Fujian Wu Qiuju Shanghang Experimental Primary School

Teaching content: PEP compulsory education curriculum standard experimental textbook, the second volume of fourth grade mathematics, 82 pages. Teaching objectives:

1, let students guess, calculate, explore and understand the triangle relationship.

2. Grasp the significance of the triangular trilateral relationship and use it to explain the mathematical phenomena in life.

3. Cultivate students' abilities of observation, operation, cooperation, expression, abstraction, generalization, analogy and problem solving, and develop the concept of space.

Teaching emphasis: master the nature and flexible application of "the sum of any two sides of a triangle is greater than the third side". Teaching difficulty: Explore the process of discovering that the sum of any two sides of a triangle is greater than the third side. Teaching preparation: multimedia courseware, notes, experimental record sheet.

Teaching process:

First, create a situation, suspense introduction

Teacher: At ordinary times, students go to school, teachers go to work and parents go out, almost all of them will encounter the problem of Xiaoming going to school in the picture (the media presents a road map of Zhang Xiaoming going to school). How does Xiaoming go to school?

Health: There are three roads, from Xiaoming's home to the post office and then to the school. You can also go straight along the middle; You can also go to the store first and then go to school.

Teacher: Let's applaud the students who express themselves so bravely, clearly and fluently! Which road is the shortest and why?

Health: It is shortest to go straight along the middle road, according to the shortest distance between two points.

Teacher: It is very important that this student can skillfully use the mathematical knowledge of "the shortest distance between two points" to explain life problems!

Teacher: Look again, the three routes taken by Xiaoming just form two triangles (presented by the media). You can also try to explain this problem with the trilateral relationship of a triangle. What are the three sides of a triangle? You might as well guess!

Guess: the sum of two sides is greater than the third side. ...

Teacher: Success begins with guessing. Today, with curiosity, let's go into the journey of exploring and discovering the trilateral relationship of the triangle. I believe that students who have been diligent in learning, good at thinking and clever at learning will soon be able to uncover the mystery!

(blackboard writing topic).

Comments: Teacher Wu created a "real" and "familiar life situation" for students-Xiaoming's road map to school, which touched students' life complex at the "cognitive level" and then assimilated the adaptation process to the "mathematical level".

Second, operate query, verification and discovery.

(a) Hands-on experiments

Teacher: It is not enough to have a guess. Many important discoveries come from hands-on experiments. Let's do hands-on experiments, too. First of all, please listen carefully to the experimental requirements (media presentation):

1. Please take out four pieces of paper and the experimental record sheet in the envelope. Please choose any three pieces to form a triangle.

2. Complete the cooperation at the same table, with one person operating and one person assisting, and make records.

3. Do at least 3 groups of experiments.

Teacher: Did you hear the requirements of the experiment clearly? All right! Then let's see which table the two students cooperate well. Let's finish it quickly!

Students experiment, teachers check, individual guidance, and choose a group of operation experiments to go on stage.

(2) Report and arrangement

1. Student report: We will ask the students who are operating the experiment on the stage to report and see who listens most carefully!

2. Teacher's arrangement: Now the teacher has roughly arranged the students' experiments (media display) on the experimental record of triangular trilateral relations.

Is this the result of the experiment? (Yes)

Comments: It is better to say it a thousand times than to do it. Teacher Wu's sentence "Many important discoveries come from hands-on experiments, so do it" has repeatedly inspired students' desire to be mathematicians, and their strong desire to explore and enthusiasm for experiments has broken the floodgates of students' thinking. Soon, the students said that the wonderful thing came from "personal experience", and they knew the true face of Lushan Mountain for the first time-"When the sum of two sides is less than or equal to the third side, a triangle cannot be enclosed".

(3) Further exploration

(1) Counterexample

Teacher: Suppose three pieces of paper are three sides of a triangle. We use negative examples to select the first two groups to study. Why can't the three sides of the first two groups of experiments form a triangle? Please discuss in groups before expressing your opinions.

Health 1: In the first two groups, the short side is too short and the long side is too long to form a triangle. (Guide)

Students look at the picture and say the whole sentence)

The sum of the top two short sides of Sheng 2: 1 group is shorter than the bottom long side.

Health 3: In the second group, the two short sides are as long as the long sides and form two parallel lines.

