Abstract:
Devices and methods for hands-on learning of mathematical parabolic concepts are provided. For example, a device for teaching mathematics includes a base having a plurality of channels for accepting vertical rods of varying lengths; a lever movably attached to the base at a pivot, the lever passing through the plurality of channels; and a plurality of vertical rods, each of the plurality of vertical rods located at least partially in one of the plurality of channels and resting on the lever; wherein the plurality of vertical rods are sized and arranged so that the tops of the vertical rods illustrate a parabolic function; and wherein moving the lever about the pivot moves the plurality of rods within the plurality of channels and illustrates changes to the parabolic function represented by the vertical rods.

Description:
BACKGROUND 
     Technical Field 
     This disclosure is related to the field of physical teaching models and in particular to physical models for mathematical concepts. 
     Related Art 
     Principles of algebra can often be somewhat challenging to visualize and comprehend for students. Many students (or users, used interchangeably throughout this specification) have difficulty understanding equations and their graphical representation, especially quadratic polynomials such as parabolic equations. 
     Often teachers rely on coordinate grids (on graphing paper or chalkboards or dry-erase boards) to illustrate the geometric characteristics of parabolas. However, such drawings are by their nature static, and teachers cannot easily provide a visual understanding of how changes to the coefficients of the polynomial equation changes the shape of the parabola without drawing multiple graphs. This can be difficult for students to follow. Continuous efforts are being made to improve teaching techniques and devices to aid students. 
     SUMMARY 
     Teaching aids for manipulating and visualizing parabolas are disclosed herein. Providing a physical model that can be manipulated by a teacher or student can help a student visualize how the values of coefficients in a parabolic equation can change the features of the parabola. 
     In one aspect, a device for teaching mathematics includes a base having a plurality of channels for accepting vertical rods of varying lengths; a lever movably attached to the base at a pivot, the lever passing through the plurality of channels; and a plurality of vertical rods, each of the plurality of vertical rods located at least partially in one of the plurality of channels and resting on the lever; wherein the plurality of vertical rods are sized and arranged so that the tops of the vertical rods illustrate a parabolic function; and wherein moving the lever about the pivot moves the plurality of rods within the plurality of channels and illustrates changes to the parabolic function represented by the vertical rods. In a further aspect, moving the lever about the pivot illustrates changes to a coefficient b from the standard parabolic equation. Another aspect further includes a coordinate grid indicator attached to the base to provide a reference point for the parabolic function formed by the plurality of rods. 
     In another aspect, a method for teaching and/or learning concepts of parabolas includes: obtaining a base having a plurality of channels to support vertical bars sitting on a rotatable platform; positioning vertical bars in the plurality of channels on the rotatable platform such that the vertical bars are sized to resemble a parabola; rotating the rotatable platform to observe changes to the parabola of the vertical bars, where the rotation indicates changes to a coefficient b in a standard parabolic equation, y=ax 2 +bx+c. In a further aspect, the method includes, when the rotatable platform can be raised or lowered within the base, raising or lowering the rotatable platform to observe changes to the coefficient c in the standard parabolic equation. 
     In another aspect, a device for teaching mathematics includes a base having a plurality of channels for accepting vertical rods; a rotatable platform movably attached to the base at a pivot, the platform passing through the plurality of channels for supporting vertical rods; and a plurality of vertical rods, each of the plurality of vertical rods located at least partially in one of the plurality of channels and resting on the rotatable platform; wherein the plurality of vertical rods are marked and arranged so that the markings of the vertical rods illustrate at least one parabolic function; and wherein rotating the platform about the pivot moves the plurality of rods within the plurality of channels and illustrates changes to the at least one parabolic function represented by the vertical rods. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and other features and advantages of the present device, systems, and methods will become appreciated as the same become better understood with reference to the specification, claims and appended drawings wherein like reference numerals reflect like elements as illustrated in the following figures: 
         FIG. 1  illustrates a front view of an example parabolic teaching aid in accordance with an embodiment of the present disclosure; 
         FIG. 2  illustrates a front perspective view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure; 
         FIG. 3  illustrates a side perspective view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure; 
         FIG. 4  illustrates a rear perspective view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure; 
         FIG. 5  illustrates a side view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure; 
         FIG. 6  illustrates a bottom exploded view of aspects of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure; 
         FIG. 7  illustrates an exploded component view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure; 
         FIGS. 8 and 9  illustrate front views of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure, where the lever has been shifted to a low point and a high point of travel; and 
         FIG. 10  illustrates an example method for learning about parabolic functions. 
