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1、57SAT 数学讲义No pains, no gains.前言4第一章 知识点归纳51、数学运算部分（Number and Operation）5（1）阶乘5（2）数轴5（3）基础数论5（4）数列5（5）整数5（6）排列组合52、代数和方程（Algebra and Functions）6（1）因式分解6（2）指数6（3）判断二次方程有无根6（4）不等式6（5）函数6（6）正、反比例函数6（7）一次函数及图像6（8）二次函数及图像6（9）应用题73、几何（Geometry and Measurement）7（1）欧几里德几何7（2）解三角形7（3）圆7（4）立体几何7（5）坐标系7（6）图形平移

2、74、概率和统计7（1）众数8（2）中位数8（3）概率8第二章 重要定理及公式91、奇偶数运算92、等差数列与等比数列93、因式分解94、二次方程判别式（ax2+bx+c=0）95、二次函数 y=ax2+bx+c96、指数运算107、特殊角的三角值108、相似形109、平面图形的周长和面积1110、立体图形的表面积和体积11*11、圆锥曲线1112、排列组合11第三章 练习题131. 解题技巧训练132. 算术部分14（1）代数题14（2）中位数14（3）集合部分15（4）排列组合题15（5）数列部分16（6）应用题17（7）整除，最小公倍数，余数问题173. 代数问题184. 几何部分21（

3、1）平面几何21（2）立体几何部分25第四章 真题模拟（2011.5）27附录：41附录一：常见符号41附录二：SAT数学考试词汇42附录三：做题方法解析53附录四：课程规划及课后作业53前言SAT数学是SAT的一个组成部分，分数占总分的1/3。对中国 学生来说这一部分是最容易拿分的，也是最有可能得满分的。可以说，数学试题是最好对付的，因为得个不错的分数不难；也可 以说，数学试题是最不好对付的，因为许多学生想得满分。SAT数学考试共需70分钟，有3个答题区间，合计54道试题，其中44道选择题（Multiple-choice questions），10道填空题（student-produced

4、response questions）。第一个区间有20题，需要25分钟，全部为选择题；第二个区间有18题，前8题为选择题，后面10题为填空题，答题时间为25分钟； 第三个区间有16题，全部为选择题，答题时间为20分钟。选择题做对一题得1分，不做不得分，做错扣1/4的分数，填空题做对一题得1分，做错不扣分。然后所得分数查找对分表，便得到自己的分数。对分表左面的一列是考生所得的原始分数(R=C-W)，右面的 一列是考生数学部分的最终得分。C是英文Correct的缩写 ，它代表答对的题目数；W是英文Wrong的缩写，它代表答错的多项选择题数乘以1/4后再把结果四舍五入后得到的值。如果某个考生答对了

5、51道题，答错了2道多项选择题和1道非多项选择题，那么该考生的C=51，W=1(2*1/4=0.5, 四舍五入)，所以R=C- W=51-1=50。第一章 知识点归纳SAT数学共包含4部分，分别是数学运算、代数方程、几何以及概率。下面就是每个部分的知识点。1、数学运算部分（Number and Operation）包含整数、数字应用题、阶乘、数轴、平方和平方根、分数和有理数、基础数论（质数、合数、倍数、余数、约数）、比例和百分数、集合、排列组合、逻辑推理等。下面是几个需要特别注意的部分。（1）阶乘factorial计算公式。（2）数轴number axis三要素原点(origin)，正方向(p

6、ositive direction)，单位长度(unit)。（3）基础数论number质数(prime number)和合数(composite number)；公约数(common divisor)和公倍数(Common multiple)。（4）数列sequence等差数列公差、通项、求和；Arithmetic sequence: common difference, general term, Sum等差数列：通项公式 an=a1+(n-1)d；求和公式Sn=n(a1+an)2=na1+n(n-1)2d。等比数列公比、通项、求和。Geometric sequence: common di

7、fference, general term, Sum等比数列：通项公式an=a1qn-1求和公式Sn=a1(1-qn)1-q=a1-anq1-q（5）整数integer奇偶数相互之间的的加法、乘法。odd, even number addition multiplication（6）排列组合permutation combination排列Pnm；组合Cnm。；CnmCn-mn ；CmnCmnCm-1n ；2、代数和方程（Algebra and Functions）包含代数式运算、因式分解、指数、解方程和解不等式、解方程组和不等式组、绝对值、正比例和反比例函数、一次函数、二次函数、新函数定义

