People's Education Press, Grade Five, Volume One, Unit 6, 9 1 Page 92, Example 2 and Exercise.

Textbooks and learning situation analysis

The lesson "Area of Triangle" belongs to "Graphics and Geometry", which is the learning content of the first volume of the fifth grade of the Primary School Mathematics People's Education Edition. The content of the textbook is arranged after students have learned the area of rectangle three times, the understanding of triangle four times and the area of parallelogram five times. Because common polygons (including circles) can be divided into several triangles, we can not only calculate the area of each triangle before adding it, but also derive the area formula by using the triangle area formula. It can be said that the content of this lesson has the function of connecting the preceding with the following, and its core position is beyond doubt.

In the textbook of People's Education Edition, only the method of "double spelling" is provided, which guides students to transform the three triangles of right angle, acute angle and obtuse angle into the parallelogram they have learned, and at the same time, the grid diagram is abandoned. So most students will consciously think of transforming triangles into parallelograms? Or become a rectangle? Without the support of dot matrix graphics, can students successfully derive the area formula of triangle? Why are the textbooks arranged like this?

Based on the above questions and book theory, I designed a pretest to understand the starting point of this class. As shown in the figure below, among the students in Class 5 (1) who handed in the prediction paper, 22 students (84.6%) were able to independently explore and find out the area of right triangle and derive the formula, and all of them chose to study the right triangle in the grid diagram. Only 8 people (30.7%) can explore the area of acute triangle, and only 1 person (3.8%) can explore the area of obtuse triangle. After analysis, it is concluded that students have the following problems in learning ability and method selection:

Figure 1 Figure 2 Figure 3

Problem 1: weak operation ability and blunt derivation.

According to the previous test results, it is still difficult to explore how to convert the area of acute triangle into rectangle, and the obtuse triangle is the same as above. Take the right triangle area that has been successfully explored as an example, only 84.6% of the area has been successfully converted. In addition, figure 1 shows that there is an error in judging whether the converted figure is equal to the area of the original right triangle, while figure 3 does not reflect the derived result in the formula. The conclusion is correct, but I learned in the interview that I didn't know the conclusion until I previewed in advance.

Problem 2: The object of transformation is single, and the exploration is frustrated.

In the process of exploring the area of right-angled triangle, students use cut-and-paste filling and double spelling to transform, but the transformed graphics are all rectangles. Seven of the eight students who explored the area of acute triangle were converted into rectangles. When exploring the area of acute triangle and obtuse triangle, most students (as shown in Figure 2) have no way to start. Only 1 students in the class convert acute triangles and obtuse triangles into parallelograms. The reason is that most students' graphic construction ability has not reached the level of "contract composition"

Based on the above learning situation, the author thinks that the idea of transformation should run through the whole class, reason on the basis of intuitive operation, accumulate and apply basic activity experience to solve problems, and strive to improve the thinking level.

Teaching objectives

1. Through hands-on operation, the triangle is transformed into a graph whose area can already be found, and then the area calculation method of the triangle is deduced and optimized.

2. Master the calculation method of triangle area, and initially feel that the triangle area with equal base and equal height is equal.

3. Further understand the idea of "transformation" and cultivate deductive reasoning ability.

Teaching focus

Experience the derivation process of triangle area calculation formula, and can correctly use the area calculation formula for calculation.

Teaching difficulties

Infiltrate the mathematical thought of "transformation" and cultivate deductive reasoning ability.

Teaching focus

Let students experience the process of operation, cooperation and communication, inductive discovery and abstract formula.

Teaching preparation

1.ppt, triangle.

2. Pre-inspection list, 3 pieces of grid paper.

teaching process

Link 1: Clear objectives and formulate strategies. (3 minutes)

1. Cut to the chase and show me the topic.

In this lesson today, we will learn some new knowledge about triangles-the area of triangles (blackboard writing topic).

2. Recognize the bottom and stimulate interest.

What do you know about the area of triangle through preview?

(Default value: area of triangle = base × height ÷2)

3. View the method and enter the query.

How to verify the area formula of triangle?

Looking back at the method of exploring the area of parallelogram, can triangles be transformed into familiar figures?

(Default: Convert to rectangle. )

There are so many triangles, how to study them?

(Default: there are three types of triangular studies)

The courseware shows three kinds of triangles with a checkered background: right triangle, acute triangle and obtuse triangle.

(Transition: We start with a right triangle. )

The design intention is to come straight to the point, understand the starting point of students' learning, throw out the problems that need to be studied, and clarify the learning objectives. By reviewing the old knowledge, we affirm the important position of the idea of "conversion" in graphic research, and then guide students to learn the triangle area by classification through questions, infiltrating the idea of "classification verification".

