Abstract
A slope measurement and drawing aid (100) for teaching mathematics having unique indicia (130), rational-numeric slope values (184), and mathematical information (192), and a method of identifying the correct rational-numeric slope value associated with the graph of a given line, either by means of a gravity-activated indicator arm (140), or by means of the orientation of the teaching aid. For instructional purposes, when used in a classroom setting, the teaching aid can be removably held on a vertical board by magnetic (240R) or other means. When used in an individual setting, a smaller version of the teaching aid can similarly be used in a horizontal orientation on paper.
Claims
1. A mathematics teaching aid system comprising: a generally flat and planar base, said planar base having a front side and a back side with at least one straight edge; at least one pivot point reference mark substantially centrally located on said straight edge; a first coordinate line indicia parallel to the straight edge and intersecting the at least one pivot point; and a plurality of slope values indicated as a simplified numeric ratio of positive or negative whole numbers and corresponding indicia visibly displayed in predetermined positions equidistant from said pivot point reference mark on said planar base and radially disposed relative to said pivot point reference mark, each of said slope values and corresponding indicia disposed in predetermined positions corresponding to said predetermined slope values with respect to said straight edge, and ranging between values of zero and positive infinity in a positive direction and between zero and negative infinity in a negative direction, wherein the indicia of slope value of zero is positioned along a second coordinate line that is perpendicular to the first coordinate line indicia and intersects the pivot point reference mark and wherein the indicia of slope values of the positive infinity and the negative infinity are positioned along opposite ends of the first coordinate line indicia.
2. The mathematics teaching aid system of claim 1 further comprising mathematical information related to the understanding of slope or use of said mathematics teaching aid and system visibly displayed on said planar base.
3. The mathematics teaching aid system of claim 2 wherein said mathematical information comprises, but is not limited to, the symbol for infinity, any shorthand definitions for slope, any precise mathematical definition of slope, any positive or negative signs located to reinforce the concept of slope.
4. The mathematics teaching aid system of claim 1 wherein said planar base is made of suitable transparent material.
5. The mathematics teaching aid system of claim 1 wherein said planar base is of suitable size to be visible in a classroom, and substantially semicircular; whereby the radius of said semicircular base is defined by said equidistant placement of said indicia or said slope values.
6. The mathematics teaching aid system of claim 1 further comprising an attachment means for removably securing said teaching aid to a vertical presentation board.
7. The mathematics teaching aid system of claim 1 further comprising an elongated generally flat and planar indicator arm of a length sufficiently less than the radius of said semicircular base to be easily contained inside said equidistant placement of said slope values; with pivot means at said proximal end for rotationally movable attachment of said planar indicator arm to said front side of said planar base at said central reference mark.
8. The mathematics teaching aid system of claim 7 further comprising a pivot means located substantially at said pivot point reference mark on said front of said planar base for said rotationally movable attachment of said indicator arm.
9. The mathematics teaching aid system of claim 7 wherein the attachment of said indicator arm to said front side of said planar base is by means of a fastener slightly smaller than the diameter of said holes through said planar indicator arm and said planar base to thereby allow said indicator arm free rotational movement under the effect of gravity when oriented correctly.
10. The mathematics teaching aid system of claim 1 wherein the planar base is of suitable size to be used by an individual on a standard educational worksheet.
11. The mathematics teaching aid system of claim 1 further comprising mathematical information related to the understanding of slope or to the use of said mathematics teaching aid and system visibly displayed on said planar base.
12. The mathematics teaching aid system of claim 11 wherein said mathematical information includes, but is not limited to, the symbol for infinity, at least one shorthand definition for slope, positive and negative signs.
13. The mathematics teaching aid system of claim 11 further comprising a visible line imprinted substantially on and parallel to said straight edge on said planar base.
14. The mathematics teaching aid system of claim 11 further comprising a method to identify approximate said radially disposed slope values by the orientation of said slope values directly nearest to a defined mathematical reference line.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
(1) FIG. 1 is a front view of a first embodiment of an exemplary mathematical slope teaching aid according to the present invention.
(2) FIG. 2 is an exploded view of the teaching aid according to the present invention as shown in FIG. 1.
(3) FIG. 3 is an illustration of the first exemplary application of the teaching aid according to the present invention as shown in FIG. 1.
