Teaching objectives
1. Understand and master the meaning and basic properties of proportion, know the names of various parts of proportion, and know the relationship between proportion and proportion.
2. Develop students' thinking in activities such as example, classification, observation, comparison, abstraction and generalization.
3. Stimulate students' consciousness of independent participation and the spirit of active inquiry in specific practical activities, and feel the connection between mathematics and life.
Teaching emphasis: understand the meaning and basic nature of proportion.
Teaching difficulty: judging whether two ratios can form a proportion.
teaching process
first kind
First, import
1. The courseware shows the picture of the national flag. Three different scenes all have the same sign: a five-star red flag. The five-star red flag is the symbol of China people and country; These national flags are big and small. Do you know the length and width of these national flags?
2. The courseware shows the length and width of the national flag and asks questions.
The national flag at the flag-raising ceremony in Tiananmen Square: 5 meters long and 10/3 meters wide.
The national flag at the flag-raising ceremony in the playground: 2.4m long and 1.6m wide.
National flag in the classroom: 60 cm long and 40 cm wide.
These national flags vary in size. Do you want them to be as big as you want? Is there anything in it? Each national flag is different in size, but their length and width imply the same characteristics. What is this? In this lesson, we will learn the meaning and basic nature of proportion by combining the knowledge of national flag. Teacher's blackboard writing project.
Second, guide inquiry and learn new knowledge.
1, the significance of teaching proportion.
(1) P32 cases.
What is the aspect ratio of each national flag? Calculate the aspect ratio of the national flag by its name.
5: 2.4: 1.6 60:40 15: 10
What is the length-width ratio of each national flag? (All are equal)
5:** =2.4: 1.6 60:40= 15: 10 2.4: 1.6=60:40
Two expressions with equal ratios like this are called proportions.
(2) We also know that two different quantities can also form a ratio, for example:
A car travels 80 kilometers in two hours for the first time and 200 kilometers in five hours for the second time. The list is as follows:
Time (hours) 2 5
Distance (km)
Read the questions by naming the students.
Teacher: This question involves the relationship between time and distance. We use tables to express them. The first column of the table indicates time in hours, and the second column indicates distance in kilometers. How many kilometers did this car run in two hours for the first time? How many kilometers is the second five-hour drive? Ask questions and fill in the form. )
"According to this table, can you write down the ratio of distance and time between the first and second trips?" According to the students' answers, the teacher wrote on the blackboard:
The ratio of distance and time for the first trip is 80:2.
The ratio of the distance and time of the second journey is 200:5.
Ask the students to work out the ratio of these two ratios. The students call the roll and the teacher writes on the blackboard: 80:2=40, 200:5=40. Ask the students to observe the ratio of these two ratios. Ask again: What did you find? "(The ratio of these two ratios is 40, and these two ratios are equal. )
Teacher's explanation: Because these two proportions are equal, you can connect them with an equal sign to form a proportion. (Blackboard: 80:2=200:5) Two formulas with equal ratios like this are called proportions.
Pointing to the proportion formula 4.5:2.7= 10:6, he asked, "Who can tell what proportion is?" Guide students to observe that the two ratios are equal. Then write on the blackboard: two expressions with equal ratios are called proportions. Let the students read together.
"In the sense of proportion, we can know that proportion is composed of several ratios? What conditions must these two ratios meet? So, what is the key to judge whether two proportions can form a proportion? What if I can't see at a glance whether the two ratios are equal? "
According to the students' answers, the teacher concluded: Through the above study, we know that the ratio is composed of two equal ratios. When judging whether two proportions can form a proportion, the key is to see whether these two proportions are equal. If you can't see whether the two ratios are equal at a glance, you can simplify the two ratios before looking. For example, to judge whether 10: 12 and 35: 42 can form a ratio, we must first calculate 10: 12=, 35: 42=, so10:12 =. (For example, write on the blackboard while talking. )
(3) Compare the two concepts of "ratio" and "proportion".
Teacher: We learned "comparison" last semester, and now we know the meaning of "proportion". What is the difference between "ratio" and "proportion"?
