Primary school mathematics "triangle area" courseware I
Teaching material analysis:
The lesson "triangle area" is the fifth section of Unit 4, Grade 5, Beijing Normal University Edition, which belongs to the teaching category of plane graphic area calculation. Through the teaching of calculating the area of plane graphics, we can not only guide students to master the characteristics of plane graphics, grasp the internal relations between plane graphics, and truly understand the transformation ideas infiltrated in them, but also develop and utilize students' imitation ability, which is a combination of analogical thinking and creative experience.
Before learning the lesson "Area of Triangle", students have the following knowledge bases: area calculation of rectangle, square and parallelogram; The characteristics of some simple polygons, etc. The basis of students' learning method is that when learning the area of parallelogram, students have initially felt that they can use cutting, translation, rotation and other operational activities to deform the figure with equal area.
In fact, before learning this lesson, some students are not ignorant of the formula for calculating the triangle area, but just mechanical memory. They know the formula, but they don't know why. The derivation method of triangle area calculation formula is similar to that of parallelogram area calculation formula, so it is very important to further explore the equal-area deformation by using the transformation idea in this lesson, which is of great help to continue to explore the calculation of trapezoidal area, circular area and cylinder volume.
Teaching objectives:
1. Exploring and mastering the triangle area formula can correctly calculate the triangle area and solve simple practical problems with the formula.
2. Cultivate students' ability to apply existing knowledge to solve new problems.
3. Make students experience mathematical activities such as operation, observation, discussion and induction, further understand the value of reduction method, and develop students' spatial concept and preliminary reasoning ability.
4. Let students get positive emotional experience in exploration activities, and further cultivate students' interest in learning mathematics.
Teaching emphasis: explore and master the formula for calculating the area of triangle, and correctly use the formula to calculate the area of triangle.
Difficulties in teaching: discover the internal connection of graphics and deduce reasoning in transformation.
Teaching emphasis: let students experience the process of operation, cooperation and communication, induction and discovery, and abstract formula.
Preparation of teaching AIDS: courseware, parallelogram paper, two sets of identical triangular paper, scissors, etc.
Prepare learning tools: each group should prepare at least two identical right triangles, acute triangles and obtuse triangles, a parallelogram and scissors.
Teaching process:
First, create a situation to reveal the topic.
Teacher: In the first grade of our school, a group of children joined the Young Pioneers organization. The school made a 150 red scarf. Let's help calculate how much cloth to use. Are the students willing to help the school solve this problem?
Teacher: Students, what shape is the red scarf? (Triangle) Can you calculate the area of a triangle? In this class, we will study and discuss this problem together. (blackboard writing: calculation of triangle area)
Design intention: The example of using the red scarf that students are familiar with to help the school calculate how much cloth to use has aroused the students' desire to know how to find the triangle area, thus transforming the students' goal of "teaching" into the goal of "learning". ]
Second, explore exchanges and summarize new knowledge.
Teacher: Last class, we learned the calculation method of parallelogram area. How do we explore the area of parallelogram? What is the area formula of parallelogram?
(blackboard writing: parallelogram area = bottom × height)
Teacher: Last class, we converted the parallelogram into a rectangle and explored the formula for calculating the area of the parallelogram. Let's guess: can you also convert a triangle into a learned figure to find the area?
[Design Intention: Students' experience in deriving the parallelogram area formula will inevitably lead to the following questions: Can a triangle be transformed into a learned figure to find its area? Let students find the connection between old and new knowledge by themselves, and let old knowledge become the foreshadowing of new knowledge. ]
(A) group experiments, cooperative learning.
Put forward operation and query requirements.
(1) What figure did the triangle become?
⑵ What is the relationship between triangle and transformation graph?
Ask the students to take out three types of triangles prepared before class and spell, swing or cut them in groups.
(2) Students discuss the operation in groups.
Students discuss according to the questions raised by the teacher.
[Design Intention: Here, a cooperative learning scheme is designed according to the needs of students' "learning", so that students can experiment in groups and learn cooperatively, and create an opportunity for students to solve their doubts and questions. ]
(3) Show the students' cutting and spelling process and exchange reports.
Report the experiment in each group. Ask the students to paste the transformed graphics on the blackboard and then choose a representative report. )
The following situations may occur: (Put two identical triangles together)
(two acute triangles) (two obtuse triangles) (two right triangles) (two isosceles right triangles)
Through experiments, the students come to the conclusion that as long as two identical triangles can be combined into a parallelogram.
You can also cut triangles into parallelograms.
3. Summarize the process of communication and deduction, and say the letter formula.
Fill in the blanks after discussion:
(1), two identical triangles can be combined into a parallelogram; The base of this parallelogram is equal to _ _ _ _; Parallelogram is higher than _ _ _ _;
(2) The area of each triangle is equal to _ _ _ of the parallelogram area with equal base and equal height.
