Abstract:
Disclosed herein is an adapter for use with ordinary lawn and leaf bags and a method for manufacture of such adapter. The adapter has a cavity that has four sides and four corners, and a lower portion consisting of four flaps. The use of a frame, side and/or corner fasteners create an adapter where the flaps lie inside the bag and the cavity remains outside the bag to create a large surface area for filling the bag and also prevents the mouth of the bag from closing due to wind or other forces.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation-in-part of U.S. patent application Ser. No. 12/878,789, filed on Sep. 9, 2010 and entitled “Foldable Bag with Expandable Opening,” now U.S.8,517,610, and claims priority to U.S. Provisional Application No. 61/691,199, filed on Aug. 20, 2012 and entitled “An Adapter for a Free-Standing Bag,” both of which are hereby incorporated herein by reference for all purposes as if fully set forth herein. 
    
    
     FIELD OF THE INVENTION 
     The invention relates to an adapter for a free-standing bag having a cavity and a lower portion that is supported by side fasteners, corner fasteners, and/or an expandable frame, as well as methods of manufacture of such adapter. 
     BACKGROUND 
     Ordinary free standing paper bags have become very popular in all facets of day-to-day life. Some bags are often used in the retail and food industries for packing of items such as groceries and food items from restaurants and take-out services, as well in the household waste industry for disposal of leaves, grass clippings and other biodegradable items. These bags are very simple in design and therefore easy to manufacture in large quantities and in an economical manner. 
     Such paper bags are generally foldable into a relatively flat rectangular form that is compact in size, so that they can be transported and sold more conveniently. In particular, two opposing sides of the bag are typically folded inward, collapsing the cavity of the bag so the other two opposing sides are brought together and centered perpendicularly above the base of the bag. Once the two opposing sides are brought together, the base of the bag can then be pivoted around the sides so that it is substantially parallel with the sides, and together with the sides forms a substantially flat rectangular shape. Bags folded into such a flat shape are typically stacked and packaged so that they can be sold in quantity. 
     In order to use the bag for its intended purpose, the bag is generally unfolded to create a vertical standing bag with a rectangular cavity and a base as the support for the bag. The simple design provides sufficient rigidity to the bag, but also allows the bag to collapse easily due to the folds in the vertical sides of the bag. This design aspect may be particularly important in taller lawn and leaf bags. Other variants of this basic design also include handles for easy carrying of the bag containing its contents. 
     One issue with such bags, which may be particularly problematic in lawn bags, is tears in the top portion of the bag resulting from the act of loading the contents, particularly odd-shaped objects or slightly oversized items such as twigs or branches. Further, in the case of lawn bags, factors such as wind can make it difficult to load the bag when the sides of the bag collapse. The fixed opening dimension also makes it quite challenging to load items such as grass clippings and leaves. The bag can collapse upon itself resulting in partial or complete closure of the opening, which makes the act of loading the bag a frustrating experience. Furthermore, the act of transferring grass clippings into the paper bag directly using the receptacle unit that collects grass clippings generally attached behind a lawnmower can result in tears in the paper bag opening leading to spills and additional inconvenience and frustration. 
     None-the-less, this standard bag design has withstood the test of time without significant modifications to the opening dimensions and shape of the bag. One reason for this is the simple manufacturing process that easily enables production of large quantities of the existing design. A number of mechanisms and devices that can aid in filling of the paper lawn bags have been documented. Some of these are funnel-shaped devices that aid in loading the bag, (see e.g., U.S. Pat. Nos. 6,085,647 and 6,116,548), while others are rigid frames that are inserted into the bag (see e.g., U.S. Pat. No. 5,915,768) or around the mouth of the bag to keep the mouth open and prevent from collapsing (see e.g., U.S. Pat. No. 6,138,962). The variety of products that have been designed to address the explicit problem of the difficulty in loading the lawn bags is an indication of the importance of the problem and clear evidence of the need to alleviate this problem. 
     However, most of these devices or methods are items that have to be sourced separately and/or modified extensively for use with bags of other dimensions, and therefore necessitate additional expenditure of money and time. For instance, U.S. Pat. Nos. 6,085,647 and 6,116,548 demonstrate the use of a hopper mounted on a stand that can be used to fill lawn bags placed below. It can be easily concluded that a bag of a different dimension or cross-section would necessitate a different hopper and stand design and dimensions. Likewise, U.S. Pat. No. 5,915,768 demonstrates a method of creating a funnel made out of plastic material that can be used for packing yard waste into a bag. U.S. Pat. No. 6,138,962 demonstrates a method to use a frame to keep the mouth of the bag open during the act of loading. 
     Furthermore, most of these devices or methods do not have desirable characteristics for routine consumer use. For instance, U.S. Pat. No. 7,736,057 B2 and U.S. Pat. No. 7,302,978 B1 demonstrate devices that are foldable in design, but are not exactly compact. In addition, their design also limits the amount of material that can be loaded into the bag due to a rigid bottom portion that extends all the way into the base of the bag. Moreover, the material used in the manufacture of these devices is corrugated paper that can easily get wet which would cause the device to lose its structural integrity and prevent reuse of the device. Likewise, U.S. Pat. No. 5,271,589 demonstrates a method of creating a device made out of two interlocking pieces and is not compact either. This item also suffers from the same disadvantage of getting wet and is therefore designed as a disposable product. Finally, while this device can be used as an aid to fill lawn and leaf bags, it is designed for use with plastic refuse bags ideally. None of these existing solutions provide all the benefits that can be provided with a compact, light-weight, washable and reusable adapter as described in this invention. 
     Overview 
     Exemplary embodiments relate to the creation of a compact and foldable adapter that can be assembled to form an object with a cavity and a lower portion that can be inserted into ordinary, free-standing lawn and leaf bags. Upon insertion into the opening of a bag, the adapter is held in place to the opening of the bag using structural design elements such that the cavity of the adapter is outside the bag while the lower portion of the adapter is preferably inside the bag. The primary purpose of the cavity of the adapter is to form a larger surface area of the mouth of the bag to aid in easy loading of the bag. The structural design elements of the adapter are such that upon insertion of the adapter into the bag, the shape of the adapter is maintained in place and does not collapse by itself or due to external factors such as blowing wind. 
     The adapter piece is preferably created using light weight material that is fully flexible and washable. The light weight of the adapter ensures that upon insertion into, and affixing to the bag, the bag does not collapse upon itself due to the weight of the adapter. Another feature of the adapter is that it can be folded into a compact shape for easy storage after removing the structural design elements. Yet another feature of the adapter is that it is water-resistant and fully washable for reuse. The primary application of this adapter is for insertion into standard, free-standing lawn and leaf bags made out of single or double-ply paper. The shape and size of the adapter is modifiable for other similar applications. 