Second, with the help of intuition, explore at different levels.

1. Activity 1: Explore the area of right triangle independently.

(1) Show exploration and expose problems.

Pre-demonstration checklist

Default: There is an operation flow, and the exported content is wrong, incomplete or will not be exported.

(2) Cooperation and communication, thinking collision

Point at each other in the group and talk about their own ideas. If you haven't finished deriving the area of right triangle, you can modify it after listening to others' speeches.

(3)*** Show in the same body and communicate with the whole class.

In the form of * * *, all kinds of works of students in the group are presented in an orderly way.

Default value:

(* * * Homotopy presentation: No.65438 +0: Hello everyone, our group's point of view is …, please invite No.2 to speak …

The above is the operation flow, thinking and conclusion of our group. Do other groups have anything to add? )

Preset 1: cutting and compensation preset 2: cutting and compensation preset 3: doubling preset 4: doubling

Figure 4 Figure 5?

Figure 6 Figure 7

(3) Seek enlightenment and broaden your horizons.

Ask questions:

① Why not calculate the area by calculating the grid? (Default: Some are not whole cells, which makes spelling easier)

② What are the different spellings? (Default: cut and fill method, double spelling method)

③ Can it only be converted into a rectangle? (Default: it can also be converted into a parallelogram)

(4) Improve thinking based on operation?

Give everyone a right triangle exactly like the one in the picture, and form a parallelogram by operation.

Show Figure 7: Multiplying into a parallelogram.

Conclusion: The area of right triangle = base × height ÷2.

(5) Explore again and summarize the area of right triangle.

① Question: Can any right triangle be transformed in this way? Please draw an arbitrary right triangle on the square paper prepared before class to explore.

② Students operate and report their findings.

③ Induction: the area of right triangle = base × height ÷2.

Design intent

① Low floor and high ceiling. Activity 1 Provide two kinds of right triangles (one of which is a right triangle with a square background) for students to choose independently, so that students with different thinking levels can have different thinking training. The reason why the right triangle is studied first is that it is the least difficult for students to learn, and it is the easiest to transform into a rectangle, which provides a low starting point for learning. In addition, students use the grid background to operate intuitively, which reduces the abstraction of the research and realizes everyone's participation.

② Emphasis on process and expression. According to the pre-test, although students can explore the area of right triangle by "cutting and filling" and "doubling", the derivation process is often wrong. So in activity one, organize students to communicate and show in the form of * * *, and learn from each other. When communicating with the whole class in the group, pay attention to express your ideas clearly by pointing and talking in Kan Kan, and cultivate your reasoning ability.

2. Activity 2: Cooperate to explore the areas of acute triangle and obtuse triangle.

(1) Define the operation task.

Can acute triangle and obtuse triangle be solved?

Show me the study list.

acute triangle

Obtuse triangle

(2) Organize communication and release.

(presenting a typical method)?

A, acute triangle display:

Figure 8? Figure 9

Figure 10 Figure 1 1

Default method 1. Cut and paste into rectangles Figure 8

Default method 2: multiply into a rectangle? Figure 9

Follow-up: Can it only be converted into a rectangle? (Default: it can also be spelled into a parallelogram)

Default method 3: Multiply by parallelogram 10.

Ask again: what else can it be transformed into? (Default value: converted to two right triangles added)

Default method 4: Cut into two right-angled triangles 1 1.

Right triangle+right triangle:

5×4÷2-2×4÷2

=(5-2)× 4 ÷ 2

B, obtuse triangle display:

Figure 1 1 Figure 12

Question: Can it be easily cut and mended into rectangles now?

The default method is 1: rotate the obtuse triangle, put the longest side horizontally, and then cut it into a rectangle. (The bottom edge is not a whole grid, which is inconvenient)

Question: Can we learn from the first two kinds of triangles?

Default method 2: Multiply it into a parallelogram diagram 1 1 for display.

Question: What else can it be transformed into? (Subtract two right triangles)

Default method 3: large right triangle-small right triangle: as shown in figure 12.

5×4÷2-2×4÷2

=(5-2)× 4 ÷ 2

(3) summary.

Question: To explore the area of three types of triangles, which transformation method is universal? (Default value: multiplied by parallelogram)

A unified formula is obtained by induction:

Area of triangle = base × height ÷2

S=ah÷2

Design intent

(1) Make good use of squares. Activity 2 still uses the grid background, which is different from activity 1. At this time, the grid is not marked with side length 1 cm. Students at different levels have different understandings of the grid, and high-level children can understand that each grid is a unit.