Guide the students to compare the meaning and the number of terms, and finally the teacher draws the conclusion that the ratio means that there are two terms when two numbers are divided; Proportion is an equation, which means that two proportions are equal and there are four terms.
(4) Consolidate exercises.
(1) Use gestures to judge whether the two proportions on the card below can form a proportion. (Yes, just open your thumb and forefinger; You can't just cross your index fingers. )
6:3 and 12:6 35:7 and 45:9 20:5 and 16:8 0.8:0.4 and 0.3:0.6.
After the students judge, tell the basis of the judgment.
② Do P33 "Do it".
Ask students to read books without copying the questions, and write the two ratios that can constitute the proportion directly in the exercise book. The teacher will correct it while patrolling. Let them talk about how it was done and see if it was done correctly.
(3) Give the numbers 2, 3, 4 and 6, and let the students form different proportions (no integer is required).
④P36 Exercise 6, Question 65438 +0 ~ 2.
For the four numbers that can make up the proportion, write down the proportion that can make up. As long as the composition ratio can be established.
The fourth question, the four numbers given are all fractions. When writing proportional expressions, students should also write fractional forms.
2. The basic nature of teaching proportion.
(1) Name of each part of teaching proportion.
Teacher: Students can correctly judge whether two proportions can form a proportion, so what are the names of the parts of the proportion? Please open the textbook P34 to see what the proportional term, external term and internal term are.
Ask the students to point out the external and internal terms of proportion on the blackboard.
(2) The basic nature of teaching proportion.
Teacher: We know the names of the parts of proportion, so what is the essence of proportion? Now let's study it. (Write down the meaning of proportion on the blackboard: the basic properties of proportion) Please calculate the product of two internal terms and the product of two external terms in this proportion respectively. Teacher's blackboard writing:
The product of two external terms is 80×5=400.
The product of two internal terms is 2×200=400.
"What did you find?" (The product of two outer terms is equal to the product of two inner terms. ) blackboard writing: 80×5=2×200 "Is this the ratio?" Ask the students to calculate the proportional formula judged earlier in groups.
Through calculation, we find that all the proportional formulas have the same law. Who can say this law in one sentence?
Finally, the teacher summed it up and wrote it on the blackboard: proportionally, the product of two external terms is equal to the product of two internal terms. And explain the basic nature of this is called proportion.
"If the proportion is written as a fraction, what is the basic nature of the proportion?" (Pointing to 80:2=200:5) The teacher asked, rewritten as: =
"What are the two figures of this proportion of external projects? What about the internal items? "
"Because the product of two internal terms is equal to the product of two external terms, when the proportion is written in the form of a fraction, what about the product of the cross multiplication of the numerator and denominator at both ends of the equal sign?
After the students answered, the teacher stressed that if the proportion is written in fractional form, the basic nature of the proportion is that the numerator and denominator at both ends of the equal sign cross and multiply, and the product is equal.
3. Consolidate the exercises.
Before judging whether the two ratios are in direct proportion, we first judge by calculating their ratio. After learning the basic nature of proportion, we can also use the basic nature of proportion to judge whether two proportions can be proportional.
(1) Use the basic properties of the ratio to judge whether 3:4 and 6:8 can form a ratio.
(2)P34 "Do it".
Third, consolidate and deepen, expand thinking.
1. What's the difference between proportion and proportion?
Step 2 fill in the blanks
5:2=80:( ) 2:7=( ):5 1.2:2.5=( ):4
3. First apply the meaning of proportion, and then apply the basic properties of proportion, and judge that two proportions in the following group can constitute proportion.
(1) 6:9 and 9: 12 (2) 1.4:2 and 7: 10 (3) 0.5:0 .2 and:
4. Can the following four numbers form a proportion? Write down the proportion of the composition.
2, 3, 4 and 6
Fourth, the whole class summarizes and improves understanding.
What knowledge have we learned through this lesson? What is proportion? What is the basic nature of proportion? What can be done by applying the basic nature of proportion?
Fifth, classroom exercises to help digestion.
P36 ~ 37 Questions 3 ~ 6.
Sixth, extracurricular supplement, expansion and extension.
1, judge.
(1) If 3×a=5×b, then 5: a = 3: b.
(2): and:, the proportion of energy and:.
