Part I Numbers and Algebra
(A) the understanding of quantity
Positive integer, 0, negative integer
1. There is no object, which is represented by 0. 0 and 1, 2, 3 ... are natural numbers. Natural numbers are integers.
Second, the minimum digit is 1, and the minimum natural number is 0.
3. 4 degrees Celsius above zero is recorded as +4 degrees Celsius; MINUS 4 degrees Celsius was recorded as MINUS 4 degrees Celsius. "+4" is pronounced +4. "-4" is pronounced as negative four. +4 can also be written as 4.
4. Like +4, 19 and +8844 are all positive numbers. Numbers like -4,-1 1, -7,-155 are all negative numbers.
5.0 is neither positive nor negative. Positive numbers are all greater than 0, and negative numbers are all less than 0.
6. Under normal circumstances, the above sea level is represented by a positive number, and the below sea level is represented by a negative number.
Seven, usually, the profit is represented by a positive number, and the loss is represented by a negative number.
Eight, under normal circumstances, the number of people getting on the bus is represented by a positive number, and the number of people getting off the bus is represented by a negative number.
Nine, in general, income is represented by positive numbers, and expenditure is represented by negative numbers.
Ten, in general, the rise is represented by a positive number, and the decline is represented by a negative number.
Infinite decimal
1. Fractions with denominators of 10, 100, 1000 ... can be expressed in decimals. One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths. ...
Integers and decimals are numbers written in decimal notation. One, ten, hundred … and one tenth and one percent … are units of counting. The propulsion rate between every two adjacent counting units is 10.
Third, the position occupied by each counting unit is called digit. These figures are arranged in a certain order.
Fourth, the nature of the decimal: add "0" or remove "0" at the end of the decimal, and the size of the decimal remains unchanged.
5. According to the nature of decimals, it is usually possible to simplify decimals by removing the "0" at the end of decimals.
Sixth, the general method of comparing decimal sizes: first compare the numbers of integer parts, and then compare decimals, percentiles and thousandths in turn from left to right. If the number in any number is large, the decimal number is also large.
7. Rewrite a number into a number in the unit of "10,000" or "100,000,000", and add a decimal point to the right of10,000 or/kloc-0,000,000.
Eight, the general method of finding the approximate number of decimal places: 1, first clear how many decimal places to keep; 2 determine which number to look at according to needs; 3 Use the method of "rounding" to get the result.
Nine, integer and decimal places order table:
Fraction true fraction, false fraction
Firstly, the unit "1" is divided into several parts on average, and the number representing such a part or parts is called a fraction. The number representing one of them is the decimal unit of this fraction.
Second, when two numbers are divided, their quotient can be expressed by a fraction. Namely: a÷b=b/a(b≠0)
Third, the meaning of decimals and fractions can be seen that decimals are actually fractions with denominators of 10, 100, 1000 ...
Fourth, scores can be divided into true scores and false scores.
The fraction with numerator less than denominator is called true fraction. The true score is less than 1.
6. Fractions with numerator greater than or equal to denominator are called false fractions. False score is greater than or equal to 1.
Seven, numerator denominator only 1 common factor is called simplest fraction.
Eight, the basic nature of the score: the numerator and denominator of the score are multiplied or divided by the same number at the same time (except zero), and the size of the score remains unchanged.
Nine, the nature of the decimal is consistent with the basic nature of the fraction, which can be divisible by applying the basic nature of the fraction.
Percentage tax rate, interest, discount, percentage
1. A number indicating that one number is a percentage of another number is called a percentage. Percentage is also called percentage or percentage, and percentage is usually expressed as "%".
Second, the score and percentage comparison:
Third, reciprocity of fractions, decimals and percentages.
(1) Decimalize the fraction and divide it by the denominator.
(2) Take the number of components in decimal, first rewrite the scores with components of 10, 100, 1000 ... and then lower the scores.
(3) Turn the decimal point into a percentage, first move the decimal point to the right by two places, and then add hundreds of semicolons.
(4) To change the percentage into a decimal, first remove the percent sign, and then move the decimal to the left by two places.
(5) Turn the fraction into a percentage, first turn the fraction into a decimal (three decimal places are generally reserved when it is used up), and then turn the decimal into a percentage.
