EMAT 6700 Instructional
Unit
Geometry
Objective: To enhance student
learning through interactive lesson plans using technology. To have students
produce geometric formulas using technology. To have students generate concurrencies
using technology. To further advance students knowledge of Geometer’s
Sketchpad.
Prerequisites: Students do not need to
be familiar with Geometer’s Sketchpad. They should understand and be familiar
with the definitions of shapes such as triangles, circles, and rectangles
before beginning this lesson. The entire lesson plan requires a computer lab
setting.
Lesson Plan:
In this first section students produce the equation of a circle
by exploring circles in Geometer’s Sketchpad.
Students are asked to follow these directions as they explore.
 Under Graph choose Show Grid.
 Under Edit set the preferences to centimeters for
distance, degrees for angles and tenths for precision.
 Use the circle tool and make at least 5 different
circles, varying the size of each.
 Under the Graph menu choose the snap to grid option.
 Move the center of each circle to a lattice point. Move
the defining point of each circle to a lattice point.
 Select each circle and under the construct menu select
circle interior to construct it. While the interior is selected, choose a
color from the display menu. Make each circle a different color.
 Make a record of your constructions: See figure 1 below
for example.
Color of Circle

Center of Circle

Radius

RED

3,2

2

BLUE

7,3

√10

GREEN

2,2

1

YELLOW

4,2

√2

PURPLE

3,3

2

 Select each circle without selecting its area. Under
Measure select Circumference then Area.
 Is there a relationship between the center and the radius
of the circle and its circumference and area? Explore to check your
conjecture.
Appropriate
amount of time is given here for students to explore this relationship.
 Select one circle. Under Measure select Equation. Was
your conjecture correct? If not why?
 Explain how the center and the radius are incorporated
into the equation.
 Produce the correct equations for all the circles and
move each equation by its corresponding circle.
 Write the equation for a circle with center (1, 5) and
radius = 3. Now create one such circle in GSP and check yourself using the
measure tool.
 Now using the select tool, move the equations to the
upper right corner of your screen.
 Switch computer stations and match the equations to the circles.
Explain how you know your equations are on the right circles.
In the second section students generate shapes in GSP. As an
introduction students would construct a rectangle in order to learn how to
construct perpendicular and parallel lines.
Constructing a
rectangle:
 Using the segment
tool draw a segment of any size.
 Then select a point
on the segment and the segment itself. Under the construct menu choose
perpendicular line.
 Then select the other
point on the original segment and repeat the perpendicular line
construction.
 Determine the length
of the side by constructing a point on one of the perpendicular lines.
 Select this point and
the line it is on and construct another perpendicular line.
 Select the two
intersecting lines and under the construct menu choose intersection.
(NOTE: You may be able to simply click on the intersection to construct a
point there.)
 Why is this a rectangle? What are its properties? Click and
drag one of its vertices. Does it remain a rectangle?
 How would you
construct a square? A triangle? An equilateral triangle? Click on one of
the vertices and drag your shape to see if your construction is correct
and the shape remains.
Here again students are given the opportunity to explore and
construct shapes. They are encouraged to work with others. Eventually the
option of rotation under the transformation menu is introduced. Also if the
idea of creating these shapes within a coordinate system is suggested students
need to understand its limitations. You cannot drag only one point in order for
the shape to remain a square.
 Construct a segment
using the segment tool. Double click on one of the endpoint until you see
it marked by a flash of concentric circles. Now select what you want to
rotate. Under the transformation menu select the rotation option. Determine your angle of rotation and
click OK.
 You can continue to
use rotate to construct your square.
 If you used the
coordinate system to construct your square, did your click and drag test
work?
In the third section students
generate concurrencies in GSP. Students would begin by exploring sets of lines
in triangles.
 Construct a triangle
using the segment tool.
 Construct the
midpoints of these segments by selecting these segments and choosing midpoint
under the construct menu.
 Construct the
perpendiculars to the sides of the triangles through these points.
 What do you notice?
Is it true for any triangle?
 Draw a new triangle.
 Construct the
midpoints again. Construct the medians of the triangle (the segments that
connect the midpoints with the vertices opposite the midpoints).
 What do you notice?
Is it true for any triangle?
