2. The classification standard of geometry is not unique: one is to classify by column, cone and sphere. Cuboid, cube, cylinder and prism are all cylinders; Cones and pyramids are cones; The ball is a sphere. One is based on whether the surface that constitutes geometry is a plane or a surface. Cuboid, cube, prism and pyramid belong to one class, and their faces are all planes; Cylinders, cones and spheres are one kind, and their surfaces all have curved surfaces.

3. Similarities and differences between prism and cylinder: The same point is that both cylinder and prism have two bottoms. The difference is that the bottom of (1) cylinder is round, and the bottom of prism is polygonal. (2) The side of a cylinder is a curved surface, and the side of a prism is a quadrilateral.

4. Elements of graphics and their relationships: Elements of graphics are points, lines and faces, and faces have planes and faces; Lines have straight lines and curves. The relationship between them is: points move into lines, lines move into planes, and planes move into adults. Faces intersect to get lines, and lines intersect to get points. 5. The relationship between the number of vertices, edges and faces of a polyhedron: number of vertices+number of faces-number of edges =2.

6. Related concepts of prism: the intersection of any two adjacent faces is called an edge, and the intersection of two adjacent sides is called a side.

7. Three characteristics of a prism: First, all sides of the prism are equal in length; Second, the upper and lower bottom surfaces are the same figure, both of which are polygons; Third, the sides are rectangular.

8. Classification of prisms: According to the number of sides of the polygon at the bottom, prisms are divided into triangular prism, quadrangular prism and pentagonal prism.

9. Relationship between terms in a prism: A prism with a bottom with n sides, 2N vertices and 3N sides, including n sides, (N+2) sides and n sides.

10. The expanded diagram of a prism consists of two identical polygons and some rectangles. The cube expansion diagram needs to cut 7 edges and connect 5 edges. There are 1 1 kinds of expansion diagrams of cubes.

1 1. About cutting geometry: cutting geometry with a plane, and the cross-sectional shape is usually triangle, square, rectangle, trapezoid, circle, ellipse, etc. The shape of the cross section is related to the cutting geometry and the angle and direction of the cross section. The cross section of n polyhedron is a graph with at most (N+2) edges. 12. When you look at an object from different directions, you may see different figures. The figure you can see is a relative plane figure.

13. Three views refer to: front view (view from the front), left view (view from the left) and top view (view from above). 14, the front view reflects the length and height of the object, the top view reflects the length and width of the object, and the left view reflects the width and height of the object. From this, we can imagine the three-dimensional graphics reflected by the three views. The front view and the top view are equal in length; The heights of the main view and the left view are equal; The width of the top view and the left view are equal. 15, plane figure in life: 1) Polygon: A closed plane figure composed of some line segments that are not on the same line is called a polygon. Polygons are divided into triangles, quadrilaterals, pentagons and hexagons according to the number of line segments. 2) Circle: The figure formed by the rotation of a line segment around an endpoint is a circle.

16, each polygon can be divided into several triangles: an n polygon has (N-3) diagonal lines from a vertex and can be divided into (N-2) triangles. A polygon can be divided into (N- 1) triangles with (N-2) diagonal lines by connecting a point on one side of the polygon with possible vertices.

17, arc: the part between two points on the circle is called an arc.

18, sector: A figure consisting of an arc and two radii passing through the end of this arc is called a sector.

Chapter II Rational Numbers and Their Operations

1, positive number: like 3, 1. Numbers greater than 0 such as 2325 are called positive numbers.

2. Negative numbers: such as-1, -278, -2. Third class. Numbers preceded by a "-"sign are called negative numbers, and negative numbers are less than 0. 3.0 is neither positive nor negative, and 0 is the boundary between positive and negative numbers.

4. Rational Numbers: Integers and fractions are collectively called rational numbers. Integers include positive integers, zero and negative integers. Scores include positive and negative scores.

5. Classification of rational numbers: 1) Divided by symbols: positive rational numbers (including positive integers and fractions) and zero-sum negative rational numbers (including negative integers and fractions). 2) According to the definition: 1) Integer (positive integer, negative integer, zero) and Fraction (positive fraction, negative fraction).

6. When studying problems, rational numbers are usually divided into positive rational numbers, zero rational numbers and negative rational numbers for discussion. Generally, positive numbers and 0 are collectively referred to as non-negative, negative numbers and 0 are collectively referred to as non-positive numbers, positive integers and 0 are collectively referred to as non-negative integers (also called natural numbers), and negative integers and 0 are collectively referred to as non-positive integers.

