，4ac-b ;)/4a ) 当-b/2a=0时，P在y轴上;当Δ= b ;-4ac=0时，P在x轴上。 开口 3.二次项系数a决定抛物线的开口方向和大小。 当a>0时，抛物线向上开口;当a|a|越大，则抛物线的开口越小。 决定对称轴位置的因素 4.一次项系数b和二次项系数a共同决定对称轴的位置。 当a与b同号时(即ab>0)，对称轴在y轴左; 因为若对称轴在左边则对称轴小于0，也就是- b/2a当a与b异号时(即ab0, 所以b/2a要小于0，所以a、b要异号 可简单记忆为左同右异，即当a与b同号时(即ab>0)，对称轴在y轴左;当a
十个数学家的故事文档
华罗庚 有一次，他跟邻居家的孩子一起出城去玩，他们走着走着；忽然看见路旁有座荒坟，坟旁有许多石人、石马。这立刻引起了华罗庚的好奇心，他非常想去看个究竟。于是他就对邻居家的孩子说： “那边可能有好玩的，我们过去看看好吗?” 邻居家的孩子回答道：“好吧，但只能呆一会儿，我有点害怕。” 胆大的华罗庚笑着说：“不用怕，世间是没有鬼的。”说完，他首先向荒坟跑去。 两个孩子来到坟前，仔细端详着那些石人、石马，用手摸摸这儿，摸摸那儿，觉得非常有趣。爱动脑筋的华罗庚突然问邻居家的孩子：“这些石人、石马各有多重?” 邻居家的孩子迷惑地望着他说："我怎么能知道呢?你怎么会问出这样的傻问题，难怪人家都叫你‘罗呆子’。” 华罗庚很不甘心地说道：“能否想出一种办法来计算一下呢?” 邻居家的孩子听到这话大笑起来，说道：“等你将来当了数学家再考虑这个问题吧!不过你要是能当上数学家，恐怕就要日出西山了。” 华罗庚不顾邻家孩子的嘲笑，坚定地说：“以后我一定能想出办法来的。” 当然，计算出这些石人、石马的重量，对于后来果真成为数学家的华罗庚来讲，根本不在话下。 金坛县城东青龙山上有座庙，每年都要在那里举行庙会。少年华罗庚是个喜爱凑热闹的人，凡是有热闹的地方都少不了他。有一年华罗庚也同大人们一起赶庙会，一个热闹场面吸引了他，只见一匹高头大马从青龙山向城里走来，马上坐着头插羽毛、身穿花袍的“菩萨”。每到之处，路上的老百姓纳头便拜，非常虔诚。拜后，他们向“菩萨”身前的小罐里投入钱，就可以问神问卦，求医求子了。 华罗庚感到好笑，他自己却不跪不拜“菩萨”。站在旁边的大人见后很生气，训斥道： “孩子，你为什么不拜，这菩萨可灵了。” “菩萨真有那么灵吗?”华罗庚问道。 一个人说道：“那当然，看你小小年纪千万不要冒犯了神灵，否则，你就会倒楣的。” “菩萨真的万能吗?”这个问题在华罗庚心中盘旋着。他不相信一尊泥菩萨真能救苦救难。庙会散了，看热闹的老百姓都回家了。而华罗庚却远远地跟踪着“菩萨”。看到“菩萨”进了青龙山庙里，小华罗庚急忙跑过去，趴在门缝向里面看。只见“菩萨”能动了，他从马上下来，脱去身上的花衣服，又顺手抹去脸上的妆束。门外的华庚惊呆了，原来百姓们顶礼膜拜的“菩萨”竟是一村民装扮的。 华罗庚终于解开了心中的疑团，他将“菩萨”骗人的事告诉了村子里的每个人，人们终于恍然大悟了。从此，人们都对这个孩子刮目相看，再也无人喊他“罗呆子”了。正是华罗庚这种打破砂锅问到底的精神， 陈景润 陈景润一个家喻户晓的数学家，在攻克歌德巴赫猜想方面作出了重大贡献，创立了著名的“陈氏定理”，所以有许多人亲切地称他为“数学王子”。但有谁会想到，他的成就源于一个故事。1937年，勤奋的陈景润考上了福州英华书院，此时正值抗日战争时期，清华大学航空工程系主任留英博士沈元教授回福建奔丧，不想因战事被滞留家乡。几所大学得知消息，都想邀请沈教授前进去讲学，他谢绝了邀请。由于他是英华的校友，为了报达母校，他来到了这所中学为同学们讲授数学课。 一天，沈元老师在数学课上给大家讲了一故事：“200年前有个法国人发现了一个有趣的现
初三数学上下册的学习知识点总结计划与重点难点总结计划.docx