Therefore, the triangle area = _ _ _ _.
Conclusion: The area of each triangle is half that of the parallelogram.
According to the students' discussion and report, the teacher wrote the following on the blackboard:
Because: the area of triangle = the area of parallelogram ÷2.
So: triangle area = base × height ÷2 (height is the height on the base. )
[Design intention: On the basis of a large number of perceptions, through autonomous learning and courseware demonstration, students can understand the relationship between two identical triangles transformed into parallelograms more concretely and clearly. At the same time, it infiltrated the transformed mathematical thinking method, broke through the teaching difficulties and improved the classroom teaching efficiency. ]
Teacher: If S is used to represent the area of the triangle, and α and H are used to represent the base and height of the triangle respectively, can you write the formula of the area of the triangle in letters?
According to the students' answers, the teacher wrote S=ah÷2 on the blackboard.
[Design intention: Through hands-on operation, mutual discussion and communication, the triangle is transformed into the learned figure by the method of folding and cutting, and the calculation formula of triangle area is deduced. This "transformed" mathematical thinking method can help us find the direction of exploring problems, and I believe students can apply this mathematical method to explore and solve more mathematical problems in the future. ]
Third, the new knowledge and application of games.
The first level compares who has the foundation.
1, try to calculate the area of the triangle.
2. According to the conditions, find the face of the triangle.
(1) The base is 5cm high and the height is 7cm.
(2) The elevation is 13m, and the bottom is 10m.
(3) The bottom is 0.8m and the height is 1 1 decimeter.
Do the questions in groups to see who can calculate correctly.
The second level is to compare whose idea is alive.
1. Calculate the area below. What did you find? (Unit: cm)
It is concluded that the areas of two triangles with equal base and equal height are equal.
Students calculate, discuss and draw conclusions.
2. Think about it, is the following statement correct? Why?
The area of (1) triangle is half that of parallelogram. ()
(2) The area of a triangle is 20 square meters, and the area of a parallelogram with equal base and equal height is 40 square meters. ()
(3) Two triangles with equal base and height must have the same area. ()
(4) Two triangles can be combined into a parallelogram. ()
Please sit down correctly and raise your hand to tell the reason for the mistake.
The third level is to compare who uses it well.
How many square meters of cloth does it take to make 150 red scarf worn by young pioneers? (Ask students to measure the required data and then calculate)
2. Measure the base and height of the triangle in your hand and calculate its area.
When measuring, emphasize correspondence.
[Design intent: Let students learn to measure and select the required data by themselves, and apply what they have learned to solve problems flexibly. ]
Fourth, summarize and raise awareness.
1. What did you gain from this lesson? What do you want to remind everyone?
2. What solutions did you learn today?
[Design intention: Let students summarize what they have learned, that is, systematize what they have learned. ]
Triangle area in primary school mathematics courseware II
Teaching material analysis:
The area of triangle is taught on the basis of students mastering the characteristics of triangle and calculating the area of rectangle and square. Through this part of teaching, students can understand and master the calculation formula of triangle area, and use the formula to calculate the area of triangle, and at the same time deepen their understanding of the internal relationship between triangle, rectangle and square, and cultivate their practical operation ability. Further develop students' spatial concept and thinking ability, and improve students' mathematical literacy.
Analysis of learning situation:
Before learning the area of triangle, students already know the characteristics of triangle; When learning rectangular area, square area and finding the area of combined graphics, I learned to cut, complement and move, and also learned to turn unknown learning problems into known problems. Therefore, in the teaching of triangle area, students have a certain knowledge preparation and ability foundation.
Teaching objectives:
1. Go through the derivation process of the triangle area formula and understand the meaning of the formula.
2. Understand the relationship between the base and height of the triangle and the length and width of the transformed rectangle.
3. The triangle area formula will be used to calculate the triangle area.
4. Cultivate students' ability to solve simple practical problems by using what they have learned, experience the application value of mathematics, and let students feel that mathematics is around.
Teaching emphasis: derivation of triangle area formula.
Difficulties in teaching: Understanding a triangle is half the area of a rectangle with the same base (length) and the same height (width).
Teaching process:
First, the import stage
The problem of triangle size in life is generated through story scenes;
1, what is the ratio of triangle size expressed in mathematical language?
2. What methods can be used for comparison?
Summary: It is one way to compare the sizes of triangles with transparent paper, but how do you feel?
Second, the inquiry stage.
(1) Draw a triangle.
1. Each student takes out the prepared rectangular paper and draws triangles as required.
Operating instructions:
(1) Take one side of the rectangular paper as the base of the triangle.
(2) Take any point on the opposite side as the vertex of the triangle.
(3) Connect the vertex and two diagonal corners.
(4) What kind of triangle did you draw?
