教程微臣小白班

1、GRE Quantitative Reasoning（小白）趙立儀2017.2.22GRE Quantitative概述2011年ETS對(duì)GRE進(jìn)行全面改革，GRE難度大大增加，特別是2015年5月后；中國大陸有一半考生在新GRE數(shù)學(xué)部分沒有上162！就是162！改革后的GRE數(shù)學(xué)部分，增加了四分位數(shù)、百分位數(shù)、箱線圖、正態(tài)分布等高難度內(nèi)容，在概率統(tǒng)計(jì)方面難度大增；ETS出版的OG中，有一半的篇幅是在介紹數(shù)學(xué)知識(shí)點(diǎn)；太多信心滿滿的考生裸考GRE數(shù)學(xué)，結(jié)果嘞。GRE Quantitative概述GRE 分?jǐn)?shù)對(duì)申請(qǐng)的影響Harvard, Stanford, MIT, Berkeley, Princ

2、eton等大神學(xué)校錄取分?jǐn)?shù)線為330+（TOP20）;UCLA, UVA, Columbia, John Hopkins, Pennsylvania等錄取平均分為325+（TOP20-50）;TOP100的錄取分?jǐn)?shù)線為320+;數(shù)據(jù)為大致估計(jì)，會(huì)有些許誤差。Prime numberPrime factorDivisorDivisibleQuotientNumeratorDenominatorTerminate decimalVariableCoefficientQuadratic equationQuartileStandard deviationPermutationExponentDoma

3、inOriginQuadrantsInterceptPerpendicularPolygonsTrianglesQuadrilateralsVertexIsoscelesHypotenuseTrapezoidCircular cylinder質(zhì)數(shù)質(zhì)因子因子/除數(shù)可被整除的商分子分母有限小數(shù)變量系數(shù)二次方程四份位數(shù)標(biāo)準(zhǔn)差排列指數(shù)定義域原點(diǎn)象限截距垂直多邊形三角形四邊形頂點(diǎn)等腰斜邊梯形排列GRE Quantitative考試目的1. 精通算術(shù)運(yùn)算2. 精通代數(shù)方程的求解3. 具有將文字信息轉(zhuǎn)換為數(shù)學(xué)術(shù)語的能力4. 具有構(gòu)想幾個(gè)圖像以及數(shù)之間關(guān)系的能力5. 具有用直覺和非常規(guī)的方法去解決一般數(shù)學(xué)問題

4、的能力6. 具有在真實(shí)生活場(chǎng)景中運(yùn)用數(shù)學(xué)的能力GRE數(shù)學(xué)的出題總體基本假設(shè) All numbers used are real numbers. 所有的數(shù)都是實(shí)數(shù)All figures lie on a plane unless otherwise indicated. 除非題目中有專門指出，假設(shè)所有圖形都在同一平面內(nèi)All angle measures are positive 所有的測(cè)量角度都是正數(shù)All line shown as straight are straight. On the computer-based test, lines that appear “jagged” c

5、an also be assumed to be straight. 所有顯示為直線的線都可以當(dāng)作直線來處理Figures are intended to provide useful information for answering the questions. However, except where a figure is plish by a “Note” stating that the figure is drawn to scale, solve the problem using your knowledge of mathematics, not by visual me

6、asurement or estimation. 伴隨問題的圖形將為解題提供有用的信息。但是，只有在問題中指出相應(yīng)圖形是按比例畫出 drawn to scale）時(shí)，才可以用目測(cè)或者估計(jì)而得到的信息去解題。否則只能運(yùn)用你的數(shù)學(xué)知識(shí)去回答問題。GRE Quantitative題型GRE數(shù)學(xué)題型選擇題填空題單選題不定項(xiàng)選題單空式分?jǐn)?shù)式一般單選比較大小AA（空格里只能有數(shù)字，小數(shù)點(diǎn)，負(fù)號(hào)）（空格里只能有數(shù)字，負(fù)號(hào)）Quantitative Comparison questions（數(shù)量比較題）一、題目要求Questions of this type ask you to compare two qu

7、antitiesQuantity A and Quantity B, and then determine which of the following statements describes the comparison.Quantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from the information given.【ps】如果出現(xiàn)了A大于等于B的情況，應(yīng)該選什么？Multiple-choice Question

8、sSelect One Answer Choice（單項(xiàng)選擇題）These questions are multiple-choice questions that ask you to select only one answer choice from a list of five choices.解題技巧從選項(xiàng)中尋找解題線索，快速排除明顯錯(cuò)誤的選項(xiàng)沒有思路時(shí)，可以嘗試代入法，答案一定在5個(gè)選項(xiàng)當(dāng)中基于題目給的條件去做，不要腦補(bǔ)其他任何條件利用枚舉法的時(shí)候要非常注意列舉順序The figure above shows the graph of the function f defined

