CN101465080A

CN101465080A - Popular science experimental method for checking pythagorean proposition

Info

Publication number: CN101465080A
Application number: CNA2008102190698A
Authority: CN
Inventors: 姜佳男
Original assignee: Individual
Current assignee: Individual
Priority date: 2008-11-13
Filing date: 2008-11-13
Publication date: 2009-06-24

Abstract

The invention relates to a science experiment method for verifying the Pythagorean theorem. A semi-circle O1 is made by taking a line AB as the radius, and a circle O2 is made by taking a point E as the circle center and a line AE as the radius; a point C is arbitrarily selected on the semi-circle O1; a line AC extends to obtain a point G, and a line BC extends to obtain a point D; the line AB is taken as a side to make a first square frame body; the line AC and the line DC are taken as the sides to make a second square frame body, the line BC and the line GC are taken as the sides to make a third square frame body, wherein, the depths of the all the frame bodies are equal; the first square frame body, the second square frame body and the third square frame body are filled with flowing presentations, the point C is moved on the circle O1, the point D and the point G are moved on the circle O2, and the accommodating areas of the second square frame body and the third square frame body correspondingly change along with the movement of the point C, the point D and the point G; and the presentations filled in the second square frame body and the third square frame body are mutually changed or flow. The invention can be used for demonstrating the intuitively reflected Pythagorean theorem when the two right angle sides continuously change under the condition without changing one bevel side.

Description

The science popularization experimental technique of checking Pythagorean theorem

Technical field

The present invention relates to experimental teaching implement device technical field, particularly the method for the science popularization experiment of checking Pythagorean theorem.

Background technology

Pythagorean theorem is formula basic in the mathematics, just relates in school mathematics.Understand Pythagorean theorem by teaching equipment ocular demonstration and observation, to students in middle and primary schools' study with to understand Pythagorean theorem very helpful.At the Paris, FRA science hall, the device that a proof Pythagorean theorem is arranged, what this installed usefulness is the right-angle side and the hypotenuse of fixed length, its method for designing is: form three foursquare containers (contour) with right-angle side and hypotenuse respectively, wherein the container that is made of hypotenuse is filled water, the container that is made of right-angle side is empty, and when the rotation of this right-angle triangle, the water in the container that is made of hypotenuse just in time can be injected in the container that is made of right-angle side.Because 3 containers are contour, illustrate that the area that two right-angle sides form just in time equals the formed area of hypotenuse, thereby verified Pythagorean theorem.

Secondly, domestic also have several parts to relate to and prove the patented invention of Pythagorean theorem device, and for example China Patent No. is ZL200420005031.8, is called the innovation and creation of " Pythagorean theorem teaching aid ", and it mainly is made of template, and template is made up of three parts: 1.By colluding one of right-angle triangle (redness) template that long a, section chief b, chord length c form; 2.Be to be one of square (yellow) template of a by the length of side; 3.Be to be four of the square of b and congruent quadrilateral (green) templates cut apart by the length of side, all template bottom surfaces all have magnetic.Four congruent quadrilateral templates are to be that the mutually perpendicular line segment of the square central point of b is cut apart by the length of side by two, and wherein a line segment is parallel with string, and this foursquare one side overlaps with leg-of-mutton burst of limit.This invention is moved on blackboard (band iron) plane by above-mentioned two square templates and is demonstrated, and the student is familiar with very intuitively and understands " Pythagorean theorem ", has improved students'interest in learning, has strengthened teaching effect of teachers.

Though above-mentioned experimental provision can both ocular demonstration and understanding, these devices all are static, promptly can only once demonstrate to have the fixedly right-angle triangle of acute angle, and any limit of the right-angle triangle of being demonstrated all not have variation.Concerning the student that those hope are probed into, they are certain to suspect whether Pythagorean theorem is applicable to the right-angle triangle of various different acute angles.

Summary of the invention

The length of side that the present invention designs constant and two right-angle sides of hypotenuse is the right-angle triangle that constantly changes of acute angle angle in other words, thereby checking demonstration directly perceived is under the continuous situation about changing of the hypotenuse of right-angle triangle, the quadratic sum of two straight flanges equal hypotenuse square, promptly intuitively verify Pythagorean theorem.

At first we analyze and realize basic mathematic model of the present invention from the angle of mathematics geometrical principle.