7. Positive numbers and negative numbers represent quantities with opposite meanings. If a positive number represents a quantity with a certain meaning, a negative number represents a quantity with an opposite meaning. But there must be a "benchmark", which can be determined according to needs.

8. Easy-to-enter misunderstanding: Not all numbers marked with "-"are negative, but numbers marked with "+"are positive. For example, -A does not necessarily represent a negative number; when A=- 1, -A is a positive number; When A=0, it is neither positive nor negative. 9. Definition of number axis: The straight line defining origin, positive direction and unit length is called number axis.

10, number axis drawing: 1) Draw a horizontal straight line. 2) Take a point on a straight line as the origin and use this point to represent zero (mark "0" below the origin). 3) Determine the positive direction (generally, the right direction is positive), which is indicated by an arrow. 4) Choose an appropriate length as the unit length, and take a point every unit length from the origin to the right, which is expressed as 1, 2,3,4? ; From the origin to the left, take a point every unit length, which is expressed as-1, -2, -3? 1 1, the relationship between points on the number axis and rational numbers: all rational numbers can be represented by points on the number axis; But conversely, it cannot be said that all points on the number axis represent rational numbers.

12. Geometric definition of inverse number: the number represented by two points with the same distance from the origin on both sides of the number axis is called inverse number. 13. Algebraic definition of inverse number: There are only two numbers with different signs. We say that one of them is the inverse number of the other number, which is also called that the two numbers are mutually inverse numbers. The antonym of 0 is 0.

14. Representation method of inverse number: Generally speaking, the inverse number of the number A is -A, where A represents any number, which can be positive, negative or 0, and A can also represent any algebraic expression.

15, simplification of multiple symbols: the simplification of multiple symbols only considers the number of negative signs in the number, but does not consider the number of positive signs.

16. Comparison between rational number and number axis: On the number axis, the number on the right is always greater than the number on the left. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than all vectors.

17. When comparing the sizes of two numbers, when these two numbers are not sure what they are, they should generally be classified and discussed according to positive numbers, negative numbers and 0.

18. Geometric definition of absolute value: the absolute value of a number A is the distance between the point representing the number A on the number axis and the origin, and the absolute value of the number A is recorded as /A/. 19, algebraic definition of absolute value: the absolute value of a positive number is itself, the absolute value of a negative number is its inverse, and the absolute value of 0 is 0. The important property of absolute value is non-negative.

20. The comparative size rule of rational numbers: all positive numbers are greater than 0; Negative numbers are all less than 0; Positive numbers are greater than all negative numbers; Comparing the sizes of two negative numbers, the absolute value is larger but smaller.

2 1, rational number addition rule: (1) Add two numbers with the same sign, take the same sign, and add the absolute values; (2) The sum of two numbers with different signs is equal in absolute value.

Is 0; When the absolute values are not equal, take the sign of the addend with the larger absolute value and subtract the smaller absolute value from the larger absolute value. (3) When a number is added to 0, the number is still obtained.

22. Flexible application in practical calculation: 1) Add mutually opposite numbers; 2) Add the numbers with the same symbol; 3) add several numbers to get an integer; 4) Numbers with the same denominator are added.

23. The significance of rational number subtraction: knowing the sum of two addends and one of them, and finding the other addend is called subtraction. 24. the subtraction rule of rational numbers: subtracting a number is equal to adding the reciprocal of this number. That is, A-B=A+(-B)

25. Methods and steps of rational number addition and subtraction mixed operation: First, all subtractions in the mixed operation are converted into additions by using the subtraction rule; The second is to use the law of addition and additive commutative law's law of association to perform simple operations.

26. Multiplication rule of rational numbers: two numbers are multiplied, the same sign is positive, the different sign is negative, and the absolute value is multiplied. Multiply any number by 0, and the product is still 0.

27. Key memory: Multiply several numbers that are not equal to 0. The sign of the product is determined by the number of negative factors. When there are odd negative factors, the product is negative; When the number of negative factors is even, the product is positive. Then multiply by the absolute value. Multiply several numbers, one factor is 0 and the product is 0. Conversely, if the product is 0, then at least one factor is 0.

28. Multiplicative commutative law, multiplicative associative law and multiplicative commutative law are also applied in rational number multiplication.

