CN202306895U - Demonstration device of relationship between linear angle of inclination and gradient - Google Patents

Abstract

The utility model discloses a demonstration device of the relationship between a linear angle of inclination and a gradient. The demonstration device is characterized in that a plane rectangular coordinate system xOy is arranged on a plane coordinate system demonstration board, wherein O serves as the center of a circle, a unit circle that is 9cm in radius is drawn, and the unit circle is divided into 16 special angles. A parallelogram type linear rotation ruler contains a rotation main ruler, a rotation reference ruler and two adjusting rulers, wherein the rotation main ruler, the rotation reference ruler and the two adjusting rulers are fixed via detachable screws to be parallelogram-shaped. The parallelogram type linear rotation ruler is provided with a helical shaft in the central hole of the rotation main ruler. The helical shaft is equipped with a rotation metal pointer. Metal contact sheets on the metal ring of the unit circle are arranged on each special angle. Each metal contact sheet is 9cm away from the center O of the circle. A rotation metal covering is disposed on the rotation metal pointer via a screw. An insulating lead that is connected with a right side bulb pedestal and the metal ring of the unit circle, the rotation metal covering rotating on the pointer, an AA battery pedestal, and a plane coordinate system demonstration board back side circuit system together form a closed loop. The demonstration device of the utility model exhibits a dynamic characteristic, and allows the relationship between a linear angle of inclination and a gradient to be demonstrated in a visual and vivid manner. If a teacher uses the device for demonstration during teaching, students can understand easily, such that an effect of twice result with half effort is realized, and the teaching quality in a classroom is improved.

Description

The relation demonstration device of straight inclined angle and slope

Technical field

The utility model relates to a kind of demonstration device that is used for high school mathematics teaching, is the relation demonstration device of a kind of straight inclined angle and slope specifically.

Background technology

For a long time; The high school mathematics teacher in lecturing cartesian geometry about the understanding of the formation of straight inclined angle notion and slope notion and slope and tan concern this knowledge point the time; Will be there following teaching difficulty: the slope that define straight line why? Why be defined as the inclination tangent of an angle to slope? How does cross the slope formula of 2 straight line set up? In concrete teaching process; Because it is auxiliary that the suitable teaching aid of shortage is done, the teacher says that time service is dull, uninteresting, and teaching difficulty is big; The student listens and feels abstract, hard to understand class hour, and study is difficulty very.Some teacher have to draw on blackboard can only expendable still image, gives student explain straight inclined angle notion, time-consuming, has influenced teaching efficiency to a great extent.

Summary of the invention

In view of above-mentioned, the demonstration device that the purpose of the utility model provides a kind of straight inclined angle and slope relation utilizes this demonstration device, can be directly perceived, the geometric element of definite straight line under the dynamic demonstration coordinate system, make the algebra problem visualize, the number form combination.Both made things convenient for giving lessons of teacher, also be convenient to the notion of student's correct understanding straight inclined angle and slope, the straight inclined angle change and slope variation between relation, and the combining closely of straight slope and tan improved classroom teaching effect greatly.

The purpose of the utility model is achieved in that

The demonstration device of a kind of straight inclined angle and slope relation; Be to combine by plane coordinate system demonstration board, parallelogram type straight line rotation chi, screw axis and electro-optical system; For center O plane right-angle coordinate xOy being set apart from 12cm place, the plate face left side on the plane coordinate system demonstration board, has been that the unit of pressing head marks with x axle positive axis with O $\frac{\mathrm{\π}}{6},\frac{\mathrm{\π}}{4},\frac{\mathrm{\π}}{3},\frac{\mathrm{\π}}{2},\frac{2\mathrm{\π}}{3},\frac{3\mathrm{\π}}{4},\frac{5\mathrm{\π}}{6},$ Each special angle point of π.With O is the center of circle, and 9cm is that radius is drawn unit circle, with unit circle with 0, $\frac{\mathrm{\π}}{6},\frac{\mathrm{\π}}{4},\frac{\mathrm{\π}}{3},\frac{\mathrm{\π}}{2},\frac{2\mathrm{\π}}{3},\frac{3\mathrm{\π}}{4},\frac{5\mathrm{\π}}{6},$ π, $\frac{7\mathrm{\π}}{6},\frac{5\mathrm{\π}}{4},\frac{4\mathrm{\π}}{3},\frac{3\mathrm{\π}}{2},\frac{-\mathrm{\π}}{4},\frac{-\mathrm{\π}}{3},\frac{-\mathrm{\π}}{6}$ Each angle is divided into 16 special angles, and the interval of drawing [0, π] is gone up the tan image and drawn the corresponding tangent value in each special angle; Parallelogram type straight line rotation chi is divided into rotation main scale, rotation with reference to chi and two adjusting chis; Rotation main scale, rotation set into parallelogram shape with two adjusting chis by removable fasteners with reference to chi, and parallelogram type straight line rotation chi is equipped with screw axis at rotation main scale center pit; Screw axis and electro-optical system are installed in the back side of plane coordinate system demonstration board; Screw axis is equipped with the rotating metallic pointer; 6. the metal contact that 10. the unit circle quoit is gone up is set in each position, special angle; The metal contact is set in apart from center O 9cm place on the metal finger, and rotating metallic is applied mechanically screw and is installed on the rotating metallic pointer.Insulated conductor connects rotating metallic cover and No.5 cell base on positive light bulb base and unit circle quoit, the pointer, forms the closed-loop path with plane coordinate system demonstration board back side electro-optical system.