(6) First rewrite the percentage number of components into the number of components, and the quotation that can be reduced becomes the simplest score.
Fourth, memorize the reciprocity of three commonly used numbers.
Five,
1, the attendance rate indicates what percentage of the total number of people present.
2. The qualified rate indicates that the number of qualified pieces accounts for a few percent of the total number of pieces.
3. The survival rate indicates that the number of surviving trees accounts for a few percent of the total number of trees.
6. To find the percentage of one number greater than another is to find the percentage of one number greater than another.
Seven, 1, more than "1"= a few percent more than 2, less than "1"= a few percent less.
Eight, the interest earned is pre-tax interest, and the interest earned is after-tax interest.
Nine. Interest = principal × interest rate× time
X accrued interest-interest tax = earned interest
Eleven, a few fold means a few tenths, which means ten percent; A few folds means a few tenths of a point and a few tens of percent.
Twelve,
1, original price × discount = current price
2. Current price/original price = discount
3. Current price/discount = original price
Thirteen, a few percent means a few tenths means dozens of percent; A few percent means a few tenths, a few tens.
Factors and multiple prime numbers, composite numbers, odd numbers and even numbers
1.4 × 3 = 12, 12 is a multiple of 4, 12 is also a multiple of 3, and 4 and 3 are both factors of 12.
Second, the smallest multiple of a number is itself, and there is no multiple. The multiple of a number is infinite.
3. The smallest factor is 1, and the smallest factor is yourself. The number of factors of a number is limited.
Multiply of 4.5: The number in the unit is 5 or 0.
Multiply of 2: The number in a unit is 2, 4, 6, 8 or 0. A multiple of 2 is an even number.
Multiply of 3: The sum of numbers must be a multiple of 3.
5. Numbers that are multiples of 2 are called even numbers. Numbers that are not multiples of 2 are called odd numbers.
6. A number is called a prime number if it has only 1 and its own two factors.
7. If a number has other factors besides 1 and itself, it is called a composite number.
8. In the number 1-20: (1 is neither prime nor composite)
Odd numbers: 1, 3,5,7,9, 1 1 3, 15, 17, 19.
Even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Prime numbers: 2,3,5,7, 1 1, 13, 17, 19. (***8 and 77. )
Complex numbers: 4,6,8,9, 10, 12, 14, 15, 16,18,20. (*** 1 1, and 132. )
9. The smallest odd number is 1, the smallest even number is 0, the smallest prime number is 2, and the smallest composite number is 4.
10. If two numbers are multiples, the big number is the least common multiple and the decimal number is the common factor.
1 1. If two numbers only have the common factor of 1, then the common factor is 1, and the least common multiple is their product.
(2) Number of operations
Calculate regular integers, decimals and fractions
First, the calculation of integer addition and subtraction should be aligned with the same number of digits and counted from the low order.
Second, when calculating decimal addition and subtraction, the decimal point should be aligned and counted from the low place.
Third, decimal multiplication: 1, first calculate the product by integer multiplication, see how many decimal places there are in a factor * * *, then count a few from the right of the product and point to the decimal point.
2. Note: When calculating the decimal point in the product, if the number of digits is not enough, it should be preceded by 0.
Fourth, fractional division:
1, the decimal point of quotient should be aligned with the decimal point of dividend;
2. If there is a remainder, add 0 after it and continue to divide it down;
3. When the unit is not enough to quotient 1, write 0 in the integer part of quotient, and then continue to divide.
4. When the divisor is converted into an integer, the decimal point of the divisor is shifted several places to the right, and the decimal point of the dividend is also shifted several places to the right.
5. When the decimal place of dividend is less than the decimal place of dividend, 0 should be added at the end of dividend.
5. Multiply a decimal by 10, 100, 1000 ... Just move the decimal point of this decimal by one, two or three places to the right. ...
Six, a decimal divided by 10, 100, 1000 ... just move the decimal point of this decimal to the left by one, two or three places. ...
7. Fraction addition and subtraction: 1 and denominator fraction addition and subtraction, numerator addition and subtraction denominator unchanged. Add and subtract fractions with different denominators, first divide them into fractions with the same denominator, and then add and subtract.
