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高中数学双语教案

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2020-09-17 11:50
tags:高中数学教案

宜宾市高中数学期末测试卷-芜湖一中高中数学洪老师


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§1 Arithmetic 算术


The numbers 0,1,2,3……are called
whole numbers
or
integers
.
So 23 is an integer but 1.5 is not an integer.
A number greater than zero is called a
positive number.
A number smaller than zero is called a
negative number
.
0 is an integer but it is neither positive nor negative.
An integer that is divisible by 2 is an
even integer
. An integer
that is not divisible by 2 is an
odd integer
.
For any two numbers on the
number line
, the number on the left
is less than the number on the right, for example, 2<3 and -4<-3.
The absolute value of a number is the equivalent
positive
value
.
|+2|=+2,|-3|=+3
If x is an integer divisible by another integer y, then x is said
to bee a
multiple
of y, and y is said to be a
factor
or
divisor

of x.
The factors of 12 are:1,2,3,4,6,12.
The multiple of 5 are:5,10,15,20……
If p is a whole number which has no other factors besides itself
and 1,p is a
prime number
. 2, 3, 5, 7, 11,… are prime numbers.
But 1 is not considered a prime number.
The least common multiple
of two numbers is the smallest
number which is a common multiple of both numbers.



A
fraction
is a number which represents a ratio or division of
two numbers. A fraction is written in the form
a
. The number on the
b
top, a, is called the
numerator
; the number on the bottom, b, is
called the
denominator.

A fraction cannot have 0 as a denominator, since division by 0 is
not defined.
To compare the values of two fractions. Is
4
greater or less than
7
5
? First multiply 4 by 9 and get 36; then multiply 5 by 7 and get 35,
9
as shown here:

36 4
-

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5
-
35


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7
Since 36 is greater than 35,
9
5
4
is greater than .(It is not
9
7
necessary to multiply 7 by 9 as the common denominator.)



A
decimal fraction
is a fraction whose denominator is
some
power
of 10 (such as 10,100,1000 ete.)
The digits of a number are named as follows:
2,0 7 8. 1 4 6
千百十个 十百千
位位位位 分分分
数数数数 位位位
数数数
thousands’ digit 千位数
hundreds’ digit 百位数
tens’ digit 十位数
units’ digit 个位数
tenths’ digit 十分位数
hundredths’ digit 百分位数
thousandths’ digit 千分位数



Rounding off
numbers can help you get quick, approximate
answers. Examples:
Round off 538,423.98 to the nearest hundred. T he answer is
538,400.
Round off 24.76 to the nearest tenth. The answer is 24.8.
The parts of a division problem are indicated in the example below.

The
quotient
is 5

divisor
2

7
5
737dividend



35
2remainder


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The average or arithmetic
mean of 12, 3, 6 is: (12+3+6)
÷3 = 7.
If a group of numbers is arranged in order, the middle number is
called the
median
.
To find the median of 12, 3, 6, arrange them in order: 3, 6, 12.
The median is 6.
If the number of items in the group is even, the median is the
average of the two middle numbers. For example, to find the median of
6, 3, 30, 14, arrange them in order: 3, 6, 14, 30. The median is (6+14)
÷2 = 10.
In general, the median and the average of a collection of numbers
are different.



Powers, Exponents, and Roots:
If b is any number and n is a whole number greater than 0, b
n
mean
the product of n factors each of which is equal to b. Thus, b
n
=b*b*b*…
*b, where there are n copies of b.
The expression b
n
is called
the nth power of b
,n is called
the
exponent
; b is called the
base
.
An odd integer power of a negative number is negative, and an even
integer power of a negative number is positive. For example, (-2)
3
=-8,
but (-2)
4
=16.
You should know the following laws:
b
0
=1 (b≠0)
b
1
=b
b
2
*b
3
=b
(2 ?3)
=b
5

b
5
?b
(5?2)
?b
3
(b≠0)
2
b
b
3
1
(3?5)?2
?b?b?
(b≠0)
52
bb
(b
4
)
2
?b
( 4*2)
?b
8

(ab)
3
?a
3
b
3

a
3
a
3
()?
3
(b≠0)
b
b


If you raise a number d to the nth power and the result is b, then
d is called
the nth root of b .

