Mathematics courseware of the sixth grade in primary school.

How much do you know about the second volume of the sixth grade mathematics courseware? Maybe many people are not very clear about it. The following is the model essay of the second volume of the sixth grade mathematics courseware I shared. Let's have a look.

1, the meaning and basic properties of proportion

first kind

Teaching content: the significance and basic properties of P32 ~ 34 ratio.

Teaching purpose:

1, so that students can understand the meaning and basic properties of proportion and correctly judge whether two proportions can form a proportion.

2. Cultivate students' abstract generalization ability by guiding inquiry, generalization, discussion and cooperative learning.

3. Make students initially perceive that things are interrelated, changing and developing.

Teaching emphasis; The Significance and Basic Properties of Proportion

Difficulties in teaching: Use the basic nature of ratio to judge whether two numbers in a paragraph are proportional and form the proportion correctly.

Teaching process:

First, review the old knowledge, review the groundwork

1, please recall what we learned last semester. Who can say what is called comparison? And give examples to illustrate what is the former, what is the latter and what is the proportion.

The teacher wrote the examples given by the students on the blackboard and pointed out the names of the parts of the ratio.

2. We know that the quotient obtained by dividing the front and back terms of the ratio is called the ratio. Will you find the proportion? The teacher writes down the following groups of ratios on the blackboard and asks the students to find out their ratios.

12: 16 : 4.5:2.7 10:6

After calculating the ratio of each ratio, the students ask: which two ratios are equal?

(The ratio of 4.5:2.7 is equal to the ratio of 10:6. )

Teacher's explanation: Because these two ratios are equal, so are these two ratios. Let's connect them with an equal sign. (Blackboard: 4.5: 2.7 = 10: 6) What's the name of this formula that indicates that two ratios are equal? This is what we will learn in this class. (blackboard title: the meaning of proportion)

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.

Proportion can also be written as: = =

(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) 25

Distance (km) 80200

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 you make with the four numbers 0, 8, 12?

3. Please use four composite numbers within 20 to form a ratio, two of which are equal.

Proportion of the second kind of solution.

Teaching content: P35 ~ 37 solution ratio.

Teaching purpose: 1. Make students learn solution ratio's method, and further understand and master the basic nature of proportion.

2. Through cooperation, communication and practice, improve students' ability to use proportion and its basic nature.

3. Cultivate students' knowledge transfer ability and enhance their sense of cooperation.

Teaching emphasis: make students master the method of solving ratio and learn to solve ratio.

Teaching difficulties: guide students to rewrite the proportion into the form that the product of two internal terms is equal to the product of two external terms according to the basic nature of proportion, that is, the equation with unknown numbers that they have learned.

Teaching process:

First, review the old knowledge, review the groundwork

1. Last class, we learned something about proportion. Who can talk about the proportion? What is the basic nature of proportion? What can be done by applying the basic nature of proportion?

2. Judge whether two proportions in the following groups can constitute proportions? Why?

6:3 and 8: 4; And:

3. In this lesson, we will continue to learn the proportion and solution proportion. (blackboard writing topic)

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.

2, the meaning of positive proportion and inverse proportion.

Proportional quantity in first-class products.

Teaching content: P39 ~ 4 1 is in direct proportion.

Teaching requirements: 1. Make students understand the meaning of direct ratio and judge whether it is direct ratio according to the meaning of direct ratio.

2. Cultivate students' generalization ability and analytical judgment ability.

3. Cultivate students' ability to analyze problems from the perspective of development and change.

Teaching emphasis: the characteristics of proportional quantity and its judgment method.

Difficulties in teaching: Understand the proportional relationship between two variables, and discover and think about the changing law of two related quantities.

Teaching process:

First, look around and review the mat.

1, know the distance and time, and find the speed.

2. Knowing the total price and quantity, find the unit price.

3. The total amount of work, working hours and working efficiency are known.

Second, guide exploration and learn new knowledge.

1, teaching example 1:

Display: a train 1 hour travels 90 kilometers, and it travels for 2 hours 180 kilometers.

