Fraction exploration device

A fraction exploration device for teaching various mathematical principles related to fractions, comprising a base sheet having printed indicia thereon, a ruler attached pivotally to the base sheet. The printed indicia on the base sheet include intersecting grid lines which define a matrix of intersections and a series of parallelograms therebetween. Fractions are located at a plurality of the intersections of the grid lines, each fraction having a numerator, a denominator and a dividing line separating the numerator from the denominator. The dividing line of each fraction is sloped to represent a mathematical relationship of the numerator to the denominator.

BACKGROUND OF THE INVENTION

The present invention relates generally to a mathematical teaching device, and more specifically to a device that teaches various mathematical functions related to fractions.

One embodiment of the device includes a base sheet to which is attached pivotally a ruler. The base sheet is labeled with various fractions, distributed in a hierarchical relationship about the sheet. Each fraction includes a numerator, a denominator and a dividing line separating the numerator from the denominator. The dividing line of each fraction is sloped to represent a mathematical relationship of the numerator to the denominator. For example, the fraction bar may represent graphically the slope of a hypotenuse of a right triangle having a vertical side the length of the numerator, and a horizontal side the length of the denominator.

The base sheet also may be labeled with lines in a grid, the lines crossing to define grid points. The spacing of the lines in the grid may be equidistant, and the crossing of the lines may be perpendicular so that the lines define a series of squares. Alternatively, a first set of lines may be spaced at a first interval, and a second set of lines generally perpendicular and overlapping the first set may be spaced at a second interval, so that the first and second sets of lines collectively define rectangles, but not squares. As a further alternative, the lines defining the grid may be parallel, but the first and second sets need not be perpendicular to each other, so that the lines define a series of parallelograms, that may, but need not be, equilateral.

Preferably, the ruler is attached to the base sheet at a grid crossing, and this attachment defines an origin for the grid. Furthermore, the grid lines define squares, and the ruler is labeled in increments conforming to the unit length of the sides of the squares, along a reference line extending approximately from the grid origin. The ruler is rotatable about the origin, and is clear so that the relationship of the labeling on the ruler to the labeling on the grid may be observed readily. Each fraction is located at a grid point, and the fraction bar for each fraction is oriented to extend along a line from the corresponding grid point to the grid origin. The ruler includes an indicator line extending directly away from the grid origin, so that the indicator line of the ruler closely overlaps the fraction bar for a fraction at a slope approximately equal to the slope of the fraction bar, when the indicator line is aligned with the grid point for the fraction.

Yet other embodiments are envisioned within the scope of the present invention. For example, the ruler could be manufactured with a slot or other external geometry that would allow a marking pen to trace a line along the centerline of the ruler. This could be used in combination with erasable pens, or with a removable, clear, intervening sheet that could be placed between the ruler and the base sheet, so that various tracings and notations could be made by either the teacher or a student. In those embodiments in which the intervening sheet is removable, various tracings could be made for various problems, and the solutions shown on the intervening sheets could be compared by the student and teacher.

In the preferred embodiment, there are 11 vertical lines and 11 horizontal lines defining the grid. The last vertical line further is labeled with the decimal and percentage equivalent of the fractions listed along the line. Alternative quantities of lines could be used to demonstrate use of the device for other base counting systems such as base8or base16.

It is an object of the present invention to explore the interrelationship between various mathematical concepts.

Additional objects and advantages of the present invention will be understood more readily after a consideration of the drawings and the Description of the Operations.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

InFIG. 1, a fraction exploration device according to the present invention is shown, including a base sheet12and a plurality of fractions14. Each fraction14has a numerator16, a denominator18and a dividing line or fraction bar20separating numerator16from denominator18. The dividing line of each fraction is oriented to point toward a predefined focal point22.

A ruler24is attached pivotally to base sheet12at origin22, preferably by a pivot26such as a simple grommet or pin. Ruler24has an indicator line28extending to align with a plurality of fractions14as ruler24is pivoted about focal point22. For example, inFIG. 1, indicator line28of ruler24has been aligned with the fraction {fraction (10/8)}. Indicator line28also aligns with the indicia for the fraction {fraction (5/4)}, showing that {fraction (5/4)} is the numerical equivalent to {fraction (10/8)}. If the student desires, extrapolation also can be performed to determine other equivalents, such as the fact that {fraction (10/8)} is equivalent to a fraction between {fraction (4/3)} and {fraction (3/3)}, and another fraction between {fraction (2/1)} and {fraction (1/1)}. Indicator line28includes indicia30and subunit markers32so that precise measurements may be made along line28.