(4) FIG. 4 is an illustration of the second exemplary application of the teaching aid according to the present invention as shown in FIG. 1.
(5) FIG. 5 is an illustration of the third exemplary application of the teaching aid according to the present invention as shown in FIG. 1.
(6) FIG. 6 is a front view of a second embodiment of an exemplary mathematical slope teaching aid according to the present invention.
(7) FIG. 7 is an illustration of the first exemplary application of the teaching aid according to the present invention as shown in FIG. 6.
(8) FIG. 8 is an illustration of the second exemplary application of the teaching aid according to the present invention as shown in FIG. 6.
(9) FIG. 9 is a perspective illustration of the first and second embodiments of the teaching aid according to the present invention as shown in FIG. 1 and FIG. 6 being used together to conduct the first exemplary application in FIG. 3
DETAILED DESCRIPTION OF THE INVENTION
(10) For reasons explained below, and unlike known teaching aids, the teaching aid 100 may be capably used by a teacher or student, among others, to quickly measure the rational-numeric slope value of a graphed line; to estimate the slope of an imagined tangent line to a curvilinear graphed function as found in any calculus class; facilitate the rapid sketching of a derivative function ƒ′(x) based on the graphed function ƒ(x); facilitate the sketching of the second derivative function ƒ″(x) based on a sketch of a first derivative function ƒ′(x); to aid in quickly and consistently sketching a “slope-field” for certain solutions to differential equations; to estimate the tangent trigonometric ratio of a right triangle; to quickly and accurately estimate a linear regression line for a data set; in use by a practitioner trained in the field, the mathematics teaching aid may be rather universally used in a variety of applications to accomplish a wide variety slope-related measurements and drawings.
(11) FIG. 1 is a front view of an exemplary mathematical slope teaching aid 100 which may be used by teachers, students, and mathematicians, among others, to more accurately determine the numerical or slope value (typically expressed as a reduced rational number) of a line. The teaching aid may be used to measure the slope of a linear function, or estimate the slope of a perceived tangent line to a graphed curvilinear function, as well as permit the accurate drawing of a calculated slope for a line or a tangent line. The teaching aid is typically be used, but not limited to, written or projected mathematical material on a whiteboard or blackboard.
(12) As illustrated in the front view in FIG. 1, and the exploded view in FIG. 2 an embodiment of the invention 100 consists of a generally flat and planar semicircular base 120 large enough to be seen from any desk in a typical classroom. A clearly visible line 124 is imprinted along the straight edge 122 of the base 120. The base 120 has indicia 180 numerical values 182 and fractional values 184 radially disposed equidistantly from the center of the pivot point reference mark 128 that defines the semicircular perimeter of the base 130. The pivot point reference mark 128 will be the location of the hole 130. The teaching aid should be made of some suitable lightweight transparent material (e.g., acrylic plastic), an elongated indicator arm 140 (slightly shorter in length than the radius of the semicircular base 120) made of similar transparent generally flat and planar material with an index line or arrow-head 142 centrally disposed along the major axis. The indicator arm 140 has a suitable hole 152 in the proximal end to match a similar hole 150 located at the pivot point reference mark 128 on the base 120. Suitable fasteners 220A and 220B (e.g. a sleeve-nut and bolt) that movably secure the indicator arm to the front of the base and allow it to swing freely around its pivot-point 128 centrally located near the straight edge 122 of the semicircle defined by the radially disposed rational-numeric slope values 184. Rotational movement of the indicator arm 140 may typically be achieved by means of fasteners 220F and 220B that are a slightly smaller diameter than the holes provided in the base 130 and indicator arm 132. By advantage of the free movement of the planar indicator arm, when vertical, the indicator arm will find equilibrium under the force of gravity and point downward, as would a plumb bob, and clearly identify which of the radially imprinted rational-numeric slope values 180 corresponds to the inclination of the straight portion of the semicircular when aligned with a suitable mathematical graph. The invention also has some number of magnets 240L and 240R, mounted on the back surface of the base to sufficiently and temporarily secure the device 100 to a vertical board surface while in use for demonstration purposes. Also imprinted on the face of the semicircle 120 is additional slope-related mathematical information 190 useful to anyone using the teaching aid, such as, but not limited to the definition of slope as “rise/run” or (y1−y2)/(x1−x2); written and/or graphic indicators for when a slope is positive or negative; written and/or graphic indicators for when a slope is said equivalently to be “infinite,” “undefined” or have “no slope”; or any other useful indicia or information 192 that relates to the mathematics of slope.