As shown in Figure 1, be that diameter is made circle O1 with AB, be partly through making circle O2 with AE.E is a bit on circumference of another diameter vertical with AB among the circle O1.In the AEB segmental arc, make arbitrarily two string AC and the BC of circle O1, then △ ABC is a right-angle triangle.Prolong AC and hand over circle O2, prolong BC and hand over circle O2, AC=DC is then arranged, BC=GC in D in G.

Proof:

In circle O1, for string AC, obviously have: ∠ AEC=∠ ABC; In circle O2, for string AD, obviously have: ∠ AED=2 ∠ ABD=2 ∠ ABC; Therefore have: ∠ AEC=∠ DEC;

Again: AE=DE (half warp); CE=CE;

So: △ ACE ≌ △ DCE;

AC=DC in like manner can demonstrate,prove: BC=GC.

Therefore, when on the circumference of C point at O1 when mobile,, AC=DC is arranged forever, BC=GC on O2 if D, G move;

Also can draw:

AC×AC＝AC×DC；BC×BC＝BC×GC

Therefore,, suppose: AB * AB=R, AC * AC=AC * DC=P, BC * BC=BC * GC=Q according to Pythagorean theorem; Can draw: R=P+Q.Be we only need can ocular demonstration P and the area of Q situation about constantly changing under, they and equal R all the time.

According to above-mentioned model, we have proposed a kind of science popularization experimental technique of verifying Pythagorean theorem: it is characterized in that:

(1) be diameter with the line AB between an A and the some B, the top of online AB makes semicircle O1 and obtains first semicircular ring of semicircle O1;

(2) definite and some A and the equidistant some E of some B on described first semicircular ring are the center of circle with an E, are the round O2 of top making of the online AB of radius and second semicircular ring that obtains circle O2 with line AE or line BE;

(3) any selected point C on first semicircular ring; Extended line AC obtains a G to second semicircular ring, and extended line BC obtains a D to second semicircular ring;

(4) be that the below of the online AB in limit makes square framework and obtains the first square framework 3 with line AB; With line AC and line DC is that the limit makes square framework and obtains the second square framework 4, is that the limit makes square framework and obtains the 3rd square framework 5 with line BC and line GC, and wherein the degree of depth of each framework is identical;

(5) in first square framework 3, the second square framework 4 and the 3rd square framework 5, fill demonstration thing with flowability, wherein the demonstration thing quantity in the first square framework 3 equal in the second square framework 4 and the 3rd square framework 5 the demonstration thing quantity of filling and, and the demonstration thing of filling in the second square framework 4 and the 3rd square framework 5 can be in its circulation or exchange mutually between the two;

(6) transfer point C on first semicircular ring, correspondingly transfer point D and some G on second annulus, what make the second square framework 4 and the 3rd square framework 5 holds area moving and respective change with a C, some D or some G; Also exchange or mobile thereupon between the demonstration thing of filling in the second square framework 4 and the 3rd square framework 5.

Further technical scheme can also be, the demonstration thing with flowability of filling in first square framework 3, the second square framework 4 and the 3rd square framework 5 is a spheroidite.

Further technical scheme can also be, by motor driven systems, described some C of the described second square framework 4 and the 3rd square framework 5 is moved along described first semicircular ring, and described some D and some G are moved along described second semicircular ring.

According to technique scheme, we can change under the situation of the length of two right-angle sides in any continuity, the quadratic sum of two straight flanges of checking demonstration directly perceived equal hypotenuse square, checking Pythagorean theorem promptly directly perceived, guiding student is deepened the understanding to Pythagorean theorem, with seasonal student science is produced keen interest.

Because the present invention has above-mentioned advantage, can be used under the constant situation of hypotenuse the Pythagorean theorem that institute intuitively reflects when demonstrate two right-angle side continuitys and changing.

Description of drawings

Fig. 1 is the principle model figure of design technical solution of the present invention;

Fig. 2 is an embodiment synoptic diagram of using technical scheme of the present invention.

Embodiment one

As depicted in figs. 1 and 2, a kind of science popularization experimental technique of verifying Pythagorean theorem comprises following scheme step:

(4) be that the below of the online AB in limit makes square framework and obtains the first square framework 3 with line AB; With line AC and line DC is that the limit makes square framework and obtains the second square framework 4, is that the limit makes square framework and obtains the 3rd square framework 5 with line BC and line GC, and wherein the degree of depth of each framework is identical; Secondly, the four edges that constitutes the second square framework 4 is can relatively move between limit MD, limit MA, limit CD and the limit CA; The four edges that constitutes the 3rd square framework 5 is also can relatively move between limit CB, limit CG, limit NB and the limit NG.