The advantage of the utility model and the beneficial effect of generation:

The utility model demonstration device is combined closely straight slope and tan, can be directly perceived, the pitch angle of dynamic demonstration each coordinate position straight line in plane right-angle coordinate and the relation of slope, solved some mathematical problems relevant with straight slope.And, make the demonstration effect more obviously, intuitively through the utilization photoelectric technology.The utility model demonstration device; The teaching method that the utilization number form combines; Monotonicity and the compute classes problem that solves pertinency factor, curve or function related properties, discriminant function directly perceived makes abstract thinking develop into the thinking of visual pattern, help to cultivate the student by directly perceived to abstract; By specific to general good trait of thinking, form rigorous abstract thinking mode.The student can be through demonstration or hands-on, deep understands the knowledge main points, has reduced teaching difficulty, improves the quality of teaching.The utility model demonstration device also can solve the relevant problems of measurement of triangle, and simple altimeter device is processed in repacking a little.The utility model demonstration device has been filled up the blank of mathematics test apparatus, and very big promotional value is arranged aspect teaching application.

Description of drawings

Fig. 1 is the Facad structure synoptic diagram of the utility model.

Fig. 2 is the back side electro-optical system structural representation of the utility model.

Embodiment

Below in conjunction with accompanying drawing, the utility model is described further:

As shown in Figure 1: the demonstration device of a kind of straight inclined angle and slope relation, be by the plane coordinate system demonstration board 1., parallelogram type straight line rotation chi, screw axis and electro-optical system combine, 1. the plane coordinate system demonstration board is seven plywood of 46cmX46cm.For center O plane right-angle coordinate xOy being set apart from 12cm place, the plate face left side on the plate face, has been that to press 9cm be that unit head marks with x axle positive axis with O $\frac{\mathrm{\π}}{6},\frac{\mathrm{\π}}{4},\frac{\mathrm{\π}}{3},$ $\frac{\mathrm{\π}}{2},\frac{2\mathrm{\π}}{3},\frac{3\mathrm{\π}}{4},\frac{5\mathrm{\π}}{6},$ Each special angle point of π.With O is the center of circle, and 9cm is that radius is drawn unit circle, unit circle be quoit 10..With unit circle with 0, $\frac{\mathrm{\π}}{6},\frac{\mathrm{\π}}{4},\frac{\mathrm{\π}}{3},\frac{\mathrm{\π}}{2},\frac{2\mathrm{\π}}{3},\frac{3\mathrm{\π}}{4},\frac{5\mathrm{\π}}{6},$ π, $\frac{7\mathrm{\π}}{6},$ $\frac{5\mathrm{\π}}{4},\frac{4\mathrm{\π}}{3},\frac{3\mathrm{\π}}{2},\frac{-\mathrm{\π}}{4},\frac{-\mathrm{\π}}{3},\frac{-\mathrm{\π}}{6}$ Each angle is divided into 16 special angles.The interval [0, π] of drawing is gone up the tan image and is drawn the corresponding tangent value in each special angle.Parallelogram type straight line rotation chi be divided into the rotation main scale 2., rotation with reference to chi 3. and two regulate chis; The rotation main scale is long with reference to chi with rotation to be 44cm; Regulating chi for two can regulate voluntarily; Rotation main scale, rotation set into parallelogram shape with two adjusting chis by removable fasteners with reference to chi, and 4. parallelogram type straight line rotation chi is equipped with screw axis at rotation main scale center pit.The center pit aperture is 2cm.

The plane coordinate system demonstration board 1., the rotation main scale of parallelogram type straight line rotation chi 2. with rotation with reference to chi 3., 4. screw axis be installed together with electro-optical system.

Screw axis 4. and electro-optical system (Circuits System) be installed in the back side of plane coordinate system demonstration board, as shown in Figure 2: fixedly screw axis is 4. for the back side, plane coordinate system demonstration board center O position.Be used to install positive rotation main scale 2. with back side rotating metallic pointer 5., and play turning effort.10. the unit circle quoit is the center of circle with position, O place, is radius with 9cm.6. the metal contact that 10. the unit circle quoit is gone up is set in each position, special angle.5. the rotating metallic pointer is gone up the metal contact and 6. is set in apart from center O 9cm place.7. rotating metallic cover is installed in the rotating metallic pointer with screw and 5. goes up on the pointer.Insulated conductor is 8. in order to connect positive light bulb base and unit circle quoit 10., 9. 7. the rotating metallic cover reach the No.5 cell base on the pointer.