(13) FIG. 3 shows an exemplary use of the teaching aid 100. When a function is graphed on an x-y graphing axes 350 (as it might be drawn or projected on a vertical display board such as a dry-erase board), the teaching aid 100 can be used to measure the rational-numeric slope value of the function when the straight edge 122 of the teaching aid 100 is aligned with the graph 320 of a typical linear function 360 with a given slope 340 and a given y-intercept (b) 324 on the y-axis. The rational-numeric slope value 340 can be read by the indicator arm 140. Conversely, the teaching aid 100 can be used to draw the graph of a line intersecting the y-axis 322 at a known y-intercept (b) 324 with known slope 340 typically when given in slope-intercept form (e.g. y=3/2x+5). The person using the teaching aid 100 places one end the straight edge 122 of the semicircular face 120 on point (b) 324 and then manipulates the teaching aid 100 until the desired slope 340 is indicated by the gravity-activated indicator arm 140. A line may then be drawn accurately and without the need for a second point to be plotted. (Note: for most elementary applications, the teaching aid will be used when the x-y graphing axes 350 are at the same scale. For more advanced applications, a simple conversion factor can be multiplied by the slope value to give accurate results.)
(14) FIG. 4 shows an exemplary use of the teaching aid 100 being aligned along the straight edge 122 with the upper ray 420 of an exemplary acute angle 430 such that the indicated slope-value 440 is identified by the indicator arm 140. The decimal value 450 of the slope 440 can easily be divided out by hand and thereby return a tangent value 450 within a small error bounds of a calculated decimal tangent value 460 for the given angle measure 430.
(15) FIG. 5 shows an exemplary use of the teaching aid 100 to sketch a representation of the derivative function ƒ′(x) 510 that corresponds to a graph of an exemplary function ƒ(x) 500. Typically, the function ƒ′(x) 510 is plotted or otherwise placed on a board with an x-y axes 520. The teaching aid can be placed at any desired point along ƒ(x) 500 but is typically started on the left end of the graph and moved slowly along to the right tracing the outline of the graph of the function ƒ(x) 500 while the straight edge remains tangent and the slope values are noted. In this exemplary illustration of the left to right movement, four such positions are shown corresponding to four x-values, and are therefore marked on the x-axis 550 at: (a) 540, (b) 542, (c) 544, and (d) 546, as they are generated by the reading of the indicatory arm 140 at each position. At each point along the function ƒ(x) 500 the teaching aid is placed in such a way that the straight edge 122 aligns with an imagined tangent line to the function ƒ(x) 500 at that point, and the indicator arm 140 then gives the reading of the slope corresponding to the estimated tangent line at that point—thus representing its instantaneous rate of change, or derivative value. At the position corresponding to point (a) 560 a negative slope value returned by the indicator arm 140 is plotted for ƒ′(a) 570. At the position corresponding to point (b) 562 a slope value near zero is returne plotted for ƒ′(b) 572. At the position corresponding to point (c) 564 a small positive slope value is plotted for ƒ′(c) 574. At the position corresponding to point (d) 566 a negative slope value is plotted for ƒ′(d) 576. Using this method, the slope-values of a series of points (as many as desired) can be determined for the given function ƒ(x) 500, the corresponding derivative values can then be plotted on the same, or another, coordinate graphing x-y axes 550 and the approximate derivative function ƒ′(x) 510 can be rapidly and accurately sketched. Subsequently, the process can be continued, but is not shown in any figure, whereby the sketched derivative function ƒ′(x) 510 would be used to find a second derivative ƒ″(x), as is often done in calculus (e.g. when representing a graph of acceleration when given the displacement of an object).
(16) FIG. 6 is a front view of an exemplary mathematical slope teaching aid 600 which is designed to be used primarily by students, and mathematicians, among others, as in similar fashion to the first embodiments in FIG. 1, only on a personal scale. Typically the mathematical slope teaching aid 600 would be used in conjunction with mathematical slope teaching aid 100 in a classroom setting, or when a student was completing practice exercises on paper. The mathematical slope teaching aid 600 has no moving parts, but by advantage of the orientation of the semicircular base, the slope value can be closely estimated without the need of a gravity-activated indicator arm, particularly when the practitioner is familiar with the use of teaching aid 100. The teaching aid 600 may be used to measure or facilitate the drawing of a multiplicity of mathematical representations, including but not limited to, those exemplary embodiments for teaching aid 100 above. The teaching aid 600 is typically be used with, but not limited to, written mathematical material in a textbook or worksheet.