So you can't form a triangle.

(2) Appropriate expression

Teacher: (Media verification) When students say that two sides are connected, can they be said to be "the sum of two sides" (yes), and "shorter than the long side below" and "as long as the long side is less than or equal to the third side"? In simple mathematical language, it means that "the sum of two sides is less than or equal to the third side and cannot form a triangle."

(3) conjecture reveals

Guess

Teacher: Practice makes true knowledge. Students found and ruled out two situations: "the sum of two sides is less than or equal to the third side, which does not constitute a triangle". Are you sure? (OK) Oh yeah! (Humor) (Media verification) Why don't we raise the whip, pursue victory, and boldly guess again, how does the sum of the two sides form a triangle when it is related to a third party?

Health: The sum of two sides is greater than the third side to form a triangle.

Teacher: Is that true? Don't change this sentence. (Teacher writes on the blackboard)

Health: If you don't speak a little, you really can't change it. (Media Certification)

2. contradiction: 6 th division: it seems to be true! Do you think these three sides can form a triangle? (media presentation),

Health 1: Yes, because 5+ 12 > 6, 6+12 > 5. The sum of two sides is greater than the third side to form a triangle.

Health 2: No, because 5+6.

Teacher: Yes, it is impossible to enclose. Isn't this what we just verified? (impatient) So is this sentence complete? What should I say?

Health 1: The sum of the shortest two sides should be greater than the third side.

Health 2: The sum of any two sides is greater than the third side.

Teacher: Do you agree? (Agreed), yes, although 5+12 >; 6,6+12 > 5, but 5+6.

Health 1: Add the sum of the "shorter" two sides, (why) because the sum of the shortest two sides is greater than the third side,

Don't say that the sum of the two longer sides must be greater than the third side, which will definitely form a triangle.

Health 2: It can also be said that it is the sum of two sides of "arbitrary".

Teacher: What is "arbitrary"?

Health 3: Just take any two sides and add them up to be longer than the third side.

Teacher: Let's read this sentence together!

verification

Teacher: Really? Please choose your favorite group to test! If you can do oral calculation, try to do it with your mouth. Teacher: How did you quickly identify groups 5, 6 and 7?

Health 4: 5+6 > 7, so these three lines can form a triangle.

Teacher: Oh, just add it with your mouth. So just add it once?

Health 4: add if you don't believe it, 5+6 > 7; 5+7 >6; 6+7 >5. Show any two sides of a triangle.

And is larger than the third side,

Teacher: Is it necessary for students to go to so much trouble? (The teacher pretends)

Health 4: Of course not. Besides, as long as the sum of the shortest two sides is greater than the third side.

Teacher: Good point! What's your name? (Duoduo) Then let's use many methods to judge! It seems that such a good way is to give it a louder name, calling it "one plus one spirit" is better than "one stick spirit" (humor). Other groups, and so on, can also draw the same truth. Teachers, the media can quickly improve the table.

show

Teacher: Let's focus on the symbols of these formulas and see what are the similarities and differences. What does this mean?

Health: The three used in the last two groups are all greater than signs, which shows that the sum of any two sides is indeed greater than the third side to form a triangle;

Although the first two groups also have two greater than signs, as long as one is less than or equal to it, a triangle cannot be formed. It can be seen that the word "arbitrary" is very important, and the whole class should read this sentence again.

Reading problem

1. Self-study textbook

This sentence is on page 82 of the book. Please read the textbook again in your usual reading method to see if there are any problems.

2. Discrimination and interpretation

Do you have a deeper understanding of this sentence after reading the textbook? Tell us your opinion.

1: I have a deeper understanding of "only the sum of any two sides is greater than (not less than or equal to) the first one".

Three sides can form a triangle.

Teacher: If you think positively, you can also think reversely. "To form a triangle, the sum of any two sides must be greater than the third side."

Health 2: Actually, don't bother, just see that the sum of the short sides is greater than the third side. ……

Teacher: In other words, what is the key word of this sentence? -"Arbitrary", "Harmony", "Greater than" Surrounding ",then let's reread this sentence with this profound knowledge (the teacher is all ears). Grasping key words is an effective way to interpret concepts and solve problems. I hope that students will practice and use more in their usual study, and strive to be better without the best!