3 hours 270 kilometers, 4 hours 360 kilometers,

Drive 450 kilometers in five hours and 540 kilometers in six hours.

7 hours 630 kilometers, 8 hours 720 kilometers ...

(1) Display the following table and fill in the form.

Time and distance of the train.

time

Travel distance

Thinking about filling in the form: What did you find when filling in the form?

Time is changing and distance is changing, so we say that time and distance are two related quantities. (blackboard writing: two related quantities)

According to the calculation, what did you find?

The ratio of two corresponding numbers is the same or fixed, which is mathematically called determination.

The formula shows that their relationship is: distance/time = speed (certain) (blackboard writing)

(2) Teacher's summary:

Students know that time and distance are two related quantities through filling in forms and communication, and distance changes with time. As time goes on, the distance is also expanding. With the shortening of time, the distance is also shortening. That is: distance/time = speed (certain)

2, teaching example 2:

(1) Table of Total Price of Rice and Printed Fabric

Quantity 1234567 ...

The total price is 8.216.424.632.4438+0.049.257.4. ...

(2) Observe the chart and find out what rules?

Express their relationship with the formula: total price/meter = unit price (certain)

3. The abstract generalizes the meaning in direct proportion.

(1) Compare Example 1 with Example 2. Think and discuss: What are the similarities between these two examples?

(2) Two related quantities, one of which changes and the other changes. If the ratio (i.e. quotient) of two corresponding quantities is certain, these two quantities are called proportional quantities, and their relationship is called proportional relationship.

(3) Read P39 to further understand the significance of positive proportion.

(4) If X and Y are used to represent two related quantities, and K is used to represent their proportion (certain), how can the proportional relationship be expressed in letters?

X/y=k (ok)

(5) According to the meaning of direct ratio and the formula expressing direct ratio, think about it: What conditions must the two quantities that constitute the direct ratio have?

4. Read P40 case 2.

How many quantities are there in the (1) question? Which two quantities are related quantities?

(2) What is the ratio of volume to height? What's the ratio? Are you sure?

(3) What is their quantitative relationship?

(4) What do you find from the picture?

(5) Without calculation, according to the image, if the height of water in the cup is 7 cm, what is the volume of water? How high is 225 cubic centimeters of water?

Third, the class summary:

What is proportional quantity? What conditions must be met? How to judge that the quantity is proportional?

Fourth, classroom exercises:

1, P4 1 Do it.

2.P43 ~ 44 Exercise 7 1 ~ 5.

Inverse proportional quantity in the second category

Teaching content: P42 in inverse proportion.

Teaching purpose: 1. Understand the meaning of inverse proportion, and correctly judge whether two quantities are inverse proportion according to the meaning of inverse proportion.

2. By guiding students to discuss, explore, analyze and cooperate, students can further understand the relationship between things and the law of development and change.

3. The tentative idea of permeability function.

Teaching emphasis: guide students to summarize the inverse proportional quantity, that is, two numbers corresponding to two related quantities have a certain product, and then summarize the inverse proportional relationship abstractly.

Teaching difficulty: use the meaning of inverse proportion to correctly judge whether two quantities are inverse proportion.

Teaching process:

First, review and pave the way

1, are the following two quantities directly proportional? Why?

The price of buying exercise books is 0.80 yuan 1 book; 1.60 yuan, 2 copies; 3.20 yuan, 4 copies; Six copies, 4.80 yuan.

2. What are the characteristics of proportional quantity?

Second, explore new knowledge.

1, lead into a new lesson: In this lesson, we will continue to learn another feature of common quantitative relations-inverse proportional quantity.

2. Teach P42 case 3.

(1) instruct students to observe the data in the above table, and then answer the following questions:

A, what are the two quantities in the table? Are these two quantities related? Why?

B. Does the height of water change with the change of bottom area? How did it change?

C. What is the ratio of the two corresponding figures in the table? Are you sure? What is the product of two corresponding numbers? Can you find any patterns from it?