Notice the following relations:

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9??3
2?1.4
3?1.7
3
3
8?2
?8??2
25?5
0.09?0.3








Exercise
1. All the following numbers are positive
EXCEPT
(A) 5.3 (B) 0.07 (C) 0 (D) 2 (E) 5
2. The number of the factors of 24 is
(A) 9 (B) 8 (C) 7 (D) 6 (E) 5
3. The distinct prime factors of 24 are
(A) 1,2,3 (B) 2,3 (C) 2,3,6 (D) 8,3
4. How many of the positive integers less than 25 are integer multiples
of 4?
(A) 7 (B) 6 (C) 5 (D) 4 (E) 3
5. The least common denominator of
53
and is
1216
(A)
129
1
(B) (C) (D) 48 (E) 96
4848
96
6. The tenths’digit of 3,610.48 is
(A) 3 (B) 6 (C) 1 (D) 4 (E) 8

7. The units’digit of 5
4
is
(A) 20 (B) 10 (C) 5 (D) 4 (E) 0
8. Round 508.64 to the nearest tenth. The result is
(A) 510 (B) 508 (C) 508.6 (D) 508.7 (E) 509
9. 360 divided by 37, the remainder is
(A) 27 (B) 333 (C) 0 (D) 9 (E)
27

37
10. The average or arithmetic mean of 2,4,6,10 is
(A) 6 (B) 5 (C) 5.5 (D) 4 (E) 3
11. The median of 2,4,6,10 is
(A) 6 (B) 5 (C) 5.5 (D) 4 (E) 3

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12. 10
6
÷10
8
=
(A) 100 (B) 10 (C) 0.1 (D) 0.01 (E) -2
13. If 87 million =8.7*10
n
, then n =
(A) 5 (B) 6 (C) 7 (D) 8 (E) 9
14.
300?75?

(A)
225
(B) 5
3
(C) 3
5
(D) 15
3
(E) 3
75

15. Which of the following has the greatest value?
(A)
0.3
(B) 0.3 (C) 0.01п (D) (E)
16. 7 less than 5x. This means
(A) 7-5x (B) 5x-7 (C) 7<5x (D)7>5x (E) 7-5+x
17. 8 is less than 2y. This means
(A) 8-2y (B) 2y-8 (C) 8<2y (D) 2y<8 (E) 8-2+y
1
5
1
3
§2 Algebra 代数
The letters in an algebraic expression are called
variables
or
unknowns
. When a variable is multiplied by
a number, the number is called the coefficient of the variable.
Thus, in the expression 3x
2
-5y, the
coefficient
of x
2
is 3,
and the coefficient of y is –5.
The only algebraic terms which can be
add
ed end
subtract
ed are
2x
2
+3x-5x
2
=-3x
2
+3x.


like terms
. For example,
An
equation
is a statement that says two algebraic
expressions are equal.
A linear equation
or
an equation of the first
degree
is one in which variables are raised only to the first
power. For example, 2x+3=9.
A quadratic equation
is an equation which contains
squares of the unknown as well as terms of the first degree. For
example, 3x
2
+2x-4=0, x
2
-9=0。
You can solve a quadratic equation by
factoring
.
Example:
x
2
?4x?3?0

(x?3)(x?1)?0
x?3?0或x?1?0
x??3或x??1


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Exampl e:
x
2
?5x?6?0
(x?6)(x?1)?0
x?6?0或x -1?0
x?-6或x?1
Example:
x
2
?5x?6?0< br>(x?6)(x?1)?0
x?6?0或x?1?0
x?6或x?-1

