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.
Definition of knowledge point 2 1 and single item in senior one math book;
A formula consisting of the product of numbers or letters is called a monomial.
Note: A single number or a single letter is also a monomial.
2. Single coefficient:
The numerical factor in a single item is called the coefficient of the item.
Note: (1) The single coefficient can be integer, fraction or decimal. For example, the coefficient of 3x is 32 of 3.
The coefficient is1; The coefficient of 4.8a is 4.8; three
⑵ The single coefficient can be positive or negative. To determine the coefficient of the monomial, pay attention to the symbols contained in front of it.
? The coefficient of 4xy2 is 4; The coefficient of 2x2y is 4;
(3) For a monomial with only letter factor, its coefficient is 1 or-1, which cannot be considered as 0, such as? antibody
The coefficient is-1; The coefficient of ab is1;
(4) π, which stands for pi, is a fixed constant in mathematics. When it appears in the monomial, it should be regarded as a part of the coefficient, not a letter. For example, the coefficient of 2πxy is 2.
3, the number of monomials:
In a monomial, the sum of the exponents of all the letters is called the degree of the monomial.
Note: (1) When calculating the number of monomials, pay attention to the exponential sum of all letters, and don't forget that the exponent of letters is 1.
Situation. For example, the degree of the monomial 2xyz is the exponential sum of the letters z, y and x, that is, 4+3+ 1=8,
Instead of 7 times, it should be noted that the exponent of the letter Z is 1 instead of 0;
⑵ Individual indicators are only related to letter indicators, but have nothing to do with coefficient indicators.
(3) When the monomial is a single letter, its index is 1. For example, when the exponent of the monomial m is 1 and the monomial is a single constant, it is generally not discussed.
4. If the multiplication sign appears in the formula containing letters, it is usually written as "*" or omitted.
5. When writing a monomial, the number factor is written in front of the letter factor, and the number factor is converted into a false score when scoring.
Knowledge points of math book 3 in grade one. Grasp scientific knowledge as soon as possible and improve learning ability quickly. The editing teacher provides you with the knowledge points of the first grade mathematics in the new semester, hoping to bring you inspiration!
I. Objectives and requirements
1. By dealing with practical problems, it is an improvement for students to experience algebraic methods from arithmetic methods;
2. Learn how to find the equation relationship in the problem, list the equations, and understand the concept of the equations;
3. Cultivate students' ability to obtain information, analyze and deal with problems.
Second, the main points
Seeking equality from practical problems;
Establish the thinking method of solving practical problems with column equations, learn to merge similar terms, and solve ax+bx=c linear equations.
Third, difficulties.
Seeking equality from practical problems;
Analyze the known and unknown quantities in practical problems, find out the equal relationship and list the equations, so that students can gradually establish a thinking method to solve practical problems by listing the equations.
Fourth, the summary of knowledge points and concepts
1. One-dimensional linear equation: an integral equation with only one unknown number, whose degree is 1, and whose' coefficient is not zero' is a one-dimensional linear equation.
2. The standard form of one-dimensional linear equation: ax+b=0(x is unknown, A and B are known numbers, a0).
3. Conditions: A linear equation with one variable must meet four conditions at the same time:
(1) It is an equation;
(2) There are no unknowns in the denominator;
(3) The highest unknown term is1;
(4) The coefficient of the term with unknown number is not 0.
4. The nature of the equation:
Properties of the equation 1: When a number is added to both sides of the equation or the same number or the same algebraic expression is subtracted, the equation is still valid.
Property 2 of the equation: both sides of the equation expand or contract at the same time by the same multiple (except 0), and the equation still holds.
Property 3 of the equation: When both sides of the equation are multiplied (or squared) at the same time, the equation still holds.
Solving the equation is based on these three properties of the equation. One of the properties of the equation: adding a number or subtracting the same number on both sides of the equation at the same time, the equation still holds.
5. Merge similar projects
(1) Basis: Multiplication and Distribution Law
(2) Combine the unknowns with the same number of times into one item; Constants are calculated and combined into one term.
(3) When merging, the number of times is unchanged, but the coefficient is added and subtracted.
Step 6 move the project
(1) After the sign change, the term with unknown number is moved to the left of the equation, while the term without unknown number is moved to the right.
(2) Basis: the nature of the equation
(3) When you move an item from one side of the equation to the other, you must change the sign.
7. General steps to solve one-dimensional linear equation:
The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.
Universal solution:
(1) denominator: both sides of the equation are multiplied by the least common multiple of each denominator;
(2) bracket removal: first remove the bracket, then remove the bracket, and finally remove the braces; (If there is a minus sign outside the brackets, please remember to change the sign.)
(3) Moving terms: moving all the terms containing unknowns to one side of the equation and all other terms to the other side of the equation; Move the item to change the symbol.
