Inequality relation and inequality teaching plan in senior high school mathematics compulsory course.

Integrated design of inequality relation and inequality teaching plan in senior high school mathematics compulsory course.

Teaching analysis

The research of this lesson is the continuation and expansion of inequality learning in junior high school, and it is also the further development of real number theory. In the course of this lesson, students will recall the basic theory of real numbers and compare the sizes of two algebraic expressions with the basic theory of real numbers.

Through the study of this lesson, students can feel that there are a lot of inequality relations in the real world and daily life from a series of specific problem situations, and fully understand the existence and application of inequality relations. Observe, summarize and abstract the related materials of inequality relationship from the mathematical point of view, and complete the comparison process between quantity and quantity. That is, these inequality relations can be expressed by inequalities or groups of inequalities.

In the learning process of this class, some simple questions are arranged, which students can easily deal with. The purpose is to make students pay attention to the application of mathematical knowledge and methods, and at the same time stimulate students' interest in learning, and sincerely produce the desire to study inequalities with mathematical tools. According to the teaching content of this lesson, we can reproduce and recall the basic theory of real numbers, and we can compare the size of two algebraic expressions with the basic theory of real numbers.

In this kind of teaching, teachers can let students read the examples in the book, make full use of the simple tool of the combination of numbers and shapes, and directly use the one-to-one correspondence between real numbers and points on the number axis to establish the order relationship of real numbers from both numbers and shapes. On the basis of reviewing the old ones, students' understanding of inequality should be improved.

Three-dimensional target

1. Under the actual background of students' understanding of inequality, the basic theory of recalling real numbers on the number axis is used to understand the relationship between the size of real numbers and the positions of corresponding points on the number axis.

2. The difference method will be used to judge the size of real numbers and algebraic expressions, and the matching method will judge the size and range of quadratic expressions.

3. Improve students' understanding of inequality, stimulate students' interest in learning, learn new things by reviewing old ones, and experience the mystery and structural beauty of mathematics.

Important and difficult

Teaching emphasis: compare the relationship between real numbers and algebraic expressions, and judge the size and scope of quadratic expressions.

Teaching difficulty: accurately compare the size of two algebraic expressions.

Class arrangement

1 class hour

teaching process

Introduce a new course

Idea 1. Through the multimedia display of satellites and spaceships and a spectacular picture of overlapping mountains, students are brought into? Looking across the ridge into a peak, how far is it? In the vast nature and universe, students feel that there are a lot of unequal relations between the real world and daily life in specific situations, which leads to a strong desire to study unequal relations with mathematics and naturally introduces new courses.

Idea 2. (Situation introduction) List the familiar examples of students in real life, such as their height, weight, distance from school, the time of the 100-meter race, and the number of math scores, and describe the unequal relationship of some objective things by quantity. How can these unequal relations be expressed mathematically? Let students freely associate, teachers organize related materials about unequal relations, let students observe and summarize from the mathematical point of view, and let students feel that unequal relations, like equal relations, exist in the real world and daily life in large quantities. In this way, students will sincerely have the desire to study unequal relations with mathematical tools, so as to enter further inquiry learning and introduce new courses.

Promote the new curriculum

Explore new knowledge

raise a question

? 1? Recall the inequalities learned in junior high school and let the students say? Unequal relationship? With what? Inequality? How to use inequality to study and express inequality?

? 2? In the real world and daily life, there are both equal relations and a large number of unequal relations. Can you give some practical examples?

? 3? What is the relationship between any two points on the number axis and the corresponding two real numbers?

? 4? What is the relationship between any two real numbers? How to express this relationship in logical terms?

Activity: Teachers guide students to recall the concept of inequality they learned in junior high school, so that students can be clear? Unequal relationship? With what? Inequality? Similarities and differences. Unequal relations emphasize relations. Can symbols be used? & gt& lt stands for, while inequality stands for the unequal relationship between the two, available? A> bachelor of arts

Teachers and students can give examples of inequality in daily life together, so that students can fully cooperate and discuss and feel that there are a lot of inequalities in the real world. Students can further study the related contents of inequalities on the premise of understanding the actual background of some inequalities.

Example 1: The weather forecast for a certain day predicted that the maximum temperature was 32℃ and the minimum temperature was 26℃.

Example 2: Any two different points A and B on the number axis, if point A is to the left of point B, then xA

Example 3: If a number is non-negative, then it is greater than or equal to zero.

Example 4: The line segment between two points is the shortest.

Example 5: The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is smaller than the third side.

Example 6: The road sign with a speed limit of 40 km/h indicates that the driver should keep the speed V below 40 km/h when driving on the road ahead.

Example 7: The quality inspection of a brand of yogurt stipulates that the fat content in yogurt should not be lower than 2.5% and the protein content should not be lower than 2.3%.