     These as well as other aspects, advantages, and alternatives, will become apparent to those of ordinary skill in the art by reading the following detailed description, with reference where appropriate to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       An exemplary embodiment of the present invention is described herein with reference to the drawings, in which: 
         FIG. 1  shows a finished bag with the adapter piece attached; 
         FIG. 2  shows the same bag as shown in  FIG. 1  except in an expanded configuration that creates the funnel-shaped structure; 
         FIG. 3A  shows a top-down view of a connector section in an expandable upper portion of an exemplary bag; 
         FIG. 3B  shows an angular view of a connector section in an expandable upper portion of an exemplary bag; 
         FIG. 4A  shows an expandable upper portion of an exemplary bag in an expanded state; 
         FIG. 4B  shows an expandable upper portion of an exemplary bag in an unexpanded state; 
         FIG. 4C  shows a top-down view of an expandable upper portion of an exemplary bag folded to be substantially flat; 
         FIG. 5  shows the basic die-cut contiguous shape made out of a sheet of single or double-ply paper that becomes the basis of the adapter piece with all the appropriate folds; 
         FIG. 6  shows the creation of a linear adapter piece out of the original die-cut shape shown in  FIG. 5 ; 
         FIG. 7A  is a flow chart illustrating a method for manufacturing a bag, according to an exemplary embodiment; 
         FIG. 7B  is a flow chart illustrating another method for manufacturing a bag, according to an exemplary embodiment; 
         FIG. 7C  is a flow chart illustrating another method for manufacturing a bag, according to an exemplary embodiment; 
         FIGS. 8A and 8B  are block diagrams illustrating how an upper portion and lower portion of a bag may be connected, before being folded to create a cavity of the bag, according to an exemplary manufacturing method; 
         FIGS. 9A and 9B  are block diagrams illustrating how an upper portion and lower portion of a bag may be connected, before being folded to create a cavity of the bag, according to another exemplary manufacturing method; 
         FIG. 10A  shows a top-down view of an exemplary adapter piece before it is expanded; 
         FIG. 10B  shows an exemplary adapter piece in an expanded state; 
         FIG. 11A  shows an exemplary adapter piece including a plurality of side fasteners; 
         FIG. 11B  shows a front view of an exemplary side fastener; 
         FIG. 11C  shows a side view of an exemplary side fastener; 
         FIGS. 11D and 11E  show an exemplary adapter piece before and after the adapter piece is attached to a bag; 
         FIG. 12A  shows an exemplary adapter piece including a plurality of side fasteners and an expandable frame; 
         FIGS. 12B and 12C  show an exemplary expandable frame in an unexpanded and an expanded state; 
         FIG. 12D  shows a variety of exemplary configurations of an exemplary expandable frame; 
         FIG. 12E  shows an exemplary connector piece attached to an exemplary frame piece. 
         FIG. 12F  shows a top-down view of an exemplary connection end of an exemplary frame piece; 
         FIG. 13A  shows an exemplary adapter piece including an exemplary corner fastener; 
         FIG. 13B  shows an angular rearview of an exemplary corner fastener; 
         FIG. 13C  shows a side view of an exemplary corner fastener; 
         FIG. 13D  shows a dissection view of an exemplary bracket section; 
         FIG. 13E  shows an exemplary adapter piece attached to a bag; 
         FIG. 14  shows an exemplary adapter piece including a plurality of corner fasteners and an expandable frame; 
         FIG. 15  shows an exemplary corner fastener; 
         FIG. 16A  shows an exemplary adapter piece including a plurality of corner fasteners, a plurality of side fasteners, and an expandable frame; 
         FIG. 16B  shows an exemplary adapter piece attached to a bag; and 
         FIG. 17  is a flow chart illustrating a method for manufacturing an adapter, according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Exemplary embodiments of the present invention are described herein. It should be understood that the word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments. Further, those skilled in the art will understand that changes and modifications may be made to these embodiments without departing from the true scope and spirit of the invention, which is defined by the claims. 
     A bag according to an exemplary embodiment has a rectangular lower portion and an expandable upper portion that can be expanded to provide a larger opening to the bag. The lower portion preferably has inner walls that define a tubular cavity, while the foldable upper portion preferably has inner walls that define an upper cavity that varies in form depending upon whether or not the upper portion is expanded. When the upper portion of the bag is not expanded, the upper cavity preferably is tubular, and in an exemplary embodiment, is a rectangular cavity that is simply an extension of a rectangular cavity of the lower portion. However, when the upper portion is expanded, the upper cavity is preferably funnel-shaped, with a larger opening at the top of the bag. 
     As used herein, the term “cavity” shall mean any three-dimensional air-space within a structure that is enclosed on at least four sides by the structure. A cavity may be open-ended (i.e., not enclosed) on one or both ends of the structure (i.e., not enclosed by a bottom section and/or top section of the structure). A “tubular” cavity shall be understood to mean a cavity defined by side walls and having a cross-section of any shape (e.g., circular, rectangular, etc.), where the cross-sectional area is constant throughout the cavity. A “funnel-shaped” cavity shall be understood to mean a cavity defined by sloping side walls and having a cross-section of any shape, where the cross-sectional area increases/decreases throughout the cavity. 
       FIGS. 1 and 2  illustrate a free-standing bag  100  according to an exemplary embodiment. In  FIG. 1 , the triangular connector sections  130 - 136  at the corners of the bag are folded inward so that the bag has the form factor of traditional bags; i.e. a rectangular shape with a bottom panel and an opening at the top. By unfolding the triangular connector sections, a wider, funnel-shaped opening is formed.  FIG. 2  illustrates the free-standing bag  100  once the upper portion is unfolded to create the funnel-shaped opening. 
     In  FIGS. 1 and 2  the lower portion of the bag  100  includes a plurality of lower side sections and a bottom section that are connected to form a tubular lower cavity. More specifically, as shown in  FIG. 1 , the lower portion  102  preferably includes four side sections  104 - 110 , with two opposing long sides  104 ,  108  parallel to each other, and two opposing short sides  106 ,  110  that are also parallel. As shown, the four side sections  104 - 110  are connected so as to define a lower cavity within the lower portion of the bag, with the bottom section  112  connected to the four side sections  104 - 110  so as to close the lower cavity at the bottom of the bag. As further shown, the lower cavity is preferably a tubular cavity having a rectangular cross-section. 
     The upper portion of the bag includes a plurality of upper side sections, a plurality of insert sections, and a plurality of connector sections. As shown, the upper portion  120  includes four rectangular side sections  122 - 128 , with two rectangular long sides  122 ,  126  that are parallel to each other, and two rectangular short sides  124 ,  128  that are also parallel. As shown, the four connector sections  130 - 136  connect the four side sections  122 - 128  such that the upper portion can be expanded to provide a funnel-shaped opening to the bag by unfolding the connector sections  130 - 136 . Further, the upper portion  120  preferably includes four insert sections  140 - 146 , with one extending from each side section towards the lower portion of the bag. Each insert section  140 - 146  is preferably affixed by adhesive or other means to a corresponding lower side section  104 - 110 , such that a lower edge of each upper side section  122 - 128  abuts an upper edge of a corresponding lower side section  104 - 110 . 
