Geometrical instruments – Straightedge type – Plural straightedges nonadjustably fixed at right angles
Reexamination Certificate
2000-07-26
2002-01-08
Gutierrez, Diego (Department: 2859)
Geometrical instruments
Straightedge type
Plural straightedges nonadjustably fixed at right angles
C033S476000, C434S205000
Reexamination Certificate
active
06336274
ABSTRACT:
CROSS-REFERENCE TO RELATED APPLICATIONS
Not Applicable.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
Not Applicable.
BACKGROUND OF THE INVENTION
1. Field of Invention
The present invention relates to templates. More specifically, the present invention relates to a right angle template useful for teaching students of various grade levels mathematical concepts, including concepts related to algebra and calculus.
2. Description of the Related Art
The use of templates to assist in graphing mathematical formulas or functions is well-known. A template having a straight-edge for drawing a straight line on a graph is particularly well-known. Such straight edges may be used to connect plots on a graph, thereby drawing a line which represents the relationship between two variables on a graph. Such straight edges may also be used to draw a tangent line or “slope” at a point on a graph defining a polynomial function.
The use of graphs to visually define binomial and polynomial equations is frequently used to teach principles in algebra, calculus, and other mathematical fields of study. For example, students are taught in algebra that the equation of a line is defined as:
y=mx+b
where y is a value in the range of an equation, x is a value in the domain of an equation, m is the slope, and b is the “y-intercept.” Students may be asked to determine the value b in this equation by graphing a series of known values represented by (x
1
, y
1
), (x
2
, y
2
), etc. In this instance, the value of the y-intercept is determined manually by actually plotting the given x
n
and y
n
points on a graph, and extrapolating the line defined by connecting those points across x=0.
Alternatively, students may be given only one point, defined as (x
1
, y
1
), along with the value of m or b. A line could then be graphically created based upon these known values. As another exercise, a student may simply be given a line on a graph, and then asked to calculate slope by correlating pairs of x and y values. This is done using the formula:
m
=
(
y
2
-
y
1
)
(
x
2
-
x
1
)
The process of graphing lines in algebra enables students to depict real-world phenomena, such as the relationship between the height of a tree and its diameter. In this way, students can apply the concept of slope to represent rate of change in a real-world situation.
Students also learn to make predictions from a linear data set using a line of best fit. In this regard, graphing allows one to interpolate and extrapolate corresponding values of x and y which are not directly given based upon points which are given. A student can then estimate the values of the domain and range of a function at various points on a graph.
In the context of calculus, the slope of a polynomial function at a particular point f(x), defined as f′(x), represents the rate of change at a particular point in time. This is known as the derivative of a function. The derivative can typically be determined mathematically by using, for example, the limiting formula
f
′
(
x
)
=
ⅆ
x
ⅆ
y
=
lim
h
→
0
f
(
x
+
h
)
-
f
(
x
)
(
x
+
h
)
-
x
.
However, as a teaching aid, the derivative can also be determined geometrically by actually drawing a tangent line at a point, (x
n
, y
n
) and then determining (x
1
, y
1
) and (x
2
, y
2
) values. Using again the formula:
m
=
(
y
2
-
y
1
)
(
x
2
-
x
1
)
the mathematical value of the derivative, or rate of change at a point in time, can be calculated without mathematically calculating the derivative directly. Whether the value of m is positive or negative also determines whether the rate of change is increasing or decreasing.
As another exercise, the derivative of an equation can be demonstrated geometrically through the use of a secant line. For example, if a curve C has equation y=f′(x), and the student wishes to find the tangent to C at the point P(a, f(a)), then we consider a nearby point Q(x, f(x)), where x≠a, and compute the slope of the secant line PQ:
m
PQ
=
f
(
x
)
-
f
(
a
)
x
-
a
.
The student is then shown that as Q approaches P along the curve C by letting x approach a, then the value of m
pQ
approaches the actual tangent line of curve C at P. Hence, a graphical demonstration of m
pQ
as Q approaches P forms an important teaching device. This, again, is the mathematical value of the derivative f′(x) at P.
When performing the graphing exercises discussed, it is important for students to be able to accurately identify x and y values. In order to accurately identify corresponding x and y values on a graph, it is necessary that the student align a straight-edge in a direction which is exactly perpendicular to the respective x and y axis. However, those skilled in the art will understand that the process of identifying corresponding x and y values simply by using a straight-edge is not always accurate, as it is very difficult to make the required perpendicular alignments. More accurate measurements can sometimes be made by taking unusual time and care in the alignments. However, even this does not guarantee a result within what may be a required range of accuracy.
Right angle templates in the form of drafting squares, T-squares, and carpenter's squares are available. These are used by architects, draftspersons, carpenters, and the like in laying out construction plans. Such devices are disclosed in U.S. Pat. No. 5,419,054 issued in 1995 to Stoneberg, U.S. Pat. No. 5,239,762 issued in 1993 to Grizzell, U.S. Pat. No. 5,140,755 issued in 1992 to Simmons, and U.S. Pat. No. 5,090,129 issued in 1992 to Cunningham. However, these devices are cumbersome and are not practical to the algebra or calculus student who seeks to obtain quick and accurate corresponding x and y values in the classroom.
U.S. Pat. No. 4,936,020 issued in 1990 to Nablett offers a right angle template for marking a picture mat blank in preparation for subsequent cutting. In addition, U.S. Pat. No. 5,404,648 issued in 1995 to Taylor, Jr., presents a triangular shaped navigational plotter for determining the position of a ship. However, these devices are of no benefit to the math student when plotting points on a graph.
Finally, templates having radian or angular calibrations coupled with one or more straight-edges are sometimes used by math teachers. However, these do not have ruler members or straight edges at a fixed right angle to one another and, accordingly, do not provide the math student with the most efficient means for identifying corresponding values in the domain and the range of a function by allowing for perpendicular alignment of the template with both the x and y axis simultaneously.
It is clear that a need exists for a device useful for teaching students concepts of algebra, calculus, and other mathematical subjects wherein the plotting and reading of graphs is performed.
Therefore, it is an object of the present invention to provide a template which allows for the quick and accurate identification of corresponding values in the domain and the range of a function by allowing for perpendicular alignment of the template with both the x and y axis simultaneously.
It is a further object of the present invention to provide a device which will assist even elementary school students in understanding the relationship between domain and range elements on a graph.
It is another object of the present invention to provide a device comprising a right angle template which allows for the more expedient plotting of points on a graph.
It is yet another object of the present invention to provide a right angle graphing template which is both easy to use and economical to manufacture.
And yet a further object of the present invention is to provide a graphing template which can also serve as a ruler.
BRIEF SUMMARY OF THE INVENTION
Other objects and advantages of the present invention will become more apparent upon reviewing the detailed description and associated figures of the right angle graphing template. In the apparatus of the present invention, a templat
Gutierrez Diego
Jagan Mirellys
Pitts & Brittian P.C.
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