Comment: "Why can't the three sides of the first two groups of experiments form a triangle?" That sentence set off ripples in students' hearts and began to eliminate the saying that "the sum of two sides is less than or equal to the third side, so a triangle cannot be formed". The strength in my heart is like an experienced old fisherman. It is natural to boldly guess that "the sum of two sides is greater than the third side to form a triangle". When the students agreed that this sentence was impeccable, Mr. Wu said that it stimulated and broke the initial cognitive balance of the students, and recognized the true face of Lushan Mountain in the internal need of students expressing their opinions and agreeing that "arbitrary" or "shorter" was needed to be complete and strict-"The sum of arbitrary (or shorter) sides is greater than the third side, and a triangle can be formed.

Form ".

(5) in-depth promotion

Teacher: Just now, we only tried a few triangles. Is it true that three sides of every triangle in the world have such a relationship? Ask the students to draw a triangle in their notebooks at will, and then measure, calculate and compare it to see if there is such a relationship. (Teacher's Board Drawing)

Refers to student reports and collective evaluations.

Teacher: 62 students in the class, if we are given enough time, draw thousands, tens of thousands and hundreds of millions more. Close your eyes and imagine, will the result change? (no)

It seems that as long as it is a triangle, the sum of any two sides will be greater than the third side, or the sum of the shortest two sides must be greater than the third side. (The media shows it accordingly)

(6) abstract generalization

Teacher: Can we hold up a triangle like this? Is there any way to quickly represent all triangles in the world? (Student: In letters) Good idea. In mathematics, in order to facilitate communication, the letters "A, B and C" are used to represent the three sides of a triangle. (student: a+b > c; a+c & gt； b； b+ c & gt； Answer. (The teacher should write it on the blackboard accordingly and then show it to the media.)

Teacher: If there is only one formula A+B >; C, and then A and B are here.

Health 1: the shortest sides.

Health 2: Any two sides.

Comments: Just when the students thought they could let go, Mr. Wu threw out "Do all three sides of any triangle have such a relationship?" Once again, it broke the inner balance of students and improved their confidence and courage to explore the end. Students personally verified, by extension, and soon realized that "three sides of each triangle have such a relationship." Therefore, in the face of endless triangles, students naturally need letters for simple and convenient expression and communication. Students' thinking is like re-entering the no-man's land, and further appreciating the beautiful artistic conception of "infinite scenery with dangerous peaks". B.C.

Third, develop teaching materials and improve practice.

How well do the students grasp the "relationship among the three sides of a triangle"? Let's practice and test it! Are you confident to accept the challenge? (Yes)

(1) basic exercise-question 4 (media presentation) on page 86.

(2) Expanding exercises-deepening the rich knowledge contained in the fourth question on page 86 of this book (the media presents it accordingly).

1. Understanding Right Triangle

Teacher: How do you judge questions 3, 4 and 5?

Health: 3+4 > 5, so these three lines can form a triangle.

Teacher: Oh, it seems that "one plus one is better than spirit" is really clever! Then please use the "one plus one spirit" to quickly complete the other three small problems!

Teacher: Let's look back. I wonder if the students have found it? Actually, these three lines are very interesting. 3, 4 and 5 are not just three consecutive natural numbers. Can all three sides be surrounded by three consecutive natural numbers?

Health 1: Yes!

Health 2: Not necessarily, 1, 2, 3 won't work, because 1+2 equals 3.

Health 3: 0, 1, 2 is definitely not a 0!

Division 0, 1, 2; Outside 1, 2, 3, try it.

Health: 7, 8, 9 ... can be attached.

Teacher: That's true, except 0, 1, 2; Any three sides except 1, 2, and 3 are three continuous natural numbers that can form a triangle, and the triangle surrounded by three lines, such as 3, 4, and 5, is very special. Do you want to know in advance? (Thinking) (Media Play: Pythagorean Theorem-Do you know? )

2. Do you know the equilateral triangle?

Teacher: What about the triangle formed by 3, 3 and 3 lines?

I know that there are three equilateral triangles called equilateral triangles.

Teacher: As the name implies, naming it according to its meaning is really a good way to gain true knowledge! (Media introduction)

Understand isosceles triangle

Teacher: What about the triangle formed by three lines: 3, 3 and 5? Can you show it by gesture? (Media broadcast) (Make waist gestures with two index fingers)

(3) Open practice-(media broadcast) I am a small designer.