----- 初三数学知识整理与重点难点总结 第21 章二次根式 知识框图 理解并掌握下列结论： （1）是非负数；（2）；（3）； I.二次根式的定义和概念： 1、定义：一般地，形如√ā（a≥0）的代数式叫做二次根式。当a＞0 时，√ a 表示 a 的算数平方根 ,√0=0 2、概念：式子√ā（a≥0）叫二次根式。√ā（a≥ 0）是一个非负数。 II.二次根式√ā 的简单性质和几何意义 1）a≥0; √ā≥0[双重非负性] 2）（√ā）^2=a（a≥0）[任何一个非负数都可以写成一个数的平方的形式] 3)√(a^2+b^2) 表示平面间两点之间的距离，即勾股定理推论。
IV. 二次根式的乘法和除法 1运算法则 √a·√ b= √ab（ a≥ 0，b≥0） -1- ----
√a/b= √a/√ b（a≥ 0，b>0 ） 二数二次根之积，等于二数之积的二次根。 2共轭因式 如果两个含有根式的代数式的积不再含有根式，那么这两个代数式叫做共轭因式，也称互为有理化根式。 V.二次根式的加法和减法 1同类二次根式 一般地，把几个二次根式化为最简二次根式后，如果它们的被开方数相同，就把这几个二次根式叫做同类二次根式。 2合并同类二次根式 把几个同类二次根式合并为一个二次根式就叫做合并同类二次根式。 3二次根式加减时，可以先将二次根式化为最简二次根式，再将被开方数相同的进行合并Ⅵ.二次根式的混合运算 1确定运算顺序 2灵活运用运算定律 3正确使用乘法公式 4大多数分母有理化要及时 5在有些简便运算中也许可以约分，不要盲目有理化 VII.分母有理化 分母有理化有两种方法 I.分母是单项式 如:√a/ √b= √a×√ b/√b×√ b=√ ab/b
九年级数学下册重要知识点总结
初三数学下册重要知识点总结 第25章概率 1、必然事件、不可能事件、随机事件的区别 2、概率 注意：（1）概率是随机事件发生的可能性的大小的数量反映. （2）概率是事件在大量重复试验中频率逐渐稳定到的值，即可以用大量重复试验中事件发生的频率去估计得到事件发生的概率，但二者不能简单地等同. 3、求概率的方法 （1）用列举法求概率（列表法、画树形图法） （2）用频率估计概率：一方面，可用大量重复试验中事件发生频率来估计事件发生的概率.另一方面,大量重复试验中事件发生的频率稳定在某个常数(事件发生的概率)附近，说明概率是个定值,而频率随不同试验次数而有所不同,是概率的近似值,二者不能简单地等同. 第26章二次函数 1. 二次函数的一般形式：y=ax2+bx+c.(a≠0) 4．求二次函数的解析式：已知二次函数图象上三点的坐标，可设解析式y=ax2+bx+c，并把这三点的坐标代入，解关于a、b、c的三元一次方程组，求出a、b、c的值, 从而求出解析式-------待定系数法. 5．二次函数的顶点式： y=a(x-h)2+k (a≠0)；由顶点式可直接得出二次函数的顶点坐标（h, k），对称轴方程 x=h 和函数的最值 y最值= k. 6．求二次函数的解析式：已知二次函数的顶点坐标（h,k）和图象上的另一点的坐标，可设解析式为y=a(x -h)2+ k，再代入另一点的坐标求a，从而求出解析式. 8. 二次函数y=ax2+bx+c (a≠0)的图象及几个重要点的公式： 9. 二次函数y=ax2+bx+c (a≠0)中，a、b、c与Δ的符号与图象的关系： (1) a＞0 <=> 抛物线开口向上； a＜0 <=> 抛物线开口向下； (2) c＞0 <=> 抛物线从原点上方通过； c=0 <=> 抛物线从原点通过； c＜0 <=> 抛物线从原点下方通过； (3) a, b异号 <=> 对称轴在y轴的右侧； a, b同号 <=> 对称轴在y轴的左侧； b=0 <=> 对称轴是y轴； (4) b2－4ac＞0 <=> 抛物线与x轴有两个交点； b2－4ac =0 <=> 抛物线与x 轴有一个交点（即相切）； b2－4ac＜0 <=> 抛物线与x轴无交点.
