Education Edition for Instructional Designers under the Seventh Grade Mathematics
6. 1.2 square root
second kind
Teaching objectives
Knowledge and skills:
Can use a calculator to find the arithmetic square root; Understand the characteristics of infinite acyclic decimal; Can use the knowledge of arithmetic square root to solve practical problems
Process and method: Know the first irrational number 2 by origami, and know the characteristics of infinite acyclic decimal by estimating its size. Calculate the arithmetic square root with a calculator, so that students can understand that the arithmetic square root of any positive number can be found with a calculator, and then find out the law of the arithmetic square root of some numbers through some special examples, and finally let students feel the application of the arithmetic square root in real life.
Emotional attitude and values: By exploring the size of 2, we can cultivate students' estimation consciousness, understand the mathematical thought of infinite approximation in two directions, exercise students' will to overcome difficulties, establish self-confidence and improve their learning enthusiasm.
Teaching focus:
① Knowing the characteristics of infinite acyclic decimals, we can estimate the arithmetic square roots of some numbers.
(2) will use the knowledge of arithmetic square root to solve practical problems.
Teaching difficulties:
Knowing the characteristics of infinite acyclic decimals, we can estimate the arithmetic square roots of some numbers.
Teaching methods: independent inquiry, inspiration and guidance, and group cooperation.
Teaching process:
First of all, through the experiment:
How to make a big square with an area of 2 from two small squares with an area of 1?
As shown in the figure, two small squares are cut diagonally and four right triangles are put together to get a large square with an area of 2. Do you know the side length of this big square?
Let the side length of a big square be x, then x? 2. From the meaning of arithmetic square root, we can know X?
So the side length of a big square is 2.
Second, the scale of the discussion 2:
From the above experiment, we know that 2. What's its size? What are the characteristics of the numbers it represents? Let's discuss the size of 2.
Because 12? 1,22? 4 1 & lt； 2<2, so 1
Because 1.4? 1.96, 1.5? 2.25, so 1.4
Because 1.4 1? 1.988 1, 1.42? 2.0 164, so 1.4 1
Because 1.4 14? 1.999396, 1.4 15? 2.002225, so 1.4 14.
In this way, we find that its decimal place is infinite, and the decimal part is acyclic, so we become infinite acyclic decimal. 22222222, 222= 1.4 142 1356
Note: This estimation embodies the mathematical idea that two directions approach the middle infinitely. It is difficult for students to understand the first contact. The teacher should speak slowly, maybe twice. 2 =1.41421356 is an infinite acyclic decimal, but it is very abstract, so it is impossible to completely express its size. There are many numbers like this, for example, 7 and so on. Pi? It is also an infinite cyclic decimal.
Third, use a calculator to find the arithmetic square root:
Most calculators have keys, which can be used to find the arithmetic square root or approximate value of rational numbers.
Example 1. Use the calculator to find the following values:
) ( 1); (2)2 (accurate to 0.00 1
Solution: (1) Press the key in turn.
(2) Press the key 3 136 in turn? , display: 56. So what? 56 2 =, showing:1.414213562, which is an approximate value. So 2？ 1.4 14.
Note: Different brands of calculators may have different key sequences.
Fourth, explore the law:
(1) Fill in the calculation results in the table with a calculator. What rules have you found?
62.5625
(2) Use a calculator to work out 3 (the result retains 4 significant digits), and write 0.03,300 according to the law you found.
About 30,000. Can you find the value of 30 according to the value of?
Students can get the answers of (1) through the calculator, which are 0.25, 0.79 1, 2.5, 7.9 1, 25,79.1,250 in turn. From the operation results, it can be found that when the root sign is enlarged or reduced by 100 times, its arithmetic square root is enlarged or reduced by 10 times.
By who? 1.732 available.03? 0. 1732,? 17.32,30000? 173.2, the value of 30 can't be found from the value of 3, because the rule is that the square root will be expanded or reduced by 10 times, and its arithmetic square root will be expanded or reduced by 100 times, while the expansion from 3 to 30 is 10 times, so it can't be found from this rule.
Students can finish this problem independently.
Practical application of verb (abbreviation of verb);
Example 1. Xiaoli wants to use a square piece of paper with an area of 400cm and cut a piece of paper with an area of 300cm along the edge.
The length-width ratio of this rectangular paper is 3: 2. I don't know if I can cut it out, but I'm worried. Xiao Ming saw it and said, Don't worry, you can definitely cut a small piece of paper from a big piece of paper. ? Do you agree with Xiao Ming? Can Xiaoli use this paper to cut out a piece of paper that meets the requirements?
