1、SAT 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
2、 7（4）立体几何 7（5）坐标系 7（6）图形平移 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（ ） 92+=05、二次函数 .9 =2+6、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
3、）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（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
4、，合计 54 道试题，其中 44 道选择题（Multiple-choice questions），10 道填空题（student-produced 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
5、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 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
6、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(positive 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 dif
7、ference, general term, Sum等差数列：通项公式 ；=1+(1)求和公式 。=(1+)2 =1+(1)2 等比数列公比、通项、求和。Geometric sequence: common difference, general term, Sum等比数列：通项公式 =11求和公式 =1(1)1 =11（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。；CnmC n-mn ；CmnC mnC m-1n
8、；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、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
9、 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） （
10、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、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 tri
11、angle, 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, arc 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
12、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； even * 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 ；=1+(1)求和公式 。=(1+)2 =1+(1)2 等比数列：通项公式 =11求和公式 =1(1)
13、1 =113、因式分解（ ） 2=22；()2=22+2；33=()(2+2)；。3+3=(+)(2+2)4、二次方程判别式（ ）2+=0 =24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 =2+顶点为（ , ） ；2244对称轴为 x = ；2a 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3=；3=(1)3；13=3；。23=327、特殊角的三角值0 30 45 60 90Sin A 0 1/2 22 32 1Cos A 1 32 22 1/2 0Tan A 0 33 1 3 无穷大Cot A 无穷大 3 1 33 0联系：sin
14、A + cosA = 1；tan A*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 b = ac（b 为比例中项） 。合比性质a/b = c/d = (ab)/b = (cd)/d 。反比性质a/b = c/d = b/a = d/c等比性质a/b = c/d = m/n = (a+c+m)/(b+d+n)黄金分割把线段 AB 分割成 AC 和
15、 BC（AC BC） ，且 AC=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=（ -1）/2 = 0.61859、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)高/2棱形 Rhombus 边长4 两条对角线之积的 1/2圆 Circle 2r=d r210、立体图形的表面积和体积体积 Volume 表
16、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 Cylinder r2h 2r h(侧)+ 2r 2(底)球 Sphere 4r3/3 4r2圆锥 Cone r2h/3 lr/2 (l 为母线)*11、圆锥曲线12、排列组合；CnmC n-mn ；CmnC mnC m-1n ；第三章 练习题1. 解题技巧训练1 The units digit of 23333 is how much less than the hundredths digit of 10567(A) 1 (B) 2 (C)
17、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 of 8, he has 2 chips left over. If Bob finally decides to stack the chips in only 2 stac
18、ks, 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.70 C. 105 D. 140 E. 3505. If x2=y3 and (x-y)2=2x, then y could equal (A) 64 (B) 16 (C)
19、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-36s+9 and t-5s2, what are all possible values of x that satisfy the equation above?A. x22. 算术部分（1）代数题(1). Karl bought x bags of red marbles for y dollars p
20、er 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?(A) (B) (C) 3x-y (D) 2y (E) 6y23y7(2) At this bake sale, Mr. Right sold 30% of his pies to one frien
21、d. 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 the first-place finishers attended Southport High School, and 1/2 of them were girls. If 2/9 of t
22、he 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）Number of siblings per student in a preschool classNumber of siblings Number of Students0 31 62
23、23 1The 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 became equal to the median number of siblings per student. How many siblings did the new stude
24、nt 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 smallest number onlyD. Decreasing the largest number onlyE. Increasing the largest number only
25、 （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 average (arithmetic mean) of the numbers in the list?I. 7 II. 8.5 III. 11A. I only B. I and II
26、 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 is the number of values in set F, g is the number of values of in Set G, and j is the numbe
27、r 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 or Set Y with the exception of the k common members (k0). Which of the following represen
28、ts 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 possible number of campers taking climbing lessons who could swim?A. 20 B.40 C. 80 D.120 E. 200（
29、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 shown above are placed in a row so that is never at either end, how many different arrangements a
30、re 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 C. 25 D. 60 E. 120（14）In the integer 3589 the digits are all different and increase from left
31、 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 represents Peters house. Chris walks from his house to Peters house without passing the theater
32、 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 table. The cards are numbered from 1 to 4 with exactly 2 cards having each number. If a player tu
33、rns 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 experienced plumbers and 4 trainees. If a team consists of 1 experienced plumber and 2 trainees, how ma
34、ny 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 least integer of a set of consecutive integers is -25. If the sum of these integers is 26, how
35、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 sequence, each term after the first is greater than or equal to each of the terms before it. If
36、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 3 from the previous term, and every odd term in the sequence is found by adding 7 to the previ
37、ous 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 integers. How many positive integers less than 1000 are tri-factorable?（6）应用题(23) Tom and Alison are
38、 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 and the same compensation for a particular week, what was that compensation, in dollars? (Dis
39、regard 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 the coworkers fail to contribute, which of the following represents the additional amount, in
40、dollars, that each of the remaining coworkers must contribute to pay for the lunch?A. B. C. D. E. mypmy)()(py(25) In a certain store, the regular price of a refrigerator is $600. How much money is saved by buying this refrigerator at 20 percent off the regular price rather than buying it on sale at
41、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 3, the remainder is 2. If 00, what is the value of a?(15) (x-8)(x-k)= x2-5kx+mIn t
42、he equation above, k and m are constants. If the equation is true for all value of x, what is the value of m?A. 8 B. 16 C. 24 D. 32 E. 40(16) A certain function f has the property that f(x+y)=f(x)+f(y) for all values of x and y. which of the following statements must be true when a=b?I. f(a+b)= 2f(a
43、) II. f(a+b)=f(a)2 III. f(a)+f(b)=f(2a)A. None B. I only C. I and III only D. II and III only E. I, II and III（17）. The shaded region in the figure above is bounded by the x-axis, the line x=4, and the graph of y=f(x). if the point (a, b) lies in the shaded region, which of the following must be tru
44、e?I. a4 II. ba III. bf(a)(A) I only(B) III only(C) I and II only(D) I and III only(E) I, and II and III（18）The figure above shows the graphs of y=x2 and y=a-x2 for some constant a. if the length of PQ is equal to 6, what is the value of a? A. 6 B. 9 C. 12 D. 15 E. 18（19）In the figure above, ABCD is
45、a rectangle. Points A and C lie on the graph of y=px3, where p is a constant,. If the area of ABCD is 4, what is the value of p?4. 几何部分（1）平面几何（1）Each of the small squares in the figure above has an area of 4. If the shortest side of the triangle is equal in length to 2 sides of a small square, what
46、is the area of the shaded triangle?A. 160 B. 40 C. 24 D. 20 E. 16（2）. In the figure above, a shaded polygon which has equal sides and angles is partially covered with a sheet of blank paper. If x+y=80, how many sides does the polygon have?A. 10 B. 9 C. 8 D. 7 E. 6（3）The area of rectangle ABCD is 96,
47、 and AD=2/3(AB). Points X and Y are midpoints of AD and BC, respectively. If the 4 shaded triangles are isosceles, what is the perimeter of the unshaded hexagon? A. 16 B.8+6 C. 24 D. 8+16 E. 16+24222（4）In the figure above, what is the value of c in terms of a and b?A. a+3b-180 B. 2a+2b-180 C. 180-a-
48、b D. 360-a-b E. 360-2a-3b（5） The figure above shows an arrangement of 10 squares, each with side of length k inches. The perimeter of the figure is p inches. The area of the figure is a square inches. If p=a, what is the value of k?（6）. One end of an 80-inch-long paper strip is shown in the figure above. The notched edge, shown in bold, was formed by removing an equilateral triangle from the