 Draw a new triangle.
 Construct altitudes
of the triangle (the lines perpendicular to the sides but that go through
the vertex opposite that side).
 What do you notice?
Is it true for any triangle?
 Draw a new triangle.
 Construct the angle
bisectors of the triangle.
 What do you notice?
Is it true for any triangle?
When
three or more lines meet at a single point, they are said to be concurrent. The
following facts are true for every triangle:
The
medians are concurrent; they meet at a point called the centroid of the
triangle. (This point is the center of mass for the triangle. If you cut a
triangle out of a piece of paper and put your pencil point at the centroid, you
would be able to balance the triangle there.)
The
perpendicular bisectors are concurrent; they meet at the circumcenter of the
triangle. (This point is the same distance from each of the three vertices of
the triangles.)
The
perpendicular bisectors are concurrent; they meet at the circumcenter of the
triangle. (This point is the same distance from each of the three vertices of
the triangles.)
The
altitudes are concurrent; they meet at the orthocenter of the triangle.
The
angle bisectors are concurrent; they meet at the incenter of the triangle.
(This point is the same distance from each of the three sides of the
triangles.)
Triangles
are the only figures where all these concurrencies always hold.
Homework Problem: Recall that the centroid is the center of mass of a geometric
figure. How could you construct the centroid of a square?
Solution: One can use a straightedge to construct the two diagonals of
the square. The centroid is the point of their intersection. Or one can
construct the perpendicular bisectors of two consecutive sides of the square.
The intersection of these bisectors is the same centroid.
In this fourth section of the lesson plan students would be given an
entire period to explore the following problem.
In Class Problem: Draw five quadrilaterals Use one quadrilateral for each
construction below.
 Construct the eight medians of the first quadrilateral.
(There will be two medians at each vertex.)
 Construct the four midlines of the second quadrilateral.
 Construct the four angle bisectors of the third
quadrilateral.
 Construct the four perpendicular bisectors of the sides
of the fourth quadrilateral.
 Construct four altitudes in the fifth quadrilateral 
one from each vertex.
Do these constructions with a
few different quadrilaterals. Don't use only cases like squares and rectangles.
Record any observations about the constructions above. How is this different
from what you saw with triangles?
Solution: For some quadrilaterals (specifically those which can be
inscribed in a circle), the concurrency of perpendicular bisectors holds. For
all quadrilaterals, the midlines come in pairs that are parallel. For some
quadrilaterals (specifically those which can have a circle inscribed in them),
the angle bisectors are concurrent.
In this fifth section students continue their exploration of
triangle centers with the Nagel point and the Gergonne point.
 Construct a triangle. Extend the
segments of the triangle by constructing parallel lines.
 Construct the circle tangent to one of
the sides of the triangle and the line extensions of the other two sides.
This circle is called the escribed circle. First construct the center of
this circle. The center at the intersection of the exterior angle
bisectors. For example in the following diagram the red circle is the
escribed circle of triangle ABC whose center, D, is at the intersection of
the angle bisectors of angles FCA and EAC.
 Once the center is constructed, we
need to construct the radius. How would you find the defining point of
this circle?
Constructing
a line through the center of the circle and perpendicular to any of the lines
the circle will be tangent to will give us the defining point. Now that we have
a center and a defining point we can construct the circle.
NOTE:
The hide option under the Display menu can be used to unclutter
the screen and hide any lines that may interfere with future constructions.
 Repeat the escribed circle
construction for all three sides of the triangle.
 Construct a line through the center of
the escribed circle and the point of tangency of the circle and triangle.
What do you notice? Is it true for any triangle?
The
point constructed is the Nagel point.
 Construct a new triangle.
 Construct the incenter.
 Construct a circle with its center at
the incenter and tangent to all three sides of the triangle. This circle
is called the incircle of the triangle.
Again
constructing a line through the center of the circle and perpendicular to any
of the segments the circle will be tangent to will give us the defining point.
 Now construct the segment from the
point of tangency of the incircle and triangle to the opposite vertex.
What do you see? Is it true for any triangle?
This
point constructed is the Gergonne point.
Homework: Find at least one more triangle center or line and find how to
construct it in class tomorrow. Be ready to present your find to the class.