9、by f(x)=l2xl+4 for all numbers x. For which of the following functions g, defined for all numbers x, does the graph of g intersect the graph of f?A g(x)=x2B g(x)=x+3C g(x)=2x2D g(x)=2x+3E g(x)=3x2Multiple-choice QuestionsSelect One or More Answers（不定項(xiàng)選擇題）These questions are multiple-choice questions

10、 that ask you to select one or more answer choices from a list of choices. A question may or may not specify the number of choices to select. These questions are marked with square boxes beside the answer choices, not circles or ovals. Select one or more answer choices according to the specific ques

11、tion directions. If the question does not specify how many answer choices to select, select all that apply.The correct answer may be just one of the choices or as many as all of the choices, depending on the question.No credit is given unless you select all of the correct choices and no others. If t

12、he question specifies how many answer choices to select, select exactly that number of choices.解題技巧（單選的五個(gè)技巧全部保留）如果可能，將答案范圍寫在草稿紙上答案選項(xiàng)從3個(gè)到12個(gè)不等，不要為了多選而多選注意題目要求的答案?jìng)€(gè)數(shù)3. Which of the following could be the units digit of 57n, where n is a positive integer?Indicate all such digits.A. 0B. 1C. 2D. 3E. 4F. 5

13、G. 6H. 7I. 8J. 9Numeric Entry questions（數(shù)字填空題）Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxesone for the numerator and one for the denominator.To enter an integer or a decimal, either type the number in the answer box usin

14、g the keyboard or use the Transfer Display button on the calculator.First, click on the answer boxa cursor will appear in the boxand then type the number.To erase a number, use the Backspace key.For a negative sign, type a hyphen. For a decimal point, type a period.To remove a negative sign, type th

15、e hyphen again and it will disappear; the number remains.The Transfer Display button on the calculator will transfer the answer to the answer box.Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct. Enter the exact answer unless the question asks you to round your answer.To

16、 enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.The Transfer Display button on the calculator cannot be used for a fraction.Fract

17、ions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.解題技巧：確保你的答案與題目的要求是否一致（四舍五入）合理使用計(jì)算器等同的正確答案也可以給分，不要一味的追求約分 （4/9）通過常識(shí)來驗(yàn)證你的答案是否正確RESULTS OF A USED-CAR AUCTION small cars large carsNumber of cars offered 32 23Number of cars sold 16 20Project

18、ed sales total for cars offered (in thousands) $70 $150Actual sales total (in thousands) $41 $120For the large cars sold at an auction that is summarized in the table above, what was the average sale price per car?$ 大多數(shù)的計(jì)算不需要計(jì)算器，不要因?yàn)門A在那兒就非要使用一些簡(jiǎn)單的計(jì)算，盡量不要使用計(jì)算器注意運(yùn)算順序有些計(jì)算很巧妙的回避計(jì)算過程，不要過分依賴計(jì)算器（極限問題）使用Tr

19、ansfer Display時(shí)，注意近似的位數(shù) 4.4456在做分?jǐn)?shù)題的時(shí)候，計(jì)算結(jié)果有小數(shù)的時(shí)候Arithmetic算數(shù)Integer（positive/negative/0）：整數(shù)Odd integer/even integer：奇數(shù)/偶數(shù)Factor（divisor）：因子Multiple：倍數(shù)Divisible：可被整除Be divided by：被除quotient：商Remainder：余數(shù)Prime number：質(zhì)數(shù)Composite number：合數(shù)Prime factor：質(zhì)因子Prime factorization：質(zhì)因分解Least common multiple：最

20、小公倍數(shù)Greatest common divisor：最大公約數(shù)正負(fù)性 (相乘的關(guān)系) 正*正=正 正*負(fù)=負(fù) 負(fù)*負(fù)=正 奇偶性 (用2K+1和2k來帶入) 奇*奇=奇 奇+奇=偶 奇*偶=偶 奇+偶=奇 偶*偶=偶 偶+偶=偶 注意：如果n個(gè)數(shù)相乘=奇數(shù)，則n個(gè)數(shù)均為奇數(shù)質(zhì)數(shù)與合數(shù)（注意1） 質(zhì)數(shù)：因子只有1和其本身；2是唯一的偶質(zhì)數(shù) 因子中的質(zhì)數(shù)為質(zhì)因子 質(zhì)因分解： 12=22*3 30=3*2*5 LCM&GCD的求法1. If x and y are integers and xy2 is a positive odd integer, which of the following

21、 must be true? Axy is positive. Bxy is odd. Cx + y is even.2. If a2 +b2 =c2，and a,b,c are all integers. Which of the following CANNOT be the value of a+b+c？ (A) 2 (B) 1(C) -2(D) 4(E) 63. If M is the least common multiple of 90,196,and 300, which of the following is NOT a factor of M? A. 600 B. 700 C

