Seven-grade mathematics knowledge points
Axisymmetry in life
1. Axisymmetric graph: If a graph is folded along a straight line and the parts on both sides of the straight line can completely overlap, then this graph is called an axisymmetric graph, and this straight line is called an axis of symmetry.
2. Axisymmetric: For two figures, if they can overlap each other after being folded in half along a straight line, then the two figures are said to be axisymmetric, and this straight line is the axis of symmetry. It can be said that these two figures are symmetrical about a straight line.
3. The difference between an axisymmetric figure and an axisymmetric figure: an axisymmetric figure is a figure, and an axisymmetric figure is the relationship between two figures.
Connection: They are all graphs folded along a straight line and can overlap each other.
2. Two symmetrical figures must be congruent.
3. Two congruent figures are not necessarily symmetrical.
The symmetry axis is a straight line.
5, the nature of the angle bisector
1, the straight line where the bisector of the angle is located is the symmetry axis of the angle.
2. Nature: the distance from the point on the bisector of the angle is equal to both sides of the angle.
6. perpendicular bisector of line segment
1, a straight line perpendicular to a line segment and bisecting the line segment is called the midline of the line segment, also called the midline of the line segment.
2. Property: the distance between the point on the vertical line in the line segment and the two ends of the line segment is equal.
7, axisymmetric graphics are:
Isosceles triangle (1 or 3), isosceles trapezoid (1), rectangle (2), diamond (2), square (4), circle (countless), line segment (1), angle (1), etc.
8, the nature of isosceles triangle:
① The two bottom angles are equal. ② The two sides are equal. 3 "three lines in one". (4) The height on the bottom edge and the line where the bisector of the center line and the vertex is located are its symmetry axis.
9.① Equiangular equilateral ∵∠B=∠C∴AB=AC.
② "equilateral angle" ∵ AB = AC ∴∠ B = ∠ C.
10, angle bisector property:
The point on the bisector of an angle is equal to the distance on both sides of the angle.
∫OA divides equally ∠CADOE⊥AC,OF⊥AD∴OE=OF.
1 1, the nature of the middle vertical line: the distance from the point on the middle vertical line to both ends of the line segment is equal.
∫oc vertically bisects AB∴AC=BC
12, the properties of axial symmetry
1. After two figures are folded in half along a straight line, the points that can overlap are called corresponding points, the line segments that can overlap are called corresponding line segments, and the angles that can overlap are called corresponding angles. 2. Two figures symmetrical about a straight line are congruent figures.
2. If two figures are symmetrical about a straight line, the line segments connected by corresponding points are vertically bisected by the symmetry axis.
3. If two figures are symmetrical about a straight line, then the corresponding line segment and the corresponding angle are equal.
13, mirror symmetry
1. When an object is placed in front of a mirror, the mirror will change its left and right direction;
2. When placed perpendicular to the mirror, the mirror will change its up-and-down direction;
3. If it is an axisymmetric figure, when the symmetry axis is parallel to the mirror, the image in the mirror is the same as the original figure;
Through discussion, students may find the following ways to solve the problem of mutual transformation between objects and images:
(1) Take photos with a mirror (pay attention to the placement of the mirror); (2) Using the axial symmetry property;
(3) Numbers can be reversed left and right, and simple axisymmetric figures can also be made;
(4) You can see the back of the image; (5) Imagine in your mind according to the previous conclusion.
Summary of knowledge points in the second volume of junior one mathematics
Chapter 1: Lines, rays and line segments
(1) Representation methods of lines, rays and line segments
① Straight line: represented by a lowercase letter, such as straight line L, or represented by two uppercase letters, such as straight line AB.
② Ray: a part of a straight line, represented by lowercase letters, such as ray L; It is represented by two capital letters, with the endpoint in front, such as ray OA. Note: When it is represented by two letters, the endpoint letter comes first.
③ Line segment: A line segment is a part of a straight line, which is represented by lowercase letters, such as line segment A; It is represented by two letters representing the endpoint, such as line segment AB (or line segment BA).
(2) The positional relationship between a point and a straight line:
(1) point through a straight line, said the point in a straight line;
(2) The point does not pass through the straight line, which means that the point is outside the straight line.