29. Rational number division rule 1: divide two rational numbers, the same sign is positive and the different sign is negative, and divide by the absolute value. Divide 0 by any number except 0 to get 0. 30. Key memory: 0 has no countdown. The reciprocal of a negative number is the reciprocal of its absolute value. The reciprocal of a positive number is a positive number. The reciprocal of a negative number is negative. If two numbers are reciprocal, the product of these two numbers is 1.

3 1, rational number division rule 2: dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.

32. Power: Generally speaking, the operation of finding the product of n identical factors A is called power. The result of power is called power, a is called base, and n is called exponent. 33. Three problems that should be paid attention to by the power: 1) A number can be regarded as its own 1 power, and the exponent 1 is usually omitted. 2) When the radix is negative or fractional, the radix must be enclosed in brackets. 3) The power of a negative number is different from the reciprocal of the power.

34. Symbolic law of power operation: 1) The power of any positive number is positive; 2) The odd power of a negative number is negative, and the even power of a negative number is positive. 3) All positive powers of 0 are 0. Any power of 1 is 1, odd power of-1 is-1, and even power of-1 is 1.

35. Operation sequence of rational number mixed operation: first calculate the power, then calculate the multiplication and division, and finally calculate the addition and subtraction; If there are brackets, count them first. 36. Problems needing attention in the mixed operation of rational numbers: 1) The operation of rational numbers, addition and subtraction are called first-level operations, multiplication and division are called second-level operations, and multiplication and root (to be learned later) are called third-level operations. If a formula has several levels of operation, it should be operated at the third level, then at the second level, and finally at the first level. Operations at the same level are operated from left to right; When there are parentheses, the operations are performed in the order of parentheses, brackets and braces (and vice versa). 2) If there are fractions and decimals in the question, they should be converted into false fractions and fractions before calculation, subtraction into addition before operation, and division into multiplication before operation.

37. Use the non-negative sum of absolute values to find the application of letter values.

Classification of calculators: Calculators can be divided into simple calculators, scientific calculators and graphic calculators according to their functions.

Composition of calculator: The calculator panel consists of keyboard and monitor. ON the calculator keyboard, the ON key is the power-on key. Before each operation, press it to reset. DEL key is the delete key, and you can press clear when you find that there is an error in the input data; When you stop using it, press the SHIFT key first, and then press the AC key to turn off the power.

Chapter III Letter Representation of Numbers

Advantages of using letters to represent numbers: using letters to represent numbers solves the special and general relationship, and using letters to represent numbers is more general and concise. 2. The same problem, the same letter can only represent the same quantity, and different quantities should be represented by different letters. Expressions can be expressed in many ways, but the result is the same.

3. Use letters to represent operation rules and formulas, and use letters to represent quantitative relations. It is necessary to master the formula and operation law skillfully, and to analyze the meaning of the question and solve specific problems.

4. Remember the regular formula: a square is surrounded by several points, and the relationship between the total number of points and the number of edges is: s (total number of points) =(4N (number of edges)-4); A triangle consists of several points, and the relationship between each side n and the total number s is as follows: s = (3n-3); The relationship between the number of matches A and the number of squares B is as follows: A = (3b+1);

5. Algebraic expressions: like 4+3(χ- 1), χ+χ+(χ- 1), χ 5, MN, A2.

All equations are algebraic expressions. In this way, the expression formed by connecting numbers and letters representing numbers with operational symbols is called algebraic expression. Note: 1) A single number or letter is also algebraic; 2) As long as there is no equal or unequal equation, there is an algebraic expression.

6. Algebraic writing format: 1) When letters are multiplied by letters or numbers are multiplied by letters, the multiplication sign is usually omitted, and the numbers should be written before the letters; 2) When the band score is multiplied by letters, the band score should be converted into a false score and then multiplied by letters; 3) The division operation in algebraic expressions is generally written according to the writing method of fractions, with divisor as numerator, divisor as denominator and divisor converted into fractional lines; 4) In practical problems, if the algebraic expression has a unit name, if the algebraic expression is in the form of product or quotient, write the unit name after the expression. If the algebraic expression is in the form of sum or difference, the algebraic expression must be enclosed in parentheses, and then the unit name should be written after the expression.