     As noted, the upper portion  120  of the bag includes foldable connector sections  130 - 136  that allow the upper portion  120  of the bag to be expanded to create a funnel-shaped opening. To illustrate,  FIG. 3A  shows a more-detailed top-down view of one foldable connector section  134 , which connects upper side sections  126  and  128 . In  FIG. 3A , the connector section  134  is shown in an unfolded position, which provides a funnel-shaped opening to the bag. Preferably, the connector section  134  is an equilateral triangle with a corner  160  that connects a lower corner of side section  128  and a lower corner of side section  126 . The other two corners  162  and  164  are connected to only one side section (side section  126  and side section  128 , respectively). When the connector sections  130 - 136  are unfolded, upper side sections  130 - 136  angle outward from the lower side sections  104 - 110  to form the funnel-shaped opening. 
     Further, the connector section  134  preferably has a crease  150 , which allows the connector section  134  to be folded inward and upon itself, as shown in  FIG. 3B . Preferably, crease  150  bisects the connector section  134  (i.e., bisects the equilateral triangle), such that when the connection section  134  is folded inward to the upper cavity, the corners  162  and  164  of the connector section are brought together, so that edge  152  of side section  128  is substantially flush against the edge  154  of side section  154 . As such, when the connector sections  130 - 136  are folded inward, the edges of upper side sections  122 - 128  form a substantially rectangular shape, and in effect extend lower side sections  104 - 110  to form a single rectangular cavity. 
       FIG. 4A  shows a top-down view of the upper portion  120  of the bag when connector sections  130 - 136  are unfolded. The cavity defined within the upper side sections  122 - 128 , may be referred to as the upper cavity, and the cavity within the lower side sections  104 - 110  may be referred to as the lower cavity. As such the upper cavity is open at both ends, and defined as the space between a lower opening, which is defined by lower edges  172 - 178  of upper side walls  122 - 128 , and an upper opening, which is defined by the upper edges  182 - 188  of upper side walls  122 - 128  and the upper edges  190 - 196  of connector sections  130 - 136 . 
     The lower opening of the upper cavity, which is proximate to the lower cavity that is defined by lower side walls  104 - 100 , preferably has a cross-section that is substantially the same size as a cross section of the lower cavity. In other words, the rectangle formed by the lower edges  172 - 178  of the upper side sections  122 - 128  is of substantially the same dimensions and orientation as the rectangle formed by the lower side sections  104 - 110 . However, as shown, when the connector sections  130 - 136  are unfolded, the upper opening, which is distal to the lower cavity, is larger than the lower opening, as the upper edges  182 - 188  of upper side walls  122 - 128  and the upper edges  190 - 196  of connector sections  130 - 136 , together, form an octagonal opening having a greater area than the rectangular opening formed by the lower edges  172 - 178  of the upper side sections  122 - 128 . As such, when the upper portion is expanded, the upper side walls  122 - 128  angle outward from the lower edges  172 - 178 , where the upper side walls  122 - 128  connect to lower side walls  104 - 110 , thus defining a funnel-shaped upper cavity, which opens into the rectangular lower cavity. 
       FIG. 4B  shows a top-down view of the upper portion  120  of the bag when the connector sections  130 - 136  are folded inward. Once the connector sections  130 - 136  are folded, the upper opening of the upper cavity is a substantially rectangular opening. Further, since each upper side section  122 - 128  is rectangular, both the lower opening and the upper opening are substantially the same size. As such, the upper cavity is tubular, and in the illustrated embodiment, is a rectangular cavity with a cross section that is the same size as a cross section of lower cavity. Thus, when the connector sections  130 - 136  are folded inward, the upper and lower cavities preferably form one contiguous rectangular cavity. 
     Referring back to  FIGS. 1 and 2 , in a further aspect, free-standing bag  100  may include creases such that the bag folds to a substantially flat shape. In particular, when connector sections  130 - 136  are folded inward, each upper side section  122 - 128  and its corresponding lower side section  104 - 110  effectively form a single rectangular side section. Folded as such, the bag  100  has a single rectangular cavity defined by four side walls and a bottom section, and thus takes on substantially the same form as traditional bags (albeit with the connector sections extending inward to the upper cavity). Crease patterns that allow traditional bags to fold flat are well known in the art. Accordingly, standard crease patterns may be applied to the lower portion  102  of bag  100 , such as those shown by the dotted lines in the lower side sections  104 - 110  and bottom section  112  in  FIGS. 5 and 6 . As the connector sections  130 - 136  allow the upper portion to be folded so that the bag has the same form factor as a traditional bag, standard crease patterns may be extended through the upper portion  120  of the bag, as shown by the dotted lines in upper side sections  122 - 128 . These standard crease patterns, along with the creases in the connector sections  130 - 136 , allow the bag  100  to fold into a substantially flat rectangular shape, as shown by the top down view of the upper portion  120  shown in  FIG. 4C . It should be understood that other crease patterns may be employed, without departing from the scope of the invention. 
     In a further aspect, the upper portion  120  of the bag  100  (including the upper side sections, insert sections, and the connector sections) is preferably cut from a single piece of flat material. In particular, a die or other cutting technique may be used to a shape from a single piece of flat material that can manipulated to create the funnel-shaped upper portion  120 .  FIG. 5  illustrates a single piece of material cut into a shape  200  according to an exemplary embodiment, from which the upper portion  120  may be created. As shown, creases may be made in the shape  200  in order to define rectangular sections  221 - 228 , which are connected by triangular sections  230 - 236 . Further, creases may be made to define trapezoidal sections  240 - 248  which extend inward from the rectangular sections  221 - 228 . Further, each triangular section  230 - 236  includes a crease  250  that bisects the equilateral triangle. 
     Preferably, the flat shape  200  is cut, and the creases are such, that triangular sections  230 - 236  are equilateral triangles. As shown, when the rectangular sections  221 - 228  are connected by equilateral triangles, the inner edges of rectangular sections  221 - 228  form a hexagonally-shaped inner wall. More specifically, as there are five rectangular sections  221 - 228 , the inner wall defines five sides of a hexagon, with the sixth side (opposite rectangular section  246 ) being open. Furthermore, since the triangular sections  230 - 236  are equilateral, the hexagonal shape is be symmetrical; i.e. the inner edge of rectangular section  222  is of the same length and parallel to the inner edge of rectangular section  224 , the inner edge of rectangular section  221  is of the same length and parallel to the inner edge of rectangular section  228 , and the inner edge of rectangular section  226  is of the same length and parallel to the open side of the hexagon. 
     It should be understood, however, that triangular sections  230 - 236  may not be equilateral triangles. For example, the triangular sections may be isosceles triangles, in order to provide may provide a wider or narrower opening, depending on the form of the isosceles triangle. In particular, if the two equal angles of the isosceles triangle are less than sixty degrees, then the opening will be larger than if the triangular sections are equilateral, and if the two equal angles of the isosceles triangle are greater than sixty degrees, then the opening will be smaller than if the triangular sections are equilateral. Further, in embodiments where the triangular sections are not equilateral, the inner wall of the flat shape may still be hexagonal, albeit not symmetrical. For example, if the triangular sections are isosceles triangles, and if the two equal angles of the isosceles triangle are less than sixty degrees, then the inner edge of rectangular section  226  may be shorter than the opening in the inner wall that is opposite and parallel to it. Other examples are also possible. 
