The teaching goal of the fifth unit of mathematics courseware 1 in the first volume of the sixth grade of People's Education Press.
Knowledge and skills
Experience drawing circles with different tools. Know the names of the circle and its parts.
Process and method
Master the characteristics of a circle, and understand and master the relationship between the inner radius and diameter of the same or equal circle.
Emotional attitudes and values
Let students feel the beauty of mathematics and its application in life, understand the traditional knowledge of mathematics and cultivate patriotic enthusiasm.
Teaching focuses on mastering the names, characteristics and drawing methods of each part of a circle.
Teaching difficulties: mastering the names, characteristics and drawing methods of each part of the circle.
Teaching preparation and means
courseware
teaching process
Prepare lessons for the second time
First, situational introduction
Teacher: The fragments of traditional culture recited by the students just now are very wonderful. Today, the teacher also brought some related knowledge to everyone. What valuable mathematical information can you get from them? (Show the courseware).
Teacher: Look at these pictures carefully. What do they have in common?
Health: Everyone has a circle.
Health: It's all about circles.
Write on the blackboard: circle
Second, explore new knowledge independently.
(1) Draw a circle
Teacher: Some people say that because of the circle, our world has become so wonderful and magical. Then don't you want to draw this beautiful circle?
Student: Yes.
Please take out the tools for drawing circles and draw the circles you like.
Teacher: Many students drew beautiful circles by themselves, but a few students were not satisfied. Can you guess why he failed to draw them successfully?
Health: He holds the compass in the wrong way. (The compass should be held at the handle)
Health: Maybe the tip of his needle moved when he drew a circle. (The position of the needle tip must be fixed when drawing a circle)
Health: He compasses his feet step by step. Yes, the distance between two feet of compasses cannot be changed. )
(Students report and the teacher demonstrates drawing a circle with compasses. )
In fact, what my classmates just said is that we should pay attention to drawing a circle.
Now please draw a standard circle with a compass.
(B), the initial perception of the circle
Students, through your efforts to draw such a beautiful circle, what plane graphics have we learned before?
Health: Square, rectangle, triangle, parallelogram, trapezoid. (Students report and the teacher shows the corresponding courseware)
What's the difference between these figures and circles?
Health: Their sides are straight.
Yes, they are all closed figures surrounded by line segments.
Teacher: Please take out the CD from the desk and touch it. How do you feel?
Health: It's curved.
Such a curve is called a curve. A circle is a closed figure surrounded by curves. (Courseware Demonstration Circle)
(3) Self-study the concept of circle: center, radius and diameter.
As the saying goes, the circle is the most beautiful geometric figure. What do you want to know about circles?
I want to know how to find the circumference of a circle.
I want to know how to find the area of a circle.
Whether you want to find the area of a circle or the circumference of a circle, you must know the circle first. (blackboard writing: understanding of the circle)
(1) Guide the center of the learning circle
Let the students take out the disc just now, then fold it in half like the teacher, so that the upper and lower parts overlap completely and open it; Fold it in half several times from different directions. These creases are drawn with a pencil. What did you find?
Health: These creases intersect at one point.
Yes, this point is called the center of the circle and is represented by the letter O (mark the center of the circle on the blackboard when summing up).
Please mark the center of the circle by hand.
(2) Self-study radius
In fact, a circle has two important concepts: radius and diameter. How do scientists define them? The secret is hidden in Example 2 on page 56 of the math book. Please learn the relevant content by yourself and draw related concepts and important words with strokes.
Can you tell what radius is in your own words?
Health: The line segment from the center of the circle to any point on the edge of the circle is called radius.
Teacher: Any point on the edge of a circle is called any point on the circle.
Please help the teacher find the radius of the circle on the blackboard, and other students mark the radius of the circle by hand.
(3) Self-study diameter
You will know the radius by yourself. Can you find the diameter in the picture below? (Show courseware)
Why is AB not the diameter? What is this?
Health: Although it passes through the center of the circle, there is only one end on the circle, so it is not the diameter, but the radius of the circle.
Why is EF not a diameter?
Health: Not through the center of the circle.
Why is GH not the diameter?
Simply put, what are the requirements for the diameter of a circle?
Health: First, it must pass through the center of the circle; second, both ends must be on the circle; third, it must be a line segment.
(4) Explore the characteristics of the circle independently.
(1) exploration
Teacher: So far, we have discussed the problem of circle. Then do you think there is anything worth studying further?
Health: Yes (confidently).