Base sheet12also is labeled with indicia to represent intersecting grid lines. A first set of gridlines forms vertical lines34, including a base vertical line34, a first vertical line34a, continuing on to a tenth vertical line34j. Base vertical line34is created in indicia representing a dashed line to remind a user of device10that fractions on base vertical line34would require division by 0. Since the result of division by zero is an infinite number, no fractions14are shown on base vertical line34. Tenth vertical line34jincludes indicia representing subunit markers36so that precise measurements may be made along line34j.

A second set of gridlines forms horizontal lines38, including a base horizontal line38, a first horizontal line38a, continuing on to a tenth horizontal line38j. First set of lines34and second set of lines38collectively define a grid having a matrix of intersections or crossings40, and also define a series of parallelograms therebetween. The indicia for fractions14preferably are located so that each fraction bar20is bisected by a grid point40.

Furthermore, fractions14preferably are arranged in hierarchical order. Still referring toFIG. 1, base vertical line34is not labeled. The next vertical line34ais labeled with fractions14, beginning with the fraction {fraction (0/1)} and extending to {fraction (10/1)}. The next vertical line34bis labeled with fractions beginning with {fraction (0/2)} and extending to {fraction (10/2)}. This pattern is repeated up to last vertical line34j, labeled as {fraction (0/10)} through {fraction (10/10)}.

Ruler24is labeled in incremental units30along indicator line28so that a distance between grid points40that align with indicator line28may be measured. Units30preferably are spaced equal to the spacing between lines34and lines38, so that the increments on the ruler may be used to measure the hypotenuse of triangles defined by selected grid lines34and38, and indicator line28of ruler24. For example, inFIG. 2, indicator line28is marked for the fraction {fraction (10/3)}, and a triangle42is defined, with a hypotenuse44traced from near the origin to the fraction {fraction (10/3)}, a vertical side46traced along third vertical line34cand a horizontal side48traced along base horizontal line38, as shown in thicker lining. Hypotenuse44then may be measured as being between 10 and 11 units long, as marked. The markings added inFIG. 2to base sheet12and ruler24are best made with a non-permanent marker, such as a dry-erase pen for use with overhead projectors and white boards, so that device10may be reused as needed.

InFIG. 3, an alternative embodiment of the device is shown as110. Base sheet12and its indicia are as shown and described with respect toFIGS. 1 and 2. Ruler124differs in that it is formed with a slot128defining the indicator line for ruler124. Other external geometry may be used for ruler124to define similarly a stencil that allows a marking pen to trace a line along the indicator line of ruler124. This could be used in combination with erasable pens, or with a removable, clear, intervening sheet that could be placed between ruler124and base sheet12. Various tracings and notations then could be made by either the teacher or a student. In those embodiments in which the intervening sheet is removable, various tracings could be made for various problems, and the solutions shown on the intervening sheets could be compared to one another by the student and teacher.

Still referring toFIG. 3, device110is shown demonstrating the Pythagorean Theorem. Indicator line128of ruler124is aligned with the grid point40labeled with indicia representing {fraction (4/3)}, and a triangle142has been drawn to trace the lines defined therebetween. By measuring along ruler124, it will be seen that the hypotenuse of triangle142is 5. The same demonstration could be obtained by aligning the indicator line with ¾, as shown in dashed lines inFIG. 3, defining triangle142a.

A further alternative embodiment of the device of the present invention is illustrated in FIG.4. In this embodiment, a grid210is defined by numerous vertical and horizontal lines to obtain a precision of placement of fractions not available with the grid shown inFIGS. 1 through 3. The resulting grid measures approximately 100 lines234by 100 lines236. Grid210is particularly useful for representing decimal fractions such as {fraction (3.3/4.7)} or {fraction (0.33/0.47)}, and their relationship to whole number fractions such as {fraction (33/47)} (represented by triangle242), or other fractions such as ¾, {fraction (30/40)}, {fraction (4/4)}, {fraction (40/40)}, ⅘, {fraction (40/50)}, ⅗and {fraction (30/50)}.

