(Page 48~62)

The textbook is a comprehensive application of the previous ratio and proportion knowledge. Through this part of the study, on the one hand, the concepts of ratio and proportion are consolidated, on the other hand, students can experience the application of proportion in production and life, learn to solve problems with proportional knowledge, and improve their comprehensive application ability.

This section includes: scale, enlargement and reduction of graphics, and solving problems in proportion.

Specific content description and teaching suggestions

1. scale.

Scale represents the ratio of the distance on the map to the actual distance, so it can be used as a ratio. But in fact, the distance on the map is proportional to the actual distance. Finding the distance on the map or finding the actual distance according to the scale can be solved by the column proportion formula, so it can also be regarded as the application of proportion.

This section is divided into three levels of teaching: understanding scale, finding distance or actual distance on the map according to scale, and drawing with scale.

(1) Understanding scale.

Writing intention

The textbook first explains why the ratio of the distance on the map to the actual distance should be determined, clarifies its significance, and gives the concept of scale. Then, combined with the scales of two maps, the digital scale and line scale are introduced, and through an enlarged diagram of a mechanical part, students know how to express the scale that enlarges the actual distance. Finally, for the convenience of calculation, the scale is usually written as the ratio of 1. In the teaching of 1, the line scale is rewritten into numerical scale, which paves the way for the later scale calculation.

Teaching suggestion

When teaching scale, you can show a map represented by digital scale. Combined with map description, it is necessary to reduce the actual distance by a certain proportion to draw the scale. Then enlarge the scale on the map and ask the students to say what 1 in the scale means. What does 10000000 mean? Then show a map marked with digital scale and ask the students to talk about its specific meaning. In addition, show a map marked with line segment scale to let students know the line segment scale.

In order to fully understand the scale, you can also show an enlarged map, indicating that sometimes the actual distance needs to be enlarged to a certain proportion and then drawn on the map. Ask the students to find out the scale of this picture, say what it means, and realize that when the former is greater than the latter, it means enlargement.

Combined with the above three scales, it shows that the scale is usually written as the ratio of 1 for the convenience of calculation.

Teaching example 1 can be combined with the recognized line segment scale, so that students can learn how to rewrite the line segment scale into numerical scale: according to the line segment scale, write the ratio of the distance to the actual distance on the map. Because the units of the distance and the actual distance on the map are different, different units should be changed into the same units. When changing 50 kilometers into centimeters, how many zeros should be added after 50? Students are prone to make mistakes, so we should pay attention to the combination of students' mistakes so that they can master the correct methods. Finally, the scale is a ratio without a unit name.

After "doing one thing", students can make clear the way to find the scale according to the distance in the picture and the actual distance through exchanges and discussions: first, determine the first and second items of the scale according to the meaning of the scale, and write the scale, so as not to write the distance on the picture and the actual distance wrong; Then turn the two items into the same unit; Finally, the ratio is simplified to an integer ratio of 1.

(2) Example 2.

Writing intention

Teaching finds out the actual distance according to the scale and the distance on the map. The textbook gives the map and scale of Beijing subway line and the length of subway line 1 in the map, and calculates the actual length. The textbook gives a complete problem-solving process: first, set the actual distance as x cm, and then calculate the actual distance by solution ratio method according to distance/actual distance = scale on the map.

Teaching suggestion

In this example, according to the scaling relation, the unknown number is obtained by applying the equation. This method has been mastered by students when they study in solution ratio. Pay attention to the following points when teaching this example. First, when setting the unknown, because the distance on the map and the actual distance use different units, it is difficult to set which length unit should be used when setting X, so we should pay attention to guidance in teaching. How many kilometers is the actual distance required, but how many centimeters is the known distance on the map? You can first set the actual distance to x centimeters, calculate the number of centimeters of the actual distance, and then convert it into kilometers. Second, because the distance/actual distance on the map = scale, the scale here can be regarded as a constant, that is to say, the distance on the map is proportional to the actual distance, so the problem about scale can also be solved in proportion, and this problem can be pointed out in the process of solving problems.

There is no arrangement of "doing one thing" in the textbook. Teachers can do a topic of finding the distance on the map as feedback and consolidation exercises.

(3) Example 3.

Writing intention

It is to comprehensively use the relevant knowledge of scale to solve practical problems. The problem here is to draw the playground plan according to the actual length of the school playground. To solve this problem, we need to use the two things we have learned before: first, determine the scale of the plan. The second is to find the distance on the map according to the scale.

