教程微臣小白班

1、GRE Quantitative Reasoning（小白）赵立仪2017.2.22GRE Quantitative概述2011年ETS对GRE进行全面改革，GRE难度大大增加，特别是2015年5月后；中国大陆有一半考生在新GRE数学部分没有上162！就是162！改革后的GRE数学部分，增加了四分位数、百分位数、箱线图、正态分布等高难度内容，在概率统计方面难度大增；ETS出版的OG中，有一半的篇幅是在介绍数学知识点；太多信心满满的考生裸考GRE数学，结果嘞。GRE Quantitative概述GRE 分数对申请的影响Harvard, Stanford, MIT, Berkeley, Princ

2、eton等大神学校录取分数线为330+（TOP20）;UCLA, UVA, Columbia, John Hopkins, Pennsylvania等录取平均分为325+（TOP20-50）;TOP100的录取分数线为320+;数据为大致估计，会有些许误差。Prime numberPrime factorDivisorDivisibleQuotientNumeratorDenominatorTerminate decimalVariableCoefficientQuadratic equationQuartileStandard deviationPermutationExponentDoma

3、inOriginQuadrantsInterceptPerpendicularPolygonsTrianglesQuadrilateralsVertexIsoscelesHypotenuseTrapezoidCircular cylinder质数质因子因子/除数可被整除的商分子分母有限小数变量系数二次方程四份位数标准差排列指数定义域原点象限截距垂直多边形三角形四边形顶点等腰斜边梯形排列GRE Quantitative考试目的1. 精通算术运算2. 精通代数方程的求解3. 具有将文字信息转换为数学术语的能力4. 具有构想几个图像以及数之间关系的能力5. 具有用直觉和非常规的方法去解决一般数学问题

4、的能力6. 具有在真实生活场景中运用数学的能力GRE数学的出题总体基本假设 All numbers used are real numbers. 所有的数都是实数All figures lie on a plane unless otherwise indicated. 除非题目中有专门指出，假设所有图形都在同一平面内All angle measures are positive 所有的测量角度都是正数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. 所有显示为直线的线都可以当作直线来处理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. 伴随问题的图形将为解题提供有用的信息。但是，只有在问题中指出相应图形是按比例画出 drawn to scale）时，才可以用目测或者估计而得到的信息去解题。否则只能运用你的数学知识去回答问题。GRE Quantitative题型GRE数学题型选择题填空题单选题不定项选题单空式分数式一般单选比较大小AA（空格里只能有数字，小数点，负号）（空格里只能有数字，负号）Quantitative Comparison questions（数量比较题）一、题目要求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】如果出现了A大于等于B的情况，应该选什么？Multiple-choice Question

8、sSelect One Answer Choice（单项选择题）These questions are multiple-choice questions that ask you to select only one answer choice from a list of five choices.解题技巧从选项中寻找解题线索，快速排除明显错误的选项没有思路时，可以尝试代入法，答案一定在5个选项当中基于题目给的条件去做，不要脑补其他任何条件利用枚举法的时候要非常注意列举顺序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（不定项选择题）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.解题技巧（单选的五个技巧全部保留）如果可能，将答案范围写在草稿纸上答案选项从3个到12个不等，不要为了多选而多选注意题目要求的答案个数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（数字填空题）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.解题技巧：确保你的答案与题目的要求是否一致（四舍五入）合理使用计算器等同的正确答案也可以给分，不要一味的追求约分 （4/9）通过常识来验证你的答案是否正确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 $120For the large cars sold at an auction that is summarized in the table above, what was the average sale price per car?$ 大多数的计算不需要计算器，不要因为TA在那儿就非要使用一些简单的计算，尽量不要使用计算器注意运算顺序有些计算很巧妙的回避计算过程，不要过分依赖计算器（极限问题）使用Tr

19、ansfer Display时，注意近似的位数 4.4456在做分数题的时候，计算结果有小数的时候Arithmetic算数Integer（positive/negative/0）：整数Odd integer/even integer：奇数/偶数Factor（divisor）：因子Multiple：倍数Divisible：可被整除Be divided by：被除quotient：商Remainder：余数Prime number：质数Composite number：合数Prime factor：质因子Prime factorization：质因分解Least common multiple：最

20、小公倍数Greatest common divisor：最大公约数正负性 (相乘的关系) 正*正=正 正*负=负 负*负=正 奇偶性 (用2K+1和2k来带入) 奇*奇=奇 奇+奇=偶 奇*偶=偶 奇+偶=奇 偶*偶=偶 偶+偶=偶 注意：如果n个数相乘=奇数，则n个数均为奇数质数与合数（注意1） 质数：因子只有1和其本身；2是唯一的偶质数 因子中的质数为质因子 质因分解： 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: 难点：正负性、奇偶性xy20-x0，x*y*y=奇数-x, y均为奇数所以x*y无法判断正负性，但为奇数，且x+y=偶数，答案为BCSolution: 难点：奇偶性举例代数、性质应用Solution: 难点：LCM的求法和因子的判断LCM=22*32*52*72Fraction：分数Numerator：分子Denominator：分母Rational number：有理数Irrational number：无理数Common denominator：公分母Reciprocal：倒数Mixed number：带分数Decimal

