2. Classification of triangles
3. Trilateral relationship of triangle: the sum of any two sides of triangle is greater than the third side, and the difference between any two sides is less than the third side.
4. Height: Draw a vertical line from the vertex of the triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.
5. midline: in a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the midline of the triangle.
6. Angular bisector: The bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the intersection of the vertex and this angle is called the angular bisector of the triangle.
7. Significance and practice of high line, middle line and angle bisector.
8. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.
9. Theorem of the sum of interior angles of triangle: the sum of three interior angles of triangle is equal to 180.
It is inferred that the two acute angles of 1 right triangle are complementary;
Inference 2: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles;
Inference 3: One outer angle of a triangle is larger than any inner angle that is not adjacent to it;
The sum of the inner angles of a triangle is half of the sum of the outer angles.
10. External angle of triangle: the included angle between one side of triangle and the extension line of the other side is called the external angle of triangle.
1 1. The Properties of the Exterior Angle of Triangle
(1) Vertex is the vertex of a triangle, one side is one side of the triangle, and the other side is the extension line of one side of the triangle;
(2) An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it;
(3) The outer angle of a triangle is greater than any inner angle that is not adjacent to it;
(4) The sum of the external angles of the triangle is 360.
12. Polygon: On the plane, a figure composed of end-to-end line segments is called a polygon.
13. Interior angle of polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.
14. Exterior angle of polygon: the angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.
15. Diagonal line of polygon: the line segment connecting two non-adjacent vertices of polygon is called diagonal line of polygon.
16. Classification of polygons: it can be divided into convex polygons and concave polygons. Convex polygons can also be called plane polygons and concave polygons can also be called space polygons. Polygons can also be divided into regular polygons and non-regular polygons. Regular polygons have equal sides and equal internal angles.
17. Regular polygon: A polygon with equal angles and sides in a plane is called a regular polygon.
18. plane mosaic: covering a part of a plane with some non-overlapping polygons is called covering the plane with polygons.
19. Formulas and attributes
The sum formula of polygon internal angles: the sum of n polygon internal angles is equal to (n-2) 180.
20. Polygon exterior angle sum theorem;
(1) The sum of the outer angles of n polygons is equal to n180-(n-2)180 = 360.
(2) Every inner angle of a polygon and its adjacent outer angle are adjacent complementary angles, so the sum of the inner angle and outer angle of n polygon is equal to n 180.
2 1. Number of diagonal lines of polygon:
(1) Starting from a vertex of an n polygon, (n-3) diagonal lines can be drawn, and the polygon can be divided into (n-2) triangles.
(2) An n-side * * has n(n-3)/2 diagonals.
Summary of mathematical knowledge points in 2-plane rectangular coordinate system of senior one.
1. Definition: Draw two mutually perpendicular number axes on a plane, and their origins coincide to form a plane rectangular coordinate system. The horizontal axis is called the X axis or the horizontal axis, and it is customary to take the right as the positive direction; The vertical axis is called Y axis or vertical axis, and the orientation direction is positive. The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.
2. Any point on the plane can be represented by an ordered number pair, which is denoted as (a, b), where a is the abscissa and b is the ordinate.
3. The coordinate of the origin is (0,0);
The connection line with the ordinate point is parallel to the X axis;
The connecting line of the points with the same abscissa is parallel to the Y axis;
The ordinate of this point on the X axis is 0, which is expressed as (x, 0);
The abscissa of the point on the Y axis is 0, which is expressed as (0, y).
4. After the plane rectangular coordinate system is established, the coordinate plane is divided into four parts, I, II, III and IV, which are called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant respectively. The points on the coordinate axis do not belong to any quadrant.
5. The characteristics of points in several quadrants:
The first quadrant (+,+); The second quadrant (-,+);
The third quadrant (-,-); The fourth quadrant (+,-).
6.(x, y) The point symmetrical about the origin is (-x,-y);
(x, y) The point of symmetry about x is (x,-y);
The point where (x, y) is symmetrical about y is (-x, y).