The textbook shows the picture of group discussion, and it is suggested to carry out it in the form of cooperative learning. Through students' discussion, the steps to solve this problem are put forward. First determine the scale, and then determine the length and width of the playground according to the scale.

The method and result of finding the distance between two students on a long and wide picture appear in the textbook. The boy on the left finds out the distance between the length and width on the map by solving the ratio method according to the scale; The girl on the right, according to 1 cm on the map, represents the actual ground distance10 m. There are 8 10 m in 80 m and 6 10 m in 60 m. The distance between the length and width on the map is 8 cm and 6 cm respectively. This method is a little simpler than the method on the left.

Teaching suggestion

In teaching, you can show the problems, discuss the steps of solving problems in groups and start calculating.

After discussion, students choose a group to report: What is a definite scale? How to find the distance on a long and wide graph? What was the result? Can other teams evaluate whether the size they have determined is appropriate? (Judging from the distance between the length and width of the graph, let the students understand that the appropriate proportion should be determined according to the size of the graph.) Is the length and width they calculated correct?

When discussing the method of finding the distance on a long and wide graph, students should understand the algorithm and different characteristics of the method proposed by two students in the book.

Finally, let each group adjust the scale, calculate the length, width and distance of the map, and draw the plan. Pay attention to remind students to mark the scale on the map.

(4) "Do it".

Question 1 gives a map of a place, and the line segment scale is given in the map. In practice, first measure the distance on the map (the distance from Hexi Village to the bus station is 2 cm), and then calculate the actual distance between the two places.

This problem can be solved by solution ratio according to the scale, or it can be solved in this way: the distance of 1 cm on the map is equivalent to the actual distance of 600 m, because the distance between Hexi Village and the bus station is 2 cm on the map, so the actual distance between the two places is1200 m.

The second question, determining the location, is what I learned in the past. Here, the distance between each student's home and school is calculated according to the given scale, and then the positions of their three homes are marked.

(5) Explanations and teaching suggestions for some exercises in Exercise 8.

1 The problem is to rewrite the numerical scale into a line scale. Example 1 is to rewrite the line segment scale into numerical scale. Through this exercise, let the students know that the digital scale should be changed to the line scale. Finally, in the ratio of the distance on the map to the actual distance, the unit of the actual distance should be rewritten as the required unit.

Question 3. Find the proportion of ladybug pictures. The title gives the actual length of ladybug. We have to measure its length on the map first, and then calculate the scale with the distance on the map. This is an enlarged picture of ladybug, so the last item on the picture scale is 1. Please remind the students to pay attention to this when practicing.

Question 9: Let the students measure the length and width of the house on the spot, then calculate their distance on the map according to the proportion, and finally draw the floor plan of the house. When determining the scale, students should be reminded to determine according to the size of the plan.

In the question 10, the appropriate scale should be determined according to the size of the given plan. It is more appropriate to use the distance of 1 cm on the map to represent the actual distance of 200 m.

When it comes to drawing with scale, students should be reminded to indicate the scale in the drawing.

2. Magnification and reduction of graphics.

Writing intention

The enlargement and reduction of graphics is the practical application of proportion. Through the study of this part of the content, students can understand the phenomenon of enlargement and reduction from a mathematical point of view, and know that after the figure is enlarged or reduced according to a certain proportion, only the size has changed, but the shape has not changed, so as to realize the similar characteristics of the figure change, and can enlarge or reduce the simple figure according to a certain proportion on the square paper.

Firstly, the textbook presents some zoom-in and zoom-out phenomena in life in the form of pictures: taking pictures, reading with a magnifying glass, and zooming in on charts, people and shadows with a projector, so that students can get a preliminary understanding of the zoom-in and zoom-out phenomena in life.

Then, the characteristics of graphic enlargement and reduction are further studied through example 4. In the textbook, students are first required to draw three simple enlarged plans of plane graphics on square paper according to the ratio of 2∶ 1, so that students can understand through drawing that to enlarge a graphic according to a certain proportion, only the edges of the graphic can be enlarged according to a certain proportion. Then ask the students to observe the pictures before and after enlargement. By comparison, they realized that the size of the figure had changed before and after enlargement, but the shape had not changed. On this basis, let the students reduce the three enlarged figures by the ratio of 1∶3, and realize that after a figure is reduced by a certain ratio, the figure becomes smaller, but the shape remains the same. Finally, the textbook comprehensively understands from two aspects, and it is concluded that after all sides of the graph are enlarged or reduced in the same proportion, the obtained graph only changes in size and shape.