8. Comparison of fractional size: 1 Compared with denominator fraction, molecular macromolecules are smaller. 2 compare the scores of different denominators, and divide them first and then compare them; If the numerator is the same, the denominator is big and small.
9. Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is denominator.
Ten, the number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B.
Four operational relationships
Two laws
1. The quotient of division remains unchanged: the dividend and divisor are multiplied or divided by the same number (except 0) at the same time, and the quotient remains unchanged.
Second, the product invariance law of multiplication: if one factor is multiplied by a few and another factor is divided by a few, then their products remain unchanged.
Simple calculation
First, the operation law:
Second, the reciprocity of multiplication and division. (hint: the symbol is opposite; Multiply two numbers to get "1". )
Third, the method of finding the divisor.
① Rounding method. (2) the law. ③ Tailing method.
4. Comparison between product and factor, quotient and bonus:
magnitude relation
Three. Formulas and equations
Use letters to represent numbers.
1. In a formula containing letters, when a number is multiplied by letters, letters and letters, the middle multiplication sign can be recorded as ""or omitted. When omitting the multiplication sign between numbers and letters, write the numbers before the letters.
2.2a and a2 have different meanings: 2a means the addition of two A's, a2 means the multiplication of two A's ... namely: 2a = a+a, a2= a×a A.
Third, numbers in letters:
① Use letters to represent any number: for example, X=4 a=6.
② Use letters to express common quantitative relations, such as s=vt.
(3) use letters to express the algorithm: for example, a+b = b+a.
④ Use letters to express the calculation formula: S=ah.
Equality and equality
1. Equations with unknowns are called equations.
Second, the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation.
Third, the process of finding the solution of the equation is called solving the equation.
Four, the relationship and difference between equation and equation:
Verb (abbreviation of verb) The basic property of equation (1): Add (or subtract) the same number on both sides of the equation at the same time, and the result is still an equation.
Sixth, the basic properties of the equation (2): both sides of the equation are multiplied (or divided) by a number that is not equal to zero at the same time, and the result is still an equation.
Seven, the general steps to solve application problems with column equation:
(1) Find out the meaning of the problem, find out the unknown, and express it with X.
(2) Find out the equal relationship between quantity and quantity in the application problem and list the equations.
③ Find the solution of the equation.
(4) Check or check and write the answer.
(d) Positive proportion and inverse proportion
Ratio and proportion
First, the relationship and difference between ratio and proportion:
Second, the relationship and difference between ratio, fraction and division:
Third, the difference between seeking ratio and simplifying ratio:
Fourth, simplify the ratio:
① The simplified method of integer ratio is: divide the first term and the last term of the ratio by their common divisor.
② Decimal ratio is simplified by converting it into integer ratio first, and then simplifying it by integer ratio simplification method.
③ The simplified method of fractional ratio is to multiply the front and rear terms of the ratio by the least common multiple of the denominator.
5. Scale: We call the ratio of the distance on the map to the actual distance the scale of this map.
6. Scale = distance on the map: actual distance Scale = distance on the map/actual distance.
Inverse ratio
1. ratio: two related quantities, one changes and the other changes. If the ratio (that is, quotient) of the corresponding two numbers in these two quantities is certain, these two quantities are called proportional quantities, and the relationship between them is called proportional relationship.
Inverse proportion: two related quantities, one of which changes and the other changes accordingly. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called inverse proportional quantities, and their relationship is called inverse proportional relationship.
Third, the difference between positive proportion and inverse proportion:
Part II Space and Graphics
(A) the understanding and measurement of graphics
Quantitative measurement
A unit of length is used to measure the length of an object. Commonly used length units are: kilometers, meters, decimeters, centimeters and millimeters.
Second, the length unit:
Third, the area unit is used to measure the size of the surface or plane figure of an object. Public area units: square kilometers, hectares, square meters, square decimeters and square centimeters.
Four, the calculation of land area, usually in hectares. The square with a side length of 100 meters covers an area of 1 hectare.
Five, measuring a large area of land, usually in square kilometers. Square land with side length 1000 m, area 1 km2.