勤奋数学家小故事简短
勤奋数学家小故事简短 【篇一：勤奋数学家小故事简短】 高斯念小学的时候,有一次在老师教完加法后,因为老师想要休息,所以便出了一道题目要同学们算算看,题 目是：1+2+3+ .+97+98+99+100 = 老师心里正想,这下子小朋友一定要算到下课了吧!正要借口出去时, 却被高斯叫住了!原来呀,高斯已经算出来了,小朋友你可知道他是如何算的吗?高斯告诉大家他是如何算 出的：把 1加至 100 与 100 加至 1 排成两排相加,也就是说：1+2+3+4+ .+96+97+98+99+100 100+99+98+97+96+ .+4+3+2+1 =101+101+101+ .+101+101+101+101 共有一百个101相加,但算式重 复了两次,所以把10100 除以 2便得到答案等于从此以后高斯小学的学习过程早已经超越了其它的同学, 也因此奠定了他以后的数学基础,更让他成为——数学天才!自己可以浓缩一下 【篇二：勤奋数学家小故事简短】 最佳答案1、数学家的故事——苏步青 苏步青1902年9月出生在浙江省平阳县的一个山村里.虽然家境清贫,可他父母省吃俭用,拼死拼活也要供他上学.他在读初中时,对数学 并不感兴趣,觉得数学太简单,一学就懂.可是,后来的一堂数学课影响 了他一生的道路. 那是苏步青上初三时,他就读浙江省六十中来了一位刚从东京留学归 来的教数学课的杨老师.第一堂课杨老师没有讲数学,而是讲故事.他说：“当今世界,弱肉强食,世界列强依仗船坚炮利,都想蚕食瓜分中国.中华 亡国灭种的危险迫在眉睫,振兴科学,发展实业,救亡图存,在此一举.‘天 下兴亡,匹夫有责’,在座的每一位同学都有责任.”他旁征博引,讲述了数学在现代科学技术发展中的巨大作用.这堂课的最后一句话是：“为了 救亡图存,必须振兴科学.数学是科学的开路先锋,为了发展科学,必须 学好数学.”苏步青一生不知听过多少堂课,但这一堂课使他终身难忘. 杨老师的课深深地打动了他,给他的思想注入了新的兴奋剂.读书,不仅 为了摆脱个人困境,而是要拯救中国广大的苦难民众；读书,不仅是为 了个人找出路,而是为中华民族求新生.当天晚上,苏步青辗转反侧,彻 夜难眠.在杨老师的影响下,苏步青的兴趣从文学转向了数学,并从此立 下了“读书不忘救国,救国不忘读书”的座右铭.一迷上数学,不管是酷暑 隆冬,霜晨雪夜,苏步青只知道读书、思考、解题、演算,4年中演算了 上万道数学习题.现在温州一中（即当时省立十中）还珍藏着苏步青 一本几何练习薄,用毛笔书写,工工整整.中学毕业时,苏步青门门功课 都在90分以上. 17岁时,苏步青赴日留学,并以第一名的成绩考取东京高等工业学校, 在那里他如饥似渴地学习着.为国争光的信念驱使苏步青较早地进入 了数学的研究领域,在完成学业的同时,写了30多篇论文,在微分几何
人教数学九年级下册初三下册数学知识点归纳
人教数学九年级下册初三下册数学知识点归纳 学好数学的关键就在于要适时适量地进行总结归类，接下来WTT就为大家整理了这篇(九年级)初三下册数学知识点归纳，希望可以对大家有所帮助。 二次函数及其图像二次函数(quadratic function)是指未知数的最高次数为二次的多项式函数。二次函数可以表示为 f(x)=ax^2+bx+c(a不为0)。其图像是一条主轴平行于y轴的抛物线。 一般的，自变量x和因变量y之间存在如下关系： 一般式 y=ax+bx+c(a0,a、b、c为常数)，顶点坐标为(-b/2a，-(4ac-b2)/4a) ; 顶点式 y=a(x+m)2+k(a0,a、m、k为常数)或y=a(x-h)2+k(a0,a、h、k为常数)，顶点坐标为(-m，k)对称轴为x=-m，顶点的位置特征和图像的开口方向与函数y=ax2的图像相同，有时 题目会指出让你用配方法把一般式化成顶点式; 交点式 y=a(x-x1)(x-x2) [仅限于与x轴有交点A(x1，0)和 B(x2，0)的抛物线] ;