       FIG. 7A  is a flow chart illustrating a method for manufacturing a bag, according to an exemplary embodiment. As shown, the method involves cutting a flat shape (such as the shape shown in  FIG. 5 ), from a single flat piece of material  200 , as shown by block  702 . The flat shape is then creased so as to define, within the flat shape: (1) five rectangular sections (e.g. sections  221 - 228 ), (2) four triangular connector sections that connect the rectangular sections (e.g., sections  230 - 236 ), and (3) five trapezoidal sections extending from the rectangular sections towards a center of the flat shape (e.g., sections  240 - 248 ), as shown in step  704 . Preferably, each triangular section comprises a central crease dividing the triangular section and allowing the triangular section to be folded upon itself. Accordingly, as shown in step  706 , the method further involves folding the triangular sections along the central creases such that the rectangular sections are linearly aligned along a common axis to form a single collective rectangular shape with the triangular sections extending outward from the single rectangular shape (e.g., as shown in  FIG. 6 ). As shown by step  708 , once triangular connector sections  230 - 236  have been folded so that a shape such as that shown in  FIG. 6  is formed, the method involves folding and connecting two opposing ends of the flat rectangular shape (e.g., the end of the shape defined by sections  222  and  241 , and the end of the shape defined by sections  221  and  240 ), such that the shape forms an expandable upper portion of the bag as described herein. This upper portion may then be connected to a lower portion to form the bag, as shown by block  710 . 
     More specifically, in order that the piece of material  200  having the shape shown in  FIG. 5  be formed into an upper portion  120 , which is expandable to provide a funnel-shaped opening, the following may be performed. First, the triangular sections  230 - 236  may be folded inward along creases  250 , such that rectangular sections  221 - 228  are linearly aligned along a common axis, as shown in  FIG. 6 . The shape  200  may then be folded at right angles along the creases  290 , and rectangle  221  may be connected to rectangle  222  to form a single rectangle of the same dimensions as rectangle  226 . By so doing, shape  200  is now arranged as upper portion  120 . Specifically, rectangles  222 - 226  are arranged to serve as upper side sections  122 - 128  in upper portion  120 , and the single rectangle formed by rectangles  221  and  222  is arranged to serve as upper side section  122 . Further, the trapezoidal sections  240 - 246  are arranged to serve as insert sections  140 - 146 , and the triangular sections  230 - 236  are arranged to serve as connector sections  130 - 136 . Additionally, creases  250  and  290  allow the triangular sections to expand to define a funnel-shaped cavity, and fold inward to define a rectangular cavity. 
     In the manufacturing of an upper portion from a single piece of material, it should be understood that once rectangles  221  and  222  connected to form an upper side section, the upper side section is preferably of the same dimensions as rectangle  226 , which serves as the opposing upper side section of the upper portion of the bag. However, the combined width of rectangles  221  and  222  is preferably greater than the length of rectangle  226 , so that rectangles  221  and  222  can be overlapped in order that the rectangles be adhered to each other to form an upper side section. Further, it should be understood that the widths of rectangles  221  and  222  may vary, so long as rectangles  221  and  222  can be connected to form an upper side wall having the same dimensions as rectangle  226 . Further, it is possible that rectangles  221  and  222  may be replaced by a single rectangular section that can be connected to a triangular section in order to form the upper portion of the bag. 
       FIG. 7B  is a flow chart illustrating another method for manufacturing a bag, according to an exemplary embodiment. In  FIG. 7B , steps  752 - 756  are carried out in the same or similar manner as steps  702 - 706  of  FIG. 7A . However, as shown by step  758 , the single collective rectangular shape having the triangular extending outward is connected to a second flat rectangular piece of material to form a larger rectangular shape, with the single collective rectangular shape having the triangular extending outward forming an upper portion and the second flat rectangular piece of material forming a lower portion. Then after the upper and lower portion have been connected, the larger rectangular shape is folded and connected to form a bag with an expandable upper portion, as shown by step  760 . 
       FIGS. 8A and 8B  show how an upper portion and lower portion of the bag may be connected, before being folded to create a cavity of the bag, using a method such as that shown in  FIG. 7B . More specifically, as shown in  FIG. 8A , when the single piece of material  200  for the upper portion of the bag is folded as shown in  FIG. 6 , the trapezoidal sections  241 - 248  (i.e., insert sections) may be adhered to a second flat piece of material  300 . The second flat piece of material  300  may be folded along creases  390 A- 390 D (and possibly along other creases not shown) and adhered along sides  310  and  320  to form the lower portion of the bag. Preferably, the trapezoidal sections  241 - 248  are adhered such that an edge  292  formed by the rectangular sections  221 - 226  is substantially flush against an upper edge  392  of the piece of material  300  for the lower section, and such that creases  290 A- 290 D of the upper portion of the bag align with creases  390 A- 390 D of the lower portion, as shown in  FIG. 8B . Alternatively, both piece  200  and piece  300  may be folded and adhered so as to form the upper and lower portion of the bag, respectively, before the trapezoidal sections  241 - 248  are adhered to piece  300 . 
     In an alternative embodiment, an exemplary paper bag including an upper portion and lower portion that function as described herein, may be formed from one or more pieces of material having the same form as the piece or pieces from which standard bags are formed, with the addition of four tabbed connector pieces. In particular, as shown in  FIG. 9A , four slits  902 A-D may be cut in a piece of flat material  900 . Further, four tabbed connector sections  910  may be created. Each tabbed connector section  910  includes a foldable triangular connector  950  as described herein and two tabs  952 A and  952 B extending from the triangular connector section  950 . The four tabbed connector sections  910  may then be inserted in slits  902 A-D, and tabs  952 A and  952 B adhered to piece  900 , as shown in  FIG. 9B . The piece  900  may then be folded along creases  990 A- 990 D (and possibly along other creases not shown) and adhered along sides  910  and  920  to form the bag such that when the foldable triangular connectors  950  are folded, the connectors  950  extend inward to the bag. 
     In another aspect of the invention, the invention may take the form of an expandable adapter piece, which provides a larger funnel-shaped opening for standard bags. The expandable adapter piece preferably takes the form of the upper portion of a bag as described herein. For example, the adapter piece may take the form of an upper portion of a bag as illustrated in  FIGS. 1 ,  5 , and  6 , which includes insert sections, such as insert sections  240 - 248 . The insert sections may be inserted along the side walls of a standard bag to connect the adapter piece to the standard bag. The insert sections may additionally or alternatively include other means for connecting the adapter piece to the standard bag. For instance, the insert sections may include adhesive for adhering the connector sections to the standard bag. Other variations for connecting the adapter piece to a standard bag are also possible. 
       FIGS. 10A and 10B  depict an adapter piece  1000  according to an exemplary embodiment.  FIG. 10A  is a top-down view of the adapter piece  1000  before it is expanded, while  FIG. 10B  depicts the adapter piece  1000  after it is expanded. The adapter piece  1000  includes a plurality of creases  1020 - 1026 , a plurality of insert sections  1040 - 1046 , a plurality of upper side sections  1060 - 1066 , and a plurality of connector sections  1080 - 1086 . 