Teacher: Well said, in fact, if nothing else, there are many rich laws about the center, diameter and radius. Do you want to do it yourself? (think! Students have disks, rulers, compasses and so on, which are our research tools. Later, I will ask my classmates to make a discount, measure, compare and draw a picture. I believe everyone will make new discoveries. Two small suggestions: First, don't forget to record your group's conclusions or even any small findings on the research paper during the research process, and then we will communicate together.
(Later, with the beautiful music, the students began to study in groups, recorded the research results on the "research discovery list" provided by the teacher, and communicated in the group first. )
(2) Report
Teacher: It's not good to fund research. We must be good at communicating with everyone and sharing our findings, don't you think?
Health: Yes.
Below, let's share your findings! The teacher collected some representative findings.
Display found 1: A circle has countless radii.
Teacher: Can you tell me how you found it?
Health: Our group found it by folding. Fold a circle in half first, then in half, in half, and so on. After expanding, you will find that there are many radii on the circle.
Health: Our group made this discovery by drawing pictures. As long as you keep drawing, you will draw countless radii in the circle.
Student: Our group didn't discount it or draw it. We just came up with it.
Teacher: Oh? Can you be specific?
Health: Because the line segment connecting the center of the circle and any point on the circle is called the radius of the circle, and there are countless points on the circle (pointing to the disk when talking), there are countless such line segments. Doesn't that mean there are countless radii?
Teacher: It seems that each group has made the same discovery in different ways. There are at least countless diameters. Do you need to explain why again?
Health: No, because the reason is the same.
Teacher: What's new about radius or diameter?
Show Discovery 2: All radii or diameters are equal in length.
Teacher: Can you tell me what you think?
Health: Our group discovered it through quantity. Draw a few radii on the circle at will, and then measure. It turns out that they are all equal in length and diameter.
Health: Our group is on sale. If you fold a circle in half, you will find that all the radii overlap, which means that all the radii are equal. The diameter and length are equal, and the reason should be the same.
I think that since the center of the circle is at the center of the circle, the distance from the center of the circle to any point on the circle should be equal, which also shows that the radius is equal everywhere.
Health: I have a little supplement about this discovery. Because different circles have different radii. So it is necessary to add "in the same circle" to make this discovery accurate.
Teacher: What do you think of his supplement?
Health: That makes sense.
Teacher: It seems that only by communicating with each other and supplementing each other can we make our findings more accurate and perfect. Did you find anything new?
Discovery 3: In the same circle, the diameter is twice the radius.
Teacher: Please tell the original group how you found it.
Health: We measured it by hand.
Teacher: Is there any different way?
Health: We folded it by hand.
Health: We can also think about the meaning of radius and diameter. Since it is called "radius", it should naturally be half the diameter length. ...
Teacher: It seems that everyone's imagination is really rich.
Health: Our group also found that the size of a circle is related to its radius. The longer the radius, the bigger the circle, the shorter the radius and the smaller the circle.
Teacher: The size of a circle is related to its radius, so what is its position related to?
Health: It should be related to the center of the circle. Where the center of the circle is fixed, the position of the circle is there.
Students still have many wonderful discoveries to show. It doesn't matter, then please cut out what you just found after class and post it on the math corner at the back of the classroom, so that the whole class can share it together, ok?
Health: OK.
Third, expand applications.
Do it after class.
Fourth, summary:
Students, after nearly forty minutes of hard work, what new gains have you made?
Homework design exercise 13 2 questions
blackboard-writing design
Understanding of circle
d=2r r=
Teaching objectives of unit 5 of the first volume of mathematics courseware 2 for sixth grade of People's Education Press
Knowledge and skills
Let the students know what a circle is. Understand and master the meaning and approximate value of pi.
Process and method
Cultivate and develop students' concept of space, and cultivate students' ability to abstract and solve simple practical problems.
Emotional attitudes and values
By understanding Zu Chongzhi's contribution to pi, we can see through patriotism.
Teaching focus
Understand and master the formula for calculating the circumference.
Teaching difficulties
Understand and master the formula for calculating the circumference.
Teaching methods and learning methods
Visual demonstration method
Teaching preparation and means
Slides or multimedia courseware.
teaching process
Prepare lessons for the second time
First of all, the introduction of passion
1. The animal kingdom is holding an animal sports meeting. It's too noisy. Do you want to see it?
2. A little goat and a sika deer are running on the circular track and the square track respectively. Let's guess who ran the longest distance in the end.
Second, explore new knowledge.
(1) Review the circumference of a square and guess what the circumference of a circle might have to do with it.