Area unit of intransitive verbs: (100)
Unit of volume is used to measure the space occupied by objects. Commonly used unit of volume are: cubic meter, cubic decimeter (liter) and cubic centimeter (milliliter).
Eight. Unit of volume: (1000)
Nine, the commonly used quality units are: ton, kilogram, gram.
X. mass unit:
Eleven, commonly used time units are:
Century, year, quarter, month, ten days, day, hour, minute and second.
Twelve. Time unit: (60)
Thirteen, high-level unit name rewritten into low-level unit name should be multiplied by the rate; The name of the low-level unit should be rewritten as the name of the high-level unit, divided by the forward speed.
Fourteen, commonly used units of measurement are expressed in letters:
Understanding, perimeter and area of plane graphics
First, connect two points with a ruler to get a line segment; Extending one end of the line segment indefinitely can get a ray; Extend both ends of a line indefinitely and you can get a straight line. Line segments and rays are both parts of a straight line. A line segment has two endpoints and its length is limited. A ray has only one endpoint, a straight line has no endpoint, and both rays and straight lines are infinitely long.
Second, two rays from a point form an angle. The size of the angle is related to the size of both sides, and has nothing to do with the length of the sides. The unit of measurement of angle size is ().
Third, the classification of angles: angles less than 90 degrees are acute angles; An angle equal to 90 degrees is a right angle; An angle greater than 90 degrees and less than 180 degrees is an obtuse angle; The angle equal to 180 degrees is a flat angle; An angle equal to 360 degrees is a fillet.
Four, two straight lines intersecting at right angles are perpendicular to each other; Two straight lines that do not intersect in the same plane are parallel to each other.
5. A triangle is a figure surrounded by three line segments. Each line segment constituting a triangle is called an edge of the triangle, and the intersection of every two line segments is called the vertex of the triangle.
6. Triangle can be divided into acute triangle, right triangle and obtuse triangle according to angle.
According to different sides, it can be divided into equilateral triangle, isosceles triangle and arbitrary triangle.
7. The sum of the internal angles of a triangle is equal to 180 degrees.
8. In a triangle, the sum of any two sides is greater than the third side.
Nine, in a triangle, there is at most one right angle or at most one obtuse angle.
X. A quadrilateral is a figure surrounded by four sides. Common special quadrangles are parallelogram, rectangle, square and trapezoid.
Xi。 A circle is a curved figure. The distance from any point on the circle to the center of the circle is equal, and this distance is the length of the radius of the circle. The line segment passing through the center of the circle with both ends in the circle is called the diameter of the circle.
Twelve, there are some graphics, folded in half along a straight line, and the graphics on both sides of the straight line can completely overlap. This graph is an axisymmetric graph. This straight line is called the axis of symmetry.
Thirteen, the sum of all the edges of a figure is the perimeter of the figure.
Fourteen, the size of the surface of the object or the closed plane figure is called their area.
Fifteen. Derivation of calculation formula for plane figure area;
What is the derivation process of 1 parallelogram area formula?
① A parallelogram can be transformed into a rectangle by cutting and translating.
② The length of rectangle is equal to the base of parallelogram, the width of rectangle is equal to the height of parallelogram, and the area of rectangle is equal to the area of parallelogram.
③ Because: rectangular area = length× width, so: parallelogram area = bottom× height. Namely: S=ah.
2 the derivation process of triangle area formula?
① Two identical triangles can form a parallelogram.
② The base of a parallelogram is equal to the base of a triangle, the height of a parallelogram is higher than that of a triangle, and the area of a triangle is equal to half the area of a parallelogram with equal base and equal height.
(3) Because the parallelogram area = base × height, and the triangle area = base × height ÷2. Namely: S=ah÷2.
3 the derivation process of trapezoidal area formula?
① Two identical trapezoids can be used to form a parallelogram.
② The base of parallelogram is equal to the sum of the upper and lower base of trapezoid, the height of parallelogram is higher than that of trapezoid, and the area of trapezoid is equal to half of that of parallelogram.
③ Because: parallelogram area = bottom × height, so: trapezoid area = (upper bottom+lower bottom) × height ÷2. Namely: S=(a+b)h÷2.