     In the present embodiment, the adapter piece  1000  may be made of a flexible material which does not degrade when exposed to water. Further, the adapter piece  1000  may be washable and/or otherwise reusable. In other embodiments, the adapter piece  1000  may be made of a flexible material or a washable, water-proof material. Other materials such as paper or cardboard, which may or may not be degradable, are also possible. 
     Further, the adapter piece  1000  may be made out of a single piece of material which is cut, folded, and/or adhered to itself to form the shape that includes insert sections  1040 - 1046 , upper side sections  1060 - 1066 , and connector sections  1080 - 1086  It is also possible that the adapter piece  1000  may be made out multiple pieces of material. For example, each insert section  1040 - 1046 , each side section  1060 - 1066 , and/or each connector section  1080 - 1086  could be a separate piece of material, and the separate pieces could be glued or otherwise attached to form the adapter piece  1000 . 
     In the illustrated embodiment, the plurality of connector sections  1080 - 1086  are between the plurality of upper side sections  1060 - 1066 . For example, the connector section  1082  is between the upper side section  1060  to the upper side section  1062 . The plurality of insert sections  1040 - 1046  are connected to the plurality of upper side sections  1060 - 1066  at the plurality of creases  1020 - 1026 . For example, the insert section  1040  is connected to the upper side section  1060  at the crease  1020 . 
     In the illustrated configuration, the adapter piece  1000  includes a cavity  1010 . The cavity  1010  includes a lower opening, defined by the plurality of creases, and an upper opening, defined by the outer edge of the expanded plurality of upper side sections  1060 - 1066  connected by the plurality of connector sections  1080 - 1086 . In the illustrated embodiment, the lower opening may be of approximately the same size as the opening at the upper end of a bag for which the adapter piece is intended. 
     When the adapter piece  1000  is expanded, the upper opening is larger than the lower opening, as depicted in  FIG. 10B . More specifically, as shown in  FIG. 10B , the plurality of insert sections (only insert sections  1040  and  1042  are shown) are directed downwards at the plurality of creases (only creases  1020  and  1022  are shown), thereby defining a lower portion that can be inserted into or placed around an upper opening of a bag. The plurality of upper side sections  1060 - 1066  connected by the plurality of connector sections  1080 - 1086 , which defines the cavity  1010 , are folded upwards at the plurality of creases  1020 - 1026 , away from the plurality of insert sections  1040 - 1046 . As such, insert sections  1042  to  1046  may be inserted into a bag (e.g., a paper lawn bag), in order to provide an expandable opening via which contents may be placed into the bag (e.g., lawn waste). 
       FIG. 11A  shows another adapter piece  1100 , according to an exemplary embodiment. In particular, the adapter piece  1100  includes a plurality of horizontal-restraining straps and a plurality of side fasteners.  FIGS. 11B and 11C  depict a side fastener  1120 .  FIG. 11B  shows a front view of the side fastener  1120 , while  FIG. 11C  shows a side view of the side fastener  1120 . The side fastener  1120  includes an upper piece  1130  and a lower piece  1140 . The upper piece  1130  is defined by an upper edge  1132  and a lower edge  1134 . In the present embodiment, the upper piece  1130  may be shaped in a trapezoidal manner such that the upper edge  1132  is wider than the lower edge  1134 . In other embodiments, the upper piece may be shaped in other shapes without departing from the present invention. 
     The upper piece  1130  is connected to the lower piece  1140  at an angle  1150  at the lower edge  1134 . The angle  1150  may be an acute angle, as in the illustrated example. In an exemplary embodiment, the angle  1150  may be, 30 degrees, for example. In other exemplary embodiments the angle between the upper piece  1130  and the lower piece  1140  may be 45 degrees or 60 degrees. Further, it should be understood that other angles are possible, without departing from the scope of the invention. 
     Each of the plurality of upper side sections (only upper side sections  1060  and  1062  are shown) may include at least one horizontal-restraining strap, which is configured to receive one of the plurality of side fasteners. In the illustrated embodiment, the ends of horizontal-restraining straps  1102 - 1108  are attached to the side section  1062  in such a manner that the middle of each restraining strap can be pulled away from the side section to which it is attached to create a gap between the horizontal-restraining strap and the upper side section. A side fastener, such as side fasteners  1120  and  1122 , may then be inserted between each restraining straps  1102 - 1108  and the side section to which it is attached. Similarly, the ends of horizontal-restraining strap  1110  are attached to the upper side section  1060  in such a manner that an air gap is possible in between the middle of horizontal-restraining strap  1110  and upper side section  1060 , such that the side fastener  1124  may be inserted. 
     When a side fastener  1120  is inserted into a horizontal-restraining strap (e.g., horizontal-restraining strap  1102 - 1110 ), the side fastener  1120  may provide structural rigidity to the adapter piece  1100 . For example, the side fastener  1120  is shaped and sized with a decreasing width, such that the bottom of the side fastener is narrower than the gaps between the horizontal-restraining straps  1102  and  1104  and the side section, and the top of the side fastener  120  is wider than the gaps. As such, when the side fastener  120  is inserted into these gaps, the horizontal-restraining straps  102  and  104  may support the side fastener  1120 , and thus may hold the side fastener between the straps and the upper side section  1062  in a substantially stationary position. The side fastener  1120  may be said to held in a substantially stationary position in that minor fluctuations of its position are possible. In this configuration, the side fastener  1120  may be said to be “removably connected” to the adapter piece  1100 . 
     In practice, the lower piece  1140  may be inserted first into the horizontal-restraining strap  1102  and then into the horizontal-restraining strap  1104 . Next, the upper piece  1130  first engages the horizontal-restraining strap  1104  and then the horizontal-restraining strap  1102 . As noted, the shape of the horizontal-restraining straps  1102  and  1104  together with the width of the upper piece  1130  cause the upper piece  1130  to become removably connected with the horizontal-restraining straps  1102  and  1104  and the upper side section  1062 . This configuration helps to prevent external forces, such as wind or gravity, from pulling the side fastener  1120  downwards and out of horizontal-restraining straps  1102  and  1104 . 
       FIGS. 11D and 11E  depict the adapter piece  1100  before and after it is attached to a bag  1190 , according to an exemplary embodiment. The bag  1190  includes an opening  1192  at an upper end  1194 . In the present embodiment, the bag  1190  may be a free-standing bag for lawn clippings. The adapter piece  1100  is inserted into the opening  1192 , such that the plurality of insert sections (only insert sections  1040  and  1042  are depicted) are located on the inside of the bag  1190 , while the plurality of side fasteners (only side fasteners  1120 - 1124  are shown) are located on the outside of the bag  1190  at the upper end  1194 . The plurality of side fasteners may act as clamps, which attach the bag  1190  to the plurality of insert sections. That is, the bag  1190  may be placed between the plurality of side fasteners and the plurality of insert sections, thereby removably attaching the adapter piece  1100  to the bag  1190 . 
       FIG. 11A  shows an exemplary arrangement of horizontal-restraining straps. However, other arrangements and combinations are also possible. For example, while  FIG. 11A  depicts the upper side section  1062  including horizontal-restraining straps  1102 - 1108 , in another embodiment, the upper side section  1062  includes only horizontal-restraining straps  1102  and  1106 . In yet another embodiment, the upper side section  1062  includes only one horizontal-restraining strap located in the middle of the upper side section  1062 . Other variations and different configurations of restraining straps are possible as well. 
