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.
Knowledge points of triangle in junior high school mathematics
I. Objectives and requirements
1. Know the triangle, know the meaning of the triangle, know the sides, internal angles and vertices of the triangle, and express the triangle in symbolic language.
2. Experience the practical activities of measuring the side length of a triangle and understand the unequal relationship among the three sides of the triangle.
3. Know how to judge whether three line segments can form a triangle, and use it to solve related problems.
4. The interior angle theorem of triangle can be deduced from the properties of parallel lines.
5. Some simple practical problems can be solved by applying the triangle interior angle sum theorem.
Second, the main points
Theorem of sum of interior angles of triangle;
In order to understand the concept of triangle, three bars can be expressed in symbolic language.
Third, difficulties.
The reasoning process of triangle interior angle sum theorem;
Identify all triangles without repetition or omission in specific graphics;
Judging whether three line segments can form a triangle by the unequal relationship of three sides of a triangle.
Fourth, the knowledge framework.
Verb (abbreviation of verb) summary of knowledge points and concepts
1. triangle: A figure composed of three line segments that are not on the same line and are connected end to end is called a triangle.
2. Classification of triangles
3. Trilateral relationship of triangle: the sum of any two sides of triangle is greater than the third side, and the difference between any two sides is less than the third side.
4. Height: Draw a vertical line from the vertex of the triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.
5. midline: in a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the midline of the triangle.
6. Angular bisector: The bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the intersection of the vertex and this angle is called the angular bisector of the triangle.
7. Significance and practice of high line, middle line and angle bisector.
8. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.
9. Theorem of the sum of interior angles of triangle: the sum of three interior angles of triangle is equal to 180.
It is inferred that the two acute angles of 1 right triangle are complementary;
Inference 2: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles;
Inference 3: One outer angle of a triangle is larger than any inner angle that is not adjacent to it;
The sum of the inner angles of a triangle is half of the sum of the outer angles.
10. External angle of triangle: the included angle between one side of triangle and the extension line of the other side is called the external angle of triangle.
1 1. The Properties of the Exterior Angle of Triangle
(1) Vertex is the vertex of a triangle, one side is one side of the triangle, and the other side is the extension line of one side of the triangle;
(2) An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it;
(3) The outer angle of a triangle is greater than any inner angle that is not adjacent to it;
(4) The sum of the external angles of the triangle is 360.
12. Polygon: On the plane, a figure composed of end-to-end line segments is called a polygon.
13. Interior angle of polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.
14. Exterior angle of polygon: the angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.
15. Diagonal line of polygon: the line segment connecting two non-adjacent vertices of polygon is called diagonal line of polygon.
16. Classification of polygons: it can be divided into convex polygons and concave polygons. Convex polygons can also be called plane polygons and concave polygons can also be called space polygons. Polygons can also be divided into regular polygons and non-regular polygons. Regular polygons have equal sides and equal internal angles.
17. Regular polygon: A polygon with equal angles and sides in a plane is called a regular polygon.
18. plane mosaic: covering a part of a plane with some non-overlapping polygons is called covering the plane with polygons.
The seventh grade mathematical formula daquan (next semester)
1 x number of shares per share = total number of shares ÷ number of shares = 2 1 multiple × multiple ÷ 1 multiple = multiple ÷ multiple = 1 multiple 3 speed × time = distance \
5 working efficiency × working time = total workload ÷ working efficiency = working time ÷ total workload ÷ working time = working efficiency 6 addend+addend = sum-one addend = another addend 7 minuend-subtree = difference minuend-difference = subtree+subtree = minuend 8 factor × factor = product ÷.
C perimeter s area a side length perimeter = side length ×4 C=4a
Area = side length × side length S=a×a 2 cubic v: volume a: side length surface area = side length× side length× ×6 S Table =a×a×6
Volume = side length × side length × side length V=a×a×a 3 rectangle
C perimeter s area A side length perimeter = (length+width) ×2 C=2(a+b) area = length× width S=ab 4 cuboid
V: volume s: area a: length b: width h: height (1) surface area (length× width+length× height+width× height )× 2s = 2 (AB+AH+BH) (2) volume = length× width× height V=abh 5 triangle S area A bottom H high area.