Analysis: Students generally think that a large piece of paper can be cut into a small piece of paper. Correct this misunderstanding through calculation and explanation.
Solution: Set a rectangular piece of paper with a length of 3xcm and a width of 2cm.
According to the relationship between side length and area, we can get: 3x? 2x？ 300，6x？ 300，x？ 50，x？ 22 ? The length of a rectangular piece of paper is 3 cm. Because it is 50 ~ 49, it is ~ 7, thus ~ 2 1.
That is, the length of rectangular paper is greater than 2 1cm, while the side length of square paper is only 20cm, so the length of rectangular paper will be greater than that of square paper.
I can't agree with Xiao Ming Xiaoli can't cut a rectangular piece of paper that meets the requirements with this square piece of paper.
Six, in-class exercises:
1. Use the calculator to find the following values:
(1) (2).2036 (3) (accurate to 0.0 1)
2. Estimated size:
(1) and 12 (2)? 1 and 0.5 2
3. Known 2? 1.4 14, and find the values of 0.02, 0.0002, 200 and 20000.
Seven. Course summary
1, when the root sign increases or decreases, its corresponding arithmetic square root also increases or decreases accordingly, so we can use the clamp value method to find the approximate value of the arithmetic square root;
2. The approximate value of the arithmetic square root of any positive number can be obtained with a calculator;
3. What is the law between the expansion (or contraction) of the radical sign and the expansion (or contraction) of its arithmetic square root?
4. What is the infinite acyclic decimal?
Eight, homework
Exercise 6, 1, question 3 and question 5 on page 47 of the textbook.
Teaching reflection:
Application of Geometry Sketchpad in Junior High School Mathematics Teaching
abstract:
Geometric sketchpad is an information technology software that provides services for mathematics teaching under the development of modern science and technology. Because it provides a good teaching environment for mathematics teaching, the combination of numbers and shapes is widely used in junior high school mathematics teaching, helping teachers to make courseware related to mathematics teaching more conveniently, improving students' interest in learning, and cultivating students' thinking ability, independent learning consciousness and innovative spirit. In the concrete teaching practice, teachers can create vivid and concrete teaching environment by using the geometry sketchpad, so that the fuzzy and abstract mathematics teaching becomes intuitive, vivid and concrete, so as to better grasp the concrete mathematics knowledge. This paper will briefly discuss the application of geometry sketchpad in junior high school mathematics teaching.
Key words:
Geometric drawing board; Junior high school mathematics; Classroom teaching; Trial teaching
With the continuous development of science and technology, multimedia technology is more and more applied to modern teaching, and now multimedia technology has become an important means and tool to assist education and teaching. Geometric sketchpad is a simple and powerful teaching software, which can not only draw and calculate, but also adapt to the teaching content of modern courses, meet the teaching requirements of quality education, help to simplify the complex, create a vivid and concrete teaching environment for students, help students understand teaching texts and improve their spirit of innovation and exploration. It is an effective teaching method to use it in junior high school mathematics classroom teaching, and teachers can effectively improve classroom teaching efficiency through the display of this high-tech technology. Therefore, how to apply the Geometer's Sketchpad to junior high school mathematics classroom teaching has become a topic discussed by the majority of mathematics teachers. This paper will briefly discuss the application of geometry sketchpad in junior high school mathematics classroom teaching.
First, follow the people-oriented principle and avoid usurping the role of the host.
Although the geometric sketchpad is a good teaching method, teachers need to realize that it is only an auxiliary means of classroom teaching and should be followed in the process of design and production? People-oriented? The teaching principle of students' view, instead of mechanically copying the classroom teaching content into the geometry drawing board, makes it unique and follows the book, usurping the role of the host of junior high school mathematics classroom. Therefore, when making geometry sketchpad, teachers should analyze the contents of classroom teaching in detail, arrange students' practical activities fairly and reasonably, and organize students' sketchpad exercises, questions and answers and other related activities to connect with the teaching content. Like teaching? What is the positional relationship between circles? In the first class, teachers can review and lead in with the help of the Geometer's Sketchpad, show several positional relationships between points and circles with the courseware, then organize students to think and answer, then show a lot of mathematical knowledge theories and definitions with the Geometer's Sketchpad, and get three positional relationships between points and circles, and then show several positional relationships between straight lines and circles with the Geometer's Sketchpad courseware. Without the help of multimedia teaching technology, it is impossible to show a lot of information to students in a short time. Finally, import with the help of geometric sketchpad? What is the positional relationship between circles? In order to enhance the teaching effect, teachers can use the geometric sketchpad to make it? Annular eclipse? This animation effect creates a relaxed and happy learning atmosphere for students and promotes them to learn better and faster.