     
    
    
     DETAILED DESCRIPTION 
     The use of physical devices that can be manipulated (manipulatives) as a means of teaching mathematical concepts relies on a constructivist educational paradigm, which can build upon a student&#39;s physical intuitions and broaden understanding of more abstract ideas. As such, a device for modeling parabolas is provided according to an aspect of the disclosure. 
     A parabola is generally represented in one of two equation forms. The first is the “standard form,” i.e. y=ax 2 +bx+c, where x and y are the variables and a, b, and c are coefficients for a given equation. The second is the “vertex form,” i.e. y=a(x−h) 2 +k, again with x and y as variables and a, h, and k are coefficients for a given equation. The vertex form is so called, because the coordinates (h, k) describe the vertex of the parabola resulting from the equation on a standard Cartesian (x, y) coordinate graph. 
     In general a teaching aid is provided that can be manipulated to show the relationships between parabola shapes and alignment on a standard Cartesian (x, y) coordinate graph when the equation coefficient values are changed. In the standard form of the equation, a parabola has an axis of symmetry that is defined by the line, x=−b/2a. Varying the magnitude of a changes the overall width of the parabola, about the axis of symmetry. Compared to a parabola with |a|=1, any parabola with |a|&lt;1 will have a broader shape and a flatter curve at the vertex because the value of y for each coordinate pair on the graph will be reduced by a factor equal to the a coefficient. Similarly, for any parabola with |a|&gt;1, the parabola will have a narrower shape and a more pointed curve at the vertex. The value of a cannot be zero without losing the parabolic shape and leaving a straight line; positive values of a produce a parabola that opens upward, and negative values of a produce a parabola that opens downward. Additionally, as indicated, changing h and k alter the location of the parabola&#39;s vertex. 
       FIG. 1  illustrates a front view of a parabolic teaching aid  100  in accordance with an embodiment of the present disclosure. The teaching aid includes a base  102  with a movable lever  104  and a plurality of vertical bars  106  of varying heights that can be placed to resemble a parabolic curve. As illustrated in  FIG. 1 , there are nine vertical bars  106  whose top end points  108  can generally indicate points along a virtual parabola (illustrated by the dotted line  105 ). The number and spacing of the vertical bars may be altered in various aspects to include more or less than those pictured. The figures represent an example embodiment only. In an aspect, each vertical bar  106  rests on or otherwise can be moved by lever  104 . Level  104 , in an aspect, can be raised and lowered with respect to a pivot point  312  (see  FIG. 3 ). Throughout the figures, dashed arrows with circled numbers indicate the perspective of other figures—where the numbers indicate the figures. The discussion herein may refer to a standard (x, y) coordinate system, where the y-axis is a vertical axis and the x-axis is a horizontal axis; with respect to the teaching aid  100 , the x-axis—in some aspects—may be considered the horizontal line represented by the top of base  102 . It should be noted that these axes may be illustrated, such as on the base, or imaginary and need not be fixed with respect to any one aspect of the teaching aid  100 . 
       FIG. 2  illustrates a front perspective view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure. From this view, a plurality of channels  210  can be seen having holes at the top of base  102 , each of these channels being capable of accepting one of the plurality of vertical bars  106 .  FIG. 3  illustrates a side perspective view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure, which also shows a portion of the back. Pivot point  312  can be seen from this angle. The pivot  312  may include a pin, a nail, a screw, a bolt, or other fastening mechanism which will allow lever  104  to rotate about the pivot  312 . 
       FIG. 4  illustrates a more rear perspective view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure. From this perspective a vertical riser  414  is visible through which pivot  312  connects to lever  104 . Optionally, in an aspect, the vertical riser  414  may include a channel  416  which would allow the pivot to be raised and lowered with respect to the base  102 . Movement of this pivot point up or down, without changing any other feature of the manipulative, would illustrate a change in the value of k in the vertex equation of a parabola or a change in the value of c in the standard equation. It serves to raise or lower the vertex in parallel with the y-axis (or more precisely along the parabola&#39;s axis of symmetry). 
