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.
Junior one mathematics knowledge points
Application of one-dimensional linear equation
1. Types of applied problems for solving linear equations of one variable
(1) Explore the problem of regularity;
(2) Quantity;
(3) Sales problem (profit = selling price-purchase price, profit rate = profit purchase price ×100%);
(4) engineering problems (① workload = per capita efficiency × number of people × time; (2) If a job is completed in several stages, the sum of workload in each stage = total workload);
(5) Travel problem (distance = speed × time);
(6) the problem of equivalent transformation;
(7) Sum, difference, multiplication and division;
(8) Distribution problem;
(9) Competition score;
(10) Current navigation problem (downstream speed = still water speed+current speed; Water velocity = still water velocity-water velocity).
2. The basic idea of solving practical problems by using equations:
First, find out the unknown quantity and all known quantities in the problem through examination, set the required unknown quantity as X directly or indirectly, and then use the formula containing X to express the related quantity, find out the equation between them, and solve it to get the answer, that is, set, column, solution and answer.
List five steps of solving application problems by linear equations of one variable.
(1) Examination: Carefully examine the questions, determine the known quantity and the unknown quantity, and find out the equivalent relationship between them.
(2) Assumptions: Assumptions about the unknown (X). According to the actual situation, it can be directly unknown (ask whatever you want) or indirectly unknown.
(3) Column: list the equations according to the equivalence relation.
(4) Solution: Solve the equation to obtain the value of the unknown quantity.
(5) Answer: Check whether the unknown value is correct and write a complete answer.
Math methods and skills in junior one.
1. Please summarize the learning methods.
Yue: "Learning mathematics, like learning other things, requires research methods. The method I recommend to you is: study in advance, develop associations, sum up more and find out what is reasonable.
Please talk about the benefits of studying in advance.
First of all, studying in advance can tap one's own potential and cultivate self-study ability. After studying in advance, you will find that you can solve many problems independently, which is very helpful to improve your self-confidence and cultivate your interest in learning. "
Secondly, it is enough to eliminate the "hidden danger" of new knowledge. Studying in advance can find out what's wrong with your understanding of new knowledge on the existing basis. On the contrary, if you listen to others directly. It seems that I can reach this level of understanding from the beginning. Practice has proved that this is not the case.
Thirdly, some contents in advanced learning were not fully understood at that time, but after careful consideration, even if they were left behind, the brain would subconsciously "process". When the teacher's progress reaches this content, we will have a second understanding, which will be much deeper.
Finally, studying in advance can improve the quality of lectures. After studying in advance, we find that most of the new knowledge is completely understandable. Only a few places need help from others. In this way, you can concentrate on understanding "these places", that is, "good steel is used in the cutting edge". In fact, there is not much time to concentrate in a class.
3. Please talk about association and summary.
Yue: Association and summarization run through the whole process of learning. The understanding of every kind of knowledge must have a cognitive basis. The process of finding cognitive basis is association, and cognitive basis is a summary of previous knowledge. The more concise, clear and reasonable the previous summary, the easier it is to associate. In this way, new knowledge can be integrated into the original knowledge structure, laying the foundation for the next association. Association and summarization are particularly effective in solving problems. Maybe you didn't know this before, but your ability to solve problems is very strong, which shows that you are smart and you used this method unconsciously. If you can clearly understand this, your ability will be stronger.
4. So how do we preview?
Say: "Let's talk about the goal of learning first: (1) Understand the background of knowledge generation and find out the process of knowledge formation.
(2) Know the position and function of knowledge sooner or later: (3) Summarize the laws of cognitive problems (or tell which laws were used in previous cognitive problems).
Let's talk about the specific method first: (1) Understanding of the concept. Mathematics is highly abstract. It is usually understood by concrete things. Sometimes with literal meaning: sometimes with other disciplines. Sometimes it is tacit to understand the field of concepts with the help of graphics. You must try to understand the concept before you do the problem.
(2) Preview the formula theorem, which is a summary of the most used "laws". Such as: complete square formula, Pythagorean theorem, etc. The proof of formula derivation theorem often contains rich mathematical methods and quite useful law of solving problems. Such as the proof of the theorem of bisector of triangle interior angle. You should first deduce the formula or prove the theorem yourself. If you can't do it, you should refer to others' practice. Whether you do it yourself or watch others, you should talk about how you came up with it.
(3) For the treatment of examples and exercises, see (2) above and Article 5 below.
Article on important knowledge points of senior one mathematics in People's Education Edition;
★ The arrangement of mathematics knowledge points in the first day of the People's Education Edition
★ Summary of knowledge points of the first-year mathematics people's education edition
★ People's Education Edition Senior One Mathematics Knowledge Points
★ 202 1 Summary of Mathematics Knowledge Points in Grade One
★ Summary of the knowledge points in the first volume of mathematics in the first grade education edition.
★ People's Education Edition, the first volume of the seventh grade, mathematical knowledge points
★ What are the key knowledge points of junior high school mathematics?
★ 202 1
★ Encyclopedia of seventh grade mathematics knowledge points
★ People's Education Edition, Grade One Mathematics, Book Two, review and summarize knowledge points, and prepare for the senior high school entrance examination.