(3) In a proportion, the two external terms are 7 and 8 respectively, so the sum of the two internal terms must be 15.
2. How many proportions can be formed by using the four numbers of 6, 8, 10 and 12 respectively?
3. Please use four composite numbers within 20 to form a ratio, two of which are equal.
Second lesson
First, review and pave the way
1. Last class, we learned proportion. Please judge which two proportions can make up the proportion. (Courseware demonstration)
2:3 0.5:0.2 0.6:0.8 1/3: 1/ 10 3: 1.2 4:6 2/3: 1/5 3/5:4/5
Discuss communication: What is proportion? Were those students' judgments right just now? How did you know?
2. Fill in the blanks and explain the reasons.
1:3=( ):( ) 3:8=9:( )
Because there are many ratios equal to 1:3, the answer to this question is not unique, as long as the ratio is 1/3. 5: 3 = 9: (24), according to the basic properties of the ratio, the product of the inner term is 8×9=72, and the product of the outer term should also be 72,72 ÷ 3 = 24, so 24 is filled in the brackets.
3. Introduce the unknown term in 3:8=9 by the title: () It can also be expressed by x, and write 3: 8 = 9: x. Finding the unknown term in the ratio like this is called solution ratio. Teacher's blackboard writing project.
Second, guide exploration and learn new knowledge.
1, what is the solution ratio?
We know that there are four items in the ratio * * *. If we know any three items, we can find another unknown item in this ratio. Finding the unknown term in the proportion is called the solution ratio. Solution ratio should be solved according to the basic nature of proportion.
2. Teaching example 2.
(1) Set the unknown term as X. Solution: Let the height of this model be x meters.
(2) List the proportion according to its meaning: X:320= 1: 10.
(3) Ask students to point out the external and internal terms of this ratio, and explain which three items they know and which one to seek.
According to the basic nature of proportion, what form can it become? 3x=8× 15 .
What has this become? (equation. )
Teacher's explanation: in this way, the solution ratio becomes a solution of the equation, and the value of unknown x can be obtained by using the previously learned method of solving the equation. Solution ratio should also be written as "solution:" because solving equations should be written as "solution:".
(4) The students said that the teacher wrote the solution ratio on the blackboard.
Teacher: From the process of solving the ratio just now, we can see that the ratio can be changed into an equation according to the basic properties of the ratio, and then the unknown X can be obtained by solving this equation. ..
3. Teaching example 3.
Example 3: Solution ratio =
Q: "What's the difference between this ratio and Example 2?" (This ratio is in fractional form. )
Can this fractional proportion be solved by equation according to the basic properties of proportion?
After the students answered, the teacher explained that when writing equations, the product of unknowns is usually written on the left side of the equal sign, and then the blackboard is: 1.5x = 2.5x6.
Ask the students to fill in the solution process in the textbook. After solving it, let them say how they solved it.
4. Summarize the process of solution proportioning.
Just now, we learned the proportion. Let's think back, what should we do first? (It becomes an equation according to the basic properties of proportion. )
What should we do after it becomes an equation? (According to the method of solving equations learned before. )
As can be seen from the above process, which step is new knowledge in the process of solution comparison? (It becomes an equation according to the basic properties of proportion. )
5.P35 "Do it". Students answer independently. When correcting, let the students say how to do it.
Third, consolidate and deepen, expand thinking.
P37 Question 7.
Fourth, the whole class summarizes and improves understanding.
What is the solution ratio? What is the basis of the solution? What should I pay attention to in the writing format of Xiebi?
Fifth, classroom exercises to help digestion.
P37 ~ 38 Question 8 ~ 1 1.
Sixth, extracurricular supplement, expansion and extension.
1, P38 question 12, 13.
2,4: 8 = 1 2: 24. If the second term decreases1,how much does the fourth term decrease to make the ratio stand?
3. Make a ratio with two ratios. Both internal terms of the known ratio are 15. Please find out the two external terms of this ratio and write down the ratio.
4. The four terms of a ratio are all integers greater than 0, and the ratio of its two ratios is, the first term is 3 times smaller than the second term, and the third term is 3 times that of the first term. Please write down this ratio.