Solving travel problems with equations is the ending part of solving simple equations in Unit 5 of the first volume of the fifth grade of People's Education Press. Before that, students have learned the meaning of equations and the methods of solving equations, as well as the sum and difference times of solving problems with equations. Students have been exposed to the movement of an object before. This lesson will lead students to explore the movement process of two objects.

Classroom design

1, for example 1

(1) Encountered a problem: Xiao Lin rode 250m per minute and Xiao Yun rode 200m per minute. Xiaolin's home and Xiaoyun's home are 4.5km apart. At nine o'clock on Sunday morning, they set off from home by bike. When did they meet?

(2) Rear-end collision: The express train and the local train depart from A and Mile Mile respectively, with a distance of 120 km and the same driving direction. The express train is 80 kilometers per hour and the local train is 40 kilometers per hour. How many hours does it take for the express train to catch the local train?

2. (Please ask two students to take the stage to demonstrate the activity 1 and explain why they exercise like this, and other students will supplement it. )

? Luo: "I understand this. Xiao Lin's speed is 250m per minute, Xiao Yun's speed is 200m per minute, 4.5km, which is the distance between their two families. They left home at 9 o'clock and began to walk. Walking in the opposite direction means walking face to face until they meet. "

Health: "the most important thing is that they exercise for the same time!" " "

(Students demonstrate Activity 2 and explain why they exercise like this, and other students supplement)

Catalpa: "It must be an express train chasing a slow train."

Rui: "You can't catch the local train, so the starting point of the express train is behind the local train."

"They are heading in the same direction."

? Please use pictures to show the movement process of objects and complete them independently.

(Some students drew a line graph):

(Example 1 The first (1) topic is students drawing)

(Some students drew a histogram):

(Example 1 The first (1) topic is students drawing)

Some students use the histogram model they have learned before to construct the relationship between the total amount and each part in their minds, and the information in the question is roughly expressed by charts.

(Example 1 Student Painting in the Second Section)

? Carol: "I use a triangle to show where they started and where they met, and then show their speed."

? Park: "He also indicated their moving direction with arrows."

I can't wait to add: "You can mark the meeting place with a small flag."

Plate three, find the equivalence relationship with line segment diagram

Analysis example (1):

Method 1

Luo: "Xiao Lin's speed × time = Xiao Lin's distance, Xiao Yun's speed × time = Xiao Yun's distance, and the distance between them is the total distance."

Teacher: "Can any students tell me what the equivalence relation Luo worked out is?"

? Small: "the distance between Xiao Lin and Xiao Yun = the total distance"

Method 2

Yong: "I calculated that the total distance they move per minute is 250+200, and they move 450 meters per minute."

Teacher: "What is 250+200?"

Health: "The total distance of * * * lines per minute"

? The student added: "Speed and"

Teacher: "Is this analysis correct? Can any students briefly summarize the equivalence relationship? "

? Health: "(Xiao Lin's speed+Xiao Yun's speed) × time = total distance"

Sheng added: "Speed and time = total distance"

? Xiao: "In fact, one of Yong and Luo is positive thinking and the other is reverse thinking. But they all use the same equivalence relation. "

Analysis example (2):

Health: "I marked the starting point of the express train and the local train respectively, and marked the place where they caught up with small flags."

At this point, students draw the starting point and ending point of the express train by hand in the online line segment diagram, and other students quickly find the distance of the express train.

Teacher? : "I found the distance of the express train. Can you find the distance of the local train in the line segment? "

Health pointed out.

Teacher: "Besides the distance between the express train and the local train, what is another known quantity?"

? Health: "the distance to catch up"

Teacher: "What kind of relationship can you see between these three quantities through the line segment diagram?"

Student: "You can find the total quantity and the quantity of each part in the line segment."

Rui: "The distance of the express train-the distance of the local train = the distance of catching up"

Teacher: "Do any students use other quantitative relations?"

Rui: "I assume that the local train is equal to stopping at the same place, and the speed of the express train MINUS the speed of the local train is the speed of the express train."

Teacher: "What does he mean? Can someone explain it to other students again? "

Xin: "In the past, the speed of the express train was 80 km/h and the speed of the local train was 40 km/h. Now, assuming that the local train is not moving, the express train is 80-40 = 40 km/h."

Teacher: "Rui made it very clear, Xin understood it at once, and Xin repeated it clearly." But I have a question. What is the speed of the express train minus the local train? "

? Health: "speed difference"

Teacher: "How to express the quantitative relationship between Rui and Yong?"

Jia: "Speed difference × time = distance"

Students perceive the direction and trajectory of the object in the demonstration, thus successfully constructing the basic model of the travel problem, and then refining it, we can quickly find the relationship between the total amount and each part. It is found that the problems of meeting and catching up mainly lie in the direction of their movement and the change of the relationship between their movement distance and the original distance. In this lesson, we will focus on breaking through the problem of meeting.