7. Distance from point to two axes: the distance from point P(x, y) to X axis is ︱ y ︳;
The distance from the point P(x, y) to the y axis is ︱x︳.
8. The coordinates of the points on the bisector of the first and third quadrants are (m, m);
The coordinates of the points on the bisectors of the second and fourth quadrants are (m, -m).
Unequal and unequal groups
(1) inequality
A formula connected by an inequality symbol (,≥, ≤, ≦) is called inequality.
(2) the essence of inequality
① symmetry;
② Transitivity;
③ monotonicity of addition, that is, additivity of inequality in the same direction;
④ Monotonicity of multiplication;
⑤ Multiplicity of positive inequality in the same direction;
⑥ Positive inequalities can be multiplied;
⑦ Positive inequalities can be squared;
(3) One-dimensional linear inequality
A formula connected by an inequality symbol contains an unknown number whose degree is 1, whose coefficient is not 0, and whose left and right sides are algebraic expressions is called one-dimensional linear inequality.
(4) One-dimensional linear inequalities
The group of one-dimensional linear inequalities consists of several one-dimensional linear inequalities with the same unknowns.
Point, line, surface and body knowledge points
Composition of 1. Geometry
Point: The point where straight lines intersect is the point, which is the most basic figure in geometry.
Line: The intersection line between faces is a line, which can be divided into straight lines and curves.
Face: Surrounding the body is the face, which is divided into plane and curved surface.
Volume: Geometry is also called volume for short.
2. Point to line, line to surface, surface to body.
Representation of points, lines, rays and line segments
In geometry, we often use letters to represent figures.
A dot can be represented by capital letters.
Lowercase letters can represent a straight line.
A ray can be represented by an endpoint and another point on the ray.
The endpoint of a line segment can be represented by two capital letters.
note:
(1) indicates points, lines, rays and line segments, and the points, lines, rays and line segments should be marked before the letters.
(2) Lines and rays have no length, but line segments have length.
(3) A straight line has no endpoint, a ray has one endpoint, and a line segment has two endpoints.
(4) The positional relationship between points and straight lines can be divided into two types:
The point is on a straight line, or a straight line passes through the point.
② The point is outside the straight line, or the straight line does not pass through this point.
Type of horn
Acute angle: An angle greater than 0 and less than 90 is called an acute angle.
Right angle: An angle equal to 90 is called a right angle.
Oblique angle: an angle greater than 90 and less than180 is called obtuse angle.
Boxer: An angle equal to 180 is called a boxer.
Excellent angle: more than180 and less than 360 is called excellent angle.
Bad angle: more than 0 and less than 180 is called bad angle, and acute angle, right angle and obtuse angle are all bad angles.
Fillet: An angle equal to 360 is called a fillet.
Negative angle: the angle formed by clockwise rotation is called negative angle.
Positive angle: the angle of counterclockwise rotation is positive angle.
Angle 0: An angle equal to zero.
Complementary angle and complementary angle: if the sum of two angles is 90, it is complementary angle, and if the sum of two angles is180, it is complementary angle. The complementary angles of equal angles are equal, and the complementary angles of equal angles are equal.
Inverse vertex angle: When two straight lines intersect, there is only one common vertex, and both sides of the two corners are opposite extension lines. These two angles are called antipodal angles. Two straight lines intersect to form two pairs of vertex angles. The two opposite angles are equal.
There are also many kinds of angle relationships, such as internal dislocation angle, congruent angle and internal angle of the same side (in the three-line octagon, it is mainly used to judge parallelism).
Summary of Mathematics Knowledge Points in Junior One 3 Positive and Negative Numbers
The concepts of ⒈, positive number and negative number
Negative number: a number less than 0. Positive number: a number greater than 0. 0 is neither positive nor negative.