22、. 900 D. 2,100 E. 4,900Solution: 難點(diǎn)：正負(fù)性、奇偶性xy20-x0，x*y*y=奇數(shù)-x, y均為奇數(shù)所以x*y無法判斷正負(fù)性，但為奇數(shù)，且x+y=偶數(shù)，答案為BCSolution: 難點(diǎn)：奇偶性舉例代數(shù)、性質(zhì)應(yīng)用Solution: 難點(diǎn)：LCM的求法和因子的判斷LCM=22*32*52*72Fraction：分?jǐn)?shù)Numerator：分子Denominator：分母Rational number：有理數(shù)Irrational number：無理數(shù)Common denominator：公分母Reciprocal：倒數(shù)Mixed number：帶分?jǐn)?shù)Decimal

23、point：小數(shù)點(diǎn)Base：底數(shù)Exponent：指數(shù)Terminate decimal有限小數(shù)Repeat decimal 循環(huán)小數(shù)Infinite decimal 無限小數(shù)Units, tens, hundreds, thousandsTenths, hundredths, thousandths3/5 的分母加s- three fifthsSquare root 平方根Cube root 立方根Fourth root 四次方根Ratio ：比率-ratio A to B：A：BProportion：成比例的（directly：正比）2. 注意在比較類里面變化的基數(shù)是什么？ 第一類： 自我

24、變化類 公式： （|新-舊|）/ 舊 例子： 從2到3增加了百分之多少？ 從3到2減少了百分之多少？ 第二類： 兩個(gè)不同的數(shù)之間的比較 公式： （|先-后|）/ 后 例子： 1比2少百分之多少？ 2比1多百分之多少？指數(shù)運(yùn)算規(guī)則 x-a=1/xa (xa)*(xb)=xa+b (x)a)b=xab xa/xb=xa-b x0=1 (xy)a=xayaIf the tens digit x and the units digit y of a positive integer n are reversed, the resulting integer is 9 more than n. What

25、 is y in terms of x ?A. 10 xB. 9 xC. x + 9D. x 1E. x + 1真題搶鮮!Solution: 難點(diǎn)：數(shù)論知識(shí)的應(yīng)用&tens、unit的釋義n=xy=10*x+y*1，n=yx=10*y+x*1n=n+9Solution: 難點(diǎn)：at most: 至多，平方根的判斷 易錯(cuò)點(diǎn)：開放之后的正負(fù)性，答案選EAlgebra(代數(shù))Variable 變量Like terms 同類項(xiàng)Constant 常數(shù)Coefficient 系數(shù)Identity 恒等式Linear equation 線性方程Quadratic equation 二次方程linear in

26、equalities 線性不等式Functions 函數(shù)Interests 利息coordinate geometry 坐標(biāo)幾何straight lines 直線graph of functions 函數(shù)的圖像2xy2+3xy+6z-21xz-xy+0.5 is called an algebraic expression 代數(shù)表達(dá)式2xy2, 3xy, 6z, -21xz, -xy, 0.5稱為terms（代數(shù)）項(xiàng)3xy和-xy稱為like terms or similar terms（同類項(xiàng)）0.5稱為constant term 常數(shù)項(xiàng)coefficient系數(shù)：2xy2這一項(xiàng)的系數(shù)是2，

27、 -21xz的系數(shù)是-212xy2+3xy+6z-21xz-xy+0.5 =1稱為equation等式，等式的solution解就是滿足等式的值。如x=2是x2-2x=0的一個(gè)解y=f(x)稱為一個(gè)函數(shù)Domain定義域?yàn)楹瘮?shù)有定義的所有x值Range值域?yàn)楹瘮?shù)所有可能的取值如y=f(x)=x2-1,那么定義域就是所有實(shí)數(shù)，值域?yàn)閥-1graphs of functionsy=ax+b 的圖像是一條直線，斜率為a, y軸的截距為b（截距可正可負(fù)可為0）y=ax2+bx+c , 其中a0的圖像是一個(gè)拋物線parabola如y=x2-2x-3的頂點(diǎn)vertex是（1, -4），x=1是對(duì)稱軸lin

28、e of symmetry，開口向上opens upward(x-a)2+(y-b)2=r2的圖像是一個(gè)圓circle，圓心center為（a, b), radius半徑為ry=f(x)和y=g(x)的圖像交點(diǎn)就是f(x)=g(x)的解上加下減&左加右減壓縮shrunk &拉伸stretchedWEWE陳琦GRE團(tuán)隊(duì) 80313頻道Geometry(幾何)line and angles 線和角度Polygons 多邊形Triangles 三角形Quadrilaterals 四邊形Circles 圓Three-dimensional Figures 三維圖形 如圖所示，A,B,C為point點(diǎn)，

