Objectives:
- Students will be able to translate,
reflect, rotate, and resize (dilate or constrict) any geometric shape
with respect to any given line segment and point.
Materials needed:
- Teacher will need projector and Geogebra program
- Students will need paper, pencil, ruler,
compass, and protractor
Time required:
- Takes about 45 minutes total-- 5 min for
preparation and 10 min per
transformation.
What students should already know how to do:
- What line segements, triangles, angles,
and circles are
- How to draw each of these with a ruler,
protractor, and compass
Procedure using Geogebra:
- Set up projector connected to computer in order to project a new
page in the Geogebra program.
- Have students gather all their necessary materials--students will
draw these transformations as you go.
- Construct a triangle ABC, a line segment DE, and a point F.
- Define translation:
Shifting an object without rotating or resizing it. Every point of the
shape must move the same distance in the same direction.
- Translate triangle ABC
in the direction and distance of segment DE.
- Draw 3 segments the length and direction of DE from each of the
triangle vertices.
- Connect the endpoints of these three segments to create the
translated triangle.
- Draw a line through segment DE to create line DE.
- Define reflection:
Flipping an object over a line to create a mirrored image. The
reflection is the same size, and each point on the object is the same
distance from the central line as its corresponding point on the
reflection.
- Reflect triangle ABC
through line DE.
- Draw line segments from each vertex to line DE, perpendicular
to line DE. (Note that right angles are created.)
- Double the length of all three line segments on the other side
of line DE.
- Connect these endpoints to create the reflected triangle.
- Define resizing:
Making a shape bigger (dilation)
or smaller (contraction).
The two shapes are similar, meaning the angles are the same and the
sides are proportional to each other.
- Constrict triangle ABC
by a factor of 1/2 about point F.
- Connect each of the vertices of the triangle to point F.
- Find the midpoints of each of these line segments.
- Connect the three midpoints to create the constricted triangle.
(Note that the sides of the constricted triangle are half the length of
each corresponding side of the original triangle.)
- Define rotation:
Turning an object around a center point. The distance from the center
to any point on the shape stays the same while turning. (Each point
makes a circle around the center.)
- Rotate triangle ABC 108
degrees counterclockwise about point F.
- Connect each of the vertices of the triangle to point F.
- Using a compass, construct three different circles using point
F as the center and these line segments as radii.
- Using a protractor, measure 108-degree angles counterclockwise
around each circle for each radius.
- Draw a new radius to create the angle, and mark its
intersection with the circle for all three segments.
- Connect the new points on the three circles to create the
rotated triangle.
Assessment:
On a test, give four different pictures of a graph each with one of the
following:
- Given: shape and line segment. Have them translate the shape in
the direction/distance of the line segment.
- Given: shape and line. Students must reflect the object across
the line.
- Given: shape and a point. Students must dilate/constrict the
object by a factor of __ about the point.
- Given: shape and a point. Students must rotate the object
__degrees clockwise about the point.
Extension/homework:
- For homework:
Draw and label
triangle ABC, line segment DE, and point F. Then
perform the following:
(a) Translate triangle ABC in the
direction and distance of line segment DE.
(b) Reflect the result of part (a) through
line DE.
(c) Dilate the result of part (b) by a
factor of 3/5 about point F.
(d) Rotate the result of part (c) by 117
degrees counterclockwise about point F.
- Have students use the Applet Planar
transformations. They
can access this interactive applet at home and walk through the
steps of the construction on their own to reinforce what they learned
in class and use it to help them on their homework assignment.
Modifications
- If projector and Geogebra
cannot be used, work through the construction on the black/white board
with the students, using a yardstick, large compass, and large
protractor.
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Michelle Beckman
beckman.michelle@gmail.com
Updated December 10, 2008.