AU2018101269A4

AU2018101269A4 - Trigonometry tool

Info

Publication number: AU2018101269A4
Application number: AU2018101269A
Authority: AU
Inventors: John Lawton
Original assignee: Objective Learning Materials Pty Ltd
Current assignee: Objective Learning Materials Pty Ltd
Priority date: 2018-01-30
Filing date: 2018-08-30
Publication date: 2018-10-04
Anticipated expiration: 2026-08-30
Also published as: ZA201900590B

Abstract

A drawing template 1 for overlying paper. The template includes one or more first portions 3 against which a pencil is abuttable to define on the paper at least a portion of a circle having a centre and a radius. The template further includes one 5 or more second portions against which the pencil is abuttable to define on the paper at least a portion of at least one of a sine wave running, along an axis substantially coincident with the centre, and having an amplitude substantially equal to the radius, and a cosine wave running, along an axis substantially coincident with the centre, and having an amplitude substantially equal to the 10 radius. At least part, of the portion of the at least one of the sine wave and the cosine wave, corresponds to at least part of the portion of the circle. E~I gn co u in

Description

TRIGONOMETRY TOOL

FIELD

The invention related to trigonometry.

BACKGROUND

Trigonomic functions such as sine (opposite + hypotenuse) and cosine (adjacent * hypotenuse) are important mathematical functions. One method for teaching is by reference to a unit circle including a radial line representing the hypotenuse and sweeping about this circle in the anti-clockwise direction from the right-hand (3 o'clock) position. Utilising this approach, some students are able to gain a practical and intuitive understanding of these functions.

Most students, and some students in particular, learn best from tangible (hands on) learning experiences. Various mechanical devices such as Scotch yokes can be utilised to teach sine and cosine functions. Such devices can be relatively costly and cumbersome and consume valuable classroom time in that they must be retrieved from storage and subsequently packed away each time that they are used.

With the foregoing in mind, the present invention aims to provide improvements in and for teaching and learning trigonometry, or at least to provide an alternative for those concerned with trigonometry.

SUMMARY

One aspect of the invention provides a drawing template, for overlying paper, including one or more first portions against which a pencil is abuttable to define on the paper at least a portion of a circle having a centre and a radius; and one or more second portions against which the pencil is abuttable to define on the paper at least a portion of at least one of a sine wave running, along an axis substantially coincident with the centre, and having an amplitude substantially equal to the radius; and a cosine wave running, along an axis substantially coincident with the centre, and having an amplitude substantially equal to the radius; and at least part, of the portion of the at least one of the sine wave and the cosine wave, corresponding to at least part of the portion of the circle.

Preferably the one or more first portions include a curved portion running about at least 30° of the circle. Most preferably the one or more first portions are configured to enable at least 50% of the circle to be drawn without moving the template.

The one or more second portions may include a curved portion running along at least 30° of the at least one of the sine wave and the cosine wave. Preferably at least 90°, of the portion of the at least one of the sine wave and the cosine wave, corresponds to at least part of the portion of the circle.

The template may include one or more third portions against which a pencil is abuttable to define on the paper the axis of the at least one of the sine wave and the cosine wave. The template may include defined portions against which a pencil is abuttable to mark on the paper, along the at least one of the sine wave and the cosine wave, locations corresponding to 30°, 45° and 60°.

The template may include first markings quantifying an angle about the portion of the circle. Optionally there are second markings quantifying an angle along the portion of the at least one of the sine wave and the cosine wave. Preferably third markings run parallel to the axis of the at least one of the sine wave and the cosine wave to relate the portion of the circle to the portion of the at least one of the sine wave and the cosine wave. The template may include markings quantifying a magnitude of the at least one of the sine wave and the cosine wave.

Preferably a period of the at least one of the sine wave and the cosine wave is substantially equal to a circumference of the circle.

Another aspect of the invention provides a method of defining on paper at least a portion of circle and at least a portion of a trigonomic wave corresponding to the circle, including placing on the paper the template; and abutting a drawing implement against the one or more first portions; and abutting a drawing implement against the one or more second portions. BRIEF DESCRIPTION OF DRAWINGS.

Figure 1 is a front view of a portion of a template.