       FIG. 5  illustrates a side view of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure.  FIG. 6  illustrates a bottom exploded view of aspects of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure. Channels  210  for accepting the vertical bars  106  are seen in base  102 . As can be seen in this figure, channels  210  may be generally connected but are shaped to accept the vertical bars  106  (not pictured). In other aspects, each channel may be fully separate, some channels may be open to each other, or there may be combinations of the same or the like. An example of the lever assembly is also illustrated with the lever  104  attached to pivot  312  and vertical riser  414 . As illustrated here, the lever is angularly movable about pivot  312 . In an aspect, pivot  312  may include sufficient rotational resistance with lever  104  that the lever remains in place when not acted on by a user. In another aspect, the lever may be more freely movable. 
       FIG. 7  illustrates another exploded view with all of the components of the parabolic teaching aid of  FIG. 1 . Illustrated are base  102 , lever  104 , a plurality of vertical bars  106  (including at least one short bar  106   s  and one or two long bars  106   l ), pivot  312 , vertical riser  414 , a base cap  716 , and a lever handle  718 . It should be noted that base cap  716  may alternatively be a top cap  716 , as the teaching aid as illustrated can be assembled in a variety of manners and still accomplish the purposes as disclosed herein. 
     In an aspect, each component may be made of plastic and formed by, for example, injection molding, machining, or 3D printing technology. Other generally rigid materials may also be used for some or all of the components illustrated, including glass, wood, metal, and the like. In an aspect, vertical rods  106  are provided in pairs of varied lengths so as to mirror a parabola as illustrated. In some aspects, an odd number of vertical bars  106  may include one unpaired vertical bar, such as a short bar  106   s.  In order to simulate an open down parabola (one with a negative a value in the standard form equation, y=ax 2 +bx+c), an extra short bar  106   s  may be included and one long bar  106   l  may be removed from the model, so that the channel  210  closest to the pivot  312  may accept a long bar  106   l  and shorter and shorter vertical bars are placed in channels that are farther from the pivot. The outermost channels  210  may then preferably include short bars  106   s.  Additionally, several extra pairs of vertical bars  106  that are longer and/or shorter than those illustrated may be supplied in some aspects to allow for parabolic shapes with greater magnitudes of the coefficient a. In an aspect, the vertical bars  106  and/or the channels  210  may be color-coded to provide simple assembly of one or more types of parabola graphs. In other aspects, various text or symbol markings may serve similar purposes. For example, in an aspect, each length of vertical bar  106  may have a different color and the channels  210  may have one or more color markings that coincide with the vertical bar colors. In another aspect, instructions may be included to let students know how to quickly assemble parabolas with different a, b, and c values. 
     In another alternative aspect, vertical bars  106  may include markings or color coding along their length to represent different values of a, such that when properly assembled, a student could observe multiple parabolas (such as one up to a blue color and another up to a green color and another up to a red color and so on, for example) at a time. In one aspect, such color coding could even allow all vertical bars to be the same height. The user would then look at the color coding or other markings on the vertical bars, rather than the overall height of the bars to understand the parabola changes. An added advantage of having vertical bars of the same height is that the bars  106  could simply be flipped over to represent a change from positive to negative a values, or vice versa. 
       FIGS. 8 and 9  illustrate front views of the parabolic teaching aid of  FIG. 1  in accordance with an embodiment of the present disclosure, where the lever has been shifted to a low point and a high point of travel. No particular angle is required or intended to be limiting in various aspects. In the example as illustrated,  FIG. 8  shows the lever  104  (using handle  718  as a guide) moved downward to an angle of approximately 45° below horizontal, and  FIG. 9  shows the lever  104  and handle  718  moved upward approximately 45° above horizontal. In general, the lever  104  can move smoothly between these two extremes. In various implementations, the lever may move between extremes that are greater or less than 45° while remaining within the spirit of the disclosure here. In  FIG. 8 , the vertex  807  of virtual parabola  105  can be seen to be shifted to the right of center. In  FIG. 9 , the vertex  807  of virtual parabola  105  can be seen to be shifted to the left of center. 
     In another aspect, pivot  312  may include a cog-like mechanism such that the lever “clicks” into one or more discreet settings, such as at parallel and at either extreme and/or angles in between. This may allow for easy study of visual representation of specific values of a, b, and c (standard equation) or h and k (vertex equation), in some aspects. 
