Summary and induction of mathematics knowledge points in seventh grade
Summary of seventh grade mathematics knowledge points 1
1. rational number:
(1) Any number that can be written in form is a rational number. 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; π is not rational number;
(2) 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;
2. Number axis: The number axis is a straight line that defines the origin, positive direction and unit length.
3. The opposite number:
(1) There are only two numbers with different signs, and we say that one of them is opposite to the other; The antonym of 0 is still 0;
(2) Note: The inverse of a-b+c is-A+B-C; The inverse of a-b is b-a; The inverse of a+b is-a -a-b；;
4. Absolute value:
(1) The absolute value of a positive number is itself, the absolute value of 0 is 0, and the absolute value of a negative number is its inverse; Note: the absolute value means the distance between the point representing a number on the number axis and the origin;
(2) The absolute value can be expressed as:
The problem of absolute value is often discussed in categories;
(3)a| is an important non-negative number, that is | A | ≥ 0; Note: |a|? |b|=|a？ b|，
5. Rational number ratio: (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 seventh grade mathematics knowledge points II
Binary linear equations
1. Binary linear equation: contains two unknowns, and the degree of the unknowns is 1. Such an equation is a binary linear equation. Note: Generally speaking, a binary linear equation has countless solutions.
2. Binary linear equations: two binary linear equations are put together.
3. Solution of binary linear equations: The values of two unknowns that make the left and right sides of two equations of binary linear equations equal are called the solutions of binary linear equations. Note: Generally speaking, binary linear equations have only solutions (that is, common * * * solutions).
4. Solutions of binary linear equations;
(1) substitution elimination method; (2) addition, subtraction and elimination;
(3) Note: It is the key to judge how to solve the problem simply.
5. Application of linear equation. ※:
(1) For an application problem, the more unknowns are set, the easier it may be to establish the equation, but the more troublesome it may be to solve the equation, otherwise it will be difficult to establish the equation and easy to solve.
(2) For equations, if the number of equations is equal to the number of unknowns, the value of the unknowns can generally be obtained;
(3) For equations, if the number of equations is one less than the number of unknowns, then the value of the unknowns cannot be found, but the relationship between any two unknowns can always be found.
One-dimensional linear inequality (group)
1. Inequality: A formula that connects two algebraic expressions with inequality is called inequality.
2. The basic properties of inequality:
The basic property of inequality is 1: add (or subtract) the same number or the same algebraic expression on both sides of inequality, and the direction of inequality remains unchanged;
The basic properties of inequality 2: both sides of inequality are multiplied (or divided) by the same positive number, and the direction of inequality remains unchanged;
The basic property of inequality 3: both sides of inequality are multiplied by (or divided by) the same negative number, and the direction of inequality should be changed.
3. Solution set of inequality: the value of the unknown quantity that can make the inequality hold is called the solution of this inequality; The set of all solutions of an inequality is called the solution set of this inequality.
4. One-dimensional linear inequality: an inequality with only one unknown number, degree 1 and coefficient not equal to zero is called one-dimensional linear inequality; Its standard form is ax+b0 or ax+b0, (a0).
5. Solution of one-dimensional linear inequality: The solution of one-dimensional linear inequality is similar to the solution of one-dimensional linear equation, but we must pay attention to the application of inequality property 3; Note: when representing the solution set of inequality on the number axis, we should pay attention to the empty circle and real point.
Summary of seventh grade mathematics knowledge points 3
Addition and subtraction of algebraic expressions
I. Algebraic expressions
1, the formula that connects numbers or letters representing numbers with operation symbols is called algebraic expression. A single number or letter is also algebraic.
2. Replace the letters in the algebraic expression with numerical values, and the result calculated according to the operational relationship in the algebraic expression is called the value of the algebraic expression.
Second, algebraic expressions.
1, single item:
(1) An algebraic expression consisting of the product of numbers and letters is called a monomial.
(2) The numerical factor in a single item is called the coefficient of the item.
(3) The sum of the indices of all the letters in the monomial is called the number of times of the monomial.
2.polynomial
The sum of (1) monomials is called a polynomial.
(2) Each monomial is called a polynomial term.
(3) Items without letters are called constant items.
3. Ascending order and descending order
(1) Arranging polynomials according to the exponent of x from large to small is called descending power arrangement.
(2) Arranging polynomials according to the exponent of x from small to large is called ascending power arrangement.
Third, the addition and subtraction of algebraic expressions
The theoretical basis of 1. Algebraic addition and subtraction is: the rule of removing brackets, the rule of merging similar items, and the multiplication distribution rate.
Rules for removing brackets: If there is a "ten" in front of brackets, remove brackets and the "+"in front of them, and all items in brackets will remain unchanged; If there is a "one" in front of the bracket, remove the bracket and the "one" in front, and change the symbols of everything in the bracket.
2. Similar items: items with the same letters and the same letter index are called similar items.
Merge similar projects:
(1) The concept of merging similar terms: merging similar terms in polynomials into one term is called merging similar terms.
(2) Rules for merging similar items: when the coefficients of similar items are added, the result will be taken as the coefficient, and the index of letters will remain unchanged.