23、point：小数点Base：底数Exponent：指数Terminate decimal有限小数Repeat decimal 循环小数Infinite decimal 无限小数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. 注意在比较类里面变化的基数是什么？ 第一类： 自我

24、变化类 公式： （|新-旧|）/ 旧 例子： 从2到3增加了百分之多少？ 从3到2减少了百分之多少？ 第二类： 两个不同的数之间的比较 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？指数运算规则 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真题抢鲜!Solution: 难点：数论知识的应用&tens、unit的释义n=xy=10*x+y*1，n=yx=10*y+x*1n=n+9Solution: 难点：at most: 至多，平方根的判断 易错点：开放之后的正负性，答案选EAlgebra(代数)Variable 变量Like terms 同类项Constant 常数Coefficient 系数Identity 恒等式Linear equation 线性方程Quadratic equation 二次方程linear in

26、equalities 线性不等式Functions 函数Interests 利息coordinate geometry 坐标几何straight lines 直线graph of functions 函数的图像2xy2+3xy+6z-21xz-xy+0.5 is called an algebraic expression 代数表达式2xy2, 3xy, 6z, -21xz, -xy, 0.5称为terms（代数）项3xy和-xy称为like terms or similar terms（同类项）0.5称为constant term 常数项coefficient系数：2xy2这一项的系数是2，

27、 -21xz的系数是-212xy2+3xy+6z-21xz-xy+0.5 =1称为equation等式，等式的solution解就是满足等式的值。如x=2是x2-2x=0的一个解y=f(x)称为一个函数Domain定义域为函数有定义的所有x值Range值域为函数所有可能的取值如y=f(x)=x2-1,那么定义域就是所有实数，值域为y-1graphs of functionsy=ax+b 的图像是一条直线，斜率为a, y轴的截距为b（截距可正可负可为0）y=ax2+bx+c , 其中a0的图像是一个抛物线parabola如y=x2-2x-3的顶点vertex是（1, -4），x=1是对称轴lin

28、e of symmetry，开口向上opens upward(x-a)2+(y-b)2=r2的图像是一个圆circle，圆心center为（a, b), radius半径为ry=f(x)和y=g(x)的图像交点就是f(x)=g(x)的解上加下减&左加右减压缩shrunk &拉伸stretchedWEWE陈琦GRE团队 80313频道Geometry(几何)line and angles 线和角度Polygons 多边形Triangles 三角形Quadrilaterals 四边形Circles 圆Three-dimensional Figures 三维图形 如图所示，A,B,C为point点，

29、L为line直线，AB,BC,CD为line segment 线段，线段AB的两个端点endpoints 为A和B。线段AB的长度length为8，BC长度为6，CD长度为6。相同长度的线段我们称为congruent line segments。线段BD的中点midpoint为CPerpendicular 垂直Right angle 直角如图所示当两条直线相交的时候，形成了4个角angles，每个角都有一个共同的顶点vertex P。 APC和 BPD称为对顶角opposite angles或者vertical angles，对顶角相等。锐角acute angle 为小于90的角，如 APC。

30、钝角obtuse angle 为大于90的角，如 APD。三角形内角和为180polygon多边形内角和为180 (n-2)等边三角形equilateral triangle， 等腰三角形isosceles triangle直角三角形right triangle，直角边legs，斜边hypotenuse勾股定理Pythagorean theorem（毕达哥拉斯定理）WEParallelogram 平行四边形Rectangle 长方形Rhombus 菱形Square 正方形Trapezoid 梯形Circle 圆Center 圆心 Radius 半径Diameter 直径Congruent ci

31、rcles 全等圆Circumference 周长Chord 弦Concentric circles 同心圆arc 弧圆外接四边形circumscribed圆内接三角形inscribedTangent line：切线；the point of tangent ：切点弧长的求法：arc/2r=/360扇形面积的求法：S/r2=/360Frequency Distribution 频率分布图Bar Graphs 条形图Circle Graphs 饼状图Histograms 直方图Scatterplots 散点图Time Plots 时间曲线图区分题目给的图表是哪一种毫无意义，主要看清横纵坐标或者图

32、形代表的意义算术平均数（Average or Arithmetic Mean or Mean): 所有数据之和除以数据个数中数（Median): 将所有数据从小到大排列，取中间的数或中间两个数的算术平均数众数（Mode): 一组数据中出现频率最高的数。一组数据中可能不止一个众数极差（Range): 一组数据中最大数与最小数之差方差（Variance): 一组数据中每个数与算术平均数之差的平方和的算术平均数标准方差（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 正态分布 The standard normal distribution 标准正态分布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：难点：百分位数78-56=22，于是在26.5和37.1之间有大约22%的数即8500*22%=1870，选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.一个正态分布有两个参数：m（平均数）和d（标准差）M决定了分布的中心，d决定了分布的高矮胖瘦（d越大越矮胖，反之越高瘦）其中有一个特殊的正态分布我们称为stan