8、、应用题。（1）因式分解factorization常见因式分解的公式（平方差、完全平方perfect square、立方差The cubic difference、立方和cubic sum等）；十字相乘法。（2）指数exponent指数的运算。（3）判断二次方程有无根用判别式discriminant。（4）不等式inequality不等式两边同除负数的情况；解含有绝对值的不等式。（5）函数function概念，定义域domain，值域range设在某个变化过程中有两个变量x和y，如果对于x在某个取值范围内的每一个值，按照某一规则，y都有唯一确定的值与x对应，那么就称y是x的函数。（6）正、反比

9、例函数proportional function, inverse proportional function解析式及图像。（7）一次函数及图像linear function截距intercept，平行，垂直（8）二次函数及图像quadratic function三种解析式以及解析式中系数与图像的关系。（9）应用题抓关键词（带数字或者数学运算词句）（常用数学表达OG p252）；看清问题；列出算式。3、几何（Geometry and Measurement）包含线角、三角形（等边、等腰、直角）、四边形（平行四边行、矩形、正方形）的面积和周长、正多边形（内角和、周长、面积）、圆、立体几何、坐标系

10、、图形平移。（1）欧几里德几何补角supplementary angle、余角complementary angle、同位角corresponding angle、内错角alternative inner angle、同旁内角same-side interior angles；三种三角形、三种四边形、正多边形内角和。acute triangle, obtuse triangle, rectangular triangle（2）解三角形特殊角的三角值、勾股定理Pythagorean theorem。（3）圆直径半径、面积、周长、弧长直线与圆相切radius, area, perimeter, a

11、rc length（4）立体几何圆、圆柱、圆锥、棱锥、棱柱等图形的半径、表面积、体积。sphere, cylinder, cone, pyramid, prism（5）坐标系平面直角坐标系、两点间距、中点公式。（6）图形平移左加右减。4、概率和统计包含数据解释（圆图、线图、海拔图和象形图）、统计初步（平均值、众数、中位数、加权平均数）、初等概率、几何概率、排列组合。（1）众数mode可以为多个。（2）中位数median将一组数据从小到大排列，外置处在最中间的数据。（3）概率独立事件与非独立事件（例题OG p300）。第二章 重要定理及公式1、奇偶数运算even + even = even； e

12、ven * even = even；even + odd = even； odd * odd = odd；odd + odd = even。 odd * even = even。2、等差数列与等比数列等差数列：通项公式 an=a1+(n-1)d；求和公式Sn=n(a1+an)2=na1+n(n-1)2d。等比数列：通项公式an=a1qn-1求和公式Sn=a1(1-qn)1-q=a1-anq1-q3、因式分解a-b2=a2-b2；(a±b)2=a2±2ab+b2； a3-b3=a-ba2+ab+b2；a3+b3=a+ba2-ab+b2。4、二次方程判别式（ax2+bx+c=0

13、）=b2-4ac大于0，有俩实根；小于0，无实根；等于0，一个实根。5、二次函数 y=ax2+bx+c顶点为（-b2a,-b2-4ac4a）；对称轴为x =-b2a；a正数，抛物线开口向上，a负数，则向下，c为y轴上的截距。6、指数运算a3=a×a×a；a-3=(1a)3；a13=3a；a23=3a2。7、特殊角的三角值0°30°45°60°90°Sin A01/222321Cos A132221/20Tan A03313无穷大Cot A无穷大31330联系：sin²A + cos²A = 1；tan A

14、*cot A = 1，tan A =sin A/cos A。互余三角值：sin(90°- A) = cos A, cos(90°-A) = sin A;tan(90°-A) = cot A, cot(90°- A) = tan A。8、相似形基本性质a：b = c：d <=> ad = bc。特例a：b = b：c <=> b² = ac（b为比例中项）。合比性质a/b = c/d => (a±b)/b = (c±d)/d 。反比性质a/b = c/d => b/a = d/c等比性质a/

15、b = c/d = m/n => (a+c+m)/(b+d+n)黄金分割把线段AB分割成AC和BC（AC > BC），且AC²=AC*BC，则叫做把线段AB黄金分割，C点成为AB的黄金分割点，AC/AB=（5-1）/2 = 0.6189、平面图形的周长和面积周长Perimeter面积Area三角形Triangle三边之和(底×高)/2正方形Square边长×4边长的平方矩形Rectangle(长+宽)×2长×宽平行四边形Parallelogram(长+宽)×2底×高梯形Trapezoid四边之和(上底+下底)&#