     To assist in understanding the graphing of a parabola, a separate coordinate grid indicator (see indicator  920  in  FIG. 9 ) may be provided in some aspects. In general, the coordinate grid indicator would illustrate the x- and y-axes and may illustrate gradations for positive and negative values. In an aspect, the coordinate grid indicator may include gridlines as well. In an aspect, the coordinate grid indicator may comprise a generally clear or at least partially transparent rigid plastic sheet with the x- and y-axes printed or etched into it. A coordinate grid indicator may attach to the front of base  102  in a fixed or movable manner in various aspects such that the vertical bars  106  can be seen through the grid indicator to illustrate various values for the modeled parabola in relation to the grid indicator. It will be understood that a wide variety of materials, and designs can be used to accomplish the purpose of the coordinate grid indicator. For example, in another aspect, the coordinate grid indicator may attach to the back of base  102  with the vertical bars  106  partially obscuring a user&#39;s view of the grid indicator. In such an embodiment, the grid indicator need not be transparent and could be made of an opaque material. Including a movable coordinate grid indictor can allow a simple representation of how changing the value of c in the standard equation affects the graphing of a parabola—it simply raises or lowers the parabola with respect to the x-axis. As discussed above, in an aspect where the pivot  312  is movable along a vertical channel  416 , the same lesson can be taught with a fixed grid indicator. 
     With an example of the parabolic teaching aid now described, the manipulation of this teaching aid and its relation to understanding parabolas will be set forth in more detail. As described above, one of the simplest parabolas can be represented by the equation y=x 2 . While many parabolic functions can be represented, the following description will use this simple equation as a representative example. In this case, for the standard and vertex equations, a=1 and b=c=h=k=0. The physical representation of a graph of this equation using the teaching aid described may be best illustrated in  FIGS. 1, 2, and 4 . In order to show this graph with the teaching aid, first the lever  104  is placed in the horizontal position. The center vertical bar  106  will be a short bar  106   s,  having a length that puts its top point at an imaginary x=y=0 coordinate. As described, this may be actually shown by positioning a grid coordinator indicating that the x- and y-axes cross at the point where the top of the short bar  106   s  rests. In another aspect, this may be represented by having a short bar  106   s  whose length puts its top generally at an equal height as the top of base  102 . In an aspect, each set of outward channels  210  then may represent a gradation of 1 in the positive or negative x direction. As such, the channels  210  immediately next to the one that has accepted short bar  106   s  should accept vertical bars  106  that are longer than short bar  106   s  in order to represent a value of y=1 (because these channels would represent x=−1 and x=1). The next closest out channels  210 , which represent x=−2 and x=2, would then accept vertical bars that are longer again and of a length to represent y=4. Similarly, the next closest out channels  210 , which represent x=−3 and x=3, would then accept vertical bars that are longer again and of a length to represent y=9. The outermost channels  210  would then accept long bars  106   l  that represent y=16. Of course, this is a representative example only and other vertical bar lengths may be used to illustrate different functions, with higher or lower values of a. In one example, lower values of a can be illustrated by spacing the vertical bars farther apart, such as by skipping every other channel  210  or by using adjacent vertical bars  106  whose change in length varies more gradually. Conversely, larger values of a can be shown by using higher variances in adjacent vertical bar lengths or by skipping some bar lengths and moving the longest bars  106   l  to channels that are closer to the pivot  312  and leaving the channels farther out empty. Additionally, it should be noted that as long as the parabolic shape is maintained, the lengths of the vertical rods  106  may not strictly correspond to easily identified integer lengths on a coordinate system. Furthermore, negative values of a, which change the parabola to a downward shape can also be illustrated by reversing the lengths of the vertical rods to show a long rod  106   l  in a central channel, and shorter and shorter vertical rods being placed in the channels  210  out to short rods  106   s,  for example. 
     It should also be noted that values of h and k can also be varied by changing the initial setup of the teaching aid. For example, the point (h, k) represents the vertex of the parabola. For higher and lower values of h, the vertical bars  106  can be shifted to the left or the right in channels  210 . Additionally, in some aspects a coordinate grid indicator may shift left or right. To change values of k, in some aspects a coordinate grid indicator may be raised or lowered and/or the pivot  312  may be raised or lowered when it is movable within a vertical channel  416 . 