In the embodiment of grid210shown inFIG. 4, each grid point could be labeled with a fraction indicating whole number fractions ranging from {fraction (1/100)} through {fraction (100/1)}, if desired. This labeling has not been applied toFIG. 4because it would be so small that it would be difficult to read the fractions with the naked eye. If such labeling is to be used, the device preferably would be manufactured on a scale of at least two-feet by two-feet that would allow human-readable fractions at each grid point.

From the foregoing identification of the elements of the fraction exploration device, numerous mathematical procedures may be performed. One such procedure demonstrates the whole-number relationship between various fractions, as discussed with respect to FIG.1. Another demonstrates the length relationship between a hypotenuse and sides of a triangle, as discussed with respect toFIGS. 2 and 3.

Numerous other mathematical operations and demonstrations may be performed with the present device, as will be understood readily by those having skill in the art. The type of concepts that the student can explore using the present device ranges from the simple to the fairly complex. It starts with the study of fractions, moves on to geometry and number theory, and includes linear algebra. The device can be used to help students understand elementary concepts or do extensions into deeper mathematical subjects. The following are examples of these concepts.

1. Show the equivalence of two fractions.

2. Indicate when a fraction is in lowest terms.

3. Show what a fraction is when reduced to lowest terms.

4. Show a geometric representation of lowest terms.

5. Show when one fraction is less (or greater) than another.

6. Allow exploration of similar right triangles.

7. Explore ratio and proportion.

8. Change fractions to decimals.

9. Find fractions close in value to a given decimal.

10. Change fractions to percent.

11. View the graph of linear functions of the form y=mx.

12. Explore the concept of the slope of the line y=mx.

15. Work with the Pythagorean Theorem.

16. View addition of fractions in another way.

17. View multiplication of fractions in another way.

Answers to the following lesson questions may be explored with the fraction exploration device.

1. Equivalence: When are two fractions equal? Why are two fractions equal? How do we tell two fractions are equal?

2. Lowest terms: What does it mean for a fraction to be in lowest terms? How is that manifested on fraction exploration device10?

3. Visible from origin: What does “visible from the origin” mean? What property do fractions that are visible from the origin have?

4. Similar triangles: What are “similar triangles?” How are they represented on fraction exploration device10?

5. Ratio and proportion: What does the ratio a/b mean? How is it represented on fraction exploration device10? How do you solve a proportion like a/b=x/c on fraction exploration device10?

6. Less than: When is one fraction less than another? How do we tell when one fraction is less than another?

7. Greater than: When is one fraction greater than another? How do we tell when one fraction is greater than another?

8. Changing fractions to decimals: What does a decimal mean? How do we change a fraction to a decimal?

9. Finding fractions close to decimals: Is every decimal equivalent to a fraction? How do we estimate a fraction that is close to a decimal? Can we tell which fraction is closer in value to a decimal?

10. Percent: How do you find percent on the device? How do we convert from fractions to decimals or percent?

11. Line: What lines from analytic geometry can be represented on fraction exploration device10? What does the “m” in the equation y=mx correspond to on fraction exploration device10?

12. Slope: What is slope? How does it relate to fraction exploration device10?

13. Pic's Theorem: What is Pic's Theorem and how is it used? Proof?

14. Determinants: What is a “determinant” and how does it relate to an integral lattice?

15. Pythagorean Theorem: What is the Pythagorean Theorem? How can it be checked on fraction exploration device10?

16. Adding fractions: What is the manifestation of adding fractions on fraction exploration device10?

17. Multiplying: What is the manifestation of multiplying fractions on fraction exploration device10?

18. Farey Numbers: What is the definition of a Farey series? Give examples of Farey series. Give properties of the Farey Numbers. What is the relationship of fraction exploration device10and Farey Numbers?

While the present invention has been shown and described with reference to the preferred embodiment, it will be apparent to those skilled in the art that other changes in form and detail may be made therein without departing from the spirit and scope of the invention as defined in the appended claims.