Transcript Document

Using GeoGebra
in Analitic geometry
Svetlana Maletin
Verica Govedarica
High school "Jovan Jovanovic Zmaj"
Novi Sad, Serbia
Two views: the algebra window and the geometry window
an expression in the algebra window corresponds
to an object in the geometry window and vice versa.
Line.
• y=kx
• discuss: how the
parameter k influeces
the direction of line
• y=kx+n
• discuss how the
parameter n influences
the y-axis segment
• parallel lines ( k2=k1 )
Line.
• x/m + y/n =1
• perpendicular lines
• y = k x + n1
• y = -1/k + n2
( k2 = -1/k1 )
A lot of tasks...
Circle.
•
•
•
•
•
•
Circle c(O,r)
center: point C(p,q)
radius: r
----------------------equation:
(x-p)2 + (y-q)2 = r2
Intersection of circle and line.
• Intersection of circle c(O,r) and line y = k x + n
for diferent k and n
• tangents to circle paralel with fixed line
Intersection of circle and line.
• tangents on c perpendicular to a fixed line
Intersection of circle and line.
• tangents through point Mc on c ,
for diferent place of point M
• tangent through point Ac on c ,
for diferent place of point A on c
Ellipse.
• major axis = 2a, minor axis = 2b
• a is semimajor axis, b is semiminor axis
• circle is a special case of an ellipse
Ellipse.
• ellipse - the locus of points on a plane where the sum of
the distances from any point on the curve to two fixed
points is constant.
• The two fixed points are called foci (plural of focus).
• F1(-c,0), F2(c,0)
c2 = a2 - b2
Ellipse.
• Moving point Mc,
• equation:
2
d=d1+d2= const.
2
x
y
 2 1
2
a
b
Hyperbola.
• Hiperbola - the locus of points on a plane where the
difference of the distances from any point on the curve to
the two fixed points is constant.
• The two fixed points are called foci (plural of focus).
• F1(-c,0), F2(c,0)
c2 = a2 + b2
Hyperbola. Equation. Asymptote.
• Equation:
x2
y2
 2 1
2
a
b
• Asymptote of hyperbla
y
b
x
a
y
b
x
a
Parabola.
• equation: y2 = 2 p x
• focus: F( p/2, 0)
• directrix: x = - p/2
Reasons for introduction GeoGebra into teaching
• GeoGebra is a simple and interesting tools suitable for
teaching Analitic geometry.
• Using the algebra window and the geometry window, the
students get a clear view of the things that they are
learning.
• GeoGebra is especially usefull for the first encounter with
conics.
• GeoGebra is helpfull to teachers for making a lot of tasks
with ease.
Test. Results of group A
• Group A learned Analitic Geometry on the clasic way,
without using GeoGebra.
12
0-20
7%
81-100
17%
10
21-40
17%
8
61-80
20%
6
4
41-60
39%
2
0
number of students
81-100
61-80
41-60
21-40
0-20
5
6
12
5
2
Test. Results of group B
• Group B learned Analitic Geometry using GeoGebra.
12
0-20
0%
10
21-40
33%
81-100
40%
8
6
41-60
7%
4
61-80
20%
2
0
number of students
81-100
61-80
41-60
21-40
0-20
12
6
2
10
0
Problems with using GeoGebra in teaching
• Problems may occur when working with large groups of
students, because of some of them can't concentrate.
• GeoGebra can't be used allone, becase the students must
learn to use equations and finsh tasks by themselves