     More complex changes to the parabola can then be seen by raising and lowering the lever  104 . Moving the lever from the horizontal starting position (as shown in  FIGS. 8 and 9 ) maintains the same general curvature (i.e., the a value remains constant) of the parabola, but changes the value of b in the standard equation. Moving the lever also has the effect of shifting the vertex (h, k) along an unseen identical but inverted parabola. In other words, changing only the value of b has the effect of moving the position of the axis of symmetry left (positive b values) or right (negative b values), as well as moving the vertex, where the new vertex positions track the curve formed by an inverted version of the same parabola. The effect that the b coefficient has on the parabola may be most effectively visualized by the shifting sequence of parabola movements as the lever  104  is continuously moved by a user. In an aspect, a coordinate grid indicator may also have an indication of an inverted parabola that matches the parabola defined by the vertical rods, so that the user can watch the vertex trace the path of that parabola. 
       FIG. 10  illustrates one example method  1000  for learning about parabolas using a physical teaching aid as described herein. Many other methods for learning about parabolas may be used with the same physical teaching aid. Starting with block  1030 , a student or other user obtains a base  102  having a lever  104  attached to it at a pivot  312 , wherein the lever can hold a plurality of vertical bars  106  of varying lengths. At block  1032 , the student chooses a value for the coefficient a of the standard equation. If the value is positive, the process continues to block  1034 , where a short vertical bar  106   s  is placed in a central channel of the base  102 . While this may be a middle channel of the base  102 , such as to represent that the axis of symmetry is y=0, in an aspect, this need not be the case; the short vertical bar may be placed left or right of the center to represent other axes of symmetry. At block  1036 , the user places larger and larger vertical bars in channels working out from the short vertical bar&#39;s position. As described above, a user may use instructions or color-coding or other coding to understand the placement of the bars in an aspect. In another aspect, the vertical bars may have numbers on them to indicate values, and a user may need to solve the parabolic equation with specific values, given by an instructor for example, to decide which vertical bars  106  should be placed in which channels. 
     Conversely, if a negative value of a is chosen, the process proceeds to block  1038  where a long vertical bar  106   l  is placed in a central channel of the base  102 . Just as with respect to the positive values above, this may be a middle channel of the base  102 , such as to represent that the axis of symmetry is y=0, but this need not be the case. At block  1040 , the student places shorter and shorter vertical bars  106  in channels working out from the long vertical bar&#39;s position. As with block  1036 , a user may use instructions or color-coding or other coding to understand the placement of the bars in an aspect. In another aspect, the vertical bars may have numbers on them to indicate values, and a user may need to solve the parabolic equation with specific values, given by an instructor for example, to decide which vertical bars  106  should be placed in which channels. 
     In either case, once the user has constructed their parabola in the teaching aid, at block  1042 , the student can raise and lower the lever  104 , which allows them to observe the shift in the parabola based on changes to the coefficient b in the standard equation at block  1044 . Optionally, in an aspect, the student may also be able to raise and lower the pivot  312  at block  1046 , which will allow the student to observe the shift in the parabola abased on changes to the coefficient c in the standard equation at block  1048  (raising the pivot point indicates an increase in the value of c). Alternatively, in an aspect with a movable coordinate grid indicator, moving the grid coordinator up and down can also illustrate changes to the coefficient c—although it will be noted that moving the coordinate grid indicator up with respect to the parabola of the vertical bars actually represents lessening the value of c. 
     Another advantage of the teaching aid as described herein is that the lever  104  and a coordinate grid indicator could also be manipulated to illustrate mathematical changes in linear formulas, y=ax+b, for younger or more remedial students or for comparison between linear and quadratic/parabolic equations. 
     Thus, methods and devices for mathematics learning have been described. Note that references throughout this specification to “one embodiment” or “an embodiment” or “one aspect” or “an aspect” mean that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present disclosure. Therefore, it is emphasized and should be appreciated that two or more references to “an embodiment” or “one embodiment” or “an alternative embodiment” (or similar uses of “aspect”) in various portions of this specification are not necessarily all referring to the same embodiment. Furthermore, the particular features, structures or characteristics being referred to may be combined as suitable in one or more embodiments of the disclosure, as will be recognized by those of ordinary skill in the art. Additionally, alternatives other than those specifically described herein will be understood to fall within the scope of the teachings herein. For example, lever  104  may be replaced by a knob at the pivot to rotate a platform on which the vertical bars may rest. Alternatively, gears and a cranking mechanism may also supply the rotational movement suggested herein. While the present disclosure is described above with respect to what is currently considered its preferred embodiments, it is to be understood that the disclosure is not limited to that described above.