Chapter 2: the distance between two points
(1) Distance between two points: The length of the line segment connecting two points is called the distance between two points.
(2) There is a certain distance between any two points on the plane, which refers to the length of the line segment connecting these two points. When learning this concept, pay attention to the last two words "length", that is, it is a quantity with size, which is different from a line segment, which is a figure. The length of a line segment is the distance between two points. It can be said that it is a line segment, not a distance.
Chapter 3: Cubes
(1) The general method to solve this kind of problem is to fold the paper according to the diagram, or directly imagine it on the basis of understanding the unfolded diagram.
(2) It is the key to solve this kind of problem to distinguish the geometric expansion diagram from the real object, and to establish the concept of space by combining the transformation between the three-dimensional figure and the plane figure.
(3) There are 1 1 cases in the cubic expansion diagram. After analyzing various situations in the plane expansion diagram, carefully judge which two surfaces are relative.
Chapter 4: Solving one-dimensional linear equations.
Definition: The value of an unknown quantity that makes the left and right sides of a linear equation equal is called the solution of a linear equation.
Substituting the solution of the equation into the original equation, the left and right sides of the equation are equal.
13, solving a linear equation:
1. General steps for solving linear equations with one variable
Removing the denominator, removing brackets, moving terms, merging similar terms, and converting the coefficient into 1 are just the general steps to solve the linear equation with one variable. According to the characteristics of the equation, all the steps are to gradually transform the equation into the form of x = a.
2. When solving a linear equation with one variable, first observe the form and characteristics of the equation. If there is a denominator, generally go to the denominator first; If there are both denominators and brackets, and the denominator can be eliminated after the items outside the brackets are multiplied by the items inside the brackets, the brackets should be removed first.
3. When solving an equation similar to "ax+bx=c", merge the left side of the equation into one term according to the method of merging similar terms, that is, (A+B) x = C.
The equation is gradually transformed into the simplest form of ax=b, which embodies the idea of reduction.
When the coefficient of ax=b is changed to 1, the calculation should be accurate. Once it is clear whether the two sides of the equation are divided by a or b, especially when a is a fraction; Second, we must accurately judge symbols. The same sign X of A and B is positive, and the different sign X of A and B is negative.
The first volume of seventh grade mathematics knowledge points
rational number
★ Classification of rational numbers
1. If divided by definition, rational numbers can be divided into integers (positive integers; Negative integer; 0) and score (positive score, negative score).
If divided by positive and negative, rational numbers can be divided into positive rational numbers (positive integer; Positive fraction), 0, negative rational number (negative integer; Negative score).
2. All rational numbers can be expressed by fractions, and π is not rational.
number axis
★ 1. Definition of number axis: The straight line defining origin, positive direction and unit length is called number axis.
Corresponding thing
1. Only two numbers with different symbols are called reciprocal. (The reciprocal of 0 is 0)
absolute value
The distance from point A to the origin on the 1. number axis represents the absolute value of a. ..
★2. Nature of absolute value: non-negative.
3. The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0.
Size of rational number
1. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers.
2. Two negative numbers, the larger one has the smaller absolute value.
Addition of rational numbers
1. Add two numbers with the same sign, take the same sign, and then add the absolute values.
2. Add the numbers of two different symbols with different absolute values, take the addend symbol 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. Add a number to 0 and you still get the number.
3. In addition to rational numbers,
Addition exchange rate: two numbers are added, and the position of the exchange addend remains unchanged.
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.
Rational number subtraction
Subtracting a number is equal to adding the reciprocal of this number.
★ Multiplication of rational numbers
Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value. Multiply any number by 0 to get 0.
Reciprocal: Two numbers whose product is 1 are reciprocal.
Multiplication exchange law: Multiplication exchange law multiplies two numbers, and the position of exchange factor and product remains unchanged.
Multiplication and association law: when three numbers are multiplied, the first two numbers are multiplied first, or the last two numbers are multiplied first, and the product is unchanged.
Multiplication and distribution law: a number multiplied by the sum of two numbers is equivalent to multiplying this number by these two numbers respectively, and then adding the products.
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