(3) Steps to merge similar projects:
A. find similar projects accurately.
B. Reverse the distribution law, and add the coefficients of similar items together (with brackets) to keep the letters and the indexes of letters unchanged.
C. write the results after the merger.
(4) Note:
A. If the coefficients of two similar items are opposite, the result after merging similar items is 0.
B. Don't leave out items that can't be merged.
C. As long as there are no more similar terms, it is the result (which may be a single term or a polynomial).
Note: The key to merging similar items is to correctly judge similar items.
3, several general steps of algebraic expression addition and subtraction:
(1) List algebraic expressions: enclose each algebraic expression in parentheses and then connect it with a plus sign and a minus sign.
(2) Open brackets according to the rules for opening brackets.
(3) Merge similar items.
4, the general steps of algebraic evaluation:
Algebraic simplification of (1)
(2) Substitution calculation
(3) For some special algebraic expressions, "whole substitution" can be used for calculation.
A preliminary understanding of graphics
A, three-dimensional graphics and plane graphics
1, cuboids, cubes, spheres, cylinders and cones are all three-dimensional figures. In addition, prisms and pyramids are also common three-dimensional figures.
2. Rectangular, square, triangle and circle are all plane figures.
3. Many three-dimensional graphics are surrounded by some plane graphics, which can be expanded into plane graphics by proper cutting.
Second, points and lines
1, there is a straight line after two, and there is only one straight line.
2. The line segment between two points is the shortest.
3. The line segment AB at point C is divided into two equal line segments AM and MB, and point M is called the midpoint of line segment AB. Similarly, line segments have bisectors and quartiles.
4. The figure formed by the infinite extension of line segments in one direction is called ray.
Third, the angle
The 1. angle is a graph composed of two rays with a common endpoint.
2. Rotate around the endpoint until the end edge and the start edge of the angle form a straight line, and the formed angle is called a flat angle.
3. Rotate around the endpoint until the ending edge and the starting edge overlap again, and the angle formed is called fillet.
4. Degrees, minutes and seconds are commonly used units of angle measurement.
Divide a fillet into 360 equal parts, each equal part is an angle of one degree, and record it as1; Divide the angle of 1 degree into 60 equal parts, each part is called an angle of 1 minute, and it is recorded as1'; Divide the angle of 1 into 60 equal parts, and each part is called 1 sec, and it is recorded as 1 ".
Fourth, the comparison of angles.
Starting from the vertex of an angle, the ray that divides the angle into two equal angles is called the bisector of the angle. Similarly, there is the so-called bisector.
Verb (abbreviation for verb) complementary angle and complementary angle
1. If the sum of two angles is equal to 90 degrees (right angle), the two angles are said to be complementary.
2. If the sum of two angles is equal to 180 (flat angle), it is said that the two angles are complementary.
3. The complementary angles of equal angles are equal.
4. The complementary angles of equal angles are equal.
Six, the intersection line
1, definition: When two straight lines intersect and one of the four angles formed is a right angle, then the two straight lines are perpendicular to each other. One of the straight lines is called the perpendicular of the other straight line, and their intersection point is called the vertical foot.
2. Note:
(1) The vertical line is a straight line.
⑵ The four angles formed by two straight lines with vertical relationship are all 90.
(3) Verticality is a special case of intersection.
(4) Vertical symbols: a⊥b, AB⊥CD.
3. Draw a known straight line with countless vertical lines.
4. There is one and only one straight line perpendicular to the known straight line.
5. Of all the line segments connecting points outside the straight line and points on the straight line, the vertical line segment is the shortest. Simply put: the vertical line is the shortest.
6. The length from a point outside a straight line to the vertical section of the straight line is called the distance from the point to the straight line.
7. One vertex has a common * * *, one side has a common * * *, and the other side is an extension line opposite to each other. Such two angles are called adjacent complementary angles.
There are four pairs of adjacent complementary angles when two straight lines intersect.
8. One vertex has a common * * *, and both sides of the corner are opposite extension lines. These two angles are called antipodal angles. Two straight lines intersect and have two opposite angles. The vertex angles are equal.
Seven, parallel lines
1. In the same plane, if two straight lines have no intersection, they are parallel to each other, and it is recorded as: a ∨ b.
2. Parallelism axiom: After passing a point outside a straight line, there is one and only one straight line parallel to this straight line.
3. If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other.
4, determine the method of two straight lines parallel:
(1) Two straight lines are cut by the third straight line. If congruent angles are equal, two straight lines are parallel. To put it simply: the same angle is equal and two straight lines are parallel.
(2) Two straight lines are cut by a third straight line. If the internal dislocation angles are equal, two straight lines are parallel. To put it simply: the internal dislocation angles are equal and the two straight lines are parallel.
(3) Two straight lines are cut by a third straight line. If they are complementary to each other, the two straight lines are parallel. To put it simply: the internal angles on the same side are complementary and the two straight lines are parallel.
5, the nature of parallel lines
(1) Two parallel lines are cut by a third straight line and have the same angle. To put it simply: two straight lines are parallel and have the same angle.
(2) Two parallel lines are cut by a third line, and the internal dislocation angles are equal. To put it simply: two straight lines are parallel and their internal angles are equal.
(3) The two parallel lines are cut by the third straight line and complement each other. Simply put, two straight lines are parallel and complementary.