Figure 2 is a front view of a template.

DESCRIPTION OF EMBODIMENTS

Figure 1 illustrates a portion of a template 1 incorporating features by which the template becomes a learning aid for learning trigonomic functions.

The template 1 is formed of transparent polycarbonate. Whilst other sheet materials may also be convenient, transparent materials are preferred.

The template 1 includes a first portion in the form of circular cutout 3 against which a pencil is abuttable to draw a circle on a page underlying the template. In this example, the portion 3 is shaped to enable a complete circle to be drawn, although other variations on the theme are possible. Other shapes are possible, e.g. the portion 3 might take the form of an arcuate slot through which 90° of the circle could be defined on the underlying page. Other variants are possible. Indeed, three points on the page are sufficient to define a circle.

For the avoidance of doubt, as the wording and similar wording is used herein, the edge of the circle 3 is an edge against which a pencil is abuttable, although other drawing implements (e.g. pens) could be used. Vice versa, an edge intended to provide an abutment for a pen is an edge against which a pencil is abuttable.

The template 1 further includes a second portion 5 in the form of a serpentine slot 5 along which a pencil can be moved to draw a sine wave corresponding to and in alignment with a unit circle drawn with the aid of the circle 3. As such, the drawn sine wave runs along a horizontal axis 7 coincident with a centre 3a of the circle drawn with the aid of the circle 3. In this case, a marking in the form of a horizontal line 7a runs along the axis 7. Holes 7b are spaced along the axis 7 at 30° intervals to enable these angles to be marked off along the sine wave. Other defined features for marking off the angles, e.g. notches, are possible. The defined features need not be along the axis 7a, albeit that that location is preferred. For example, notches may be marked off along an edge of the slot 5.

In this particular example, a circle is provided for marking off the 45° point along the sine curve, and the slot 5 passes through the axis 7 at the 180° mark so that there is no separate circle 7b for this location. In another example, a circle 7b may be provided at the 180° mark along a short break in the slot 5.

Each of the holes 7b is associated with a respective marking 7c quantifying the angle along the horizontal axis.

Markings 9 run parallel to the axis 7 to relate the magnitude of the sine curve to the corresponding locations around the unit circle. In this case, the markings 9 are solid lines and there are four markings 9 above the axis 7 corresponding to sine values of 0.5, V2/2, V3/2 and 1.0. The magnitude of the sine wave is conveyed by markings 11. In this particular example, the sine curve extends beyond 180° and includes markings below the axis corresponding to the markings 9, 11 above the axis.

The template 1 can be used to provide a tangible learning experience. As a first step, a student might first rule a horizontal line and a vertical line intersecting the horizontal line. The template 1 is then placed on the page and maneuvered to align the crossed pair of lines with angular markings 13 about the circle 3 and thereby orient the circle 3 concentrically to the intersection of the two lines. The student can then draw in a portion of the unit circle (say, 30°, (π/6)) by abutting a pencil (or other drawing implement) with the circle 3 and moving it about the appropriate portion of that circle. A student can then draw the corresponding portion of the sine wave by placing a pencil into the left-hand end (i.e. the 0,0 point) of the slot 5 and maneuvering the pencil along the curve until the 30° point is reached.

Through this process the student mechanically (as opposed to electronically) generates a graphical illustration that the end point of the curve they have just drawn corresponds in height to the height of the point about the unit circle and a height of 0.5 (from the markings 11). With ongoing experimentation, students can gain an intuitive understanding of the sine curve and its intimate link to the unit circle. A corresponding arrangement can be used for teaching and learning cosine. For this purpose, a dedicated device may be produced. Alternatively, the template 1 might be adapted to serve both purposes. By way of example, a vertical axis and cosine curve akin to the axis 7 and sine curve 5 may run vertically down from the circle 3.

Preferred forms of the device at least predominantly consist of sheet material and do not exceed the boundaries of an A4 sheet and are thereby adapted to being conveniently packed away with a student's papers.

