Summary and induction of mathematics knowledge points in the second day of junior high school
Chapter 1 Pythagorean Theorem
Definition: If the two right-angled sides of a right-angled triangle are A and B, respectively, and the hypotenuse is C, that is, the sum of squares of the two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse.
Judgment: If three sides of a triangle meet the requirements of a +b = c, then the triangle is a right triangle. Definition: Three positive integers satisfying a +b =c are called Pythagorean numbers.
Chapter II Real Numbers
Definition: Any finite decimal or infinite cyclic decimal is a rational number. Infinitely cyclic decimals are called irrational numbers (rational numbers can always be expressed by finite decimals or infinitely cyclic decimals).
Generally speaking, if the square of a positive number X is equal to A, then this positive number X is called the arithmetic square root of A. In particular, we stipulate that the arithmetic square root of 0 is 0.
Generally speaking, if the square of a number x is equal to a, then this number x is called the square root of a (also called quadratic square root), and a positive number has two square roots; 0 has only one square root, which is 0 itself; Negative numbers have no square root. The operation of finding the square root of a number is called square root, where a is called square root.
Generally speaking, if the cube of a number X is equal to A, then this number X is called the cube root of A (also called cube root). The cube root of a positive number is a positive number; The cube root of 0 is 0; The cube root of a negative number is a negative number. The operation of finding the cube root of a number is called square root, where a is called square root. Rational numbers and irrational numbers are collectively called real numbers, that is, real numbers can be divided into rational numbers and irrational numbers.
Every real number can be represented by a point on the number axis; On the contrary, every point on the number axis represents a real number. That is, there is a one-to-one correspondence between real numbers and points on the number axis.
On the number axis, the point on the right represents a larger number than the point on the left.
Chapter III Translation and Rotation of Graphics
Definition: In a plane, a figure moves a certain distance along a certain direction, and such figure movement is called translation. Translation does not change the shape and size of the graph.
After translation, the line segments connected by the corresponding points are parallel and equal; The corresponding line segments are parallel and equal, and the corresponding angles are equal.
In a plane, turning a figure around a fixed point by an angle in a certain direction is called rotation. This fixed point is called the rotation center and the rotation angle is called the rotation angle. Rotation does not change the size and shape of the graph.
The angle formed by the connecting line of any pair of corresponding points and the rotation center is the rotation angle, and the distance between the corresponding points and the rotation center is equal.
The fourth chapter discusses the properties of quadrilateral.
Definition: If two straight lines are parallel to each other, the distance between any two points on one straight line and the other straight line is equal, which is called the distance between parallel lines.
Parallelogram: Two sets of quadrangles with parallel opposite sides. The opposite sides are equal, the diagonal lines are equal, and the diagonal lines are equally divided. Two groups of parallelograms with parallel opposite sides are parallelograms, two groups of parallelograms with equal opposite sides are parallelograms, and one group of parallelograms with parallel and equal opposite sides are parallelograms.
Diamond: A group of parallelograms with equal adjacent sides (the nature of parallelograms). Four sides are equal, two diagonal lines are perpendicular to each other, and each diagonal line bisects a set of diagonal lines. A set of parallelograms with equal adjacent sides is a rhombus, a parallelogram with vertical diagonal is a rhombus, and a quadrilateral with equal four sides is a rhombus.
Rectangle: A parallelogram with a right angle (the nature of a parallelogram). Diagonal lines are equal and all four corners are right angles. A parallelogram with right angles is a rectangle, and a parallelogram with equal diagonal lines is a rectangle.
Square: A group of rectangles with equal adjacent sides. A square has all the properties of parallelogram, rhombus and rectangle. A set of rectangles with equal adjacent sides is a square, and a diamond with right angles is a square.
Trapezoid: a quadrilateral with parallel opposite sides and non-parallel opposite sides. A quadrilateral whose opposite sides are parallel and whose opposite sides are not parallel is a trapezoid. Isosceles trapezoid: A trapezoid with two equal waists. The two internal angles on the same base are equal and the diagonal lines are equal. Two trapezoid with equal waist are isosceles trapezoid,
Two trapeziums with equal internal angles on the same base are isosceles trapeziums.
Right trapezoid: A trapezoid with a vertical waist and bottom. The trapezoid with vertical waist bottom is a right-angled trapezoid.
Polygon: On a plane, a closed figure composed of several line segments that are not on the same line is called a polygon. The sum of the internal angles of an N-polygon is equal to (n-2)× 180.