Note: ① The letter A can represent any number. When a represents a positive number, -A is a negative number; When a stands for negative number, -a is positive number; When a represents 0, -a is still 0. (If the judgment topic is: the number with a positive sign is positive and the number with a negative sign is negative, this statement is wrong. For example, +a, -A cannot make a simple judgment. )
② Sometimes "+"can be added before positive numbers, and sometimes "+"can be omitted. Therefore, the positive sign omitting "+"is a positive sign.
2. Quantities with opposite meanings
If a positive number means a quantity with a certain meaning, a negative number can mean a quantity opposite to a positive number, such as:
8℃ above zero means:+8℃; 8 degrees below zero means 8 degrees below zero.
The meaning of 3 and 0
(1)0 means "none", for example, there are 0 people in the classroom, which means there is no one in the classroom;
(2)0 is the dividing line between positive and negative numbers, and 0 is neither positive nor negative. For example:
(3)0 represents the exact quantity. For example, 0℃ and the benchmark in some topics, such as taking sea level as the benchmark, 0 meters is sea level.
rational number
1, the concept of rational number
(1) Positive integers, 0 and negative integers are collectively called integers (0 and positive integers are collectively called natural numbers).
(2) Positive and negative scores are collectively referred to as scores.
(3) Positive integers, 0, negative integers, positive fractions and negative fractions can all be written in the form of fractions, and such numbers are called rational numbers.
Understanding: Only numbers that can change the number of components are rational numbers.
① π is an infinite acyclic decimal, which cannot be written in fractional form and is not a rational number.
(2) Finite decimals and infinite cyclic decimals can be converted into component numbers, both of which are rational numbers.
(3) Integers can also be converted into component numbers, and component numbers are also rational numbers.
Note: After the introduction of negative numbers, the range of odd and even numbers is also expanded. For example, -2, -4, -6 and -8 are even numbers, and-1, -3 and -5 are also odd numbers.
Summary of mathematics knowledge points in the first day of junior high school 4. The solution of one-dimensional linear inequality;
The solution of one-dimensional linear inequality is similar to the solution of one-dimensional linear equation, and its steps are as follows:
1, denominator;
2. Remove the brackets;
3. Mobile projects;
4. Merge similar projects;
5. The coefficient is 1.
Second, the basic properties of inequality:
1, the same algebraic expression is added (or subtracted) on both sides of the inequality, and the direction of the inequality remains unchanged;
2. Both sides of the inequality are multiplied by (or divided by) the same positive number, and the direction of the inequality remains unchanged;
3. When both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality will change.
Third, the solution of inequality:
The value of the unknown quantity that can make the inequality hold is called the solution of the inequality.
Fourth, the solution set of inequality:
All the solutions of an inequality with unknowns constitute the solution set of this inequality.
Basic properties of verbs (abbreviation of verb) in solving inequalities;
Property 1: Add (or subtract) the same number (or formula) on both sides of the inequality, and the direction of the inequality remains unchanged.
Property 2: Both sides of the inequality are multiplied by (or divided by) the same positive number, and the direction of the inequality remains unchanged.
Property 3: Both sides of the inequality are multiplied by (or divided by) the same negative number, and the direction of the inequality changes.
Common inspection methods
(1) Investigate the solution of linear inequality in one variable;
(2) Investigate the essence of inequality.
Misunderstanding reminder
Ignore the problem of unequal sign change direction.
Induction of key knowledge points in junior high school mathematics
Arithmetic of rational number multiplication
1, the commutative law of multiplication: AB = BA
2. The law of multiplicative association: (ab) c = a (BC);
3. Distribution law of multiplication: a(b+c)=ab+ac.
monomial
Algebraic expressions that contain only the product of numbers and letters are called monomials.
Note: The monomial is composed of coefficients, letters and indices of letters.
multinomial
The sum of 1 and several monomials is called a polynomial. Where each monomial is called a term of this polynomial. Items without letters in polynomials are called constant terms. The degree of the term with the highest degree in a polynomial is called the degree of the polynomial.
