The key and difficult knowledge point of mathematics in the sixth grade graduation exam of primary school: trip problem
Basic concepts:
The travel problem studies the movement of objects and the relationship among speed, time and distance.
Basic formula:
Distance = speed × time; Distance ÷ time = speed; Distance/speed = time
Key questions:
Determine the position and direction during the movement.
Meeting problem: speed sum × meeting time = meeting distance (please write other formulas)
Catch-up problem: catch-up time = distance difference ÷ speed difference (write other formulas)
Flow problem: downstream flow range = (ship speed+current speed) × downstream time.
Upstream stroke = (ship speed-water speed) × upstream time
Downstream speed = ship speed+current speed
Current speed = ship speed-water speed
Still water velocity = (downstream velocity+upstream velocity) ÷2
Water velocity = (downstream velocity-upstream velocity) ÷2
Running water problem: the key is to determine the speed of the object, refer to the above formula.
Bridge crossing problem: the key is to determine the moving distance of the object, refer to the above formula.
Main methods: line drawing method.
Basic question:
Given any two quantities of distance (meeting distance and catching distance), time (meeting time and catching time) and speed (speed sum and speed difference), find the third quantity.
The induction of mathematics knowledge points in the sixth grade
First, the characteristics of the circle
1. Circle is a plane figure surrounded by a closed curve in a plane.
2. Features of the circle: beautiful appearance and easy rolling.
3. center o: the point of the center is called the center. The center is generally represented by the letter o.
After the circle is folded in half for many times, the intersection of creases is at the center of the circle, that is, the center of the circle. The center of the circle determines the position of the circle.
Radius r: The line segment connecting the center of the circle and any point on the circle is called radius. In the same circle, there are countless radii, all of which are equal. The radius determines the size of the circle.
Diameter d: The line segment whose two ends pass through the center of the circle is called the diameter. The same circle has countless diameters, and all the diameters are equal. The diameter is the longest line segment in a circle.
The inner diameter of the same circle or equal circle is twice the radius: d=2r or r=d÷2.
4. Equal circles: circles with equal radii are called concentric circles, and equal circles can be completely overlapped by translation. Concentric circles: Two circles with coincident centers and unequal radii are called concentric circles.
5. The circle is an axisymmetric figure: if a figure is folded in half along a straight line, the figures on both sides can completely overlap, and this figure is an axisymmetric figure. The straight line where the crease lies is called the symmetry axis.
Figures with symmetry axis: semicircle, sector, isosceles trapezoid, isosceles triangle and angle.
A figure with two axes of symmetry: a rectangle.
A figure with three axes of symmetry: an equilateral triangle
A figure with four axes of symmetry: a square
Figures with or without symmetry axis: circles and rings.
Step 6 draw a circle
(1) The distance between two feet of a compass is the radius of a circle. (2) Draw a circle: fix the radius, center of the circle and make a circle.
Second, the circumference of the circle:
The length of a curve around a circle is called the circumference of the circle, and the circumference is represented by the letter C.
1, the circumference of a circle is always more than three times the diameter.
2. Pi: The ratio of the circumference to the diameter of a circle is a fixed value, which is called Pi and is expressed by the letter π.
Namely: pi = circumference ÷ diameter ≈3. 14.
Therefore, the circumference of a circle (c)= diameter (d)×π- circumference formula: c = π d, c = 2π r.
Pi π is an infinite acyclic decimal, and 3. 14 is an approximation.
3. Circumference change law: how many times does the radius expand, how many times does the diameter expand, and the circumference expansion multiple is the same as the radius and diameter expansion multiple.
4, semicircle circumference = half circumference+diameter = π r+d.
Review methods of mathematics in the sixth grade of primary school
First of all, we should make clear the purpose and task of review and proceed from reality.
Review must not be a simple mechanical repetition. It is necessary to sort out the basic knowledge of mathematics learned in primary schools through the review system, clarify the key points and key points of knowledge, and clarify the internal relations between knowledge, so that students' four operational abilities, preliminary logical thinking ability and spatial concept can be further improved on the original basis.
Through review, students can systematically master the basic knowledge about integers, decimals, fractions, percentages, ratios and simple equations, and can correctly and quickly calculate the four points of integers, decimals and teaching, thus improving their calculation ability. Further mastering a common measurement unit can skillfully calculate the perimeter, area and volume of some geometric shapes, and can simply measure your land and calculate earthwork, thus cultivating students' spatial concept. Being able to master common quantitative relations and methods to solve practical problems, improve students' ability to solve practical problems with arithmetic methods and equations, and cultivate students' ability to solve practical problems with logical thinking.