重要概念：a，b，c为常数，a0，且a决定函数的开口方向，a0时，开口方向向上，a0时，开口方向向下。a的绝对值还可以决定开口大小,a的绝对值越大开口就越小,a的绝对值越小开口就越大。 牛顿插值公式(已知三点求函数解析式) y=(y3(x-x1)(x-x2))/((x3-x1)(x3-x2)+(y2(x-x1)(x- x3))/((x2-x1)(x2-x3)+(y1(x-x2)(x-x3))/((x1-x2)(x1-x3) 。由此可引导出交点式的系数a=y1/(x1*x2) (y1为截距) 求根公式 二次函数表达式的右边通常为二次三项式。 求根公式 x是自变量，y是x的二次函数 x1,x2=[-b((b^2-4ac))]/2a (即一元二次方程求根公式)(如右图) 求根的方法还有因式分解法和配方法 在平面直角坐标系中作出二次函数y=2x的平方的图像， 可以看出，二次函数的图像是一条永无止境的抛物线。 不同的二次函数图像 如果所画图形准确无误，那么二次函数将是由一般式平移得到的。 注意：草图要有 1本身图像，旁边注明函数。
(完整版)关于数学家的励志故事大全【三篇】
关于数学家的励志故事大全【三篇】 了解一些数学家的故事，以及数学史,很有好处.学生通过感人、有趣的数学家的历史事例，以及一些数学的重大事件，有助于了解数学的发生和发展。下面是无忧考网整理分享的关于数学家的励志故事大全，欢迎阅读与借鉴。 【欧姆与欧姆定律】 乔治·西蒙·欧姆生于德国埃尔兰根城，父亲是锁匠。父亲自学了数学和物理方面的知识，并教给少年时期的欧姆，唤起了欧姆对科学的兴趣。16岁时他进入埃尔兰根大学研究数学、物理与哲学，由于经济困难，中途缀学，到1813年才完成博士学业。欧姆是一个很有天才和科学抱负的人，他长期担任中学教师，由于缺少资料和仪器，给他的研究工作带来不少困难，但他在孤独与困难的环境中始终坚持不懈地进行科学研究，自己动手制作仪器。 欧姆对导线中的电流进行了研究。他从傅立叶发现的热传导规律受到启发，导热杆中两点间的热流正比于这两点间的温度差。因而欧姆认为，电流现象与此相似，猜想导线中两点之间的电流也许正比于它们之间的某种驱动力，即现在所称的电动势。欧姆花了很大的精力在这方面进行研究。开始他用伏打电堆作电源，但是因为电流不稳定，效果不好。后来他接受别人的建议改用温差电池作电源，从而保
证了电流的稳定性。但是如何测量电流的大小，这在当时还是一个没有解决的难题。开始，欧姆利用电流的热效应，用热胀冷缩的方法来测量电流，但这种方法难以得到精确的结果。后来他把奥斯特关于电流磁效应的发现和库仑扭秤结合起来，巧妙地设计了一个电流扭秤，用一根扭丝悬挂一磁针，让通电导线和磁针都沿子午线方向平行放置;再用铋和铜温差电池，一端浸在沸水中，另一端浸在碎冰中，并用两个水银槽作电极，与铜线相连。当导线中通过电流时，磁针的偏转角与导线中的电流成正比。他将实验结果于1826年发表。1827年欧姆又在《电路的数学研究》一书中，把他的实验规律总结成如下公式：S=γE。式中S表示电流;E表示电动力，即导线两端的电势差，γ为导线对电流的传导率，其倒数即为电阻。 