Triangle height = area × 2 ÷ base Triangle base = area× 2 ÷ height 6 parallelogram S area A base H height area = base× height s=ah 7 trapezoid
S area A upper bottom B lower bottom H height area = (upper bottom+lower bottom) × height ÷2 s=(a+b)× h÷2 8 circle.
S area c perimeter ∏ d= diameter r= radius (1) perimeter = diameter x ∏ = 2 x ∏× radius c = ∏ d = 2 ∏ r.
(2) Cylinder with area = radius × radius ×∏ 9
V: volume h: height s; Bottom area r: bottom radius c: bottom perimeter
(1) lateral area = perimeter of bottom× height (2) surface area = lateral area+bottom area× 2 (3) volume = bottom area× height (4) volume = lateral area ÷2× cone with radius of 10.
V: volume h: height s; Bottom area r: bottom radius
Volume = bottom area × height ÷3 Total number ÷ Total number of copies = formula (sum+difference) ÷2= large number (sum-difference) ÷2= decimal and multiple problems.
Sum ÷ (multiple-1)= decimal × multiple = large number (or sum- decimal = large number) difference multiple problem.
Difference ÷ (multiple-1)= decimal× multiple = large number (or decimal+difference = large number) tree planting problem
1 The problem of planting trees on unclosed lines can be divided into the following three situations:
(1) If trees are to be planted at both ends of the non-closed line, then: number of plants = number of segments+1= total length ÷ plant spacing-1 = total length ÷ (number of plants-1).
2 If you want to plant trees at one end of the unclosed line and not at the other end, then:
Number of plants = number of segments = total length/plant spacing = plant spacing × plant number = total length/plant number
(3) If no trees are planted at both ends of the non-closed line, then:
Number of plants = number of nodes-1= total length ÷ plant spacing-1 total length = plant spacing × (number of plants+1) plant spacing = total length ÷ (number of plants+1)
2 The quantitative relationship of planting trees on the closed line is as follows: number of trees = number of segments = total length ÷ plant spacing = plant spacing × number of trees = total length ÷ number of trees profit and loss problem.
(Profit+Loss) ÷ Difference between two distributions = number of shares participating in distribution.
(Big profit-small profit) ÷ Difference between two distributions = number of shares participating in distribution.
(big loss-small loss) ÷ The difference between the two distributions = the number of shares participating in the distribution meets the problem.
Encounter distance = speed and x Meet time = Meet distance ÷ Sum of speed and speed = Meet distance ÷ Meet time tracking problem.
Catch-up distance = speed difference × catch-up time = catch-up distance ÷ speed difference = catch-up distance ÷ catch-up time flow.
Downstream velocity = still water velocity+countercurrent velocity = still water velocity-water velocity = (downstream velocity+countercurrent velocity) ÷2 Water velocity = (downstream velocity-countercurrent velocity) ÷2 Concentration problem
Solute weight+solvent weight = solution weight ÷ solution weight × 100%= concentrated solution weight × concentration = solute weight ÷ concentration = solution weight profit and discount problem profit = selling price-cost.
Profit rate = profit/cost × 100%= (selling price/cost-1)× 100%.
Up and down amount = principal × up and down percentage
Discount = actual selling price/original selling price × 100% (discount
1km =1000m1mm =1mm =10cm1mm =10cm/kloc-0. 8+00000 m2 1 m2 = 100 m2/m2 = 1 00 m2 1 m2 = 100 m2 (volume) product unit conversion 1 m3 =
Junior high school mathematics learning methods
First, take the initiative to preview
The purpose of preview is the process of actively acquiring new knowledge, which is helpful to mobilize the initiative of learning. Before explaining new knowledge, it is an important means to read the teaching materials carefully and develop the habit of previewing actively.