Second, create a learning environment with the help of geometric sketchpad.
Mathematics is a practical and comprehensive subject, which requires students to have certain logical thinking ability and spatial imagination ability, which are gradually cultivated in the process of continuous practice. In mathematics teaching, with the help of geometry sketchpad software, students can move and observe graphics at will, and make bold guesses and verifications, thus strengthening students' intuitive understanding of graphics, enriching geometry experience and further improving their ability to understand and prove knowledge. It can be seen that the application of geometry sketchpad in junior high school mathematics classroom teaching can effectively help students improve their cognitive ability and enhance their mathematics learning ability. The appearance and application of this kind of teaching software makes the original abstract and boring mathematical knowledge intuitive and clear, makes students who hate mathematics find it lively and interesting, not only provides many conveniences and conditions for students' mathematics learning, but also makes them devote themselves to classroom teaching, truly become the main body of learning and dare to pursue knowledge. In addition, it also helps to build students' confidence in learning, and regards learning mathematics as a happy thing, so that middle schools can do and learn. Through the application of the teaching software Geometer's Sketchpad, students can directly copy the mathematical concepts and learning contents that need to be studied repeatedly at home for repeated learning, providing families with learning difficulties with an opportunity to re-recognize learning and truly implement the new curriculum reform. All for the development of every student? Teaching philosophy, promote the progress and development of all students.
Third, use the geometric sketchpad to reveal the theorem connection.
Generally speaking, there are certain connections and differences between different concepts or related objects of mathematical knowledge. Using the dynamic function in the geometry sketchpad, we can reveal the connections and differences between different mathematical concepts to a certain extent, and also present the movement change process between them more conveniently and quickly, which has a positive role in promoting students to better learn the essence of mathematical concepts and obtain correct concepts, thus helping to develop and cultivate students' cognitive level and understanding ability. Like a professor? Axisymmetric graphics? In this section, we can use the dynamic function of the geometric sketchpad to deepen students' understanding of this knowledge point, help students understand the properties and concepts of axisymmetric graphics more intuitively and vividly, and help students distinguish their connections and differences from the related knowledge points of central symmetry.
Fourth, use the geometric sketchpad to help find problems.
Geometric sketchpad provides a good teaching environment and conditions for mathematics classroom teaching, which is convenient for students to actively discover and explore. This kind of teaching software can produce vivid animation effect in just a few minutes, drag the mouse at will, measure the angle dynamically and change the shape of the graph at will. It is applied to junior high school mathematics classroom teaching to create a dynamic learning environment for students to discover problems and explore constantly, so as to improve students' mathematics literacy and promote their all-round development. Like teaching? Similar triangles's conclusion? In the first class, teachers can design and analyze experimental courses, so that students can find and summarize through continuous exploration, and finally draw experimental conclusions, thus creating good practical opportunities for students, enriching curriculum knowledge, allowing students to actively participate in classroom practice, actively explore and actively learn, and laying a solid foundation for efficient mathematics classroom construction.
In a word, the flexible use of geometry sketchpad in junior high school mathematics classroom teaching can not only improve students' rational understanding of mathematics learning, but also help teachers break through the difficulties in teaching and optimize classroom teaching effect. You can also create a good teaching situation to stimulate students' interest in learning, improve students' initiative in learning, and establish a correct sense of mathematics learning, which will help to cultivate and develop students' inquiry ability and logical thinking ability and lay a solid foundation for students' future study.
Author: Zhong Conggui Unit: Huachong Middle School, Shuyang County
References:
[1] Chang Jiajie. Geometric sketchpad? Research on the application of mathematics in junior high school [D]. Ningxia University, 20 15.
[2] Zhao Shengchu, Du Weiwei, Lu Xiumin. Practice and Exploration of Geometry Sketchpad in Junior Middle School Mathematics Teaching [J]. China Audio-visual Education, 2012,03:104-107.
[3] Weng Juanjuan. Research on the Effectiveness of Geometry Sketchpad in Junior Middle School Mathematics Teaching [D]. Suzhou University, 20 10.
[4] Li Yuquan. The Effective Application of Geometry Sketchpad in Junior High School Mathematics Teaching [J]. Science and Education Guide (Later Period), 2015,05:134-135.