7. Column algebra: it refers to the expression of the quantitative relationship described in written language in the problem with formulas containing letters and operational symbols, which is called column algebra. 8. Matters needing attention in column algebra: 1) Carefully examine the questions, and correctly convert the words expressing quantitative relations into corresponding operations. Such as sum, difference, product, quotient, square, reciprocal, big, small, more, less, increase, increase to, expand, shrink, multiple, fraction, ratio, division, etc. , are commonly used words to express quantitative relations. 2) Pay attention to the operation sequence indicated by the language description, and generally read before writing. 3) In complex problems, make clear the operation order of quantitative relations, correctly use brackets representing operation procedures, divide levels, and gradually list algebraic expressions. 4) Pay attention to the difference between sum of squares and square of sum, cube of sum and cube of sum, division and division.

9. The practical meaning of algebraic expression: it is to give specific meaning to the letters and operation symbols in algebraic expression. It should be noted that the quantitative relationship in practical problems must be consistent with the quantitative relationship expressed by algebraic expressions.

10, the relationship between various practical problems: 1) Let a unit number of three digits be χ, a decimal digit be у and a hundredth digit be z, then the three digits can be expressed as:100z+1у+χ. 2) Two two-digit numbers are multiplied, and the numbers on the ten digits of the two numbers are the same. If the sum of the numbers on each digit is 10, there is (10a+b) (10a+c) =100a (a+1)+BC.

1 1. Algebraic evaluation: The process of replacing letters in algebraic expressions with numerical values and calculating the results according to the operations specified in algebraic expressions is called algebraic evaluation. 12, the value of algebraic expression: generally, it is not a fixed quantity, but changes with the change of the value of letters in algebraic expression. In addition, the value of the algebraic expression must be calculated according to the operation specified in the algebraic expression.

13, evaluation method of algebraic expression: 1) Replace letters in algebraic expression with numerical values, which is called "replacement" for short. 2) Calculation result according to the algebraic expression, referred to as "calculation" for short.

14, the nonnegativity of absolute value, reciprocal, reciprocal, square and absolute value and the application of substitution evaluation method in algebraic evaluation. 15. Algebraic term: the part separated by each operator in an algebraic term is called an algebraic term.

16, coefficient of algebraic term: the number factor before each letter is called the coefficient of the term. The coefficient includes the symbol before it. If one term in the algebraic expression contains only the letter factor, its coefficient is 1 or-1. 17, constant term: Algebraic term without letters is called constant term.

18. Similar items: items with the same letters and the same letter index are called similar items.

19. Precautions for judging similar items: 1) There are two conditions for judging whether several items are similar: first, they contain the same letters; Second, the indexes of the same letters are the same, and these two conditions must be met at the same time. 2) Similar terms have nothing to do with the arrangement order of coefficients and letters. 3) Special attention: Several constant terms are also similar.

20. Merging similar items: Merging similar items into one item is called merging similar items. When merging similar items, the coefficients of similar items are added, and the results are taken as coefficients, and the letters and the indexes of letters remain unchanged.

2 1. Similar item merging steps: 1) Find out the similar items accurately; 2) Using the law, the coefficients of similar items are added together, and the index of letters remains unchanged; 3) Calculate the sum of all the coefficients by rational number addition, and write the combined result. 4) The results of merging similar items should be arranged according to the descending power or ascending power of a letter.

22. The meaning of removing brackets: When there are brackets in modern number operations, it is often necessary to remove the brackets first, so that the operation can proceed smoothly.

23. Rules for removing brackets: 1) There is a "+"before brackets. After removing the brackets and the "+"sign in front of them, the symbol of the original brackets remains unchanged. 2) There is a "-"sign before the brackets. After deleting the brackets and the "-"sign in front of them, the symbols of the items in the original brackets will change. 24. When comparing the sizes of two numbers (or algebraic expressions), you can use the difference to compare the size with 0. When the difference is greater than 0, the minuend is larger; When the difference is less than 0, the minuend is relatively small. 25. The sequence of dismantling the support: dismantle the support layer by layer from the inside out; Remove the support layer by layer from the outside to the inside; Both the inner and outer brackets are deleted. 26. The mathematical methods used to explore the law are: classified discussion; Transformation method; Induction.

Chapter IV Plane Figures and Their Positional Relations

1. Line segment: A line segment has two endpoints. The length can be measured.

2. Ray: A ray is formed by the infinite extension of a line segment in one direction. A ray has an endpoint. The length cannot be measured. 3. Straight line: A straight line is formed by the infinite extension of line segments in two directions. A straight line has no end points, so its length cannot be measured.