Therefore, cultivate self-study ability, learn to read books under the guidance of teachers, and preview with teachers' carefully designed thinking problems. For example, if you teach yourself an example, you should find out what the example is about, what the conditions are, what you want, how to answer it in the book, why you answer it like this, whether there is a new solution and what the steps are. Grasp these important problems, think with your head, go deep step by step, and learn to use existing knowledge to explore new knowledge independently.
Second, active thinking.
Many students just listen and can't think actively in the process of listening, so when they encounter practical problems, they don't know how to apply what they have learned to answer them. The main reason is that you didn't consider the trouble caused in class. In addition to following the teacher's thinking, we should also think more about why we define it like this, and what are the benefits of solving problems like this. Taking the initiative to think can not only make us listen more carefully, but also stimulate our interest in some knowledge and help us learn more. Rely on the teacher's guidance to think about the way to solve the problem; The answer is really not important; What matters is the method!
Third, be good at summing up laws.
Generally speaking, there are rules to follow in solving mathematical problems. When solving problems, we should pay attention to summing up the law of solving problems. After solving each exercise, we should pay attention to reviewing the following questions:
(1) What is the most important feature of this problem?
(2) What basic knowledge and graphics are used to solve this problem?
(3) How do you observe, associate and transform this problem to achieve transformation?
(4) What mathematical ideas and methods are used to solve this problem?
(5) Where is the most critical step to solve this problem?
(6) Have you ever done a topic like this? What are the similarities and differences between solutions and ideas?
How many solutions can you find to this problem? Which is the best? What kind of solution is a special skill? Can you sum up under what circumstances?
Put this series of questions through all aspects of problem solving, gradually improve and persevere, so that children's psychological stability and adaptability to problem solving can be continuously improved, and their thinking ability will be exercised and developed.
Fourth, broaden the thinking of solving problems.
Math problem solving should not be limited to this topic, but should be generalized, think more and think more. After solving a problem, think about whether there are other simpler methods that can help you broaden your mind and have more choices in the process of doing the problem in the future.
5. There must be a wrong book.
Speaking of wrong questions, many students feel that they have a good memory and can remember them without wrong questions. This is an "illusion", and everyone has this feeling. When the problems increase and the learning content deepens, you will find yourself at a loss. Therefore, wrong questions can record your own shortcomings at any time, which is helpful to strengthen the knowledge system and improve learning efficiency. Many schoolmasters got high marks because they used the wrong textbooks on their own initiative.
Six, five aspects of thinking
"1×5" learning method is to do a problem and think from five aspects, which can be combined with "summing up the law" and "expanding thinking" mentioned above. These five aspects are:
What is the knowledge point of this question?
(2) Why do you want to do this?
How did I come up with it?
What else can I do? Is there any other way?
⑤ A topic is changeable. See how many forms it has.
Don't feel trouble. The hardest thing to cultivate study habits is the first month, just like a rocket launch. The hardest part is the ignition and take-off stage. Therefore, once you have developed good math study habits and ways of thinking, you will be very relaxed in your future study.
Seven, independently complete the homework.
Nowadays, many students use some applications to help them finish their homework. They can find the answer by looking for photos, or copy other students' homework. This can be divided into two situations. One is for quick photos and speed. If you do this often, you will form a bad habit of reviewing questions, which is easy to be clear at a glance and careless. Another is for convenience, which will lead to students' fear of trouble. Once the topic is a little difficult, they will start to get upset and confused. So everyone must form the good habit of finishing homework independently.
Sort out the related articles about the knowledge points of the first volume of mathematics in senior one;
★ Induction of knowledge points in the first volume of junior high school mathematics.
★ arrangement of key knowledge in the first volume of junior one mathematics
★ Summarize the knowledge points in the first volume of junior high school mathematics.
★ Summary of knowledge points in the first volume of junior high school mathematics
★ Summarize the knowledge points of the first volume of mathematics in grade one.
★ Knowledge points in the first volume of first grade mathematics
★ Summary of mathematical knowledge points in the first volume of the first day of junior high school.
★ Summary of the knowledge points in the first volume of mathematics in the first grade education edition.
★ Complete collection of knowledge points in the first volume of junior high school mathematics.
★ Summarize the knowledge of the first volume of seventh grade mathematics.
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