16、215;高/2棱形Rhombus边长×4两条对角线之积的1/2圆Circle2r=dr210、立体图形的表面积和体积体积Volume表面积Surface Area棱镜Rectangular Prism长×宽×高2(长×宽+长×高+宽×高)立方体Cube棱长的立方6×棱长×棱长圆柱Cylinderr2h2r h(侧)+ 2r2(底)球Sphere4r3/34r2圆锥Coner2h/3lr/2 (l为母线)*11、圆锥曲线12、排列组合；CnmCn-mn ；CmnCmnCm-1n ；第三章 练习题1. 解题技巧训练1

17、The units digit of 23333 is how much less than the hundredths digit of (A) 1 (B) 2 (C) 3 (D) 4 (E) 52. What is the units digit of 1597365?3. Bob has a pile of poker chips that he wants to arrange in even stacks. If he stacks them in piles of 10, he has 4 chips left over. If he stacks them in piles o

18、f 8, he has 2 chips left over. If Bob finally decides to stack the chips in only 2 stacks, how many chips could be in each stack?A. 14 B. 17 C. 18 D. 24 E. 344. If x and y are two different integers and the product 35xy is the square of an integer, which of the following could be equal to xy?A. 5 B.

19、70 C. 105 D. 140 E. 3505. If x2=y3 and (x-y)2=2x, then y could equal (A) 64 (B) 16 (C) 8 (D) 4 (E) 26. For positive integers p, t, x and y, if px=ty and x-y=3, which of the following CANNOT equal t?A. 1 B. 2 C. 4 D. 9 E. 257. If 3t-3>6s+9 and t-5s<12, and s is a positive integer less than 4, t

20、hen t could be any of the following EXCEPT A. 6 B. 8 C. 10 D.12 E. 328. If n and p are integers greater than 1 and if p is a factor of both n+3 and n+10, what is the value of p?A. 3 B. 7 C. 10 D. 13 E. 309. If x is a positive integer greater than 1, and x3-4x is odd, then x must be (A) Even (B) odd

21、(C) prime (D) a factor of 8 (E) divisible by 810.If the graph above is that of f(x), which of the following could be f(x)A. f(x)= B. f(x)= C. f(x)=/x/+3 D. f(x)=|x+3| E. f(x)=|3x|11. xy=x+y. If y>2, what are all possible values of x that satisfy the equation above?A. x<0, B. 0<x<1 C.0<

22、;x<2 D.1<x<2 E. x>22. 算术部分（1）代数题(1). Karl bought x bags of red marbles for y dollars per bag, and z bag of blue marbles for 3y dollars per bag. If he bought twice as many bags of blue marbles as red marbles, then in terms of y, what was the average cost, in dollars, per bag of marbles? (

23、A) (B) (C) 3x-y (D) 2y (E) 6y(2) At this bake sale, Mr. Right sold 30% of his pies to one friend. Mr. Right then sold 60% of the remaining pies to another friend. What percent of his original number of pies did Mr. Right have left?(A) 10% (B) 18% (C) 28% (D) 36% (E) 40%(3) At a track meet, 2/5 of th

24、e first-place finishers attended Southport High School, and 1/2 of them were girls. If 2/9 of the first-place finishers who did NOT attend Southport High School were girls, what fractional part of the total number of first-place finishers were boys?(A) 1/9 (B) 2/15 (C) 7/18 (D) 3/5 (E) 2/3（2）中位数（4）N

25、umber of siblings per student in a preschool classNumber of siblingsNumber of Students03162231The table above shows how many students in a class of 12 preschoolers had 0, 1, 2, or 3 siblings. Later, a new student joined the class, and the average (arithmetic mean) number of siblings per student beca

26、me equal to the median number of siblings per student. How many siblings did the new student have?A. 0 B. 1 C. 2 D. 3 E. 4（5）In a set of eleven different numbers, which of the following CANNOT affect the value of the median?A. Doubling each numberB. Increasing each number by 10C. Increasing the smal