Example:
4x
2
?9?0
(2x?3)(2x?3)?0
2x?3?0或2x-3?0
22
x?-或x?
33





A
ratio
is a comparison of two numbers by division. The
a
ratio of a to b is written as a:b or .
b
A
proportion
is a statement that two ratios are equal.
210
For example,2:3=10:15, or
?
.
315
In the proportion a:b=c:d, the terms on the outside (a and
d) are called the
extremes
, and the terms on the inside (b
and c) are called the
means.
Two variables, a and b, are
directly proportional
, if
they satisfy a relationship of the form a=kb, where k is a number.
Two variables, a and b, are
indirectly proportional
,
if they satisfy a relationship of the form ab=k, where k is a
number.
Sometimes the word “directly” is omitted. So “a and b are
proportional” means: a and b are directly proportional.


An
inequality
is a statement that one expression is not
equal to another.

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To find the solution set for an inequality, you must remember
these principles.
1. You can add the same quantity to or subtract the same
quantity from both sides of an inequality without altering
its solution set.
If 5>3, then 5+2>3+2. If 5>3, then 5-4>3-4.
2. You can multiply or divide both sides of an inequality by
the same negative quantity without altering its solution
set, you must change the direction of the inequality.
53
Iif 5>3, then 5(-2)<3(-2). If 5>3, then .
?
?4?4




Exercise
1. We know that
9?3,

9
≠-3. If
y
2
?9,
then y=
(A) +3 only (B) –3 only (C) +9 only (D) –9 only
(E) +3 or -3
2. If P = EI , and E = IR, find P in terms of E and R.
E
2
RE
R

(A)

(B)E
2
R

(C)
2

(D)

(E)

R
R
E
E
3.
If 5x+2<2x+5, then all of the following may be a value
of x
EXCEPT
(A) 0 (B) 1 (C) –1 (D) –2
4. A bag of chicken feed will feed 18 chickens for 54 days.
For how many days will it feed 12 chickens?
(A) 36 (B) 38 (C) 54 (D) 72 (E) 81
5. 7x-5y = 13; 2x-7y =m 26; 9x-12y =?
(A) 13 (B) 26 C) 39 (D) 42 (E) 52
10
6. How many solutions does the equation a-3 = have?
a
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4











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What’s the Chinese for the following word?
1. integer__________ 2. numerator__________ 3. round
off__________4. decimal__________5.
fraction__________ 6. positive__________ 7.
multiple__________ 8. power__________ 9.
remainder__________ 10. median__________ 11.
denominator__________ 12. odd integer__________ 13.
exponent__________ 14. arithmetic mean__________ 15.
quotient__________ 16. prime number__________ 17.
absolute value__________ 18. divisor__________ 19.
negative__________ __________
How to describe “
b
n
”___________________________

n
b
”_______________
















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§Geometry
An acute angle has a measure greater than 0°but less than
90°.
A right angle has a measure of 90°.
An obtuse angle has a measure greater than 90°but less
than 180°
A straight angle(which looks like a straight line ) has
a measure of 180°.
Two angles that together have a measure of 90°are called
complementary angles.
Two angles that together have a measure of 180°(that is,
two angles that together form a straight line) are called
supplementary angles.
If two lines intersect, the four angles formed are either
equal or supplementary. The angles opposite one another
are called vertical angles and are equal. The angles next
to one another are supplementary.
If two parallel lines are intersected by a third line,
called a transversal, the eight angles formed are either
equal or supplementary.