2. Large group communication.
3. Guess: Let the students guess the area of the triangle according to their own drawings, which is a fraction of the whole rectangular area.
4. Observe the special relationship between the drawn triangle and rectangle.
5. Draw a height on the base of the known triangle and observe how much the area of the triangle is drawn.
(2) Experiment
1, tangent triangle.
Operating instructions:
(1) Cut out the triangle you drew.
(2) Put the rest into the cut triangle.
Thinking: Is the remaining triangle as big as the cut triangle?
(3) Fill in the experimental report.
2. Students communicate after completing the report.
(3) induction
According to the students' experiments, we come to the conclusion that:
The area of a right triangle is half that of the corresponding rectangle.
The area of an acute triangle is half that of the corresponding rectangle.
The area of an obtuse triangle is half that of the corresponding rectangle.
(1) Please summarize it in one sentence.
(2) Mathematically: triangle area = corresponding rectangular area /2.
(3) According to the area formula of rectangle, the area formula of triangle is deduced.
(4) The area formula of triangle is expressed by letters.
Third, the application stage:
1, teaching example 1
2. Calculate the area of three triangles in the import stage.
(1) Measure the bottoms and heights of the three triangles respectively and make records.
(2) Calculate the area of each triangle.
(3) communication.
Expand: Find two triangles with the same area in the following figure. Why?
Fourth, summary.
What did we learn in this class? 2. What conditions do I need to know to calculate the triangle area?
"Area of Triangle" Courseware III of Primary School Mathematics
First, the teaching objectives
Knowledge and skills
Let students go through the process of exploring the formula of triangle area calculation and master the method of triangle area calculation, which can solve the corresponding practical problems.
(2) Process and method
Through operation, observation and comparison, we can develop students' spatial concept, infiltrate and transform ideas, and cultivate students' ability to analyze, synthesize, abstract and solve practical problems.
(3) Emotional attitudes and values
Let students get positive emotional experience in exploration activities, and further cultivate students' interest in learning mathematics.
Second, the difficulties in teaching
Teaching emphasis: explore and master the formula of triangle area calculation.
Difficulties in teaching: understanding the derivation process of triangle area calculation formula and experiencing the idea of transformation.
Third, teaching preparation.
Multimedia courseware, learning tool bag (each group has two identical right triangles, acute triangles and obtuse triangles) and a red scarf.
Fourth, the teaching process
(1) Review and pave the way to stimulate interest and introduce new ideas.
1, review old knowledge.
(1) Calculate the area of each graph below.
(2) create a situation.
Students, please look at the red scarf on your chest. What shape is it? Want to cut a red scarf, do you know how big red cloth to use? To find the size of the required red cloth is to find out what this triangle is.
2. Review and introduce new ideas.
(1) Review: Do you still remember the area calculation formula of parallelogram? How is it derived?
(2) Introduction: If you know the calculation formula of triangle area, you can directly calculate the size of red cloth needed for cutting red scarf. In today's class, we will learn the area of a triangle. (blackboard title: area of triangle)
The design intention is to review the old knowledge, realize the convenience of calculating the graphic area with formulas, review the derivation process of the parallelogram area calculation formula, awaken the students' related activity experience, and prepare for the teaching of deducing the triangle area calculation formula later. At the same time, students are familiar with the introduction of red scarf into new lessons, and realize that math problems come from life, which stimulates their interest in learning.
(2) Actively explore and derive formulas.
1, operation conversion.
(1) Question: Since a parallelogram can be converted into a rectangular derived area formula, can a triangle be converted into a rectangular derived area formula?
(2) Students operate in groups and teachers patrol for guidance.
Students' operation presupposition: If students can't use the cut-and-paste method to convert the triangle into the learned figure when they only use one triangle, the teacher can guide them to change their way of thinking in time and try to use two identical triangles.
(3) student statements.
Default spelling 1: Make a parallelogram with two identical acute triangles.
Default spelling 2: Make a rectangle or parallelogram with two identical right triangles (take a rectangle as an example).
Default spelling 3: Make a parallelogram with two identical obtuse triangles (take one of them as an example).
(4) Think about it: You all spell differently, but we can find out what figure you can spell as long as the two triangles are exactly the same.
Students observed that some use two identical acute triangles as parallelograms, some use two identical right triangles as rectangles or parallelograms, and some use two identical obtuse triangles as parallelograms. Although the selected triangles are different, the spelling results are different, but as long as two identical triangles are used, a parallelogram can be spelled.
2. Observe and think.
(1) What do you find by observing the parallelogram and the original triangle?
(2) Students report after independent thinking: the base of a triangle is equal to the base of a parallelogram, the height of a triangle is higher than that of a parallelogram, and the area of a triangle is half that of a parallelogram.
3. Summarize the formula.
(1) Can you write the formula for calculating the triangle area yourself?
(2) Summarize the formula.