Teaching suggestion

(1) Show the picture on page 56 of the textbook and ask the students to say what the picture reflects. What is a magnifying object? What is a shrinking object? This shows that there are many phenomena of enlargement and contraction in life. Now let's learn "Zooming in and out of graphics".

(2) Teaching example 4.

Example 4 illustrates the significance of graphic magnification of 2∶ 1, and lets students know that graphic magnification of 2∶ 1 means that each side of the graphic is magnified twice as much as the original one. Then ask the students to draw enlarged figures. When drawing a right triangle, students can be guided to think that the hypotenuse of a right triangle cannot directly see how many squares it has. Is it enough to enlarge the two right sides to twice the original size? After drawing, students can count or measure, and find that the length of the hypotenuse after enlargement is twice as long as that before enlargement. Then ask the students to observe and compare the original picture with the enlarged picture to see what changes have taken place. Combined with specific graphics, through discussion and communication, it is known that after a graphic is enlarged at the ratio of 2∶ 1, the length of each side of the graphic is enlarged to twice the original length, but the shape of the graphic remains unchanged.

Then the question arises: If the edge of the graph is reduced by 1∶3, what will happen to the enlarged graph? Let the students discuss. It is concluded that the figure is reduced, but the shape remains the same, and each side of the reduced figure is reduced to the original length respectively.

On this basis, guide students to draw the conclusion that "all sides of the figure are enlarged or reduced in the same proportion, but the size has changed and the shape has not changed."

Do it independently, communicate how to think and operate, and correct mistakes in time.

3. Solve problems in proportion.

This part of the content is mainly about the positive and negative ratio, which students have actually touched before, but only answered by normalization and induction. Here, we mainly learn to answer with proportional knowledge. Through solving, students can further skillfully judge the positive and negative proportional quantity, deepen their understanding of the concept of positive and negative proportional quantity, and make better preparations for applying proportional knowledge to solve some problems in middle school mathematics, physics and chemistry. At the same time, because the solution is based on the meaning of positive and negative proportion, it can also consolidate and deepen the understanding of the simple equation.

(1) Example 5.

Writing intention

Teaching applies the meaning of direct proportion to solve problems. The textbook leads to the practical problem of asking for water fee from Aunt Zhang's conversation with her. In order to strengthen the connection between knowledge, let students answer first with the learned methods, and then learn to answer with proportional knowledge.

In order to highlight the idea of solving problems with proportional knowledge, the textbook emphasizes two main points with color words. First of all, we should judge what proportion the two quantities in the topic are, and then list the necessary equality relationship of the proportion formula.

The process of solving with proportional knowledge is relatively complete, that is, setting unknowns and listing equation solutions. Finally, the textbook is expanded: let students think about how to ask about water consumption if they know the water fee.

Teaching suggestion

The key to solving the problem of positive-negative ratio with proportional knowledge is to let students correctly find out two related quantities, judge which proportion they are in, and then list the equations according to the meaning of positive-negative ratio. Therefore, some quantitative relations can be given before teaching, so that students can judge what proportion and what basis.

After introducing Example 5, I asked: Have you ever learned to answer such questions? Can you answer that? Ask the students to answer by themselves and exchange solutions. Further explanation: This kind of question can be answered with the knowledge of proportion, and today we will learn to answer with the knowledge of proportion.

The following questions can be asked for students to think and discuss:

What are the two quantities in the (1) problem?

(2) what is their ratio? On what basis do you judge?

③ According to this proportional relationship, can you list the equations?

Through discussion and exchange, students can make it clear that because the water price is certain, the water fee is directly proportional to the tonnage of water used. In other words, the ratio of water fee to water consumption of the two companies is equal. Then set the unknown number, list the equation according to the meaning of positive proportion, and then solution ratio worked out the unknown number. After the solution, students can also check whether the unknown X meets the meaning of the question. The test method is to substitute the obtained number into the original equation (that is, the equation) to see if the equation is established. Substituting the obtained 16 into the equation, the left formula = = 1.6, the right formula = = 1.6, and the left formula = right formula, that is, their ratios are equal, which accords with the meaning of the question, so the solution is correct.

