Rational Numbers in Chapter 1 of Directory
Chapter II Addition and subtraction of algebraic expressions
Chapter III One-variable Linear Equation
Chapter IV Preliminary Geometry
Chapter 1 Rational Numbers 1. 1 Positive Numbers and Negative Numbers
① Positive Numbers: Numbers greater than 0 are called positive numbers. (When necessary, sometimes "+"is added before the positive number. )
② Negative number: The previously learned numbers other than 0 with a negative sign "-"in front of them are called negative numbers. It has the opposite meaning to a positive number.
③0 is neither positive nor negative. 0 is the boundary between positive and negative numbers and is the only neutral number.
Note: Find out the quantities with opposite meanings: North and South; Things; Up and down; Left and right; Rise and fall; High and low; Growth decline, etc.
1.2 rational number
1, rational number (1) integer: positive integers, 0 and negative integers are collectively called integers; (2) scores; Positive score and negative score are collectively called scores;
(3) Rational Numbers: Integers and fractions are collectively called rational numbers.
2. Definition of number axis (1): Numbers are usually represented by points on a straight line, which is called number axis;
(2) Three elements of the number axis: origin, positive direction and unit length;
(3) Origin: Take any point on the straight line to represent the number 0, and this point is called the origin;
(4) Relationship between points on the number axis and rational numbers: All rational numbers can be represented by points on the number axis, but not all points on the number axis represent rational numbers.
3. Antiquities: Only two numbers with different symbols are called reciprocal. (Example: the reciprocal of 2 is-2; The reciprocal of 0 is 0)
4. Absolute value: (1) The distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A, and it is recorded as |a|. Geometrically, the absolute value of a number is the distance between two points.
(2) The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0. Two negative numbers, the larger one has the smaller absolute value.
Addition and subtraction of rational number 1.3
(1) rational number addition rule:
1, two numbers with the same sign are added, the same sign is taken, and the absolute values are added.
2. Add two different symbols with different absolute values, take the symbol of the addend with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value. Two opposite numbers add up to 0.
When a number is added to 0, you still get this number.
Commutative law and additive associative law
2 rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number.
Multiplication and division of rational number 1.4
(1) rational number multiplication rule: 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 to get 0;
Two numbers whose product is 1 are reciprocal.
Multiplicative commutative law/associative law/distributive law
2 rational number division rule: dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number;
Divide two numbers, the same sign is positive and the different sign is negative, and divide by the absolute value;
Divide 0 by any number that is not equal to 0 to get 0.
1.5 power of rational number
1, the operation of finding the product of n identical factors is called power, and the result of power is called power. In the n power of a, a is called the base and n is called the exponent. The odd power of a negative number is negative and the even power of a negative number is positive. Any power of a positive number is a positive number, and any power of 0 is 0.
2. Mixed arithmetic of rational numbers: multiply first, then divide, and finally add and subtract; Operation at the same level, from left to right; If there are brackets, do the operation in brackets first, and then follow the brackets, brackets and braces in turn.
3. Using scientific counting method, the number greater than 10 is expressed as the n power of a× 10. Note that the range of a is1≤ a.
Chapter II Addition and subtraction of algebraic expressions 2. 1 algebraic expressions
1, monomial: a formula consisting of the product of numbers and letters. Coefficient, degree of monomial. A monomial is an algebraic expression of the product of numbers or letters. A single number or letter is also a monomial. Therefore, the key to judge whether an algebraic expression is a monomial depends on whether the numbers and letters in the algebraic expression are products, that is, the denominator does not contain letters. If the expression contains addition and subtraction, it is not a monomial.
2. Single factor: refers to the numerical factor in a single item;
3. Number of monomials: refers to the sum of the indices of all the letters in the monomials.
4. Polynomial: the sum of several monomials. The key to judge whether an algebraic expression is a polynomial is whether each term in the algebraic expression is a monomial. The degree of each monomial, constant term and polynomial is the highest degree among polynomials. The degree of polynomial refers to the degree of the highest term in polynomial, here is the highest term, and its degree is 6; Polynomial term refers to each monomial in polynomial. Pay special attention to the polynomial term, including the attribute symbol before it.
5. Numbers are represented by letters or quantitative relations are represented by columns. Please note that each term of monomial and polynomial is preceded by a symbol.
6. Monomial and polynomial are collectively called algebraic expressions.
2.2 Addition and subtraction of algebraic expressions
1. Similar items: items with the same letters and the same index. It has nothing to do with the coefficient before the letter (≠0).
2. Similar items must meet two conditions at the same time: (1) contains the same letters; (2) the same number of letters, both are indispensable. Similar terms have nothing to do with coefficient size and alphabetical order.