① blackboard writing formula: area of triangle = base × height ÷2.
② The calculation formula of triangle area is expressed by letters.
(3) Review and summarize.
We already know that the area of a triangle is equal to the base times the height and then divided by 2. Looking back, how was it deduced?
(2) Teacher's summary: When we can't convert a triangle into a learned figure, we choose two identical triangles to put together. No matter two identical acute triangles, right triangles or obtuse triangles, they can finally be combined into a parallelogram. Through observation and thinking, it is found that the base of the original triangle is equal to the base of the inlaid parallelogram, the height of the original triangle is higher than that of the inlaid parallelogram, and the area of the original triangle is half of that of the inlaid parallelogram. In today's learning process, students still use the method of converting the area of unknown triangle into the area of known parallelogram, which is very good! In the future study, if you encounter similar problems again, I hope you can continue to solve them in this way.
This part of design intention has designed three levels of teaching: operation transformation, observation and thinking, and formula generalization. First, ask questions, so that students can use the transformed ideas to operate with questions. From my own display and thinking, I found that two identical triangles can be combined into a parallelogram, thus finding the equal relationship between them; In the final summary, let students review the process of deducing the formula, which not only cultivates students' review and reflection ability, but also further infiltrates and transforms students' thoughts.
(C) consolidate the use and solve problems
1, example 2 on page 92 of the textbook.
(1) Show examples and ask questions.
(2) Understand the meaning of the topic and describe the content of the topic.
In your own words, what does the title mean?
② According to the picture and text, the students describe that the bottom of the red scarf is 100cm and the height is 33cm, so find its area.
(3) Collect information and clarify the problem.
Question: What mathematical information did you get from the topic? Ask for what?
2 thinking: What are the requirements for the area of the red scarf?
③ Summary: The required area of a red scarf is actually the area of a triangle with a base of 100cm and a height of 33cm.
(4) Students answer independently.
(5) Students' reports, teachers' writing on the blackboard and standardized writing.
(6) Help students to establish a certain concept of space by comparing the physical objects with the calculated results.
2. Complete the "hands-on" exercise.
(1) Complete the "Doing" question on page 92 of the textbook 1.
① Students finish independently.
2 deskmates talk to each other about how they did it.
(2) Complete the second question of "Do something" on page 92 of the textbook.
① Students finish independently.
② Class communication: What is the base and height of this triangle? How to calculate its area?
(3) Complete the third question of "Do something" on page 92 of the textbook.
① Students finish independently.
2 deskmates talk to each other about how they did it.
③ Class communication: How to calculate this problem?
Design Intention Example 2 echoes the research questions put forward at the beginning of the class, which not only consolidates the application of triangle area calculation formula, but also cultivates students' ability to solve practical problems. Then, completing the "do-do" exercise after class can help students further understand the area formula.
(D) variant exercises, internalization to improve
1, basic exercise.
Complete exercise 20 on page 93 of the textbook, question 1.
(1) Students do it independently.
(2) deskmates tell each other how they calculate.
(3) Classroom communication: Can you tell me the meaning of each traffic warning sign? How to calculate its area? Make a gesture the size of a traffic warning sign. (At the same time, carry out safety education to help students establish the concept of space. )
2. Improve your practice.
Complete exercise 20, question 3 on page 93 of the textbook.
How to calculate the area of these three triangles? What do you need to know? (Measure the base and height of each triangle first, and then calculate it with the formula. )
(2) Students finish independently.
(3) Class communication: What is the base and height of each triangle? How to calculate the area of a triangle?
The design intention is to consolidate students' understanding and application of the triangle area calculation formula through layered exercises, and to educate students on traffic safety at the same time.
(5) Summarize the class and talk about the gains.
1. What did you learn in this class today? How did you learn it?
2. Today, we derived the formula for calculating the triangle area, and learned to use the formula to solve practical problems in life. When deducing the calculation formula, we choose to put two identical acute triangles, right triangles or obtuse triangles together and transform them into known parallelogram areas to study. Then, through observation and comparison, the equivalent relationship between triangle and parallelogram before and after transformation is found, and the calculation formula of triangle area is deduced to be equal to the base multiplied by the height divided by 2. Today, students still use transformed ideas to solve the problems they encounter, and finally use formulas to solve practical problems in life smoothly.
3. Introduce mathematics knowledge.
(1) Students, do you know? Today, our ancestors discovered the formula for calculating the triangle area that we worked out together.
(2) Students, the mathematicians in ancient China were great, but the teacher thought you were great! Didn't we also find the calculation method of triangle area? In fact, only one triangle can be transformed into a parallelogram, and the formula for calculating the area of the triangle is derived. Interested students can try it after class!
(6) Homework exercises
1. Class assignment: Exercise 20, Question 2.
2. Homework: Exercise 20, Question 4.