2. Items with the same letter and the same letter index are called similar items. Several constant terms are similar.
Methods of improving mathematical thinking
Transform thinking
Transforming thinking is both a method and a kind of thinking. Transformational thinking refers to changing the direction of the problem from one form to another from different angles when encountering obstacles in the process of solving problems, and seeking the best way to make the problem simpler and clearer.
Creative thinking
Innovative thinking refers to the thinking process of solving problems with novel and original methods. Through this kind of thinking, we can break through the boundaries of conventional thinking, think about problems with unconventional or even unconventional methods and perspectives, and get unique solutions.
Cultivate the habit of questioning.
In family education, parents should always guide their children to ask questions, learn to question and reflect, and gradually develop habits.
After the child comes home from school, let the child review what he learned that day: how did the teacher explain and how did the students answer? When the child answered, he then asked, "Why?" "What do you think?" Inspire the child to tell the process of thinking and try to let him make his own evaluation.
Sometimes, you can deliberately make some mistakes, so that children can discover, evaluate and think. Through such training, children will gradually form independent opinions on thinking and develop the habit of questioning.
Summary of Mathematics Knowledge Points in Senior 15 (1) All numbers that can be written in form are rational numbers. Positive integers, 0 and negative integers are collectively referred to as integers; Positive and negative scores are collectively called scores; Integers and fractions are collectively called rational numbers. Note: 0 is neither positive nor negative; -a is not necessarily negative, and +a is not necessarily positive; P is not a rational number;
(2) Classification of rational numbers: ① Integer ② Fraction
(3) Note: among rational numbers, 1, 0 and-1 are three special numbers with their own characteristics; These three numbers divide the numbers on the number axis into four areas, and the numbers in these four areas also have their own characteristics;
(4) Natural number 0 and positive integer; A & gt0a is a positive number; A & lt0a is negative;
A≥0a is positive or 0a is non-negative; a≤0？ A is negative or 0a is not positive.
Rational number ratio size:
(1) The greater the absolute value of a positive number, the greater the number;
(2) Positive numbers are always greater than 0 and negative numbers are always less than 0;
(3) Positive numbers are greater than all negative numbers;
(4) The absolute values of two negative numbers are larger than the size, but smaller;
(5) Of the two numbers on the number axis, the number on the right is always greater than the number on the left;
(6) large number-decimal number >; 0, decimal-large number < 0.
Summary of Mathematics Knowledge Points in Senior One 6 I. Relevant Concepts of Equation
1. Equation: An equation with an unknown number is called an equation.
2. One-dimensional linear equation: contains only one unknown (element) X, and the exponents of the unknown X are all 1 (degree). This equation is called one-dimensional linear equation. For example,1700+50x =1800,2 (x+1.5x) =1
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.
Note: The solution of the (1) equation and the solution of the equation are different concepts. The solution of the equation is essentially the result of the solution, which is a numerical value (or several numerical values), and the meaning of solving the equation refers to the process of finding the solution of the equation or judging that the equation has no solution. (2) The test method of the equation solution is to substitute the unknown value into the left and right sides of the equation to calculate its value, and then compare the values on both sides to draw a conclusion.
Second, the nature of the equation.
Properties of equation (1): Add (or subtract) the same number (or formula) on both sides of the equation, and the results are still equal.
The properties of the equation (1) are expressed in the form of a formula: if a=b, then a c = b c
Property of the equation (2): If both sides of the equation are multiplied by the same number, or divided by the same number that is not 0, the results are still equal. The property (2) of the equation is expressed in the form of a formula: if a=b, then ac = bc; if a=b(c≠0), then ca=cb.
Third, the law of shift term: moving the symbol on one side of the equation to the other side is called shift term.
Fourth, the rule of removing brackets.
1. The factors outside the brackets are positive numbers, and the symbols of the items after removing the brackets are the same as those of the corresponding items in the original brackets.