3. Merge similar items: Merge similar items in polynomials into one item. Reduction of criminal law, combination method and distribution method are all acceptable.
4. Rules for merging similar items: after merging similar items, the coefficients of the obtained items are the sum of the coefficients of similar items before merging, and the letter part remains unchanged;
5, the law of brackets: brackets, look at the symbol: it is a plus sign, the same sign; This is a negative sign and a positive sign.
6. The general steps of algebraic expression addition and subtraction:
One trip, two searches, three combinations.
(1) If you encounter brackets, first remove the brackets according to the rules for removing brackets. (2) Combine similar projects. (3) Merge similar items.
Chapter III One-Dimensional Linear Equation 3. 1 One-Dimensional Linear Equation
1, the equation is an unknown equation.
2. The equations all contain only one unknown (element) X, and the exponent of the unknown X is 1 (degree). Such an equation is called a linear equation with one variable.
Note: To judge whether an equation is a linear equation, we should grasp three points:
1) The formula of unknown quantity is algebraic expression (equation is integral equation);
2) The simplified equation contains only one unknown number;
3) The number of unknowns in the ranking equation is 1.
3. Solving the equation is to find the value of the unknown quantity that makes the left and right sides of the equation equal, and this value is the solution of the equation.
4. Properties of the equation: 1) Adding (or subtracting) the same number (or formula) on both sides of the equation at the same time, the results are still equal;
2) Both sides of the equation are multiplied by the same number at the same time, or divided by the same number that is not 0, and the results are still equal.
Note: When using nature, be sure to pay attention to the simultaneous changes on both sides of the equal sign; We must pay attention to the number 0 when using property 2.
3.2, 3.3 Solving linear equations with one variable
In the actual process of solving equations, the following steps may not be fully used, and some steps need to be used repeatedly. Therefore, we should pay attention to the following points when solving the equation:
(1) Denominator removal: multiply both sides of the equation by the least common multiple of each denominator, and do not omit items without denominator; The numerator is a whole, and brackets should be added after removing the denominator; Denominator removal and denominator rounding are two concepts that cannot be confused;
(2) Remove brackets: first remove brackets, then remove brackets, and finally remove braces; Don't omit the items in brackets; Don't get the symbols wrong;
③ Shift the term: move the term containing the unknown to one side of the equation, and all other terms to the other side of the equation (shift the term to change the sign);
(4) Merging similar terms: don't lose terms, the solution equation is homomorphic deformation, each step is an equation, and it can't be written in the form of energy equivalence like calculation or simplification;
⑤ The coefficient is changed to 1: the letter and its exponent invariant coefficient are changed to 1, and the unknown coefficient A on both sides of the equation is divided to get the solution of the equation. Don't confuse numerator and denominator.
3.4 Practical problems and linear equations of one variable
I. Conceptual carding
(1) The general steps of making a linear equation to solve practical problems are as follows: (1) Examining questions, paying special attention to keywords and their meanings, and finding out the relevant quantitative relations; ② Set the unknown number (focus unit); ③ List the equations according to the equality relation; ④ Solve this equation; ⑤ Test and write the answer (including the company name).
⑵ Equivalence relations and typical examples in some fixed models refer to the application of linear equations with one variable.
Second, the way of thinking (summarize the mathematical thinking methods commonly used in this unit)
(1) modeling idea: through the analysis of the quantitative relationship in practical problems, abstract it into a mathematical model and establish a linear equation.
⑵ Equation thought: The idea of solving practical problems with equations is equation thought.
⑶ Transformation idea: In essence, the process of solving a linear equation with one variable is to transform the unknown coefficient into 1 by removing the denominator, brackets, shifting terms and merging similar terms, constantly replacing the original equation with new and simpler equations, and finally gradually transforming the equation into the form of x=a, which embodies the transformation idea of turning "unknown" into "known".
⑷ The idea of combining numbers with shapes: When solving problems with a series of equations, the quantitative relationship is analyzed with the help of line segments and charts, so that the quantitative relationship in the problem can be displayed intuitively, which embodies the superiority of combining numbers with shapes.
5. Classification idea: In the process of solving the letter coefficient equation and the absolute value symbol equation, classification discussion is often needed, and in the process of solving practical problems related to scheme design, attention should also be paid to the application of classification idea in the process.
Third, the research of mathematical thinking method.
1. When solving a linear equation with one variable, you need to know what deformation each step has made and what problems you need to pay attention to.
2. When looking for the quantitative relationship of practical problems, we should be good at using intuitive analysis methods, such as tabular method, linear analysis method and graphic analysis method.