27、lest number onlyD. Decreasing the largest number onlyE. Increasing the largest number only （6）. The least and greatest numbers in a list of 7 real numbers are 2 and 20, respectively. The median of the list is 6, and the number 3 occurs most often in the list. Which of the following could be the aver

28、age (arithmetic mean) of the numbers in the list?I. 7 II. 8.5 III. 11A. I only B. I and II only C. I and III only D. II and III only E. I, II and III（3）集合部分(7) Set F consist of all of the prime numbers from 1 to 20 inclusive, and set G consist of all of the odd numbers from 1 to 20 inclusive. If f i

29、s the number of values in set F, g is the number of values of in Set G, and j is the number of values in F or G, which of the following gives the correct value of f(j-g)?A. 4 B. 8 C. 10 D. 11 E. 18(8) Set X has x members and set Y has y members. Set Z consists of all members that are in either Set X

30、 or Set Y with the exception of the k common members (k>0). Which of the following represents the number of members in set Z?A. x+y+k B. x+y-k C. x+y+2k D. x+y-2k E. 2x+2y-2k(9) Of the 240 campers at a summer camp, 5/6 could swim, if 1/3 of the campers took climbing lessons, what was the least po

31、ssible number of campers taking climbing lessons who could swim?A. 20 B.40 C. 80 D.120 E. 200（4）排列组合题(11)Mr. Jones must choose 4 of the following 5 flavors of jellybean: apple, berry, coconut, kumquat, and lemon, How many different combinations of flavors can Mr. Jones choose?（12） If the 5 cards sho

32、wn above are placed in a row so that is never at either end, how many different arrangements are possible?（13） As shown above, a certain design is to be painted using 2 different colors. If 5 different colors are available for the design, how many differently painted designs are possible?A. 10 B. 20

33、 C. 25 D. 60 E. 120（14）In the integer 3589 the digits are all different and increase from left to right. How many integers between 4000 and 5000 have digits that are all different and that increased from left to right?（15）. On the map above, X represents a theater, Y represents Chriss house, and Z r

34、epresents Peters house. Chris walks from his house to Peters house without passing the theater and then walks with Peter to the theater and then walks without walking by his own house again. How many different routs can Chris take?（16）In a certain game, 8 cards are randomly placed face-down on a tab

35、le. The cards are numbered from 1 to 4 with exactly 2 cards having each number. If a player turns over two of the cards, what is the probability that the cards will have the same number?（17）The Acme Plumbing Company will send a team of 3 plumbers to work on a certain job. The company has 4 experienc

36、ed plumbers and 4 trainees. If a team consists of 1 experienced plumber and 2 trainees, how many different such teams are possible?（18）If p, r, m, n, t and s are six different prime numbers greater than 2, and N=p*r*s*m*n*t, how many positive factors, including 1 and N, does N have?（5）数列部分(19) The l

37、east integer of a set of consecutive integers is -25. If the sum of these integers is 26, how many integers are in this set? A. 25 B. 26 C.50 D. 51 E. 52(20) 1,2,2,3,3,3,4,4,4,4.All positive integers appear in the sequence above, and each positive integer k appears in the sequence k times. In the se

38、quence, each term after the first is greater than or equal to each of the terms before it. If the integer 12 first appears in the sequence as the nth term, what is the value of n?(21) The first term of a sequence of numbers is 2. Subsequently, every even term in the sequence is found by subtracting

39、3 from the previous term, and every odd term in the sequence is found by adding 7 to the previous term. What is the difference between 77th and 79th terms of this sequence?A. 11 B. 7 C. 4 D. 3 E. 2(22) A positive integer is said to be “tri-factorable” if it is the product of three consecutive intege

40、rs. How many positive integers less than 1000 are tri-factorable?（6）应用题(23) Tom and Alison are both salespeople. Toms weekly compensation consists of $300 plus 20 percent of his sale. Alisons weekly compensation consists of $200 plus 25 percent of her sales. If they both had the same amount of sales

41、 and the same compensation for a particular week, what was that compensation, in dollars? (Disregard dollar sign when gridding your answer)(24) To celebrate a colleagues graduation, the m coworkers in an office agreed to contribute equally to a catered lunch that costs a total of y dollars. If p of

42、the coworkers fail to contribute, which of the following represents the additional amount, in dollars, that each of the remaining coworkers must contribute to pay for the lunch?A. B. C. D. E. (25) In a certain store, the regular price of a refrigerator is $600. How much money is saved by buying this