       FIG. 12A  shows another adapter piece  1200 , according to an exemplary embodiment. In particular, the adapter piece  1200  includes a plurality of vertical-restraining straps and an expandable frame  1220 . The expandable frame  1220  provides structural rigidity to the adapter piece  1200 .  FIGS. 12B-C  depict the expandable frame  1220  in more detail. The expandable frame  1220  includes a plurality of frame connector pieces  1230  and a plurality of frame pieces  1240 . In the present embodiment, each frame connector piece  1230  is rounded through an arc of ninety degrees and includes two receiving ends  1232 . In other embodiments, the frame connector pieces may be straight and include multiple straight sections so as to connect the frame pieces at 90 degrees (or at another angle) to each other. In the present embodiment, the receiving ends  1232  may be both hollow and open. In other embodiments, the receiving ends  1232  may be filled and closed. Each frame piece  1240  includes a receiving section  1250  and an expansion section  1260 . The receiving section  1250  includes a connection end  1252  and an insertion end  1254 ; the insertion end  1254  may be both hollow and open. The expansion section  1260  includes a base portion  1262 , which includes a juncture end  1264  and a telescoping end  1266 , and a telescoping portion  1270 . The cross section of the telescoping portion  1270  is smaller than the cross sections of both the base portion  1262  and the insertion end  1254 . In other embodiments, the frame connector pieces  1230  may have a squared-off edge, rather than the illustrated rounded edge. It should be understood that other shapes are possible, without departing from the scope of the invention. Likewise, in other embodiments, the frame pieces  1240  may have a rectangular, a square-like, a hexagonal, or a pyramidal cross-section as opposed to the illustrated circular cross-section. It should be understood that other cross-sections are possible, without departing from the scope of the invention. 
     The plurality of vertical-restraining straps are attached to the adapter piece  1100 . In the present embodiment, each of the plurality of upper sides sections (only upper side sections  1060  and  1062  are shown) and each of the plurality of insert sections (only insert sections  1040  and  1042  are shown) include an end of at least one vertical-restraining strap. For example, referring to  FIG. 12 , one end of both vertical-restraining straps  1210  and  1212  is attached to the upper side section  1062  and the other end of both vertical-restraining straps  1210  and  1212  is attached to the insert section  1042 . In other embodiments, the plurality of vertical-restraining straps can be attached only to the plurality of upper side sections or only to the plurality of insert sections. 
     Referring to  FIG. 12B , the telescoping portion  1270  is connected to the base portion  1262  at the telescoping end  1266 . The telescoping portion  1270  is inserted into the receiving section  1250  at the insertion end  1254 . Thus, the expansion section  1260  is connected to the receiving section  1250 , thereby creating one of the plurality of frame pieces  1240  whose length is expandable up to the length of the telescoping portion  1270 . A first frame piece is connected to a second frame piece using one of the plurality of frame connector pieces  1230 . The connection end  1252  of the first frame piece is inserted into one of the two receiving ends  1232 . Similarly, the juncture end of the second frame piece is inserted into the second of the two receiving ends  1232 . Similarly, a third and fourth frame piece are connected to the first and second frame pieces, thereby creating the expandable frame  1220 . It should be understood that the expandable frame  1220  may be in any shape, such as a hexagon or a triangle, without departing from the scope of the invention. 
     Referring to  FIG. 12 , the expandable frame  1220  is attached to the adapter piece  1200  using the plurality of vertical-restraining straps. The plurality of frame pieces are inserted into the plurality of vertical-restraining straps, for example, vertical-restraining straps  1210 - 1216 . In the present embodiment, the dimensions of the expandable frame  1220  may be approximately equal to the dimensions of the lower opening of the cavity  1010 . The expandable frame  1220  is attached to the adapter piece  1200 , such that plurality of side fasteners, for example, side fasteners  1120 - 1124 , are located between the adapter piece  1200  and the expandable frame  1220 . 
     In operation, the length of each frame piece  1240  is expandable up to the length of the telescoping portion  1270 .  FIG. 12C  depicts a frame piece  1240  in an expanded configuration. In one embodiment, the frame pieces  1240  may be expanded by applying longitudinal forces in opposite directions against the receiving section  1250  and the expansion section  1260 . In another embodiment, the frame pieces  1240  may be expanded by applying a longitudinal force against the receiving section  1250  in a direction away from the expansion section  1260 . In other embodiments, the frame pieces  1240  may be expanded by applying a longitudinal force against the expansion section  1260  in a direction away from the receiving section  1250 . Consequently, as depicted in  FIG. 12D , the size of the expandable frame  1220  can be expanded or contracted in response to fluctuations in the size of the lower opening of the upper cavity  1010 . It should be understood that other configurations are possible, without departing from the scope of the invention. 
       FIGS. 12E and 12F  show an embodiment of the frame connector piece  1230  and the frame piece  1240 , according to an exemplary embodiment. In particular, the frame connector piece  1230  further includes at least one L-shaped key slot at each of the two receiving ends  1232 , and the frame piece  1240  further includes at least one lock key at the connection end  1252  and at least a second lock key at the juncture end (not shown in  FIG. 12E  or  12 F). In the present embodiment, the frame connector piece  1230  includes two L-shaped key slots at each of the two receiving ends  1232  (only one L-shaped key slot  1280  is shown in  FIG. 12E  at each of the two receiving ends  1232 ), and the frame piece  1240  includes two lock keys at both the connection end  1252  and the juncture end  1264  (only one lock key  1282  at the connection end  1252  is shown in  FIG. 12E ).  FIG. 12F  depicts a top-down view of the connection end  1252  including two lock keys  1282 . It should be appreciated that the juncture end  1264  is configured in a similar manner as the depicted connection end  1252 . 
     The at least one L-shaped key slot is cut out of the receiving end  1232 , for example, as the L-shaped key slot  1280  is depicted in  FIG. 12E . The at least one lock key is connected to the frame piece  1240  at the connection end  1252 , and the at least second lock key is connected to the frame piece  1240  at the juncture end. 
     In practice, as shown in  FIG. 12E , the frame piece  1240  may be inserted into the frame connector piece  1230  when the two lock keys (only one lock key  1282  is shown) are aligned with the two L-shaped key slots at the receiving end  1232  (only one L-shaped key slot  1280  is shown). The frame piece  1240  is then twisted or otherwise turned, such that the two lock keys  1282  removably connect the frame piece  1240  to the frame connector piece  1230 . 
     In other embodiments, the plurality of frame connector pieces  1230  further include internal threading located at the two receiving ends  1232 , and the plurality of frame pieces  1240  further include external threading located at the connection end  1252  and the juncture end  1264 , such that a frame piece is connected to a frame connector piece by twisting the frame piece into the internal threading of a receiving end. The internal threading may be comparable to the internal threading of a standard threaded nut, while the external threading may be comparable to the external threading of a standard threaded bolt. The connection of the frame piece to a frame connector piece may be comparable to a standard threaded bolt being screwed into a standard threaded nut. 