4. Representation method of line segment: (1) Use two end letters on the line segment to represent a line segment. (2) Use lowercase letters to represent a line segment. 5. Representation method of rays: (1) rays are represented by letters representing endpoints and letters representing points on rays. Endpoint letters must be written in front. 6. Representation method of straight line: (1) Take any two points on the straight line and use capital letters representing the two points to represent the straight line. (2) Use lowercase letters to represent straight lines.

7. Connection and difference between line segments, rays and straight lines: Connection means that line segments, rays and straight lines are all straight lines, and line segments can extend in one direction to get rays, while line segments can extend in two directions to get straight lines. So we can know that both the ray and the line segment are part of a straight line, and the line segment is part of the ray. This is the connection between the three. The difference is that a straight line can extend infinitely in two directions, a ray can extend infinitely in one direction, and the line segment itself cannot extend. A straight line has no endpoint, a ray has one endpoint, and a line segment has two endpoints.

8. Basic properties of straight lines: there is only one straight line after two points (it can also be said that two points determine a straight line), which is also a straight line axiom.

9. The nature of the line segment (axiom): In all the connecting lines between two points, the line segment is the shortest, which can be referred to as between two points and the line segment is the shortest.

10, the distance between two points: the length of the line segment between two points is called the distance between these two points. Distance refers to the length of a line segment, which is a value, not the line segment itself.

1 1. Compare the lengths of two line segments: (1) Overlapping method: Compare on the same straight line. (2) Measurement method: Measure the length of line segments with a scale, and then compare them.

12, the midpoint of the line segment: point M divides the line segment AB into two equal line segments AM and BM, and point M is called the midpoint of the line segment AB. The midpoint of a line segment divides the line segment into two equal line segments, which is equal to half the length of the original line segment. The original line segment is twice as big as the two segmented lines.

13. Definition of an angle: An angle is a graph formed by a ray rotating around an endpoint from the starting position to the ending position. In other words, an angle consists of two rays with a common endpoint, and the common endpoint of the two rays is the vertex of the angle. These two rays are called the edges of the angle. The two basic elements of an angle are its vertex and its edge.

14. Representation of angle: (1) is represented by three capital letters. The letter of the vertex of the corner is written in the middle. The letters of the dots on the corner are written on both sides and can be interchanged. (2) Expressed in capital English letters. The premise of this representation is that there is only one angle with a point as its vertex, otherwise it cannot be represented in this way. (3) expressed in numbers. ④ Represented by lowercase Greek letters.

15. Angle measurement: a protractor for measuring angles. Note: (1) is centered (vertex to center). (2) Coincidence (one side coincides with the zero scale line on the scale). (3) Reading (reading online on the other side).

Chapter 5 One-variable linear equation

1. Equation: An equation with an unknown number is called an equation.

2. An equation must meet two conditions: one is an equation and the other is an unknown, and both are indispensable.

3. Solution of the equation: The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation. The solution of a linear equation is also called a root.

4. Unary linear equation: In an equation, there is only one unknown, and the exponent of the unknown is 1 degree. Such an equation is called a one-dimensional linear equation. 5. One-dimensional linear equation must meet three conditions: first, there is only one unknown; Second, the number of unknowns is1; Third, it is an integral equation, which is indispensable.

6. The general steps to solve the time series equation of the application problem: 1) Set the unknown number, and set what is generally sought in the simple problem as X (other quantities are also acceptable). 2) Analyze the relationship between known quantity and unknown quantity, and find out the equivalence relation. 3) The quantities on the left and right sides of the equivalence relation are expressed by an algebraic expression containing х. 7. The basic property of the equation 1: the same algebraic expression is added (or subtracted) on both sides of the equation at the same time, and the result is still an equation.

8. Basic properties of Equation 2: When both sides of the equation are multiplied by the same number (or divided by the same number that is not 0), the result is still an equation.

9. Matters needing attention in applying the basic properties of the equation: 1) When applying the property 1, you must pay attention to adding (or subtracting) the same number or the same equation on both sides of the equation at the same time, and pay special attention to "simultaneous" and "identical". 2) When applying Property 2, we should not only pay attention to multiplying (or dividing) the same number on both sides of the equation at the same time, but also pay attention to the fact that both sides of the equation cannot be divisible by 0, because 0 cannot be divisible.