3. There are two aspects in the test of the application problem of solving the column equation: (1) Whether the test result is the solution of the equation;
⑵ is to judge whether the solution of the equation conforms to the actual meaning in the topic.
Fourth, application (common equivalence relation)
Trip problem: s=v×t
Engineering problem: total workload = work efficiency × time.
Profit and loss problem: Profit = price-cost.
Interest rate = profit/cost × 100%
Price = list price × discount quantity × 10%
Savings profit: interest = principal× interest rate× time.
Sum of principal and interest = principal+interest
The fourth chapter geometry preliminary 4. 1 geometry
1, geometric figure: the figure obtained by objects of various shapes is called geometric figure.
2. Three-dimensional figures: All parts of these geometric figures are not on the same plane.
3. Plane figures: All parts of these geometric figures are on the same plane.
4. Although solid figure and plane figure are two different geometric figures, they are interrelated.
Some parts of three-dimensional graphics are plane graphics.
5. Three views: left view, front view and top view.
6. Development diagram: Some three-dimensional figures are surrounded by some plane figures, and they can be developed into plane figures by cutting their surfaces properly. Such a plane figure is called the expanded figure of the corresponding three-dimensional figure.
7, (1) referred to as geometry; What surrounds the body is the surface; Faces intersect to form a line; Lines intersect to form points;
2 points without size, line and surface straight;
(3) Geometry consists of points, lines, surfaces and bodies;
(4) Slowly moving into a line, the line moves into a surface, and the surface moves into a main body;
5] Point: It is the basic element of geometry.
4.2 Lines, rays and line segments
1, axiom of straight line: there is a straight line after two points, and there is only one straight line. That is, two points determine a straight line.
2. When two different straight lines have a common point, we say that the two straight lines intersect, and this common point is called their intersection.
3. The point where a line segment is divided into two equal line segments is called the midpoint of this line segment.
4. Axiom of line segment: Of all the connecting lines between two points, the line segment is the shortest (between two points, the line segment is the shortest).
5. The length of the line segment connecting two points is called the distance between these two points.
6. Representation of straight line: The straight line as shown in the figure can be recorded as straight line AB or straight line M. 。
(1) To describe the right picture in geometric language, we can say:
Point P is outside the straight line AB, and points A and B are on the straight line AB.
(2) As shown in the figure, the point O is on both the straight line M and the straight line N, which we call a straight line.
M and n intersect and the intersection point is o.
7. Take the point o on the straight line, divide the straight line into two parts, remove one part of one side, and keep the point 0 and the other part to get a ray, as shown in the figure, which is a ray and recorded as ray OM or ray A. Huludao Yingba Education Alliance/18342389605.
Note: Light has an endpoint and extends infinitely in one direction.
8. Take two points A and B on a straight line, divide the straight line into three parts, remove the parts on both sides, keep points A and B and a part in the middle, and get a line segment. The graph is a line segment, marked as line segment AB or line segment A. 。
Note: A line segment has two endpoints.
4.3 jiao
1. Definition of angle: A graph composed of two rays with a common endpoint is called an angle. The common endpoint is the vertex of the angle, and the two rays are the two sides of the angle. As shown in the figure, the vertex of the angle is O, and the two sides are rays OA and OB respectively.
2. The angle has the following expressions:
① It is represented by three capital letters and the symbol "∞". The three capital letters are the vertex and any point on both sides, and the letter of the vertex must be written in the middle. The corner as shown above can be written as ∠AOB or ∠BOA.
(2) Expressed in capital letters. This letter is the vertex. The angle shown above can be expressed as ∠ O, and when two or more angles are the same vertex, they cannot be expressed in capital letters.
(3) represented by numbers or Greek letters. In the corner, near the vertex of the corner.
Draw an arc and write Greek letters or numbers. The two corners as shown in the figure are marked as ∠ and ∠ 1 respectively.
2. The angle measuring system in degrees, minutes and seconds is called an angle system. The degrees, minutes and seconds of an angle are all in hexadecimal.
1 degree =60 minutes 1 minute =60 seconds 1 fillet =360 degrees 1 flat angle = 180 degrees.
3. bisector of an angle: Generally speaking, starting from the vertex of an angle, the ray that divides the angle into two equal angles is called the bisector of this angle.
4. If the sum of two angles is equal to 90 degrees (right angle), they are called complementary angles, that is, each angle is the complementary angle of another angle;
If the sum of two angles is equal to 180 degrees (flat angle), the two angles are said to be complementary angles, that is, each angle is the complementary angle of the other angle.
5. The complementary angles of the same angle (equal angle) are equal; The complementary angles of the same angle (equal angle) are equal.
6. Orientation: Generally speaking, the direction of object movement is described based on due south and due north.
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