1. By comparing the lengths of two runways, can you calculate their perimeters? If students talk about the shape of a corner or a line, please follow the trend: a square is surrounded by four such line segments, and a circle is surrounded by a smooth curve. )
2. (Answer the perimeter of the square) Q: How did you work it out? (The circumference of a square = the side length ×4. The teacher's blackboard c=4a.) What do you say about the circumference and side length of a square? (4 times, 1/4) Teacher, the circumference of a square is always 4 times its side length, which is a fixed number. )
3. Can the circumference of a circle be calculated? If you know the formula, can you work it out? It seems necessary to study the calculation method of circumference. Let's study the circumference together. (Title on the blackboard: circumference)
4. Guess: What do you think the circumference might be related to?
(2) Metrological verification
1, the teacher asked: Can you think of a good way to measure its circumference?
① Health 1: Put the circle on the edge of the ruler and roll it for one week, and measure the circumference of the circle by rolling. Teachers and students cooperate to demonstrate the circumference of teaching AIDS.
② Wrap a rope around a circle, and then measure the length of the rope to get the circumference of the circle.
2.① Students begin to measure and verify the conjecture. Students experiment in groups, and write down their perimeters and diameters, and fill them in the table in the book.
② Observe the data and find out by comparison.
Question: Observe, what do you find? (The diameter of a circle changes, so does its circumference. The smaller the diameter, the shorter the circumference; The larger the diameter, the longer the circumference. The circumference of a circle is related to its diameter. )
3. Compare the data and reveal the relationship.
The circumference of a square is four times the length of its side, so is there a fixed multiple relationship between the circumference and the secret diameter of the circle? Guess, the circumference of a circle may be several times the diameter.
Students begin to calculate: divide the circumference of each circle by its diameter and fill in the third column of the table in the book.
Q: How many times are these perimeters related to the diameter (more than three times)? Finally, teachers and classmates concluded that the circumference of a circle is always more than three times the diameter, and the blackboard is more than three times. How much is more than three times? Guide students to read.
(3) Introduce pi
1, Teacher: The circumference of any circle is more than three times its diameter. This is a fixed number. We call it pi, which is expressed by the letter ∏. Write with your fingers.
2. How was pi discovered? Please read the small materials in the textbook and tell and educate the students in moral education.
3. Summary: As early as 1500 years ago, Zu Chongzhi calculated that pi was between 3. 1.4 1.5926 and 3. 1.4 1.5927, which was a whole thousand years earlier than that of other nationalities. This is a great contribution of the Chinese nation to the history of world mathematics. Today, the students found out for themselves.
The circumference of a circle is always a little more than three times its diameter. How did we calculate just now? The division of two numbers can also be said to be the ratio of two numbers, so the result is the ratio of the circumference to the diameter of a circle. We call the ratio of the circumference to the diameter of a circle π, which is expressed by the letter π. This ratio is fixed, but the difference of the results we get now is mainly caused by the error of measuring tools and methods. What is the value of pi? Tell me what you know. (Emphasize π≈3. 14, pay attention to approximation when speaking, and calculate according to the exact value when writing, with an equal sign. )
(4) Derive the formula
1. Can you calculate the circumference of a circle up to now? How to calculate?
2. If C stands for circumference and D stands for diameter, how to write the letter formula? (Blackboard: c=πd) Tell you the diameter directly. Can you find the circumference of a circle? The circumference of a circle is π times the diameter, which is a fixed number.
3. Can you find the circumference of a circle by knowing the radius? How many times the circumference is its radius?
Third, use formulas to solve problems.
Courseware demonstration example 1
Trial calculation C=2πr
2× 3.14× 33 = 207.24cm ≈ 2m
1 km =1000m
1000÷2=500 (circle)
Answer: (omitted)
Fourth, expand applications.
1. The diameter of a circular desktop is 0.95 meters. What's its circumference? (Figures shall be kept to two decimal places)
2. The maximum radius of the vase is 15cm. How long is this week? The diameter of the vase mouth is 16 cm. What is the circumference of the vase mouth? The diameter of the bottom of the vase is 20 cm. What is the circumference of the bottom of the vase?
Verb (abbreviation of verb) summary
What do you want to say to everyone through this class?
The operation design is 1, and the minute hand length of the clock face is 10 cm. How many centimeters does the needle tip pass in one revolution?
2. The diameter of the fountain is10m. It is necessary to enclose the fountain with two stainless steel railings. What is the total length of two stainless steel circles?
blackboard-writing design
perimeter of a circle
Example 1, C=2πr
2× 3.14× 33 = 207.24cm ≈ 2m
1 km =1000m
1000÷2=500 (circle)
Answer: (omitted)