Research content:
Concept, analytical formula, image, property and application of power function.
Research process:
1. Find the definition domain of the function according to the analytical formula of the function;
2. Draw the image of the function;
3. Discuss the range, monotonicity and parity of functions with images and analytical expressions.
The above process of learning power function is suitable for the basic process of learning a specific function (such as exponential function, logarithmic function, trigonometric function, etc.). ).
Research methods:
1. From concrete to abstract, from special to general
The textbook summarizes the * * * properties of analytical formula through the * * properties of bases and exponents in five-resolution function, and obtains the definition of power function.
2. Combination of tradition and modernity
Three junior high school students have learned the five power functions given in the textbook, and students can draw pictures by tracing points only by drawing images with indices of 3 and 1/2.
Teachers can use new technical means to draw five function images in the same coordinate system, which is convenient for students to observe their * * * and personality and lay the foundation for obtaining properties.
3. Unity of Algebra and Geometry
Guide students to understand the essence of function from two angles: function image and analytical formula. Domain, parity and so on can be obtained from the analytical formula; In turn, fields and parity are used to help draw pictures.
4. The selection of examples and exercises is a supplement and perfection to the teaching content.
It is not rigorous to generalize that function property from the function image. Through examples, combined with what I have learned before, I conduct reasoning and proof in a strict sense. It is not only a continuation of what we have learned before, but also a supplement to the follow-up content.
Teaching comprehension:
1. When the power exponent is negative, the X-axis and the Y-axis are asymptotes of the power function, where the meaning of the asymptote is the same as that of the hyperbolic asymptote.
2. The textbook suggests learning quintic power function, so can we expand the teaching content in the teaching process?
In this paper, the image and properties of a power function are derived, and its * * * and personality characteristics are analyzed. It can be classified according to whether the power exponent is greater than 1, equal to 1, greater than 0 and less than 1, and equal to 0 and less than 0.
3. Pay attention to the guiding role of exercises after class.
In the second problem of power function exercise, we should pay attention not to compare the size directly with the properties of real numbers, but to choose a suitable power function model and use the monotonicity ratio of power functions. This idea often appears in the function ratio.
4. Study the mapping function of ideas.
Through the study of power function, students can experience, appreciate and feel the basic content, process and method of studying function. This research idea is applied to the study of other functional models to achieve the degree of applying what you have learned.
5. Learn mathematical knowledge, pay attention to the historical background of mathematical knowledge, and clarify the ins and outs of knowledge.
The history of power function generation
In 263, Liu Hui annotated the Nine Chapters of Arithmetic. Under the rule of finding the rectangular area in the chapter of "Square Field", he wrote: "This is called Tian Mi". He also said that the product of length multiplied by width is called power. This is the first time that power appears in mathematical literature. In the chapter of Pythagorean, Liu Hui expressed Pythagorean theorem as: "Pythagorean power is combined with chord power." The power here refers to the square area or the result of the square area.
More than 300 years later, Li attached great importance to Nine Chapters Arithmetic, but he disagreed with Liu Hui's use of notional words in this way. In the Ming dynasty, some math books didn't use power characters at all.
1607, Matteo Ricci and Xu Guangqi jointly translated Euclid's Elements of Geometry, in which Xu Guangqi re-used the dynamic words. He said: "The number of times you raise yourself is called power." This is the first time to define the concept of power.
On the other hand, the concept of power is also influenced by foreign countries. 159 1 year, the French mathematician Veda once expressed "power" in Latin in his algebraic masterpiece Introduction to Analytical Methods, which was later translated into English. 1935, China published "Mathematical Terminology", which translated "force" into "power" and defined this term from then on.