Beyond the mentioned features related to trigonometry, the drawing template may have other pencil-abutting edges to define other shapes such as polygons, circles and ellipses. A preferred form of the device incorporates a protractor. As such, the device may be utilised by students younger than those conventionally taught trigonometry. In Australia, trigonometry is taught in advanced Year 10 classes. Exposing younger students to the mentioned circle and wave portions will help build familiarity with the concepts and challenge students at earlier school years to consider the trigonomic functions. This can foster an appetite in junior and middle school students for more advanced mathematics.

Figure 2 illustrates a template 101 including a trio of circle-curve pairs 3, 5; 103, 105; 203, 205.

The pair 3, 5 is substantially as described in Figure 1. The pair 103, 105 illustrates the curve y=sin(x/2), that is, wherein the angle about the inner circle is scaled by a factor of 0.5. To assist with the intuitive understanding of this concept, axis 107 is positioned in register with the axis 7 and the length of the axis (i.e. the amount of the markings 107c) is adjusted by an amount corresponding to the factor (i.e. an amount corresponding to 0.5 in this case).

The curves defined by portions 5, 105 each have an amplitude corresponding to the radius of their corresponding circle. The pair 203, 205 shows an alternate approach that may be separately useful. The portion 203 is to draw a unit circle. The portion 205 is for drawing the curve y=2sin(x), i.e. for drawing a curve having an amplitude factor of 2. Advantageously, the period of the curve corresponds to the circumference of the circle.

The invention is not limited to the described examples. Rather, the invention is defined by the claims; e.g., whilst the circle 3 defines the full 360° of the circle and the slot 5 defines the corresponding full 360° of the sine wave, in another variant the first portion may be adapted to define only, say, the first 90° of the unit circle and the second portion might be adapted to define the corresponding first 90° of the sine wave.

Claims

1. A drawing template, for overlying paper, including one or more first portions against which a pencil is abuttable to define on the paper at least a portion of a circle having a centre and a radius; and one or more second portions against which the pencil is abuttable to define on the paper at least a portion of at least one of a sine wave running, along an axis substantially coincident with the centre, and having an amplitude substantially equal to the radius; and a cosine wave running, along an axis substantially coincident with the centre, and having an amplitude substantially equal to the radius; and at least part, of the portion of the at least one of the sine wave and the cosine wave, corresponding to at least part of the portion of the circle.

2. The template of claim 1 wherein the one or more first portions include a curved portion running about at least 30° of the circle.

3. The template of claim 1 or 2 wherein the one or more first portions are configured to enable at least 50% of the circle to be drawn without moving the template.

4. The template of claim 1, 2 or 3 wherein the one or more second portions include a curved portion running along at least 30° of the at least one of the sine wave and the cosine wave.

5. The template of any one of claims 1 to 4 wherein at least 90°, of the portion of the at least one of the sine wave and the cosine wave, corresponds to at least part of the portion of the circle.

6. The template of any one of claims 1 to 5 including one or more third portions against which a pencil is abuttable to define on the paper the axis of the at least one of the sine wave and the cosine wave.

7. The template of any one of claims 1 to 6 include defined portions against which a pencil is abuttable to mark on the paper, along the at least one of the sine wave and the cosine wave, locations corresponding to 30°, 45° and 60°.

8. The template of any one of claims 1 to 7 including first markings quantifying an angle about the portion of the circle; and second markings quantifying an angle along the portion of the at least one of the sine wave and the cosine wave.

9. The template of any one of claims 1 to 8 including third markings running parallel to the axis of the at least one of the sine wave and the cosine wave to relate the portion of the circle to the portion of the at least one of the sine wave and the cosine wave.

10. The template of any one of claims 1 to 9 including markings quantifying a magnitude of the at least one of the sine wave and the cosine wave.

11. The template of any one of claims 1 to 10 wherein the at least one of the sine wave and the cosine wave includes the sine wave.

12. The template of any one of claims 1 to 11 wherein the at least one of the sine wave and the cosine wave includes the cosine wave.

13. The template of any one of claims 1 to 12 wherein a period of the at least one of the sine wave and the cosine wave is substantially equal to a circumference of the circle.

14. A method of defining on paper at least a portion of circle and at least a portion of a trigonomic wave corresponding to the circle, including placing on the paper the template of any one of claims 1 to 12; and abutting a drawing implement against the one or more first portions; and abutting a drawing implement against the one or more second portions.