The teacher further instructed: It is of course good to find that the mathematics around us is good, which shows that students have entered the subject of mathematics, but at the same time, we can observe, summarize and abstract these quantities from the perspective of mathematics, which is what everyone who studies mathematics must do. So, what knowledge can we use to express these unequal relationships? It is easy for students to think of inequalities or groups of inequalities to represent these unequal relationships. Then inequality is a formula that connects two algebraic expressions with unequal symbols, such as -7.

The teacher instructed the students to express the above seven examples with inequalities. For example 1, if t is used to represent the temperature of a certain day, then 26℃? t？ 32℃. Example 3, if X represents non-negative, then X? 0. Example 5, |AC|+|BC| >|AB|, as shown below.

|AB|+|BC| >|AC| 、|AC|+|BC| >|AB| 、|AB|+|AC| >|BC|。

| AB |-| BC | & lt； |AC| 、| AC |-| BC | & lt； |AB| 、| AB |-| AC | & lt； |BC|。 You can also swap the positions of the minuend and the minuend.

Example 6, if the speed is expressed by V, then V? 40 km/h example 7, f? 2.5%，p？ 2.3%. For example 7, the teacher should guide students to pay attention to the fat content and protein content in yogurt at the same time, to avoid writing F? 2.5% or p? 2.3%, this is not right. But it can be expressed as f? 2.5% and p? 2.3%.

The teacher asked the students to answer the above questions in turn, and then gave two conclusions in the textbook with the projector.

Discussion results:

(1)(2) omitted; (3) For any two points on the number axis, the real number corresponding to the right point is greater than the real number corresponding to the left point.

(4) For any two real numbers A and B, when a=b, a >；; b，a0？ a & gtb； a-b=0？ a = b； a-b & lt； 0? a

Application example

Example 1 (example in this section of the textbook 1 and example 2)

Activity: Make students familiar with the basic methods of comparing the sizes of two algebraic expressions: difference and collocation.

Comments: There are two examples in this section that are solved by factorization and collocation method. These two methods are often used in algebraic deformation and students should master them.

Variant training

1. If f(x)=3x2-x+ 1 and g(x)=2x2+x- 1, the relationship between f(x) and g(x) is ().

A.f(x)>g(x) B.f(x)=g(x)

Cost and freight price

A: A.

Analysis: f (x)-g (x) = x2-2x+2 = (x-1) 2+1? 1 & gt； 0,? f(x)>g(x)。

2. known x? 0, compare the sizes of (x2+ 1)2 and x4+x2+ 1.

Solution: from (x2+1) 2-(x4+x2+1) = x4+2x2+1-x4-x2-1= x2.

∵x？ 0, x2>0. Thus (x2+1) 2 >; x4+x2+ 1。

Example 2 Compare the sizes of the following groups (A? b)。

(1)a+b2 and 2 1A+ 1B (A >: 0, b>0);

(2)a4-b4 and 4a3(a-b).

Activity: Comparing the sizes of two real numbers is often determined by judging their difference signs according to the relationship between the operational properties of real numbers and the order of sizes. This example can be done by students independently, but students should be guided to have sufficient reasons in the final symbolic judgment and reasoning, which cannot be ignored.

Solution: (1) a+B2-21a+1b = a+B2-2aba+b =? a+b？ 2-4ab2？ a+b？ =? a-b？ 22? a+b？ .

∵a & gt； 0, b>0 and a? b，？ a+b & gt； 0，(a-b)2 & gt； 0.a-b？ 22? a+b？ & gt0, that is, A+B2 > 2 1a+ 1b。

(2)a4-B4-4a 3(a-b)=(a-b)(a+b)(a2+B2)-4a 3(a-b)

=(a-b)(a3+a2 b+ab2+B3-4a 3)=(a-b)[(a2 b-a3)+(ab2-a3)+(B3-a3)]

=-(a-b)2(3 a2+2 ab+B2)=-(a-b)2[2 a2+(a+b)2]。

∫2 a2+(a+b)2？ 0 (take an equal sign if and only if a=b=0),

A again? b，？ (a-b)2 >0，2 a2+(a+b)2 & gt； 0.？ -(a-b)2[2 a2+(a+b)2]& lt； 0.

? a4-B4 & lt； 4a3(a-b)。

Comments: Comparing dimensions is often used as a difference method, and the general step is to make a difference deformation judgment symbol. The common means of deformation are decomposition factor and formula, the former will? Poor? Become? Product? , the latter will? Poor? Turn into one or more completely flat roads. And then what? Or both.

Variant training

Given x>y and y 0, compare the sizes of xy and 1.

Activity: compare the size relationship between any two numbers or expressions, as long as the size relationship between their difference and 0 is determined.

Solution: xy- 1=x-yy.

∵x & gt； y，？ x-y & gt； 0.

When y < 0, x-YY

When y>0, x-YY > 0, namely xy-1>; 0.？ xy & gt 1.