43、 refrigerator at 20 percent off the regular price rather than buying it on sale at 10 percent off the regular price with an additional discount of 10 percent off the sale price?(A) $6 (B) $12 (C) $24 (D) $54 (E) $60（7）整除，最小公倍数，余数问题(26) When a is divided by 7, the remainder is 4. When b is divided by

44、 3, the remainder is 2. If 0<a<24 and 2<b<8, which of the following could have a remainder of 0 when divided by(A) (B) (C) a-b (D) a+b (E) ab(27) The alarm of Clock A rings every 4 minutes, the alarm of Clock B rings every 6 minutes, and the alarm of Clock C rings every 7 minutes. If the

45、 alarms of all three clocks ring at 12:00 noon, the next time at which all the alarms will ring at exactly the same time isA. 12:28 P.M. B. 12:56 P.M. C. 1:24 P.M. D. 1:36 P.M. E. 2:48 P.M. (28) If a, b, and c are distinct positive integers, and 10% of abc is 5, then a+b could equalA. 1 B. 3 C. 5 D.

46、 6 E. 25(29) On 5 math tests, Gloria had an average score of 86. If all test scores are integers, what is the lowest average score average score Gloria can receive on the remaining 3 tests if she wants to finish the semester with an average score of 90 or higher? A. 90 B. 92 C. 94 D. 96 E. 97(30) If

47、 is the cube of an integer greater than 1, and k2=y, what is the least possible value of y? A. 1 B. 2 C. 4 D. 6 E. 273. 代数问题(1) The height of the steam burst of a certain geyser varies with the length of time since the previous steam burst. The longer the time since the last burst, the greater the h

48、eight of the steam burst. If t is the time in hours since the previous steam burst and H is the height in meters of the steam burst, which of the following could express the relationship of t and H?A H(t)= (t-7) B. H(t)= C. H(t)=2-(t-7) D. H(t)= 7-2t E. H(t)= （2） 4) The above graph could represent w

49、hich of the following inequalities?A. y B. y< ()x C. y D. y()x E. yx-1/2（3）The change in temperature is a function of the change in altitude in such a way that as the altitude increases, so dose the change in the temperature. For example, a gain of 1980 feet causes a 60F, which of the following c

50、ould be the relationship of a and T?A. T(a)= a/300 B. T(a)= a-330 C. T(a)=330/a D. T(a)=330-a E. T(a)=330a（4） Let f(x) be defined as the least integer greater than x/5. Let g(x) be defined as the greatest integer less than x/5. What is the value of g(18)+f(102)? A. 21 B. 22 C. 23 D. 24 E. 25（5）Radio

51、active substance T-36 dose not stay radioactive forever. The time it takes for half of the element to decay is called a half-life. If, before any decay takes place, there is 1 gram of radioactive substance T-36, and the half-life is 7 days, how much remains after 28 days? A. 7-28 B. 2-4 C. 2-2 D. 1-

52、28 E. 22(6) Luke purchased an automobile for $5000, and the value of the automobile decreases by 20 percent each year. The value, in dollars, of the automobile n years from the date of purchase is given by the function V, where V(n)=5000*(0.8)n. how many years from the date of purchase will the valu

53、e of the automobile be $ 3200? A. 1 B. 1 C. 3 D. 4 E. 5(7) The cost of maintenance on an automobile increases each year by 10 percent, and Andrew paid $300 this year for maintenance on his automobile. If the cost c for maintenance on Andrews automobile n years from now is given by the function c(n)=

54、300xn, what is the value of x?A. 0.1 B. 0.3 C. 1.1 D. 1.3 E. 30(8) h(t)= c- (d-4t)2At time t=0, a ball thrown upward from an initial height of 6 feet. Until the ball hit the ground, its height, in feet, after t seconds was given by the function h above, in which c and d are positive constants. If th

55、e ball reached its maximum height of 106 feet at time t=2.5, what was the height, in feet, of the ball at time t=1?(9) If k, n, x and y are positive numbers satisfying x-4/3 = k-2 and y4/3 =n2, what is (xy) -2/3 in terms of n and k?A. B. C. D. E. 1(10) The figures above show the graphs of the function f and g. The function f is defined by f(x)=x3-4x. the function g is defined by g(x)=f(x+h)+k, where h and k are constants. What is the value o