(5) in first square framework 3, the second square framework 4 and the 3rd square framework 5, fill the beaded glass of one deck diameter about 1CM, wherein the beaded glass quantity in the first square framework 3 equal in the second square framework 4 and the 3rd square framework 5 the beaded glass quantity of filling and, and between the second square framework 4 and the 3rd square framework 5 communication passage (not drawing among the figure) is set, can allows the beaded glass of filling between these two square frameworks circulate mutually.

(6) transfer point C on first semicircular ring, accordingly on second annulus transfer point D and the some G, promptly constantly move the length of limit MD, limit MA, limit CD and limit CA in the second square framework 4, constantly move the length of limit CB, limit CG, limit NB and the limit NG of the 3rd square framework 5 simultaneously, it is square that the second square framework 4 and the 3rd square framework 5 are remained.It holds area moving and respective change with a C, some D or some G like this; The beaded glass of filling in the second square framework 4 and the 3rd square framework 5 flows by described communication passage.Can certainly adopt artificial way that the beaded glass of filling in the second square framework 4 and the 3rd square framework 5 is replaced.At this moment, we just can observe intuitively, the length of limit CA and limit CB constantly changes, correspondingly the area that holds of the second square framework 4 and the 3rd square framework 5 also constantly changes, wherein the beaded glass quantity of Tian Chonging also presents the phenomenon that this disappears and other rises, but the total quantity of the beaded glass of filling in the second square framework 4 and the 3rd square framework 5 does not change, and equals the beaded glass quantity of filling in the first square framework 3 all the time.Thereby verify Pythagorean theorem intuitively.

Above-mentioned beaded glass flows or circulation by described communication passage, is to carry out in the mode of rolling, and is not meant in the mode as liquid and flows.Be to be under the situation of demonstration medium with liquid in certain square framework, we can be understood as flowing of liquid.Secondly for realizing the present invention program, the demonstration medium can also be with analogs such as baton rounds.

In order to make presentation process lighter, can on first semicircular ring or second semi-circular track, the drive motor system be set, assist some C corresponding on the second square framework 4 and the 3rd square framework 5, put D or put G along moving on first semicircular ring or second semi-circular track.

Claims

A kind of science popularization experimental technique of verifying Pythagorean theorem; It is characterized in that:

(1) be diameter with the line AB between an A and the some B, the top of online AB makes semicircle O1 and obtains first semicircular ring of semicircle O1;

(2) definite and some A and the equidistant some E of some B on described first semicircular ring are the center of circle with an E, are the round O2 of top making of the online AB of radius and second semicircular ring that obtains circle O2 with line AE or line BE;

(3) any selected point C on first semicircular ring; Extended line AC obtains a G to second semicircular ring, and extended line BC obtains a D to second semicircular ring;

(4) be that the below of the online AB in limit makes square framework and obtains the first square framework (3) with line AB; With line AC and line DC is that the limit makes square framework and obtains the second square framework (4), is that the limit makes square framework and obtains the 3rd square framework (5) with line BC and line GC, and wherein the degree of depth of each framework is identical;

(5) in the first square framework (3), the second square framework (4) and the 3rd square framework (5), fill demonstration thing with flowability, wherein the demonstration thing quantity in the first square framework (3) equal the demonstration thing quantity of filling in the second square framework (4) and the 3rd square framework (5) and, and the demonstration thing of filling in the second square framework (4) and the 3rd square framework (5) can be in its circulation or exchange mutually between the two;

(6) transfer point C on first semicircular ring, correspondingly transfer point D and some G on second annulus, make the second square framework (4) and the 3rd square framework (5) hold area with the mobile of a C, some D or some G respective change; Also exchange or mobile thereupon between the demonstration thing of filling in the second square framework (4) and the 3rd square framework (5).
2. the science popularization experimental technique of checking Pythagorean theorem according to claim 1: it is characterized in that: the demonstration thing with flowability of filling in the first square framework (3), the second square framework (4) and the 3rd square framework (5) is a spheroidite.
3. the science popularization experimental technique of checking Pythagorean theorem according to claim 1: it is characterized in that: pass through motor driven systems, described some C of the described second square framework (4) and the 3rd square framework (5) moved along described first semicircular ring, described some D and some G are moved along described second semicircular ring.