10. Use the equation to compare the sizes of two unknowns: it can be used as a difference comparison method, if A-B > 0, then A > B；; If a-b < 0, then a < b；; If A-B=0, then A =B b B. At the same time, note that when subtracting an algebraic expression from both sides by using the property of equation 1, you should pay attention to enclose this algebraic expression in brackets. 1 1. moving term rule: after changing the sign, any term in the equation can be moved from one side of the equation to the other. This kind of deformation is called shifting term, and this law is called shifting term law.

12, the key explanation is: 1) The basis of the shift term is: the basic property of the equation is1; 2) The term shift must be to move an item in an equation from one side of the equation to the other, instead of exchanging some items on the left or right side of the equation; 3) When moving items, change the sign. The same sign cannot move items.

13. The general steps of solving a linear equation with one variable: the basic idea is to transform the equation to one side of the equation, to the other side of the constant term, and finally to "transform" the equation into the form х = a. Step: 1) denominator: multiply the least common multiple of each denominator on both sides of the equation (using the basic property of the equation 2); 2) Remove the brackets: first remove the brackets, then remove the brackets, and finally remove the braces (using the distribution method); 3) Shift the term: move all the terms containing the unknown to one side of the equation, and all the other terms to the other side of the equation (using the basic properties of the equation1); 4) merging similar terms: transforming the equation into Aх=B(A≠0) (using the rule of merging similar terms); 5) coefficient transformation to 1: divide the unknown coefficient a on both sides of the equation to get the solution of the equation х=B/A (using the basic property 2 of the equation).

14, there are three common mistakes in solving the formula: 1) moving the item and forgetting the number; 2) For items without denominator, multiplication is omitted when denominator is removed; 3) When naming, forget to use parentheses when the molecule is not a polynomial.

15, the number relation in the calendar: the difference between two adjacent numbers in each row is 1, and the difference between two adjacent numbers in each column is 7; The difference between two adjacent numbers in the upper left and lower right direction is 8, and the difference between two adjacent numbers in the upper right and lower left direction is 6.

16. Rationality of solutions of linear equations with one variable: When solving practical problems with a series of equations, we should pay attention to verifying whether the solutions still conform to the practical problems. If so, this is the required solution. If not, there is no solution to the problem.

17, the problem of shape change: This kind of problem is common in the following situations: 1) The shape has changed, but the volume has not changed. At this time, the equal relationship is that the volume is equal before and after the change. 2) The shape and area have changed, but the perimeter has not changed. At this point, the equality relationship means that the perimeter is equal before and after the change. 3) Shape and volume are different, but the relationship between volumes can be found according to the meaning of the question, which is considered to be equal.

18, discount sales related concepts: cost price: purchase price, the price when buying in the store. List price: the price indicated when the store sells. Selling price: the actual price of a commodity when it is sold. Profit rate: the ratio of profit to commodity cost price.

19. discount sales related formula: 1) profit = selling price-cost price (purchase price); 2) Profit rate = profit/cost price *100%; 3) selling price = cost price+profit = cost price ×( 1+ profit rate); Price = list price × discount quantity;

20. General steps to solve practical problems with linear equations of one variable: (1) Review: analyze what is in the problem and what to seek, and clarify the relationship between quantity and quantity; (2) Find an equivalent relationship that can express all the meanings of the application questions; (3) Assumption: Assumption is unknown, and what is generally sought is assumption х; (4) Column: According to this equation, the required algebraic expressions are listed, thus the equations are listed; (5) Solving: solving the listed equations and finding the value of the unknown quantity; (6) check: check whether the answer meets the meaning of the question; (7) Answer: Write the answer (including the company name).

2 1, equation relation: 1) road length = interval length of two adjacent trees × (number of trees-1); 2) Sailing speed along the river = still water speed+water speed; 3) The speed of sailing against the current = the speed in still water-the speed of current; 4) Downwind speed = static wind speed+wind speed; 5) Wind speed against wind = static wind speed-wind speed.

22. Circular runway problem: 1) Both parties start in the same direction at the same time: the fast one must run one more lap to catch up with the slow one; 2) Party A and Party B start from the same place and opposite directions at the same time on the circular runway: the total distance when they meet is the length of one circle of the circular runway.

23. Principal: The money deposited by the customer in the bank is called principal; Interest: the reward paid by banks to customers is called interest; Sum of principal and interest: the sum of principal and interest is called the sum of principal and interest; Interest rate: the ratio of interest to principal per period is called interest rate.