欧姆定律发现初期，许多物理学家不能正确理解和评价这一发现，并遭到怀疑和尖锐的批评。研究成果被忽视，经济极其困难，使欧姆精神抑郁。直到1841年英国皇家学会授予他荣誉的科普利金牌，才引起德国科学界的重视。 欧姆在自己的许多著作里还证明了：电阻与导体的长度成正比，与导体的横截面积和传导性成反比;在稳定电流的情况下，电荷不仅在导体的表面上，而且在导体的整个截面上运动。 人们为纪念他，将测量电阻的物理量单位以欧姆的姓氏命名。 【数学家陈景润】 陈景润是国际知名的大数学家，深受人们的敬重。但他并没有产生骄傲自满情绪，而是把功劳都归于祖国和人民。为了维护祖国的
数学家小故事十则
数学家小故事十则 一蒲丰 一天,法国数学家蒲丰请许多朋友到家里,做了一次试验.蒲丰在桌子上铺好一张大白纸,白纸上画满了等距离的平行线,他又拿出很多等长的小针,小针的长度都是平行线的一半.蒲丰说:“请大家把这些小针往这张白纸上随便仍吧！”客人们按他说的做了。 蒲丰的统计结果是：大家共掷2212次，其中小针与纸上平行线相交704次，2210÷704≈3.142。蒲丰说：“这个数是π的近似值。每次都会得到圆周率的近似值，而且投掷的次数越多，求出的圆周率近似值越精确。”这就是著名的“蒲丰试验”。 二高斯 高斯是位小学二年级的学生，有一天他的数学老师因为事情已处理了一大半，虽然上课了，仍希望将其完成，因此打算出一题数学题目给学生练习，他的题目是:1+2+3+4+5+6+7+8+9+10=？，因为加法刚教不久，所以老师觉得出了这题，学生肯定是要算蛮久的，才有可能算出来，也就可以藉此利用这段时间来处理未完的事情，但是才一转眼的时间，高斯已停下了笔，闲闲地坐在那里，老师看到了很生气的训斥高斯，但是高斯却说他已经将答案算出来了，就是55，老师听了下了一跳，就问高斯如何算出来的，高斯答道，我只是发现1和10的和是11、2和9的和也是11、3和8的和也是11、4和7的和也是11、5和6的和还是11，又11+11+11+11+11=55，我就是这么算的。高斯长大后，成为一位很伟大的数学家。高斯小的时候能将难题变成简易，当然资质是很大的因素，但是他懂得观察，寻求规则，化难为简，却是值得我们学习与效法的。 三笛卡儿 笛卡儿最杰出的成就是在数学发展上创立了解析几何学。在笛卡儿时代，代数还是一个比较新的学科，几何学的思维还在数学家的头脑中占有统治地位。笛卡儿致力于代数和几何联系起来的研究，于1637年，在创立了坐标系后，成功地创立了解析几何学。他的这一成就为微积分的创立奠定了基础。解析几何直到现在仍是重要的数学方法之一。 四冯·诺依曼 20世纪最杰出的数学家之一的冯·诺依曼。众所周知，1946年发明的电子计算机，大大促进了科学技术的进步，大大促进了社会生活的进步。鉴于冯·诺依曼在发明电子计算机中所起到关键性作用，他被西方人誉为"计算机之父".1911年一1921年，冯·诺依曼在布达佩斯的卢瑟伦中学读书期间，就崭露头角而深受老师的器重。在费克特老师的个别指导下并合作发表了第一篇数学论文，此时冯·诺依曼还不到18岁。 五杨辉 杨辉，中国南宋时期杰出的数学家和数学教育家。在13世纪中叶活动于苏杭一带，其著作甚多。他著名的数学书共五种二十一卷。著有《详解九章算法》十二卷（1261年）、《日用算法》二卷（1262年）、《乘除通变本末》三卷（1274年）、《田亩比类乘除算法》二卷（1275年）、《续古摘奇算法》二卷（1275年）。杨辉的数学研究与教育工作的重点是在计算技术方面，他对筹算乘除捷算法进行总结和发展，有的还编成了歌决，如九归口决。他在《续古摘奇算法》中介绍了各种形式的"纵横图"及有关的构造方法，同时"垛积术"是杨辉继沈括"隙积术"后，关于高阶等差级数的研究。杨辉在"纂类"中，将《九章算术》246个题目按解题方法由浅入深的顺序，重新分为乘除、分率、合率、互换、二衰分、叠积、盈不足、方程、勾股等九类。 他非常重视数学教育的普及和发展，在《算法通变本末》中，杨辉为初学者制订的"习算纲目"是中国数学教育史上的重要文献。