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In the figure, line PQ is parallel to line RS.
Then angles 1 and 2 are called corresponding angles and
are equal.
Angles 2 and 3 are called alternate interior angles and
are equal.
A triangle that has two equal sides is called an isosceles
triangle.
If all three sides of a triangle are the same length, the
triangle is called an equilateral triangle.
The sum of the measures of the three interior angles of
a triangle is 180°.
If the vertex of an angle is at the center of a circle,
then the angle is called a central angle and its measure
is equal to the measure of the arc it intercepts.
If the vertex of an angle lies on a circle, then the angle
is called an inscribed angle in the circle and its measure
is equal to half the measure of the arc it intercepts.
If the vertex of an angle lies on a circle, then the angle
is called an inscribed angle in the circle and its measure
is equal to half the measure of the arc it intercepts.
In circle O, angle POQ is a central angle. Angle POQ is
an inscribed angle.
The number of degrees of arc in a circle is 360.
The circumference of a circle with radius r is equal to
2πr.
centralangle
2
?
r
Length of arc =
360?
A line segment whose endpoints are on a circle is called
a chord. A chord which passes through the center of the
circle is a diameter. The length of a diameter is twice
the length of a radius.
A line that intersects a circle at one and only one point
is called a tangent. If a radius is drawn to the point at
which a tangent intersects a circle, the radius is
perpendicular to the tangent.





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A right triangle is any triangle that has a right angle.
The two shorter sides of a right triangle are called the
legs.
The longest side, which is the side opposite the
right angle, is called the
hypotenuse
.
The
Pythagorean theorem
states that the square
of the hypotenuse is equal to the sum of the squares of
the lengths of the legs.
The sides of a
45??45??90?
triangle are in the ratio
1
:1:
2
.
The sides of a
30??60??90?
triangle are in the ratio
1:
3:2
.
A
quadrilateral
is a
polygon
with four sides.
If the opposite sides of a quadrilateral are parallel, the
figure is a
parallelogram
.
In a parallelogram:
The opposite sides are equal.
The opposite angles are equal.
A
diagonal
divides the parallelogram into two

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congruent triangles.
The diagonals
bisect
each other.
If all the sides of a parallelogram are equal, the
figure is called a
rhombus
.
If all the angles of a parallelogram are right angles,
the figure is a
rectangle
.
If all the sides of a rectangle are equal, the figure
is a square.
A quadrilateral with two parallel sides and two sides
which are not parallel is called a
trapezoid
.
The sum of the lengths of the sides of a polygon is
called its
perimeter
.
The sum of all the angles of a polygon with n sides is
(n-2)180
?
. So the sum of the angles in a quadrilateral is
(4-2)180
?
=360
?
. The sum of the angles in a pentagon is
(5-2)180
?
=540
?
. The sum of the angles in a hexagon is
(6-2)180
?
=720
?
.


The formulas for finding the areas and volumes of
common geometric figures are show n below:
Area of a rectangle, A= bh
Area of a square, A = s
2

1
Area of a triangle, A =
bh

2
Area of a parallelogram, A = bh
Area of a circle, A =
?
r
2





The
volume
of a
rectangular solid
, V = LWH.
The volume of a
cube
, V =
s
3

The volume of a right circular
cylinder
, V=
?
r
2
h

3
The volume of a
sphere
, V =
?
r
2

4



Coordinate Geometry
The rectangular coordinate system consists of two

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perpendicular number lines that intersect at the zero
point on each.
The point at which the two number lines intersect is
known as the
origin
.
The horizontal number line is knows as the
x-axis
.
The vertical number line is knows as the
y-axis.
The numbers to the right of the origin on the x-axis
and above the origin on the y-axis are positive. The
numbers to the left of the origin on the x-axis and below
the origin on the y-axis are negative.
The coordinates of a point P( x , y ) give the location
of the point in the coordinate plane for example , the
point P( 2 , -3 ) is located in the fourth
quadrant
,2
units to the right of the y-axis and 3 units below the
x-axis.
To find the distance between two points P(
x
1
,y
1
)and
Q(
x
2
,y
2
)
, use the formula: PQ =
(x
1
?x
2
)
2
?(y
1
?y
2
)
2

To find the midpoint of a line segment PQ, with
P(
x
1
,y
1
) and
x?x
2
y< br>1
?y
2
Q(
x
2
,y
2
)
,use the formula: M (
1
,)
.
22





EXERCISE
1. A picture is 6 feet wide and 8 feet long. If its frame
has a width of 6 inches, what is the ratio of the area
of the frame to the area of the picture?
5545
(A)(B)(C)(D)(E)3.2

164512


2. If angle DBG equals
79?
and angle CBE equals
39?
, then
angle GBE equals

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(A)51?