4. the 1. positive center pit of plane coordinate system demonstration board is equipped with screw axis.4. screw axis is gone up and light bulb base is installed and forms closed-loop path (referring to Fig. 2) with plane coordinate system demonstration board back side electro-optical system.

During use; Demonstration device is fixing on the table; Plane of rotation coordinate system demonstration board is 4. back side handle of screw axis 1.; The rotation of parallelogram type straight line rotation chi, 2. the rotation main scale intersects with the tan image in rotary course, then representes pitch angle and the relation of slope and combining closely of straight slope and tan without the initial point straight line that 3. this place representes with reference to chi through true origin straight line and rotation.Simultaneously, can compare sinusoidal line, the cosine line at each angle, the situation of change of positive tangent.When the rotation main scale rotated to position, special angle, bulb was promptly light.

Illustrate:

Example 1: known 2 A (one 3,4), B (3,2), the straight line L and the line segment AB that cross some P (2, one 1) have common point, ask the slope k of straight line L and the span of pitch angle a.

Analyze: k _AP,=one 1, k _PB=3, so, one 1≤k≤3; The k value promptly have on the occasion of, negative value is arranged again; Therefore

Be divided into: so one 1≤k＜0 or 0＜k-＜3

or 0≤a＜a rctan 3

Presentation process is following:

On coordinate tie-plate face, determine A (one 3,4) with coordinate system, B (3; 2), P (2, one 1) 3 points; Fix on the P point and be rotation center with the end with reference to chi of rotation chi with it; The other end is walked around the B point from the A point, with reference to chi in the process of P point rotation, the slope variation value that main scale is corresponding is exactly the changing value of k.

Example 2, ask the codomain of function $\left(1\right),y=\frac{3\mathrm{Sin}x-3}{2\mathrm{Cos}x+10}$

Resolve: (1) $\frac{2\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="HSA00000584805200011.png"\; state="\{\{state\}\}">y</figure-callout>}}{3}=\frac{\mathrm{Sin}x-1}{\mathrm{Cos}x+5},$ Can

(cosx, sinx), the slope of the straight line of B (5,1) solves to be interpreted as some A $-\frac{5}{8}\&le;y\&le;0.$

Presentation process is following:

On coordinate tie-plate face, determine a B (1 with coordinate system; 1); Fix on B point and be rotation center with it with the end with reference to chi of rotation chi, the other end is just tangent with unit circle around arriving, and is being exactly the changing value of k with reference to the tangent corresponding slope variation value range of two points of chi and unit circle.

Example 3, known Equation f (x)=(x one 2k) ²=ax (k ∈ N) has two not wait real root, the span of realistic several a on interval [2k 1,2k+1]

Resolve: establish function f (x)=(x one 2k) ², g (x)=ax,

X ∈ [2k-1,2k+1] makes the two image in the same coordinate system, equation has two to separate and be equivalent to two images two different intersection points are arranged on [2k-1,2k+1].Y=ax should be between 0X and OB, and the coordinate that B is ordered is (2k+1,1), K _Ox＜K1≤K _Ob,

Presentation process is following:

Make para-curve f (x)=(x one 2k) according to coordinate system on the plane coordinate system demonstration board on the paper of helping attaching ²Image, other presentation process is 1 identical with example.

Claims (1)

1. the demonstration device of straight inclined angle and slope relation; Be by the plane coordinate system demonstration board 1., 4. parallelogram type straight line rotation chi, screw axis combine with electro-optical system; It is characterized in that 1. the plane coordinate system demonstration board is gone up apart from 12cm place, the plate face left side is provided with plane right-angle coordinate xOy for center O; Has been that the unit of pressing head marks each special angle point of

π with x axle positive axis with O; With O is the center of circle; 9cm is that radius is drawn unit circle; With unit circle with 0,

π, each angle be divided into 16 special angles; The interval [0, π] of drawing is gone up the tan image and is drawn the corresponding tangent value in each special angle; Parallelogram type straight line rotation chi be divided into the rotation main scale 2., rotation with reference to chi 3. and two regulate chis; Rotation main scale, rotation are bound into parallelogram shape with two adjusting chis by removable fasteners with reference to chi, and 4. parallelogram type straight line rotation chi is equipped with screw axis at rotation main scale center pit; 4. screw axis is installed in the plane coordinate system demonstration board back side 1. with electro-optical system; 5. 4. screw axis be equipped with the rotating metallic pointer; 6. the metal contact that 10. the unit circle quoit is gone up is set in each position, special angle; 5. metal finger is gone up the metal contact and 6. is set in apart from center O 9cm place; 7. rotating metallic cover is installed in the rotating metallic pointer with screw and 5. goes up, insulated conductor 8. connect positive light bulb base and unit circle quoit 10., 9. 7. the rotating metallic cover reach the No.5 cell base on the pointer, with plane coordinate system demonstration board back side electro-optical system composition closed-loop path.

Images