Teacher: Most of the roofs you usually see are the same as those just painted by your classmates. What shape is it? (isosceles triangle), let's be a small designer and design an isosceles triangle roof. This beam is 5 meters long. Which two pieces of wood can form an isosceles triangle roof with this beam?

Health 1: I choose 3 meters.

It's ok to be born 2: 4 meters.

Teacher: If we choose two 4-meter-long inclined beams, how many meters can the beams be? (Keep the whole meter)

Health 1: more than 0 meters and less than 8 meters.

Health 2: Because the whole meter should be reserved, the longest one can be 7 meters, and the shortest one can be 1 meter.

Teacher: Students close their eyes and imagine what the house looks like when the length of the beam is from 1 meter to 7 meters (the house changes from a sharp pier to a thick pier). Open their eyes to see if it is similar to the computer. (Media presentation)

Teacher: What kind of house design do you like? Why?

Health 1: I like the bungalow design, with a larger area and looks safe and elegant!

Health 2: I like the design like ice cream, sharp and exciting! ……

Teacher: It is true that you are reasonable, and so is your wife. As long as you are reasonable, you can travel all over the world!

Fourth, echo from beginning to end, sublimate and expand.

1. End-to-end echo

Teacher: Students, now you must be able to explain why Xiao Ming chose the middle straight road when he went to school with the triangular trilateral relationship. (media playback),

Health 1: The sum of any two sides of a triangle is greater than the third side, so it is closer to the middle.

Health 2: On the other hand, the third side is definitely smaller than the sum of the two sides, so it is closer to the middle.

Teacher: How did they analyze it?

Health: Both positive and negative analysis are very reasonable!

2. Variant sublimation

Teacher: But if there is greening and beautification in the middle (as shown in the picture), can Xiao Ming still walk in the middle? (No), what would you do if someone was still walking in the middle?

Health: I will tell him that it will trample on the grass and destroy the environment, which is very uncivilized.

Health: I will give more warm tips, such as "The grass is resting, please don't disturb it" and "The grass is intentional and the feet are merciful". ……

Teacher: You are really little environmental guards. Mother Earth must be very touched by this, and you must always be green because of your strong awareness of environmental protection. Life is infinite!

3. Class summary

(1) What did you gain from this lesson?

1: I know the relationship between the three sides of a triangle.

Health 2: I know that the sum of any two sides of a triangle will be greater than the third side.

Health 3: I know that it is really smart to judge whether the three lines are short or not by "one plus one is better than spirit"!

Health 4: I also know right triangle, equilateral triangle and isosceles triangle in advance.

Health 5: I also know that it is the shortest to take the middle straight. ……

(2) Do you think "the relationship between three sides of a triangle" is useful? Can you give me an example?

Health: When we cross the road, it is the shortest to go straight according to the trilateral relationship of the triangle, but if there are traffic lights, we should follow the instructions of the traffic lights, and we should not rush in for convenience, otherwise it will be very dangerous!

Teacher: Yes, we only have one life. Cultivate safety awareness from an early age! We should not only learn and apply flexibly, but also analyze concrete problems, so that mathematics can better serve our lives!

(3) What learning methods have we mainly adopted in this class?

Teacher-student summary: We mainly use guessing-experiment-guessing-verification-application.

4. Pause expansion

Please take out a 20 cm long straw that the teacher prepared for you in advance. If you give us a chance to cut the straw into triangles ourselves, where do you think the first knife will not be cut? Why?

Health 1: medium

Health 2: Anyway, you can't cut from the middle, or you can't enclose a triangle with the same length. In this way, the longer part is completed on both sides.

The shorter one is the third side.

Teacher: reasonable, reasonable! What about the second knife scissors? After the students have studied the relationship between the difference between the two sides of a triangle and the third side, I believe you can cut the triangle you want quickly and well (it will listen to you) (humor)

Attachment: blackboard design:

The relationship between the three sides of a triangle

Any (or shorter) two sides whose sum is greater than the third side can (of course) form a triangle.

a+b & gt； c；

A b

c

a+c & gt； b； b+ c & gt； a