(B)62?(C)101`?(D)108?(E)202?


3. Four equal circles of diameter one foot touch at four
points as shown in the figure. What is the area of the
shaded portion (in square feet)?

(A)1?

?
4
(B)1?
?
(C)
?
4
(D)1?4
?
(E)
?


4. Point P(4,2) is the midpoint of line OPC, where O is
at origin (0,0). The coordinates of C are
(B)(4,8)(C)(4,4)(D)(8,2)(E)(8,4)

(A)(2,1)
5. The three dimensions of a rectangular solid are p, 2p
and 3p . What is the total surface area of the solid?
(B)14P
2
(C)6P
2
(D)20P
2
(E)22P
2

(A)11P
2


















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Common Verbal Problems
1. Discount
A discount, usually expressed in percent, indicates the
part that is to be deducted from the list price. Pay
attention to the usage in English and in Chinese. For
example:
10% discount -----打九折 ;25% discount-----打七五折
Example: A bicycle originally cost $$100 and was
discounted 10%. After 3 months it was sold after being
discounted 20%. What was the single equivalent
discount?
2. Interest
Simple interest = principal
?
time
?
rate
Example: How much interest will $$10,000 earn in 9 months
at an annual rate of 6%?
3. Motion
Distance traveled = rate
?
time
Example: A man driving a distance of 90 miles averages
30 miles per hour. On the return trip he averages 45
miles per hour. What is his average speed for the round
trip in miles per hour?
4. Work
If a worker can do a job in n days, then he can do
of the job in one day.
Example: It takes John an hour to do a job that Tom can
do in 40 minutes. One morning they worked together for
12 minutes, then John went away and Tom finished the
job. How long did it take Tom to finish?
5. Mixture
Example: A dealer wishes to mix 20 pounds of nuts
selling for 45 cents per pound with some more expensive
nuts selling for 60 cents per pound, to make a mixture
that will sell for 50 cents per pound. How many pounds
of the more expensive nuts should he use?
1
n

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6. Integer
Example: Three times the first of three consecutive odd
integers is 3 more than twice the third. Find the third
integer.

EXERCISE
1. How many ounces of pure acid must be added to 20
ounces solution that is 24% acid?

( A)2
1
2
(B)5(C)6(D)7
1
2
(E)10
2. Three consecutive odd integers have a sum of 33. Find
the least of these integers.
(A) 5 (B) 7 (C) 9 (D) 11 (E) 13
3. What annual rate of interest was paid if $$5,000
earned $$300 in interest in 2 years?
(A) 2% (B) 3% (C) 4% (D) 5% (E) 6%
4. Jim started walking at 3 miles per hour. Helen
started bicycling from the same place
2
hours
later and followed the same route. If Helen was
traveling at a rate of 8 miles per hour, in how many
hours did she overtake Jim?

(A)1
1
2
(B)1
3
4
(C)2(D)2
1
4
(E)2
1

2
1
2

5. Mr. Lee can do a certain job in x hours. His son takes
twice as long to do the job. Working together, they
can do the job in 6 hours. How many hours does it take
Mr. Lee to do the job alone?

(A)7< br>1
2
(B)8
1
4
(C)8
1
2
( D)8
2
3
(E)9

6. Tony buys a radio for $$60 after receiving a discount
of 20% on the list price. What was the list price?
(A) $$72 (B) $$75 (C) $$78 (D) $$80 (E) $$82.50





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