(4) merging similar terms: transforming the equation into the form of ax=b(a0);
(5) Coefficient division 1: divide the unknown coefficient a on both sides of the equation to get the solution of equation x = b/a. 。
8. Homotopy equation
If the solutions of two equations are the same, then these two equations are called homosolution equations.
9. The same solution principle of the equation:
Add or subtract the same number or the same equation from both sides of the (1) equation, and the obtained equation is the same solution equation as the original equation.
(2) The equation obtained by multiplying or dividing the same number whose two sides are not zero is the same as the original equation.
Edit the mathematics knowledge points provided by the teacher in the new semester of grade one, hoping to bring you inspiration!
The first grade math book has 4 positive and negative knowledge points.
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 five knowledge points and concepts in the first grade math book.
1. Inequality: Use the symbol "
2. Classification of inequalities: Inequalities are divided into strict inequalities and non-strict inequalities.
Generally speaking, connected inequalities using pure greater than signs and less than signs ">" and "< are called strict inequalities, and inequalities connected by not less than signs (greater than or equal to signs), not greater than signs (less than or equal to signs)," ≥ "and" ≤ "are called non-strict inequalities or generalized inequalities.
3. Solution of inequality: The value of the unknown quantity that makes inequality valid is called the solution of inequality.
4. Solution set of inequality: All solutions of an unknown inequality constitute the solution set of this inequality.
5. Representation method of inequality solution set;
(1) is expressed by inequality: generally, an inequality with unknowns has countless solutions, and its solution set is a range, which can be expressed by the simplest inequality. For example, the solution set of x- 1≤2 is x≤3.
(2) Expressed on the number axis: The solution set of inequality can be intuitively expressed on the number axis, which vividly shows that inequality has infinite solutions. Two points should be paid attention to when expressing the solution set of inequality with the number axis: first, the boundary line should be fixed; The second is to set the direction.
6. Some of the same problem-solving principles that can be followed when solving inequalities.
The identical solution of (1) inequality F(x) F(x).
(2) If the inequality f (x)
(3) If the inequality F(x) 0, then the inequality F(x)
7. The nature of inequality:
(1) If x >；; Y, then YY; (symmetry)
(2) If x>y, y & gtz；; Then x & gtz；; (transitivity)
(3) if x>y and z is any real number or algebraic expression, then x+z >; y+z； (plus rule)
(4) If x>y, z>0, then xz & gtyz；; If x>y, z<0, and then xz.
(5) if x>y, z>0, then x ÷ z >; y \u z； If x>y, z<0, and then x \z
(6) If x>y, m>n, then X+M > Y+n (sufficient and unnecessary conditions)
(7) If x>y>0, m>n>0, then xm & gtyn
(8) if x>y>0, then the n power of x >; The n power of y (n is a positive number)
8. One-dimensional linear inequality: the left and right sides of the inequality are algebraic expressions, and there is only one unknown, and the highest order of the unknown is 1. Inequalities like this are called one-dimensional linear inequalities.
9. The general order of solving one-dimensional linear inequality:
(1) denominator (using inequality properties 2 and 3)
(2) Dismantle the bracket
(3) Shift term (using inequality property 1)
(4) merging similar projects
(5) Transform the unknown coefficient into 1 (using inequality properties 2 and 3).
(6) Sometimes it is necessary to express the solution set of inequality on the number axis.
10. Comprehensive application of linear inequality and linear function;
Generally, the function expression is obtained first, and then the inequality is simplified.
1 1. One-dimensional linear inequality group: Generally speaking, it is a combination of several one-dimensional linear inequalities about the same unknown quantity.
A set of one-dimensional linear inequalities is established.
12. Steps to solve a set of linear inequalities:
(1) Find the solution set of each inequality;
(2) Find the common part of each inequality solution set; (Generally, several axes are used)
(3) The public part is expressed in algebraic symbolic language. (it can also be said that it is a conclusion)
13. tips for solving inequality
(1) is greater than the maximum (much larger);
For example: X>- 1, X>2. The solution set of the inequality group is X>2.
(2) less than the minimum (small and small);
For example: x
(3) Crossing the middle is greater than or less than;
(4) There is no solution to the part that is not disclosed;
14. Formulas for solving inequality groups
(1) Maximize the same size.
For example, the solution set of x>2, x>3. Inequality group is X>3.
(2) Take the small as the big.
For example, x
(3) Find the middle between big and small.
For example, x
(4) Keep the change, more or less.
For example, x
15. Steps to solve practical problems by applying inequality groups
(1) Check the meaning of the problem
(2) Set unknowns, and list inequality groups according to the set unknowns.
(3) Solving inequality groups
(4) The solutions of practical problems are established by the solutions of inequality groups.
(5) Answer
16. Solving practical problems with inequality groups: its general solution is not necessarily the solution of practical problems, but should be combined with concrete analysis of real life to finally determine the result.