     In yet other embodiments, the plurality of frame connector pieces  1230  further include external threading located at the two receiving ends  1232 , and the plurality of frame pieces  1240  further include internal threading located at the connection end  1252  and the juncture end  1264 , such that a frame piece is connected to a frame connector piece by twisting the frame piece onto the external threading of a receiving end. In the present embodiment, the receiving ends  1232  may be solid and closed, as opposed to being hollow and open. 
     In other embodiments, the plurality of frame pieces  1240  may be connected to the plurality of frame connector pieces  1230  using other fastening mechanisms, without departing from the scope of the invention. 
       FIG. 13A  shows a corner of an embodiment of an adapter piece  1300 , according to an exemplary embodiment. In particular, the adapter  1300  includes a plurality of corner-restraining straps and a plurality of corner fasteners.  FIGS. 13B and 13C  depict a corner fastener  1320 .  FIG. 13B  shows an angled rearview of the corner fastener  1320 , while  FIG. 13C  shows a side view of the corner fastener  1320 . As shown, the corner fastener  1320  includes a top section  1330  and a bracket section  1340 . The top section  1330  is defined by a point edge  1332  and a base edge  1334 . The top section  1330  may be shaped in a pyramidal manner such that the base edge  1334  is wider than the point edge  1332 . In other embodiments, the top section  1330  may be shaped in a half-circular manner or any other shape in which one edge is wider than a second edge.  FIG. 13D  depicts the bracket section  1340  in greater detail. The bracket section  1340  may include a base portion  1350 , a juncture portion  1360 , and a bracket portion  1370 . The base portion  1350  is defined by a top edge  1352 , a bottom edge  1354 , and two long edges  1356 . The two long edges  1356  are bent towards one another at a 90 degree angle to form an L-shape. The juncture portion  1360  is defined by a top surface  1362 , an inside edge  1364 , and an outside edge  1366 . The juncture portion  1360  is bent at a 90 degree angle, such that the inside edge  1364  and the outside edge  1366  are L-shaped, where the inside edge L-shape is smaller than the outside edge L-shape. The bracket portion  1370  is defined by a top side  1372 , a bottom side  1374 , and two long sides  1376 . The two long sides  1376  are bent towards one another at a 90 degree angle to form an L-shape. 
     Referring to  FIGS. 13B-C , the juncture portion  1360  is connected to the base portion  1350  at the inside edge  1364  and the top edge  1352 , respectively. The juncture portion  1360  is connected to the bracket portion  1370  at the outside edge  1366  and the top side  1372 , respectively, thereby creating an air gap between the base portion  1350  and the bracket portion  1370  whose width is equivalent to the width of the top surface  1362 . The top section  1330  is attached to the bracket section  1340  at a connection angle  1390  at the point edge  1332  and the juncture portion  1360 , respectively. The connection angle  1390  may be an acute angle, as in the illustrated example. For instance, in an exemplary embodiment, the connection angle  1390  may be 30 degrees. In other exemplary embodiments, the connection angle may be 45 or 60 degrees. Further, it should be understood that other angles are possible, without departing from the scope of the invention. 
     Each of the plurality of connector sections includes at least one corner-restraining strap and each corner-restraining strap is configured to receive one of the plurality of corner fasteners. For example, referring to  FIG. 13 , a corner-restraining strap  1310  is connected to the connector section  1082 . In the current embodiment, the corner-restraining strap  1310  is attached to the connector section  1082  in such a manner that there exists an air gap between the corner-restraining strap  1310  and the connector section  1082  where corner fastener  1320  may be inserted. As shown in  FIG. 13 , the corner-restraining strap  1310  is attached to the outside of the adapter piece  1330  at the connector section  1082 . In another embodiment, the corner-restraining strap  1310  may be attached on the inside of the adapter piece  1300  at the connector section  1082 . 
     In operation, each of the plurality of corner fasteners may be inserted into at least one corner-restraining strap attached to the adapter piece  1300 . The plurality of corner fasteners may help to provide structural rigidity to the adapter piece  1300 . In the present embodiment, the corner fastener  1320  can be inserted into the corner-restraining strap  1310 . The corner-restraining strap  1310  is shaped and sized such that when the corner fastener  1320  is inserted, the corner-restraining strap  1310  temporarily secures the corner fastener  1320  to the connector section  1082  in a substantially stationary position. The corner fastener  1320  is temporarily secure in that it can be removed at any desired time and is in a substantially stationary position in that minor fluctuations in position are possible. That is to say, the corner fastener  1320  is removably connected to the corner-restraining strap  1310  and the connector section  1082 . 
     In practice, the bracket section  1340  is inserted first into the corner-restraining strap  1310 . Next, the top section  1330  engages the corner-restraining strap  1310 . The shape of the corner-restraining strap  1310 , together with the width of the top section  1330 , cause the top section  1330  to become removably connected with the corner-restraining strap  1310  and the corner section  1082 . External forces, such as wind or gravity, are unable to pull the corner fastener  1320  downwards and out of corner-restraining strap  1310 . That is, the corner fastener  1320  is unable to slip through the corner-restraining strap  1310  and thus become detached from the adapter piece  1300 . 
       FIG. 13E  depicts the adapter piece  1300  attached to the bag  1190 . In the present embodiment, the bag  1190  may be a free-standing bag for lawn clippings. The adapter piece  1300  is inserted into the opening of the bag  1190 , such that the plurality of insert sections  1040 - 1046  (only insert sections  1040  and  1042  are shown) are located on the inside of the bag  1190 , while a part of each of the plurality of corner fasteners (only corner fastener  1320  is shown) are located on both the inside and outside of the bag  1190 . For example, the base portion  1350  is located on the inside of the bag  1190 , the juncture portion rests atop a corner of the bag  1190 , and the bracket portion  1370  is located on the outside of the bag  1190 . The corner fastener  1320  acts as a clamp, which engages the bag  1190  between the base portion  1350  and the bracket portion  1370 , thereby removably attaching the adapter piece  1300  to the bag  1190 . 
     In other embodiments of the corner fastener  1320 , the top section  1330  and the bracket section  1340  may be detachably connected. The top section  1330  may further include a locking mechanism at the point edge  1332  and the juncture portion  1360  may further include a key slot. The locking mechanism may be inserted into the key slot, thereby securing the top section  1330  to the bracket section  1340 . 
       FIG. 13A  is an exemplary embodiment of an arrangement of corner-restraining straps. Other arrangements and combinations are also possible without departing from the present invention. For example, while  FIG. 13A  depicts the connector section  1082  including only the corner-restraining strap  1310 , in another embodiment, the connector section  1082  includes multiple corner-restraining straps. 