② Break through the difficulties. Activity 2 encountered a problem: the conversion object is single. How to break through? When exploring acute triangle, the author kept asking, "Can it only be transformed into a rectangle?" "What else can it be transformed into?" Break the limitations of students' thinking again and again. Guide students to think, draw a picture, multiply it into a parallelogram or divide it into right triangles. The method of exploring obtuse triangle is the same as above. Every operation and thinking of students has become the basic activity experience of learning the new triangle.

Link 3: Learn and practice skillfully, and expand the application.

(1) basic training

Figure 13

1. Look at the picture to find out the area.

A. do the math

Default 1: 3× 4÷ 2

Preset 2: 5× 2.4 ÷ 2

B. Think about it: What is the graphic area of 3×4? How about 5×2.4?

Ppt demonstration:

Figure 14 15 16 17

Design intention ① Familiar with the calculation formula of triangle area; (2) Can tell the geometric meaning of the triangle area formula.

2. How many triangles treat the shadow area in the picture below? Why are they equal? Can you also draw a triangle with the same area as them in the picture? Give it a try.

A.try to draw a picture

B. Show typical works.

(Default: 2, because they are the same height as the base. As shown in the figure below)

Figure 18 Figure 19

C. Fill in the form (default):

Bottom /cm 3 4 6 12 1 5

Height/cm 4 3 2 1 12 2.4

D. summarize and talk about your findings.

(Default: triangles with equal area and equal product of base and height)

Design Intention This topic is designed from easy to difficult, giving students enough space for graphic construction, guiding students to further understand the meaning of the formula from the perspective of function, and to understand the relationship between bottom, height and area.

(2) Expanding exercises?

Figure 20

1. How many ways can you think of to find the perimeter of a triangle?

A. do the math,

The default value is 1: 3× 4÷ 2 = 6 (square centimeter).

6×2÷2.4=5 cm

5+4+3= 12 (cm)

2: 3× 4 ÷ 2.4 = 5 (cm) is preset.

5+4+3= 12 (cm)

The default is 3: 2.4× x = 3× 4.

X=5

5+4+3= 12 (cm)

Design Intention This topic involves the area and perimeter of triangle, and comprehensively examines students' ability to solve geometric figures. The first method embodies the reverse thinking of finding the angle, and the second method embodies that the product of the base and height of a triangle with equal area is equal. Method 3 Review the calculation of triangle area by equation.

2. Find the area of ③ in the figure. ?

Question: How to find the area without bottom or height?

Figure 2 1 Figure 22

Default: (as shown above)?

The problem of design intention contains a lot of gold, which can be understood by combining with auxiliary lines. Investigate students' comprehensive ability to apply geometry knowledge.

work design

1. The area of this rectangle is 30 square centimeters. Find the shadow area.

Figure 22

2. Read the relevant information of triangle area (Zhijiang confluence operation).

3. Cooperative exploration: How to find the area of trapezoid? Can you use triangles?

Derive the area formula?

Blackboard design:

Teaching reflection

The teaching design of this lesson is based on the learning situation, guiding children to improve in the following three aspects:

First, break through the limitations of thinking and optimize methods.

Based on the teaching idea of "let different students have different development", in the process of verifying the right-angled triangle area, with the help of a simple "learning material"-square, the group first cooperates and communicates, showing the whole process of exploring the right-angled triangle area, diverging thinking, and realizing the diversification of transformation methods and objects. Then, based on this experience, explore the areas of acute triangle and obtuse triangle. Finally, the optimization method is summarized (multiplied by parallelogram and divided by 2). Students' thinking ability has been improved in purposeful operation.

Second, improve the reasoning ability, based on intuition.

Throughout the whole class, "changing ideas" runs through, focusing on reasoning with intuitive operation and clear expression in cooperation and communication. This paper attempts to use the area formula of right triangle just deduced to derive the other two kinds of triangle area formulas, which will lay the foundation for the derivation of trapezoidal area in the future.

Third, grasp the essence of the content by asking questions.

This lesson focuses on the design of teaching problems. High-order question: But there are so many triangles, how to study them? Follow-up: Can it only be converted into a rectangle? Is there any different way? What does the research on the area of right triangle and acute triangle give you? What is the meaning of the formula? ..... Promote thinking and understanding with questions.

In the teaching of this class, guided by the idea of combining operation and reasoning, I started with the study of right triangle, classified and verified, and guided students to explore and verify the conclusion step by step. Every child feels the joy of learning.