初中数学九年级上下册知识点总结
[九年级(上册) 第一章 证明(二) ※等腰三角形的“三线合一”：顶角平分线、底边上的中线、底边上的高互相重合。 ※等边三角形是特殊的等腰三角形，作一条等边三角形的三线合一线，将等边三角形分成两个全等的 直角三角形，其中一个锐角等于30o，这它所对的直角边必然等于斜边的一半。 ※有一个角等于60o的等腰三角形是等边三角形。 ※如果知道一个三角形为直角三角形首先要想的定理有： ①勾股定理：2 2 2 c b a =+（注意区分斜边与直角边） ②在直角三角形中，如有一个内角等于30o，那么它所对的直角边等于斜边的一半 ③在直角三角形中，斜边上的中线等于斜边的一半（此定理将在第三章出现） ※垂直平分线．．．．．是垂直于一条线段．．并且平分这条线段的直线．．。（注意着重号的意义） <直线与射线有垂线，但无垂直平分线> ※线段垂直平分线上的点到这一条线段两个端点距离相等。 ※线段垂直平分线逆定理：到一条线段两端点距离相等的点，在这条线段的垂直平分线上。 ※三角形的三边的垂直平分线交于一点，并且这个点到三个顶点的距离相等。（如图1所示， AO=BO=CO ） ※角平分线上的点到角两边的距离相等。 ※角平分线逆定理：在角内部的，如果一点到角两边的距离相等，则它在该角的平分线上。 角平分线是到角的两边距离相等的所有点的集合。 ※三角形三条角平分线交于一点，并且交点到三边距离相等，交点即为三角形的内心。 (如图2所示，OD=OE=OF) 第二章 一元二次方程 ※只含有一个未知数的整式方程，且都可以化为02 =++c bx ax （a 、b 、c 为 常数，a ≠0）的形式，这样的方程叫一元二次方程．．．．．． 。 ※把02 =++c bx ax （a 、b 、c 为常数，a ≠0）称为一元二次方程的一般形式，a 为二次项系数；b 为一次项系数；c 为常数项。 ※解一元二次方程的方法：①配方法 <即将其变为0)(2 =+m x 的形式> ②公式法 a ac b b x 242-±-= （注意在找ab c 时须先把方程化为一般形式） ③分解因式法 把方程的一边变成0，另一边变成两个一次因式的乘积来求解。 （主要包括“提公因式”和“十字相乘”） A C B O 图1 图2 O A C B D E F
(苏科版)初三下册数学知识点总结
（苏科版）初三下册数学知识点总结 一元二次方程的基本概念 1．一元二次方程3x2+5x-2=0的常数项是-2. 2．一元二次方程3x2+4x-2=0的一次项系数为4，常数项是-2. 3．一元二次方程3x2-5x-7=0的二次项系数为3，常数项是-7. 4．把方程3x(x-1)-2=-4x化为一般式为3x2-x-2=0. 知识点2：直角坐标系与点的位置 1．直角坐标系中，点A（3，0）在y轴上。2．直角坐标系中，x轴上的任意点的横坐标为0. 3．直角坐标系中，点A（1，1）在第一象限. 4．直角坐标系中，点A（-2，3）在第四象限. 5．直角坐标系中，点A（-2，1）在第二象限. 