29、L為line直線，AB,BC,CD為line segment 線段，線段AB的兩個(gè)端點(diǎn)endpoints 為A和B。線段AB的長(zhǎng)度length為8，BC長(zhǎng)度為6，CD長(zhǎng)度為6。相同長(zhǎng)度的線段我們稱為congruent line segments。線段BD的中點(diǎn)midpoint為CPerpendicular 垂直Right angle 直角如圖所示當(dāng)兩條直線相交的時(shí)候，形成了4個(gè)角angles，每個(gè)角都有一個(gè)共同的頂點(diǎn)vertex P。 APC和 BPD稱為對(duì)頂角opposite angles或者vertical angles，對(duì)頂角相等。銳角acute angle 為小于90的角，如 APC。

30、鈍角obtuse angle 為大于90的角，如 APD。三角形內(nèi)角和為180polygon多邊形內(nèi)角和為180 (n-2)等邊三角形equilateral triangle， 等腰三角形isosceles triangle直角三角形right triangle，直角邊legs，斜邊hypotenuse勾股定理Pythagorean theorem（畢達(dá)哥拉斯定理）WEParallelogram 平行四邊形Rectangle 長(zhǎng)方形Rhombus 菱形Square 正方形Trapezoid 梯形Circle 圓Center 圓心 Radius 半徑Diameter 直徑Congruent ci

31、rcles 全等圓Circumference 周長(zhǎng)Chord 弦Concentric circles 同心圓arc 弧圓外接四邊形circumscribed圓內(nèi)接三角形inscribedTangent line：切線；the point of tangent ：切點(diǎn)弧長(zhǎng)的求法：arc/2r=/360扇形面積的求法：S/r2=/360Frequency Distribution 頻率分布圖Bar Graphs 條形圖Circle Graphs 餅狀圖Histograms 直方圖Scatterplots 散點(diǎn)圖Time Plots 時(shí)間曲線圖區(qū)分題目給的圖表是哪一種毫無意義，主要看清橫縱坐標(biāo)或者圖

32、形代表的意義算術(shù)平均數(shù)（Average or Arithmetic Mean or Mean): 所有數(shù)據(jù)之和除以數(shù)據(jù)個(gè)數(shù)中數(shù)（Median): 將所有數(shù)據(jù)從小到大排列，取中間的數(shù)或中間兩個(gè)數(shù)的算術(shù)平均數(shù)眾數(shù)（Mode): 一組數(shù)據(jù)中出現(xiàn)頻率最高的數(shù)。一組數(shù)據(jù)中可能不止一個(gè)眾數(shù)極差（Range): 一組數(shù)據(jù)中最大數(shù)與最小數(shù)之差方差（Variance): 一組數(shù)據(jù)中每個(gè)數(shù)與算術(shù)平均數(shù)之差的平方和的算術(shù)平均數(shù)標(biāo)準(zhǔn)方差（Standard Deviation): 方差的平方根Measures of position-quartiles and percentilesInterquartile rang

33、e= the third quartile minus the first quartile The normal distribution 正態(tài)分布 The standard normal distribution 標(biāo)準(zhǔn)正態(tài)分布To find the quartiles for the list of 16 numbers 2, 4, 4, 5, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9,9 First divide the data into two groups of 8 numbers each. The first group is 2, 4, 4, 5, 7,

34、 7, 7, 7 and the second group is 7, 7, 8, 8, 9, 9, 9, 9, The second quartile, or median, is Q2=M=7. The first quartile is Q1=6(the average of 5 and 7) The third quartile is Q3=8.5 (the average of 8 and 9).2. To find the quartiles for the list of 7 numbers 2, 4, 4, 5, 6, 7, 9 First divide the data in

35、to two groups of 3 numbers each. 2,4,4 and 6,7,9 The second quartile, or median, is Q2=M=5. The first quartile is Q1=4 The third quartile is Q3=7In a distribution of 8,500 different measurements of the variable x, 26.5 isthe 56th percentile and 37.1 is the 78th percentile. Which of the followingis c

36、loses that are in the distribution such that 26.5x37.1?A 1850 B 2200 C 3500 D 4750 E 6650Solution：難點(diǎn)：百分位數(shù)78-56=22，于是在26.5和37.1之間有大約22%的數(shù)即8500*22%=1870，選AThe mean, median, and mode are equal.The data are grouped symmetrically about the mean.About two-thirds of the data are within 1 standard deviation of the mean.About 96% of the data are within 2 standard deviations of the mean.About 100% of the data are within 3 standard deviations of the mean.一個(gè)正態(tài)分布有兩個(gè)參數(shù)：m（平均數(shù)）和d（標(biāo)準(zhǔn)差）M決定了分布的中心，d決定了分布的高矮胖瘦（d越大越矮胖，反之越高瘦）其中有一個(gè)特殊的正態(tài)分布我們稱為stan