Comments: When the letter Y takes different values, the difference xy- 1 is different, so it is necessary to discuss the classification of Y.

According to the architectural design, the window area of residential buildings must be smaller than the building area. However, according to daylighting standard, the ratio of window area to building area should not be less than 10%, and the greater the ratio, the better the lighting conditions of the house. Question: When the window area and building area increase at the same time, will the lighting conditions of residential buildings get better or worse? Please explain the reason.

Activity: The key to solving the problem is to first convert the written language into the mathematical language, and then compare it before and after, using the difference method.

Solution: Let the residential window area and building area be A and B, respectively, and the added area be M, according to the requirements of Question A..

Because a+mb+m-ab=m? b-a？ b？ b+m？ & gt0, so a+MB+m > Ab。 Ab again? 10%,

So a+MB+m > ab？ 10%.

Therefore, after increasing the equal window area and floor area at the same time, the lighting conditions of the house have become better.

Comments: Generally speaking, if A and B are positive real numbers and a0, then A+MB+M >; ab。

Variant training

Known as a 1, a2,? For geometric series with everything greater than zero, the common ratio q? 1, then ()

a . a 1+A8 & gt； a4+a5 B.a 1+a8

C.a 1+A8 = a4+a5D。 A 1+A8 and A4+A5 are uncertain in size.

A: A.

Analysis: (a1+A8)-(a4+a5) = a1+a1Q7-a1Q3-a1Q4.

= a 1[( 1-Q3)-Q4( 1-Q3)]= a 1( 1-q)2( 1+q+Q2)( 1+q)( 1+Q2)。

∫{ an} Everything is greater than zero. Q>0, namely1+q > 0.

Q again? 1,? (a 1+a8)-(a4+a5)>0, that is, a1+A8 > a4+a5。

Knowledge and ability training

1. The following inequality:1A2+3 >; 2a； ②a2+B2 & gt； 2(a-b- 1)； ③x2+y2 & gt； 2xy。 The number of inequalities that are always true is ().

A.3 B.2 C. 1 D.0

2. Compare the sizes of 2x2+5x+9 and x2+5x+6.

Answer:

Analysis of 1 c:∫2 a2+B2-2(A-B- 1)=(A- 1)2+(B+ 1)2？ 0,

③x2+y2-2xy=(x-y)2？ 0.

? Only (1) hold.

2. Solution: Because 2x2+5x+9-(x2+5x+6) = x2+3 > 0,

So 2x2+5x+9 > x2+5x+6。

Course summary

1. Teachers and students complete the summary of this lesson together, from reviewing the basic properties of real numbers to comparing the sizes of two real numbers; From the inquiry and comment of examples to the subsequent variant training, let students simplify the complex, contact the old knowledge, and incorporate what they have learned in this lesson into the existing knowledge system.

2. The teacher makes the finishing point, pointing out the mistakes that are easy to make when comparing the sizes of two real numbers by using the basic properties of real numbers, and encouraging students with spare capacity to make further exploration in after-class thinking and discussion.

homework

Exercise 3? 1A group 3; Exercise 3? 1B group 2.

Design impression

1. The design of this section focuses on the optimization of teaching methods. Experience tells us that the teaching process that best reflects the teaching rules should be selected and designed in the classroom according to the specific situation, and it is not appropriate to use fixed teaching methods for a long time or copy an experimental model intact. All kinds of teaching methods can't adapt to all teaching activities well. In other words, there is no universal teaching method in the world. Flexible and changeable according to personality.

2. The design of this section focuses on difficulty control. Inequalities have a wide range of applications, which can be said to intersect with all other contents. It has always been the focus and hot spot of the college entrance examination. As the beginning of this chapter, it can be appropriately broadened, which can be regarded as a platform for students to freely explore associations, but it should not be expanded too much, so as not to bring negative effects to students.

3. The design of this part focuses on the training of students' thinking ability. Cultivating students' thinking ability and improving their thinking quality is an important subject directly faced by mathematics teachers, and it is also the main line of middle school mathematics education. Multi-solution to a problem is helpful to the divergence and flexibility of thinking and overcome the rigidity of thinking. Variant training teaching can broaden students' thinking horizons, and reflection after solving problems is helpful to improve students' critical thinking quality.

Lesson preparation materials

Amateur exercise

1. Compare the sizes of (x-3)2 and (x-2)(x-4).

2. Try to judge the sizes of the following algebraic expressions: (1)m2-2m+5 and-2m+5; (2)a2-4a+3 and -4a+ 1.

3. x>0 is known and verified as1+x2 > 1+x。

4. if x

5. Let a>0, b>0 and A? B, try to compare the size of aabb and abba.

Reference answer:

1. Solution: ∫(x-3)2-(x-2)(x-4)

=(x2-6x+9)-(x2-6x+8)

= 1 & gt； 0,

? (x-3)2 >(x-2)(x-4)。