Changed students' learning.
1. Students changed from "listening to mathematics" to "doing mathematics". By doing mathematical experiments, students' status changed from passive acceptance to active participation.
Case Pythagorean Theorem
(1) The teacher drew a right triangle with the geometric sketchpad and measured the length of the hypotenuse with the measuring function of the software. And timely inspire and induce students to think about the relationship between the three sides of a right triangle.
(2) According to the nature of the graphic area, using the experimental operation of "area division, shifting and patching", let the students "drag" the mouse to put four congruent right-angled triangles with right-angled sides A and B and hypotenuse length C into a square ABCD with side length A+B. The students found that there were two ways to put them (as shown in figure 1), and in theory, A2+B2 =.
2. Students change from "watching the demonstration" to "hands-on operation". Mathematical experiments make single media presentation a cognitive tool for students. The process and essence of mathematical thinking can be effectively revealed through instant function and animation function. Students become the subject of practice through hands-on exploration, guessing and verification.
Case Pythagorean Theorem (Figure 2)
(1) Make three squares with three sides of the right triangle ABC as side lengths, and use the "area" function of the geometric sketchpad to calculate the area S3 of the square with side length AB, and the areas S2 and S 1 of the square with side lengths BC and AC.
(2) By clicking on the animation of "Point A moves on CN" and "Point B moves on CM" (the sizes of three squares are constantly changing), let students observe the changes of dynamic graphics and data, and find out the constant quantitative relationship: S 1+S2 = S3, that is, AC2+BC2=AB2 (Pythagorean theorem), so as to think of using the area method to explain.
3. Students change from "mechanical learning" to "active exploration". Mathematical experiment teaching changes the simple teaching process from teaching guidance to student-centered teaching process through situation creation, question inquiry, collaborative learning and meaning construction.
Case sine and cosine (Figure 3)
(1) In the right triangle BAC, keep ∠A unchanged, drag point B to move on AM, and find that the values of-and-always remain unchanged.
(2) When ∠A changes, we can know from the measurement function that:-increases with the increase of ∠ a, and decreases with the increase of ∠ a. Students find that 0
Give students a chance to find problems.
The midline of a triangle
(1) After giving a beautiful lotus pond, the teacher asked: How to measure its width?
(2) Provide a measuring method: after the pond is abstracted by computer (as shown in Figure 4), just measure the length of BC. That is, choose a point A on the flat ground on one side of the pond, and then find out the midpoint D and E of line AB and AC respectively, and measure DE = 18m, and get the pond width BC = 36m.
(3) According to this, ask the students: Does this person's method make sense? What is the secret?
(4) Measure the length of three sides and the length of DE through the geometric sketchpad, and display the results on the screen. Let the students drag any vertex of the triangle and answer the following questions through observation. Let the students explore the experiment by themselves: ① What is the positional relationship between the center line DE of the triangle and each side? ② What is the equal relationship between the median line DE and the length of each side of a triangle? ③ Guess: What are the properties of the midline of the triangle? Can you prove this conjecture?
When students drag any vertex of a triangle, the position of the midline changes dynamically, and the lengths of the three sides and the midline of the triangle also change. This fully embodies the arbitrariness of triangles, guides students to pay attention to the invariant relations and invariants in the process of change, and allows students to discover laws through observation.
Promote students' understanding and memory of knowledge.
Image of case quadratic function y=ax2+bx+c
(1) The teacher draws the image of the quadratic function y=ax2+bx+c with the geometry sketchpad (as shown in Figure 5), then drags the mouse to adjust the sizes of A, B and C respectively, observes the changes of the image, and guides the students to get the image features of y=ax2+bx+c according to different changes.
(2) Ask students to discuss the image features of the following functions according to the conclusion:
y = 2 x2+3x+ 1y = 2 x2+3x- 1
y = 2 x2-3x+ 1y = 2 x2-3x- 1
y =-2 x2+3x+ 1y =-2 x2+3x- 1
y =-2 x2-3x+ 1y =-2 x2-3x- 1
Students can adjust the sizes of A, B and C in turn, observe the changes of opening size, opening direction, the position of symmetry axis and the intersection of the image and Y axis, and summarize the properties of quadratic function images. The environment provided by the Geometer's Sketchpad can free teachers from a lot of explanations, and guide students to focus on the process and the key points that should be highlighted, so that students can not only understand and remember the nature from the semantics of nature, but also when the "nature of quadratic function" appears, the nature depicted by these function images immediately emerges in their minds, thus truly grasping the nature of quadratic function.
Cultivate students' innovative thinking.
In the field of space and graphics learning, it is an important task to observe and imagine the dynamic changes of graphics, analyze and judge the inherent laws of graphics, and cultivate students' good space concept.
General review of mathematics in the third day of the case
(1) The teacher gives the experimental object (as shown in Figure 6): On the arc AB of a fan-shaped OAB with a radius of 6 and a central angle of 90, there is a moving point P, PH⊥OA, vertical foot H, and the center of gravity of △OPH is G.
(2) Ask the students to discuss whether there are equal length line segments among the line segments GO, GP and GH when the point P moves on the arc AB. If so, point out such a line segment and find out the corresponding length.
(3) When PH=x and GP = Y, students are required to solve the resolution function of Y about X. ..
(4) Ask students to solve the length PH of the line segment when △PGH is an isosceles triangle.
When students use the geometric sketchpad, it is easy to find that only GH remains unchanged by dragging the point P. Moreover, the breakthrough to find the length of line segment gh is to look at "static" from "dynamic" and find out all invariants in the process of change. Through this experiment, students' creative thinking in mathematics has been developed and their innovative consciousness has been improved.