     In another embodiment, the adapter piece  1300  further includes a plurality of horizontal-restraining straps and a plurality of side fasteners. Each of the plurality of upper side sections  1060 - 1066  may include at least one of the plurality of horizontal-restraining straps, and each horizontal-restraining strap may be configured to receive one of the plurality of side fasteners, for example, the side fastener  1120 . The adapter piece  1300  may be removably attached to the bag  1190 , such that the plurality of insert sections  1040 - 1046  are located on the inside of the bag  1190 , the plurality of side fasteners are located on the outside of the bag  1190 , and a part of each of the plurality of corner fasteners are located on both the inside and the outside of the bag  1190 . For example, after the adapter piece is attached to the bag  1190 , the lower piece  1140  of the side fastener  1120  may be located on the outside of the bag  1190 . The lower pieces  1140  acts as a clamp, which engages the bag  1190  to the side section  1042 . Further, the base portion  1350  is located on the inside of the bag  1190 , the juncture portion  1360  rests atop a corner of the bag  1190 , and the bracket portion  1370  is located on the outside of the bag  1190 . The corner fastener  1320  acts as a clamp, which engages the bag  1190  between the base portion  1350  and the bracket portion  1370 . The plurality of side fasteners and the plurality of corner fasteners thus removably attach the adapter piece  1300  to the bag  1190 . 
       FIG. 14  depicts an embodiment of an adapter piece  1400 , according to an exemplary embodiment. In particular, the adapter piece  1400  includes a plurality of vertical-restraining straps and the expandable frame  1220 . The expandable frame  1220  provides structural rigidity to the adapter piece  1400 . The expandable frame  1220  is attached to the adapter piece  1400  using the plurality of vertical-restraining straps. The plurality of frame pieces are inserted into the plurality of vertical-restraining straps, for example, vertical-restraining straps  1410 - 1414 , such that the dimensions of the expandable frame  1220  are approximately equal to the dimensions of the lower opening of the cavity  1010 . The expandable frame  1220  is attached to the adapter piece  1400 , such that each corner fastener, for example, corner fastener  1320 , is located between the adapter piece  1400  and the expandable frame  1220 . 
       FIG. 15  shows another embodiment of a corner fastener  1500  of the adapter piece  1400 , according to an exemplary embodiment. In particular, the corner fastener  1500  includes a junction post  1510 . The junction post includes a frame hook  1520 . The junction post  1510  is connected to the top section  1330  at the point edge  1332 . The junction post  1510  is connected to the juncture portion  1360  at the top surface. The expandable frame  1220  is attached to the juncture post  1510  using the frame hook  1520 . The frame hook  1520  is designed in such a manner that the expandable frame  1220  snaps or otherwise clips into the junction post  1510 , such that the expandable frame  1220  is removably connected to the corner fastener  1500 . 
     In another embodiment of the corner fastener  1320 , the corner fastener  1320  further includes a portion of the expandable frame  1220 , thereby defining a modified corner fastener. In particular, a frame connector piece  1230  may be connected to the corner fastener  1320  at the juncture portion  1360 . Two of the plurality of frame pieces  1240  may then be directly inserted into the modified corner fastener. 
       FIG. 16A  depicts a corner of an embodiment of an adapter piece  1600 , according to an exemplary embodiment. In particular, the adapter  1600  includes a plurality of horizontal-restraining straps and a plurality of side fasteners. The plurality of horizontal-restraining straps are attached to one of the plurality of upper side sections  1060 - 1066  (only upper side sections  1060  and  1062  are shown) and each of the plurality of horizontal-restraining straps is configured to receive one of the plurality of side fasteners. For example, a horizontal-restraining strap  1610  is attached to the upper side section  1062  in such a manner that there exists an air gap between the horizontal-restraining strap  1610  and the upper side section  1062  where side fastener  1120  may be inserted. Similarly, a horizontal-restraining strap  1612  is attached to the upper side section  1060  in such a manner that there exists an air gap in between the horizontal-restraining straps  1612  and the upper side sections  1060 , where side fasteners  1122  may be inserted. The plurality of side fasteners are located between the expandable frame  1220  and the adapter piece  1600 . 
       FIG. 16B  depicts a corner view of the adapter piece  1600  attached to the bag  1190 . In the present embodiment, the bag  1190  may be a free-standing bag for lawn clippings. The adapter piece  1600  may be inserted into the opening of the bag  1190 , such that the plurality of insert sections  1040 - 1046  (only insert sections  1040  and  1042  are shown) are located on the inside of the bag  1190 , the plurality of side fasteners (only side fasteners  1120  and  1122  are shown) are located on the outside of the bag  1190 , while a part of each of the plurality of corner fasteners (only corner fastener  1320  is shown) is located on both the inside and outside of the upper end of the bag  1190 . For example, the lower piece  1140  of the side fastener  1120  and a lower piece  1640  of the side fastener  1122  are located on the outside of the bag  1190 . The lower pieces  1140  and  1640  act as clamps, which engage the bag  1190  to the side sections  1042  and  1040 , respectively. Further, the base portion  1350  is located on the inside of the bag  1190 , the juncture portion rests atop a corner of the bag  1190 , and the bracket portion  1370  is located on the outside of the bag  1190 . The corner fastener  1320  acts as a clamp, which engages the bag  1190  between the base portion  1350  and the bracket portion  1370 . The plurality of side fasteners and the plurality of corner fasteners thus removably attach the adapter piece  1600  to the bag  1190 . 
       FIG. 17  is a flow chart illustrating a method for manufacturing an adapter piece  1700 , according to an exemplary embodiment. As shown at step  1702 , the method involves cutting a flat shape (such as the shape shown in  FIG. 10A ) from a single piece of material. Next, at step  1704 , a rectangular shape is excised from the flat shape so as to define a rectangular void. Then, at step  1706 , an incision is made into the flat shape at each corner of the rectangular void (as shown in  FIG. 10A ). At step  1708 , the flat shape is creased at the end of each incision so as to define a plurality of creases, for example, the plurality of creases  1020 - 1026  (as shown in  FIG. 10A ). Next, at step  1710 , the flat shape is folded at the plurality of creases so as to define a plurality of flaps, for example, the plurality of flaps could be the plurality of insert sections  1040 - 1046 . Then, at step  1712 , each corner of the flat shape is cut so as to define a pair of corner sections. The cut may be performed at a 45 degree angle at the corners of the flat shape. Additionally, the cut may be performed at any other acute angle without departing from the present invention. Lastly, at step  1714 , the corner sections are affixed together such that one corner section overlaps with the other corner section such that a cavity is formed (as shown in  FIG. 10B , for example, the cavity  1010 ). 
     In another embodiment of the method for manufacturing the adapter piece, steps  1712  and  1714  are not performed. Instead, each corner of the flat shape is sewn to itself in such a manner as to create the cavity (as shown in  FIG. 10B , the cavity  1010 ). 
     In other embodiments of the method for manufacturing the adapter piece, step  1714  is not performed. Instead, each of the corner sections is affixed to a connector section (for example, one of the plurality of connector sections  1080 - 1086 ) so as to form the cavity (as shown in  FIG. 10B , for example, the cavity  1010 ). In the present embodiment, the connector section is in the shape of a triangle. Further, in other embodiments, the connector section can take other shapes without departing from the present invention. 
     It should be understood the arrangements and functions described herein are presented for purposes of example only, and that numerous variations are possible. For instance, elements can be added, omitted, combined, distributed, reordered, or otherwise modified.