Finally, it puts forward "Uncle Wang's water fee last month 19.2 yuan. How many tons of water did their family use last month? " Ask students to apply proportional knowledge solutions and then communicate. Through revision and communication, students can make it clear that after the conditions and problems in Example 5 are changed, the proportional relationship between water fee and water tonnage in the topic has not changed, but the unknown number has changed.

(2) Example 6.

Writing intention

Learn to solve problems in inverse proportion. The arrangement idea is similar to Example 5. It is also to let students answer in the way they have learned, and then learn to answer with proportional knowledge. After answering, let the students think about how to answer the questions if they change the conditions of the questions. Finally, arrange "doing" with the same topic "Xiao Ming buys a pen", so that students can consolidate the idea of answering questions in positive and negative proportions.

Teaching suggestion

Reference can be made to the teaching of Example 5. Attention should be paid to inspiring students to make equations according to the meaning of inverse proportion, so that students can further master the characteristics of inverse proportion of two quantities and the problem-solving methods involving inverse proportion.

"Do one thing" can directly let students answer with proportional knowledge. After the answer, compare the two questions and say in a column what is the difference between the two questions and how to solve them.

Combined with "do one thing", summarize the steps of applying proportional knowledge to solve problems: 1. Analyze the meaning of the question, find out two related quantities, and judge whether it is proportional and what proportion. Second, list the equations according to the meaning of direct proportion or inverse proportion. Third, solve the equation (test after solving it) and write the answer.

4. Explanations of some exercises in Exercise 9 and teaching suggestions.

1 question, through judgment, let students further clarify that after a figure is enlarged or reduced according to a certain proportion, its edges are also enlarged or reduced according to this proportion. In this question, only the side of D is enlarged by 2∶ 1, because its four sides are twice as long as the side of A in the original picture. After judging, let the students talk about the reasons.

In question (3) of question 2, we will look at three triangles together. B and c are enlarged views of triangle a; A is a simplified figure of triangle C.

Organization and review

(Page 63 ~ 65)

This part of the textbook collates and reviews the key contents of proportional units. According to the basic requirements of this unit and the specific situation of students' learning, review teaching is carried out in a targeted manner. For some important and confusing concepts, we should pay attention to comparison and review, so that students can clearly distinguish and deepen their understanding of the concepts.

Specific content description and teaching suggestions.

1. Question 1, review the meaning of "ratio" and "proportion". Let the students talk about "What is comparison?" "What is proportion", and illustrate the connection and difference between "proportion" and "proportion" with examples.

Question 2, review the basic nature of proportion and the solution of proportion. Let the students do the exercise of solving the comparison first, and then explain what the comparison is based on.

Question 3, review the meaning of positive proportion and inverse proportion. Let the students judge first and talk about the basis of judgment. Pay attention to let students fully express themselves. Combined with this question, students can complete the third question of exercise 10.

Question 4. Review solving problems with proportional knowledge. On the basis of independent answers, what is the connection and difference between the quantitative relations of the two questions? What is the basis of the formula? Then complete questions 4 and 5 of exercise 10.

2. The "Reading Materials" on page 65 introduces Fibonacci series. Fibonacci series is named after the mathematician Fibonacci. The textbook first briefly introduces Fibonacci's life and his contribution to the development of mathematics, and then introduces the origin of Fibonacci sequence with vivid and interesting "rabbit problem". By observing the rabbit number change chart and the corresponding rabbit number change table, let the students see the relationship between the month and the rabbit logarithm, so as to understand the characteristics of Fibonacci sequence.

When learning the "rabbit problem", students will have some difficulties in understanding the birth process of rabbits, so they should be encouraged to explore by drawing pictures and other intuitive ways. For example, you can use icons such as small circles or triangles to represent small rabbits, and icons such as big circles or triangles to represent big rabbits. The number of rabbits per month can be presented by charts. In the process of presentation, students will discover the law independently, that is, except for the first two months, the rabbit logarithm of each month thereafter is the sum of the rabbit logarithm of the first two months. Using this rule, students can do further research, such as the following 13, 14, … months, a * * *, how many pairs of rabbits are there?

When teaching, we should grasp the teaching requirements. The content of "reading materials" is only to let students improve their interest in mathematics learning and feel the inner charm of mathematics through extracurricular reading and independent inquiry. Therefore, this part of the content does not need to be unified and rigid for all students.