知识点3：已知自变量的值求函数值 1．当x=2时,函数y=2x3的值为1. 2．当x=3时,函数y=1的值为1. x2 3．当x=-1时,函数y=1的值为1. x3 知识点4：基本函数的概念及性质 1．函数y=-8x是一次函数. 2．函数y=4x+1是正比例函数. 3．函数y12 x是反比例函数. 4．抛物线y=-3(x-2)2-5的开口向下. 5．抛物线y=4(x-3)2-10的对称轴是x=3. 6．抛物线y12 (x1)22的顶点坐标是(1,2). 7．反比例函数y 2 x 的图象在第一、三象限. 知识点5：数据的平均数中位数与众数1．数据
13,10,12,8,7的平均数是10. 2．数据3,4,2,4,4的众数是4. 3．数据1，2，3，4，5的中位数是3. 知识点6：特殊三角函数值 知识点7：圆的基本性质 1．半圆或直径所对的圆周角是直角. 2．任意一个三角形一定有一个外接圆. 3．在同一平面内，到定点的距离等于定长的点的轨迹，是以定点为圆心，定长为半径的圆. 4．在同圆或等圆中，相等的圆心角所对的弧相等. 5．同弧所对的圆周角等于圆心角的一半. 6．同圆或等圆的半径相等. 7．过三个点一定可以作一个圆. 8．长度相等的两条弧是等弧. 9．在同圆或等圆中，相等的圆心角所对的弧相等. 10．经过圆心平分弦的直径垂直于弦。 知识点8：直线与圆的位置关系 1．直线与圆有公共点时,叫做直线与圆相切. 2．三角形的外接圆的圆心叫做三角形的外心. 3．弦切角等于所夹的弧所对的圆心角. 4．三角形的内切圆的圆心叫做三角形的内心. 5．垂直于半径的直线必为圆的切线. 6．过半径的外端点并且垂直于半径的直线是圆的切线. 7．垂直于半径的直线是圆的切线. 8．圆的切线垂直于过切点的半径. 知识点9：圆与圆的位置关系 1．两个圆有且只有一个公共点时,叫做这两个圆外切. 2．相交两圆的连心线垂直平分公共弦. 3．两个圆有两个公共点时,叫做这两个圆相交. 4．两个圆内切时,这两个圆的公切线只有一条. 5．相切两圆的连心线切点.
六个中国数学家的故事
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苏步青1902年9月出生在浙江省平阳县的一个山村里。虽 然家境清贫，可他父母省吃俭用，拼死拼活也要供他上学。他在读 初中时，对数学并不感兴趣，觉得数学太简单，一学就懂。可是， 后来的一堂数学课影响了他一生的道路。 那是苏步青上初三时，他就读浙江省六十中来了一位刚从东京留学归来的教数学课的杨老师。第一堂课杨老师没有讲数学，而 是讲故事。他说：“当今世界，弱肉强食，世界列强依仗船坚炮利，都想蚕食瓜分中国。中华亡国灭种的危险迫在眉睫，振兴科学，发 展实业，救亡图存，在此一举。‘天下兴亡，匹夫有责 ’，在座的每一位同学都有责任。”他旁征博引，讲述了数学在现代科学技术发展中的巨大作用。这堂课的最后一句话是：“为 了救亡图存，必须振兴科学。数学是科学的开路先锋，为了发展科学，必须学好数学。”苏步青一生不知听过多少堂课，但这一堂课 使他终身难忘。