D. what does this product mean? Write the quantitative relationship between them.

(2) What did you find from it? What's the difference between this and the review questions?

A, students discuss and communicate.

B, guide the students to answer:

(3) Teachers guide students to make it clear that because the volume of water is certain, the height of water changes with the change of bottom area. The bottom area increases, the height decreases, the bottom area decreases, the height increases, and the product of the height and the bottom area is certain. We say that the height is inversely proportional to the bottom area, and the height and the bottom area are called inverse proportional quantities.

(4) If the letters X and Y are used to represent two related quantities, and K is used to represent that their products are certain, what formula can be used to represent the inverse proportion? Blackboard: x×y=k (OK)

Third, consolidate the practice.

1, think about it: What conditions should an inversely proportional quantity meet?

2. Judge whether the two quantities in each question below are inversely proportional, and explain the reasons.

(1) a certain distance, speed, time.

(2) The speed and time required for Xiao Ming to walk from home to school.

(3) The parallelogram has a certain area, a bottom and a height.

(4) Kobayashi did 10 math problems, what he did and what he didn't do.

(5) Xiao Ming takes money to buy pencils, unit price and purchase quantity.

(6) Can you give an example of inverse proportion?

Fourth, the whole class

In this lesson, we learned the inverse proportional quantity, what two quantities are inverse proportional, and how to judge whether two quantities are inverse proportional.

Verb (abbreviation of verb) classroom practice

P45 ~ 46 Exercise 7, Question 6 ~ 1 1.

Comparison of the Third Kind of Positive Proportion and Inverse Proportion

Teaching content: comparison between positive proportion and negative proportion

Teaching objective: 1. Further understand the meaning of positive proportion and inverse proportion, and make clear the connection and difference between them. Master their changing rules.

2. Enable students to correctly judge the positive and negative ratio.

3. Cultivate students' ability of analysis, comparison, abstraction and generalization, and stimulate students' interest in learning.

Teaching difficulties: the connection and difference between positive and negative proportions.

Teaching emphasis: be able to judge the positive and negative ratio.

Teaching process:

First, let's review:

Judgment: What is the relationship between the two quantities in the following groups?

1, unit price is fixed, quantity and total price.

2, the distance is certain, speed and time.

3. The side length of a square and its area.

4, a certain time, work efficiency and total work.

Second, new knowledge:

1, display topic:

2. Examples of auxiliary teaching

Show table 1

Distance (km) 5 102550 100

Time (hours) 125 1020

Table 2

Speed (km/h) 1005020 105

Time (hours) 125 1020

Discuss and communicate in groups: speak your own thoughts and fill in the blanks. Guide the students to discuss and answer.

Summarize the proportional relationship between distance, speed and time.

Speed × time = distance/time = speed/distance/speed = time

Judge:

(1) The speed is unchanged. What is the ratio of distance to time?

(2) The distance is fixed. What is the ratio of speed to time?

(3) Given a certain time, what is the ratio of distance to speed?

3. Compare the relationship between positive proportion and inverse proportion.

Similarity of positive and negative proportions: there are two related quantities, one of which varies with the other.

Difference: Positive proportion makes the change the same, one quantity expands or contracts, and the other also expands or contracts. The ratio (quotient) of every two corresponding numbers is constant, and the inverse ratio is opposite. One quantity expands (or contracts), and the other quantity contracts (expands). The product of every two corresponding quantities remains unchanged.

Third, consolidate the practice.

1, do it

Judge that one of the unit price, quantity and total price is certain, and what is the relationship between the other two quantities. Why?

The unit price is fixed, quantity and total price—

The total price is fixed, quantity and unit price—

A certain quantity, total price and unit price—

2. What is the proportion of the following related quantities? Why?

The divisor of (1) is constant and proportional to the sum.

Dividend-fixed, proportional.

(2) The preceding paragraph is definite and proportional.

(3) The latter item must be proportional to the sum.

(4) The total length, width and area of the rectangle. If the length is fixed, the width and area are positively related. Under what conditions can these three quantities form a proportional relationship, and what kind of proportional relationship is it?