(The teacher uses PPt to demonstrate and summarize):

Travel problems usually have several quantities:

Their relationship is:

This lesson will discuss the problem of meeting in detail. )

(Group discussion, representatives speak)

Teacher: "What shall we do after finding the equivalence relation with the line segment diagram?"

Student: "Find the unknown quantity."

Teacher: "Then can you find out who is the unknown?"

Health: "As can be seen from the picture, the movement time of Xiao Lin and Xiao Yun is unknown, because distance = speed × time, and speed is known, and their movement time is the same, so as long as we know the movement time, we need to know the distance between the two people."

Teacher: "What should we do after analyzing the unknown quantity?"

Health: "solution"

Teacher: "Set what?"

? Student: "suppose they meet after x minutes of exercise."

Teacher: "According to the quantitative relationship of Luo (the distance between Xiao Lin and Xiao Yun = the total distance), what kind of equation can we list?"

Health: "250x+200x=4500"

(Students solve equations independently, test and answer, and the teacher won't go into details)

? Teacher: "What should we pay attention to here? Answer directly 10 minutes? "

Health: "Ask when, what time is it?"

Health: "Plus departure time and exercise time."

Teacher: "How to make an equation according to the mediocre quantitative relationship (speed and x time = distance)?"

Health: "(250+200)x=4500"

Teacher: "Is there any other way?"

Sheng added: "It is also possible to divide the distance by the sum of the speeds equal to the time, and then use the properties of the equation to get x= 10, and 9 points plus 10 is the meeting time."

? Teacher: "This is actually using the quantitative relationship between distance, speed and time to form equations, so reverse thinking is very similar to formula method. When we learn the equation, in order to encounter the unknown quantity, we can directly bring the unknown quantity into the problem solving, which is more convenient. "

? Teacher: "Now that we have found out the time when they met here, can you make up the question next?" What quantities can be found with the found time? "

? Health 1: "How many kilometers did they walk when they met?"

Health 2: "How many kilometers did Xiaoyun walk less than Xiaolin when we met?"

Teacher-student summary: steps to solve travel problems with equations;

(1) Draw a line segment diagram and establish a model.

(2) Analyze the quantitative relationship and solve the unknown quantity.

? (3) Column equations and solve them.

(4) Test and answer

In addition, we must remember the equal relationship when we encounter problems:

(1) distance of a+distance of b = total distance.

? (2) (speed A+ speed B) × time = distance, that is, speed and× time = distance.

Consolidation exercise

? Two trains leave from two places 570 kilometers apart at the same time. Car A travels 1 10km per hour, and car B travels 80km per hour. A few hours later, two cars met.

(autonomous completion, correction process. Requirements: Draw line segments and build models. Parallel equation solving)

? Yan: "My equation is: 1 10x+80x=570, and the solution is x=3."

? Teacher: "He also indicated the unknown time on it. What quantitative relationship is the equation he listed based on? "

? Health: "distance traveled by car A+distance traveled by car B = total distance"

Variant exercise

The distance between the two places is 455 kilometers. Two cars, A and B, leave the two places at the same time, moving in opposite directions, and meet after 3.5 hours. Car A travels 68 kilometers per hour, and how many kilometers does car B travel per hour?

(done independently)

To sum up, the original design of this lesson is to directly explain the problem of meeting and chasing between two places, analyze and solve the relationship between the quantities, and then compare the two types. However, after teaching and research, it is considered that the travel problem is a complex problem in solving problems, and it cannot be guaranteed that students can fully master it in the same class. Therefore, if students want to solve problems better, they should first focus on the problems they encounter. However, if we want to make an intuitive comparison between the encounter problem and the chase problem, we can let the students encounter the problem first, and then let the course focus on the encounter problem after the students demonstrate and compare with the line chart, without the need for in-depth analysis of the chase problem. Therefore, the idea after the change of classroom design in this section is mainly divided into the following steps: ① Let students feel the travel problems in life romantically through the demonstration of students' situations; ② Students use their own methods to draw pictures, analyze the direction and trajectory of objects, intuitively perceive the travel problems in mathematics, and establish models; (3) Teachers demonstrate and summarize with PPT, and construct mathematical models with line graphs; ④ Quickly find out the equivalence relation (that is, the relation between the total quantity and each part quantity) from the line segment diagram. To solve this problem, we first need to build a model in our mind. Line charts and histograms are intuitive auxiliary tools. By making good use of these tools, we can find out the relationship between distance, speed and time in travel problems, and then we can constantly consolidate exercises and variant exercises by handling the quantitative relationship flexibly, so as to acquire the ability to solve problems.