The angle formed by the extension line with one side of the inner angle of a polygon opposite to the other side is called the outer angle of this polygon. The sum of the outer angles of a polygon is equal to 360 degrees. Triangles, quadrilaterals and hexagons can be densely laid.
Definition: On the plane, a figure rotates around a point 180. If the figures before and after rotation overlap, then this figure is called a central symmetric figure, and this point is called its symmetric center.
The line segments connected by each pair of corresponding points on the central symmetric figure are equally divided by the symmetric center.
Chapter V Determination of Location
Position representation: bearing plus distance; Coordinate; Longitude and latitude
Definition: On a plane, two mutually perpendicular book axes with a common origin form a plane rectangular coordinate system.
Usually, the horizontal position and vertical position of the two number axes are taken as the positive directions of the two number axes respectively. The horizontal axis is called X axis or horizontal axis, the vertical axis is called Y axis or vertical axis, X axis and Y axis are collectively called coordinate axes, and their common origin O is called the origin of rectangular coordinate system.
Graph changes with coordinates: up/down/left/right translation x unit length, horizontal/vertical extension x times, horizontal/vertical compression x times, magnification/reduction x times, axial symmetry about x/y axis and central symmetry about origin o.
Chapter VI Linear Functions
Definition: Generally speaking, there are two variables X and Y in a certain change process. If an X value is given and a Y value is determined accordingly, then we call Y a function of X, where X is the independent variable and Y is the dependent variable.
If the relationship between two variables X and Y can be expressed in the form of y=kx+b(k, b is a constant, k≠0), then Y is a linear function of X (X is an independent variable and Y is a dependent variable). In particular, when b=0, y is said to be a proportional function of x.
Take the values of the independent variable X and the corresponding dependent variable Y of a function as the abscissa and ordinate of a point respectively, and draw its corresponding point in the rectangular coordinate system. A graph composed of all these points is called an image of a function. The image with the proportional function y=kx is a straight line passing through the origin (0,0). In the linear function y=kx+b,
When k>0, the value of increases with the increase of this value; When k < 0, the value of decreases with the increase of the value.
Chapter VII Binary Linear Equations
Definition: An equation with two unknowns whose terms are 1 is called a binary linear equation. An equation group composed of two linear equations with two unknowns like this is called a binary linear equation group. A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation. The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations. The basic idea of solving binary linear equations is "elimination"-changing "binary" into "unitary". The solution of replacing one unknown with another is called substitution elimination method, which is called substitution method for short. The solution of adding and subtracting an unknown number with two formulas is called addition and subtraction, which is called addition and subtraction for short.
Chapter VIII Data Representation
Definition: Generally speaking, for n numbers, X 1, X2,? Xn, let's take 1/n(X 1+X2+? +Xn) is called the arithmetic average of this number, referred to as the average, and is recorded as X.
It is the weighted average of A's three test scores.
Generally speaking, data are arranged in order of size. The data in the middle position (or the average of the two data in the middle) is called the median of this set of data, and the data with the most frequent occurrence of a set of data is called the mode of this set of data.
Expanding reading: a way to improve junior high school mathematics
1, preview before class and listen carefully.
Why do you want to preview this lecture? You need to know what you don't understand at first, so listen to this question carefully in class, so that listening is more targeted, much more efficient than sitting in the classroom, and naturally the final effect is much better.
2. Brush the questions after class and summarize.
To improve math scores, we must brush the questions, and there is no need to talk about methods and skills until the amount of brush questions reaches a certain level. How to brush the questions? In fact, every day's homework is to brush the questions, so we must finish it carefully. If you have more time, you can brush the real questions of previous years. Attention! Be sure to brush the real questions. Brushing the real questions does not mean brushing the whole set. You just brush the questions that are usually deducted. When you sum up all the questions you have brushed, your level will definitely be greatly improved.
3. Ask if you don't understand and eliminate blind spots.
Many students will find a problem, that is, they understand the lecture and do a lot of problems, but they still won't encounter new problems. The fundamental reason why we can't meet new questions is that we can't extrapolate because we don't have a deep understanding of the original knowledge points. What should you do? You should solve the problem you don't understand at the first time. You can ask teachers, classmates, search software, etc. The core purpose is not to leave a blind spot of knowledge, not to leave any doubts, and to solve them at the first time, not to delay or forget.