2. The factor outside the bracket is negative, and the sign of each item is changed by the sign of the corresponding item in the original bracket after the bracket is removed.
Five, the general steps to solve the equation
1. denominator (least common multiple of denominator on both sides of the equation)
2. Parenthesis deletion (according to the rules of parenthesis deletion and distribution)
3. Move the term (move the term containing the unknown to one side of the equation, and all other terms will be moved to the other side of the equation. Moving the term will change the sign).
4. Merge (transform the equation into ax = b (a≠0))
5. Convert the coefficient into 1 (divide the coefficient a of the unknown quantity on both sides of the equation to get the solution of equation x=a(b)).
Sixth, the general steps to solve practical problems with equation thought.
1. Examination: Examination of questions, analysis of what is known and what is sought in questions, and clarification of the relationship between quantity and quantity.
2. Assumptions: Assumptions about the unknown (which can be divided into direct and indirect ways).
3. Column: List the equations according to the meaning of the question.
4. Solution: Solve the listed equations.
5. Check: Check whether the solution meets the meaning of the problem.
6. Answer: Write the answer (some units should indicate the answer)
Summary of 7 Mathematics Knowledge Points in Senior One 1. Knowledge Arrangement
Knowledge point 1: the concepts of positive numbers and negative numbers: we call numbers such as 3, 2, +0.5 and 0.03% positive numbers, all of which are numbers greater than 0; Numbers like -3, -2, -0.5 and -0.03% are called negative numbers. Are all numbers less than 0. 0 is neither positive nor negative. We can use positive numbers and negative numbers to represent quantities with opposite meanings.
Knowledge point 2: the concept and classification of rational numbers: integers and fractions are collectively called rational numbers. There are two main classifications of rational numbers:
Note: Both finite decimals and infinite cyclic decimals can be regarded as fractions.
Knowledge point 3: the concept of number axis: the straight line defining the origin, positive direction and unit length as follows is called number axis.
Knowledge point 4: the concept of absolute value:
(1) Geometric meaning: The distance from the point representing A on the number axis to the origin is called the absolute value of the number A, and it is recorded as | a |
(2) Algebraic significance: the absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of zero is zero.
Note: The absolute value of any number is greater than or equal to 0 (i.e. non-negative).
Knowledge point 5: the concept of reciprocal:
(1) Geometric meaning: The number represented by two points on both sides of the origin with the same distance is called reciprocal;
(2) Algebraic meaning: Two numbers with different signs but equal absolute values are called reciprocal. The antonym of 0 is 0.
Knowledge point 6: Comparison of rational numbers:
The basic principle of rational number size comparison: all positive numbers are greater than zero, all negative numbers are less than zero, and positive numbers are greater than negative numbers.
Comparison of rational numbers on the number axis: the number on the right is always greater than the number on the left of two numbers represented on the number axis.
Comparison between rational number and absolute value: two positive numbers, the positive number with larger absolute value is larger; Two negative numbers, the negative number with larger absolute value is smaller.
Knowledge point 7: rational number addition rule:
(1) Add two numbers with the same symbol, take the same symbol, and add the absolute values;
(2) When two numbers with different signs are added and the absolute values are equal, the sum is 0; When the absolute values are not equal, take the sign of the addend with larger absolute values and subtract the addend with smaller absolute values from the addend with larger absolute values;
(3) Adding a number to 0 still gets this number.
Knowledge point 8: rational number addition algorithm:
Additive commutative law: When two numbers are added, the position of the addend is reversed and the sum remains the same.
Law of addition and association: when three numbers are added, the first two numbers are added first, or the last two numbers are added first, and the sum is unchanged.
Knowledge point 9: rational number subtraction rule: subtracting a number is equal to adding the inverse of this number.
Knowledge point 10: rational number addition and subtraction mixed operation: according to the law of rational number subtraction, all addition and subtraction operations can be unified into addition operations, and then brackets and plus signs are omitted, and calculations are made by using the law of addition and the law of addition operation.