Draw a picture to illustrate the derivation process of the formula of circular area
① Divide the circle into several equal parts, cut it open and spell it into an approximate rectangle.
② The length of a rectangle is equivalent to half of the circumference, and the width is equivalent to the radius of the circle.
③ Because: rectangular area = length× width, so: circular area =πr×r=πr2. Namely: S=πr2.
Sixteen, the perimeter and area calculation formula of plane graphics:
Seventeen, commonly used data:
Cognition, surface area and volume of three-dimensional graphics
Cuboids and cubes have 6 faces, 12 sides and 8 vertices. Cubes are special cuboids.
Second, the characteristics of the cylinder: one side, two bottom surfaces, and countless heights.
Third, the characteristics of the cone: a side, a bottom, a vertex and a height.
4. Surface area: The sum of all the areas of a three-dimensional figure is called the surface area of this three-dimensional figure.
Verb (abbreviation of verb) Volume: The size of the space occupied by an object is called the volume of the object. The volume that a container can hold other objects is called the volume of the container.
Six, three relationships between cylinder and cone:
① Equal bottom and equal height: volume 1︰3
② Equal bottom and equal volume: height 1︰3.
③ Equal contour volume: the bottom area is 1︰3.
Seven, equal bottom and equal height cylinders and cones:
① The volume of a cone is 1/3 of that of a cylinder,
② The volume of a cylinder is three times that of a cone,
③ The volume of the cone is 2/3 smaller than that of the cylinder,
(4) The volume of a cylinder is twice as large as that of a cone.
8. Cylinders and cones with equal bottoms and equal heights: cone 1, difference 2, columns 3 and 4.
Nine, three-dimensional graphics formula derivation:
1 What figure do you get when the side of the cylinder is unfolded? What is the relationship between each part of this figure and the cylinder? (Derivation process of cylindrical lateral area formula)
① A rectangle is generally obtained when the side of a cylinder is unfolded.
② The length of the rectangle is equivalent to the circumference of the bottom of the cylinder, and the width of the rectangle is equivalent to the height of the cylinder.
③ Because: rectangular area = length × width, so: cylindrical lateral area = bottom circumference × height.
④ A square may be obtained when the side of the cylinder is unfolded.
The side length of a square = the circumference of the bottom of the cylinder = the height of the cylinder.
When we study the formula for calculating the volume of a cylinder, we can deduce it by converting the cylinder into a three-dimensional figure (approximation) that we have learned before. Please tell me the name of this three-dimensional figure and its relationship with the relevant parts of the cylinder.
(1) Divide the cylinder into several equal parts, cut it and put it together into an approximate cuboid.
② The bottom area of the cuboid is equal to that of the cylinder, and the height of the cuboid is higher than that of the cylinder.
③ Because cuboid volume = bottom area × height, cylinder volume = bottom area × height. Namely: V=Sh.
Please draw a picture to illustrate the derivation process of cone volume formula.
① Find an empty cone and an empty cylinder with equal bottom and height.
(2) Fill the cone with sand, pour it into the cylinder, and find that it is filled just three times. Pour the sand in the jar into the cone and find that it has just been poured out three times.
③ It is found through experiments that the volume of a cone is equal to one third of the volume of a cylinder with equal bottom and equal height; The volume of a cylinder is equal to three times the volume of a cone with equal bottom and equal height. Namely: V= 1/3Sh.
(b) graphics and conversion
First, the methods to change the position of graphics are translation, rotation and so on. When changing the position, the vertices, line segments and curves corresponding to each figure should be synchronously translated and rotated by the same angle.
2. Without changing the shape of a graph, only changing its size usually makes the elements of each graph, such as the length and width of a rectangle and the bottom and height of a triangle, enlarge or shrink at the same time.
3. Symmetric figures mean that the figures on both sides of the symmetry axis can completely overlap after being folded in half, but not exactly the same.
(3) graphics and location
First, in real life and situations, when facing the short distance of teaching, we usually use up, down, front and back to describe the specific location.
Second, when we face maps and orientation maps, we usually use east, west, south, north, south, east and north to describe the direction. Combined with the displayed scale, the specific distance is calculated, and the position is determined by combining the direction and distance.