(17) The mathematical slope teaching aid 600 consists of a generally flat and planar semicircular base 620 small enough to be used on a personal scale in a typical math textbook or worksheet, and made of some suitable lightweight transparent material. The semicircular base 620 is imprinted with a clearly visible line 624 that containing a pivot point reference mark 628 located at the center of the semicircle and near the straight edge 622. The semicircular base 620 has indicia 630, numerical values 632, and fractional values 634 radially disposed along the semicircular perimeter of the base to clearly indicate which of the radially imprinted rational numeric slope values correspond to the inclination of the straight edge 622 of the semicircular base when aligned with a suitable mathematical graph. Also imprinted on the face of the semicircle 620 is additional slope-related mathematical information useful to anyone using the teaching aid, such as, but not limited to the definition of slope as “rise/run” or (y1−y2)/(x1−x2); written and/or graphic indicators for when a slope is positive or negative; written and/or graphic indicators for when a slope is “infinite” or has “no slope”; or any other useful indicia that relates to the mathematics of slope.
(18) FIG. 7 shows an exemplary use of the teaching aid 600 to estimate the tangent of an exemplary angle 700 printed on a worksheet 666. To estimate the value of the tangent, the straight edge 622 of the teaching aid 600 is aligned with the upper ray 720 of the given angle 700, such that the rational-numeric slope value 740 can be identified being the closest value to the lower ray 730 of the given angle 700. The approximately equivalent decimal value of the identified rational-numeric slope value 740 can be found easily by dividing out by hand, if necessary, and return an estimated tangent value 750. This can be compared to a calculated decimal value 760 for the given angle 700.
(19) FIG. 8 shows an exemplary use of the teaching aid 600 to draw the calculated slope value for a differential equation's “slope field” 840 as given on a pre-printed page 866 of an exercise book. The given differential equation 810 is used to calculate the slope 814 at a given point 812, which is then sketched onto the given lattice point 820 provided by orienting the teaching aid so that the desired value 814 is nearest the lower edge of the worksheet 866.
(20) FIG. 9 shows an exemplary use of teaching aid 100 used in conjunction with teaching aid 600 to draw the graph of a simple linear equation 910. A board 900 contains the given equation 910 the graphed line 912, and the axes 920 while a student working at a desk is able to follow the steps being modeled by the teacher. By magnetic means the teaching aid 100 is secured to the board 900 and the teacher is free to circulate in the classroom and monitor student learning. Students use teaching aid 600 to complete the exercise pre-printed on a worksheet 930.
(21) The teaching aids 100, 600 can be economically made in a number of processes, such as cutting, stamping, molding, or machining. The teaching aid could be made of any suitable transparent material, and may be produced in a variety of sizes, with varying scale measurements and indicia. It can be made with a variety of means for allowing the gravity activated indicator arm to pivot as well as for removably securing teaching aid to a board. The placement of any means for securing the teaching aid to a board is widely flexible, and need only allow the teaching aid's easy placement and removal, and is not to be understood to be specific in location or number.
(22) While the invention has been described in terms of various specific embodiments, those skilled in the art will recognize that the invention can be practiced with modifications within the spirit and scope of the claims.
RELEVANT PRIOR ART INCLUDES
(23) U.S. Pat. No. 1,541,179 June, 1925 Parkinson
(24) U.S. Pat. No. 1,912,380 June, 1933 McCully
(25) U.S. Pat. No. 2,822,618 February 1958 Wendel
(26) U.S. Pat. No. 3,083,475 April, 1963 Lepoudre
(27) U.S. Pat. No. 1,912,380 June, 1933 McCully
(28) U.S. Pat. No. 0,078,092 April, 2008 Lin
(29) U.S. Patent Publication 0120852 May, 2008 Ramsey
(30) U.S. Pat. No. 7,942,675 May, 2011 Errthum
(31) U.S. Pat. No. 0,316,981 December, 2010 Gunasekaran