Before reviewing, we must combine the actual situation of the students in this class, determine the key points, and choose the teaching method to review. Each class should have a clear review purpose, requirements and main direction, so as to improve the review quality.
Second, determine the focus and scope of the review.
Reviewing is not simply repeating what you have learned before. Teachers must pay attention to the content of lectures and systematically sort out what you have learned. When reviewing, we should pay attention to giving full play to students' main role, mobilizing students' learning enthusiasm, inspiring students to learn by themselves, summarizing and sorting out what they have learned, and making knowledge systematic. Or inspire students to question difficulties, and teachers will guide students to solve doubts, thus promoting students' in-depth understanding of knowledge. Here are ten review points:
1) The meaning of integers and decimals, reading and writing methods, and the mutual conversion of units of measurement and names.
2) Elementary arithmetic of integers, decimals and fractions.
3) The concept, perimeter and area of plane figure.
4) Simple equation.
5) Divisibility and abacus of numbers.
6) The meaning and nature of fractions and percentages and the simplification of complex fractions.
7) Surface area and volume of three-dimensional graphics.
8) Proportion and specific gravity.
9) Solutions to various application problems and solving application problems with equations.
1 0) statistics and charts.
Third, adopt flexible review methods.
Students' initiative must be brought into play when reviewing. Encourage students to think independently. Review should not only let students simply reproduce what they have learned in mathematics. This will encourage students to learn by rote, and pay attention to promoting students' mastery and flexible use of what they have learned.
1) comparative analysis method. Some concepts, definitions, formulas and rules that students are easily confused should be gradually mastered on the basis of understanding. And through comparative analysis, help students understand their connections and differences, so as to deepen their memory.
2) autonomous reading method. The reviewed knowledge has already been learned. Teachers can choose several related textbooks for students to read independently. Teachers organize discussions around key topics, grasp key points or briefly explain what students don't understand, disperse students' thinking and cultivate students' ability to analyze problems independently.
3) sorting method. Look at the content of elementary school mathematics application problems, and there are various forms. The arrangement in the textbook is also scattered, especially the knowledge of geometry, which is abstract in content, with many concepts and formulas and complicated in calculation. So be sure to tidy it up when reviewing. Make knowledge systematic and organized. Find out the essential characteristics of all kinds of knowledge and cultivate students' logical thinking ability.
4) Inductive synthesis method. The content and knowledge of primary school mathematics are very wide. The content of each part mostly involves other parts of knowledge, which has great horizontal connection and strong knowledge mobility. Review should be from easy to difficult, from general to special, from basic to flexible, and make full use of the transfer law of knowledge to conduct a comprehensive review.
5) Focus on review. Learn about students' learning situation in time, find out the knowledge defects among students, and remedy them in time according to the specific situation. It is necessary to review and improve students' knowledge in a targeted and focused manner.
Fourth, the specific measures of review
1) Reflect on teaching and make plans. In the review, you can't repeat the knowledge step by step according to the book arrangement, lest the students eat cold rice and become dull. Teachers should systematically review the basic knowledge effectively and reasonably, internalize the knowledge structure and stimulate students to actively participate in learning activities. Therefore, the first stage of review should focus on the foundation and reflect comprehensively. At the same time, teachers should also ask each student to take notes in class. Students should write down what the teacher reviewed in class, especially the key sentences in the comprehensive blackboard writing. Teachers also check it once a week to urge students to finish it in time.
2) Special training to break through all links. In view of the knowledge points that students are prone to make general mistakes and individual mistakes, we should combine typical reflection with individual reflection, strengthen training, introduce special review methods, and break through the review ideas of all links. On the one hand, students are given special review training. On the other hand, pay attention to the reflection of unit test paper, comprehensive test paper and students' self-evaluation, and review each chapter together. Strengthen the inertia of knowledge and be flexible at this stage. On the other hand, pay attention to the examination marking and evaluation.
3) Hierarchical guidance and overall improvement. Attach importance to the hierarchical guidance of class students, develop and cultivate their individuality, and encourage students to help each other, strive together and improve together. Through these stages of review, every student will improve greatly.
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