杨老师的课深深地打动了他，给他的思想注入了新的兴奋剂。读书，不仅为了摆脱个人困境，而是要拯救中国广大的苦难民众;读书，不仅是为了个人找出路，而是为中华民族求新生。当天晚上， 苏步青辗转反侧，彻夜难眠。在杨老师的影响下，苏步青的兴趣从 文学转向了数学，并从此立下了“读书不忘救国，救国不忘读书” 的座右铭。一迷上数学，不管是酷暑隆冬，霜晨雪夜，苏步青只知 道读书、思考、解题、演算，4年中演算了上万道数学习题。现在 温州一中(即当时省立十中)还珍藏着苏步青一本几何练习薄，用毛 笔书写，工工整整。中学毕业时，苏步青门门功课都在90分以上。 17岁时，苏步青赴日留学，并以第一名的成绩考取东京高等工业学校，在那里他如饥似渴地学习着。为国争光的信念驱使苏步 青较早地进入了数学的研究领域，在完成学业的同时，写了30多篇论文，在微分几何方面取得令人瞩目的成果，并于1931年获得理学博士学位。获得博士之前，苏步青已在日本帝国大学数学系当讲师，正当日本一个大学准备聘他去任待遇优厚的副教授时，苏步青却决 定回国，回到抚育他成长的祖任教。回到浙大任教授的苏步青，生
人教版数学九年级下册知识点
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y=a(x+m)∧2+k(a≠0,a、m、k为常数)或y=a(x-h)∧２+k(a≠0,a、h、k为常数)，顶点坐标为（-m，k）对称轴为x=-m，顶点的位置特征和图像的开口方向与函数ｙ＝ax∧２的图像相同，有时题目会指出让你用配方法把一般式化成顶点式； 交点式 y=a(x-x1)(x-x2) [仅限于与x轴有交点A（x1，0）和 B（x2，0）的抛物线] ； 重要概念：a，b，c为常数，a≠0，且a决定函数的开口方向，a>0时，开口方向向上，a<0时，开口方向向下。a的绝对值还可以决定开口大小,a的绝对值越大开口就越小,a的绝对值越小开口就越大。 牛顿插值公式（已知三点求函数解析式） y=(y3(x-x1)(x-x2))/((x3-x1)(x3-x2)+(y2(x-x1)(x-x3))/((x2-x1)(x2-x3)+(y1(x -x2)(x-x3))/((x1-x2)(x1-x3) 。由此可引导出交点式的系数a=y1/(x1*x2) (y1为截距) 求根公式 二次函数表达式的右边通常为二次三项式。 求根公式 x是自变量，y是x的二次函数 x1,x2=[-b±(√(b^2-4ac))]/2a (即一元二次方程求根公式）（如右图） 求根的方法还有因式分解法和配方法 在平面直角坐标系中作出二次函数y=2x的平方的图像， 可以看出，二次函数的图像是一条永无止境的抛物线。
初三数学上下册知识点总结与重点难点总结
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二数二次根之积，等于二数之积的二次根。 2 共轭因式 如果两个含有根式的代数式的积不再含有根式，那么这两个代数式叫做共轭因式，也称互为有理化根式。 V.二次根式的加法和减法 1 同类二次根式 一般地，把几个二次根式化为最简二次根式后，如果它们的被开方数相同，就把这几个二次根式叫做同类二次根式。 2 合并同类二次根式 把几个同类二次根式合并为一个二次根式就叫做合并同类二次根式。 3二次根式加减时，可以先将二次根式化为最简二次根式，再将被开方数相同的进行合并 Ⅵ.二次根式的混合运算 1确定运算顺序 2灵活运用运算定律 3正确使用乘法公式 4大多数分母有理化要及时 5在有些简便运算中也许可以约分，不要盲目有理化 VII.分母有理化 分母有理化有两种方法 I.分母是单项式 如:√a/√b=√a×√b/√b×√b=√ab/b II.分母是多项式 要利用平方差公式 如1/√a＋√b=√a－√b/(√a＋√b)(√a－√b)=√a－√b/a－b III.分母是多项式 要利用平方差公式 如1/√a＋√b=√a－√b/(√a＋√b)(√a－√b)=√a－√b/a－b