Rotations
Grade Level: 6-8
Sunshine State Standard: MA.7.G.4.2, MA.7.G.4.3
Time: 60 minutes
Materials:
• Students: Paper, pencil, graph paper, computer with GeoGebra (if
available)
• Teachers: Computer projection, GeoGebra software.
Objectives:
• For students to understand the concept of a geometrical rotation.
• For students to be able to determine the coordinates of a rotated
point on a coordinate grid centered at the origin. (for a 90, 180, 270
and 360 degree rotation)
Vocabulary: rotation, angle of rotation, origin, center of rotation, degrees
of rotation (90, 180, 270, 360), x-coordinate, y-coordinate
Lesson Plan:
One way to start the lesson is to show the students a rotation in action.
This can be done by starting up GeoGebra and constructing a polygon and
rotating it about the center. Here are the steps in doing that.
Step 1: Create polygon
Step 2: Rotate polygon
In order to rotate an object we need to have a point of rotation. Students might
not be familiar with what is meant by a point of rotation. This can be demonstrated
to the students by using GeoGebra, show them how to rotate an image about a point
on the polygon, like a vertex or a midpoint of one side. Then show them how a
polygon is rotated about a point not on the polygon. The students should now be
told that for the remainder of the lesson we will be using the origin as the point of
rotation.
Step 3: Create a slider for the angle of rotation
Find and select the slider tool and click somewhere in the drawing pad. Select the
angle option for the slider and we can change the name to anything we want, let’s
say ‘a’. We can change the range of values and the incremental change in the angle.
Now choose the rotation tool. Click on the polygon and the point of rotation, when
the screen appears for the angle, type in ‘a’ and click ‘ok’. Go to the pointer mode
and slide the point on the slider to change the angle of rotation. Observe what
happens when the slider moves.
Angle of rotation
Point of rotation,
vertex C
Angle of rotation
Angle of rotation
Point of rotation
Point of rotation,
midpoint F
After you are convinced that the students have a good understanding of
what a rotation is, it is time to get a little more specific and see if we can
determine the coordinates of a point after it is rotated a certain number of
degrees. For the following problems we will investigate points rotated about
the origin by degrees that are multiples of 90 degrees.
This will be done in GeoGebra by setting up
an angle slider and setting the increment
to 90o, we can leave the minimum 0o and
change the maximum to any number that is
a multiple of 90 but at least 360.
Now put a point at the origin and any other point on
the grid. Select the
tool,
for this tool you need an object, Point A, a
point of rotation, Point O, and an angle of rotation, slider . You must choose those
three items in that order. When that is done, depending on where your slider is
set, you will see another point on the coordinate grid that is the rotation of your
original point. The illustration below shows what happens when Point A is rotated
about the origin (Point O) by 90, 180, 270 and 360 degrees.
Students should now be encouraged to investigate the relationship between
the coordinates of point A and Point A’. The student’s goal at the end of the
investigation is to be able to figure out the coordinates of any point rotated
about the origin rotated any number of degrees that is a multiple of 90. The
students should then be challenged to discover a way to describe how to
figure out how to figure out the coordinates of a point that is not rotated
about the origin or is not rotated a number of degrees that is a multiple of
90. Sample questions like the ones below can be asked to assess student
progress.
Sample Questions:
1. The coordinates of Point A are (3, -5). What are the coordinates of Point A’
if Point A is rotated about the origin 450 degrees?
2. The coordinates of Point A are (-4, 9). What are the coordinates of Point A’
if Point A is rotated about the origin 180 degrees?
3. The coordinates of Point A are (2, 0). What are the coordinates of Point A’
if Point A is rotated about the origin 270 degrees?
4. The coordinates of Point A are (-7, -11). What are the coordinates of Point
A’ if Point A is rotated about the origin 720 degrees?
5. A rectangle has one vertex at (4, 2) and the opposite vertex at (-4, 7).
a. Find the coordinates of the two other vertices.
b. What would the coordinates of the vertices be if the rectangle was
rotated about the origin 180 degrees?
c. What is the perimeter of the rectangle?
d. What is the area of the rectangle?
e. What would the coordinates of the vertices be if the rectangle was
reflected about the x-axis?
6. The coordinates of Point B are (-5, 12). What are the coordinates of Point B’
if Point B is rotated about the point (1, 1) 90 degrees?
7. The coordinates of Point C are (5, 0). What are the coordinates of Point C’
if Point C is rotated about the origin…
a. …30 degrees?
b. …45 degrees?
c. …60 degrees?