Download Lesson Plan 1

LESSON PLAN
TEACHER: R. Ramirez
GRADE LEVEL (S): 9, 10
SUBJECT CLASS: M$DS
TOPIC: Geometry
AIM: What is Hyperbolic Geometry?
OBJECTIVES:
After studying this section, students will be able to:
1- Identify points, lines and planes in hyperbolic Geometry.
2- Recognize and define basic properties of Hyperbolic Geometry
3- Compare and contrast basic properties of plane in Hyperbolic Geometry.
4- Use models for visualizing and explore the properties of geometry.
5- Use computer for exploring the geometric properties of the Hyperbolic
Geometry.
PERFORMANCE STANDARDS:
The student produce evidence that demonstrates understanding of geometry and
measurement concepts; that is, the student:
1- Visualizes and represents two-dimensional views of simple rectangular
three-dimensional shapes.
2- Uses many types of figures and identifies the figures by their properties.
3- Solves problems by showing relationships between and among figures.
TEACHING STRATEGIES:
Teaching strategies include: lecture, small group work, textbook reading
assignments, class discussion, writing assignments, and computer assignments.
(No knowledge of computer programming is assumed.)
MATERIAL:
Ruler, protractors, compass, balls, balloons, rubber bands, lengths of ribbon and
computers connected to the Internet.
DO NOW:
Write the definition of the following:
a) Supplementary angle.
b) Interior Angle.
c) Straight angle
MOTIVATION:
a) How can we find the measure of each angle of a triangle?
b) How can we find the sum of the measures of the angles of a
triangle?
c) If ABC // DEFG, how many degrees are in EBF?
DEVELOPMENT:
1) Elicit from students answer motivation questions.
2) From the motivation recall to the students that in Euclidean
Geometry the sum of angles of a triangle is 180.
3) States to the students that in Hyperbolic Geometry the sum of the
angles of a triangle is less than 180. Show some examples.
4) States to the students that in Hyperbolic Geometry, all the axioms
for neutral geometry hold, but the parallel postulate is restated as
follow: Through any point P not on a line l, more than one line can
be drawn through P parallel to l.
5) Let the students use the web to discover similarities and differences
among planar, and hyperbolic geometry.
6) State to the students the main differences between Euclidean and
Hyperbolic Geometry. The main differences between them are the
properties of ``straight'' lines in each geometry, as shown in Figures
1 (Euclidean geometry) and 2 (hyperbolic geometry).
Fig. #1
Fig. #2
7) Introduce the students to the Poincare software.
http://cs.unm.edu/~joel/NonEuclid/.
http://mathforum.org/sketchpad/gsp.gallery/poincare/poincare.html.
8) Explain to the students the hyperbolic plane is the plane on one
side of this Euclidean line, normally the upper half of the plane
where y > 0. In this model, lines are either:
a) The intersection of points lying on a line drawn vertical to the xaxis and the half plane, or
b) Points lying on the circumference of a semicircle drawn with its
center on the x-axis.
APPLICATIONS:
Let students use the following website www.towson.edu/non-euclidean-modules.
This web page presents a few examples of activities that are available for
students to introduce to Hyperbolic Geometry.
ACTIVITY:
1. a) Locate two points in the plane and label them P and Q. What is the
shortest path between these two points?
Stretch a piece of ribbon between the two points to indicate the shortest
path.
Q
P
b) Locate two points on the sphere. Label one point A and the other B. Use
a piece of ribbon to find the shortest path between the two points on the
sphere just as you did on the plane. Describe what this path looks like.
A
B
2. a) Draw two lines on the plane. In how many points do these two lines
Intersect?
b) Draw two great circles on the sphere. In how many points do two lines on
the sphere intersect?
B
A
SUMMARY:
1- Define Hyperbolic Geometry.
2- What is the sum of the angles of a triangle in Hyperbolic Geometry?
3- What are the differences between Hyperbolic Geometry and Euclidean
Geometry.
4- Define Hyperbolic Plane.
HOMEWORK:
Write a one to two page essay comparing and contrasting it to what we have
learned in this class.