Understanding Transformations Around the Coordinate Plane with GeoGebra
An Instructional Unit
Standard Covered: M7G2- Students will demonstrate understanding of translations, dilations,
rotations, reflections, and relate symmetry to appropriate transformations.
My students often have a hard time understanding transformations, specifically rotations and
reflections. Translations are the easiest ones to teach because the students can still see the size
and orientation of the shape and how the image has been slid over. Currently, I teach rotations in
a very low-tech way by using an overhead projector. I will lay two coordinate planes on top of
each other and rotate the top sheet that has a shape on it to literally show my students the triangle
being rotated around the coordinate plane. Not only is this cumbersome, but it also isn’t very
interesting or engaging. This instructional unit will allow me to introduce and practice reflections
and rotations using GeoGebra. Luckily, special education funds have provided my school with a
small computer lab in which my students can use for this five day unit.
Reflections
Day One
After we download the program I will allow show my class how to create points, lines, and
shapes with GeoGebra. I will give them 15 minutes to “play” with the new program so they feel
comfortable using it during the lesson. Most of my students do not have computers at home and
are not very familiar with how to use them.
We will then discuss what it means to reflect a shape. Where have they seen reflections before?
What does a reflection look like? Hopefully, this will lead to a discussion of the line of
reflection.
To introduce reflections, I would instruct my class to use the “Point” function to create three
points: A (3,4), B (4,2), and C (4,1). Then they would click the “Polygon” button to create a
triangle out of these three points. After they have created the triangle, I will provide the class in
writing with the following directions:
1.
2.
3.
4.
Click the “line” button next to the “point” button. Place your point at the origin (0,0).
Draw a line through the origin that is on top of the x axis.
Select the “transformations” button and choose “reflect object about line.”
Click on triangle ABC and the x-axis, then select “reflect object about line.” This should
create an image of triangle ABC in quadrant II.
After they have seen a reflection take place, I would further explore the concept of the line of
reflection. We would continue to experiment with the steps above using the y-axis as a line of
reflection. As we played around with reflecting over the x and y axes, I would point out the
margin to the left that displays the coordinate of each point. I would have my students write
down the coordinates of the original polygon and of the image each time they made a reflection
on the sheet I provide to them (Appendix A). As a class, we would devise a rule that can be
applied each time you reflect over the x and y-axis. Eventually, my students would see that
reflecting over the x-axis means you multiply the y coordinate by (-1) and that reflecting over the
y axis means you multiply the x coordinate by (-1). Their homework is Appendix B, which
requires them to use their notes to write the coordinates of the image after a given reflection.
These questions are similar to what they will see on the CRCT.
Day Two
To review, every student will create any quadrilateral they choose and reflect it over both the x
and y axis. They will be required to tell me the rule for reflecting over the x and y axis and make
sure they have it written in their notes. My parapro and I will be walking around during the warm
up to make sure every student is comfortable using the program and understands the directions.
I’ll select one student’s quadrilateral to use as the warm up example and we’ll all go over it.
When my students are comfortable reflecting over the x and y axis using GeoGebra, we would
hypothesize where else they could place a line of reflection. I would use my computer hooked up
to my TV to show that you can select x=2 as a line of reflection (see example below). I will
explain these new lines of reflections by telling them to find the x or y axis, and then the number
listed. So if x=2 is their line of reflection, they need to find the x axis and then find where 2 lies
on the x axis. Their line of reflection must run through this point. Each student would be given a
list of shapes to create and then a line of reflection they must use to reflect their shape (Appendix
C). Additionally, they would be required to write down the coordinates each time a reflection is
made and notice any patterns that are forming.
Rotations
Day Three
Rotations are the hardest transformation to teach and their formulas require memorizing that my
special education 7th graders have a difficult time doing. Additionally, performing rotations
requires an ability to visualize that I suspect GeoGebra will assist a great deal with. To start, we
will discuss what it means to rotate something and I’ll ask if they can think of other words that
also mean rotate. We will talk about rotation in real life, such as the hands on the clock, the Earth
around the sun, and the pencil of a compass around the sharp end of the compass. My students
will plot the following points: A (1,-1), B (3, -2), and C (2, -2) using GeoGebra. They will then
have the following directions next to their computer and will follow along as we complete them
step by step:
1. Use “polygon” function to connect the dots and create triangle ABC located in the second
quadrant.
2. Create a point D at the origin (0,0)
3. Select “rotate object around point by angle” and click in the middle of the shape to make
sure all sides are highlighted.
4. Click on the origin and a window will pop up asking how many degrees and it which
direction you would like to rotate.
5. Select 90 degrees and clockwise and note the image that forms.
6. Repeat process, but select 90 degrees counterclockwise.
As with reflections, I will have my students write down the coordinates of the original triangle
and its image. They’ll experiment with GeoGebra using Appendix D for the rest of the class
period and then I’ll ask for any rules they have found in their listings of the coordinates. This
will lead to their discovery of the following rules:
-
rotate 90 clockwise means you are switching the x and y coordinates and multiplying
y by (-1)
rotate 90 counterclockwise means you are switching the x and y coordinates and
multiplying the x by (-1)
Day Four
For the warm up, I’ll ask my class to construct a triangle and reflect it over the x axis and y = -3.
I’ll walk around the room, answer questions, and make sure each student got the correct figures.
Then, I’ll do a guided practice of 90° clockwise on my computer hooked up to the TV so
everyone can see. I’ll ask the class to construct any triangle they choose and rotate it
90°counterclockwise.
After we’ve reviewed what we’ve done this week, I’ll ask them about their general concept of
180° and 360° and have a brief class discussion. I’ll give them their printout of how to rotate
polygons, and ask the class to rotate 180° (see example below), 270°, and 360°. They will have
to fill out Appendix E as they go, which will keep track of the original coordinates and the
coordinates of each image. They will be asked to hypothesize about a possible rule we could
create for each rotation and will hopefully come up with the following:
-
rotate 180 degrees means you are switching both coordinates
rotate 270 degrees means you are rotating 90 degrees in the opposite direction
rotate 360 degrees means your shape is exactly where it started, since there are 360
degrees in a circle
The rest of the class will be spent as they choose, whether it is reflecting or rotating shapes.
Day Five
The class will have to use GeoGebra to complete their Friday quiz. The following instructions
will be on the whiteboard and on a small sheet next to their computer when they come in:
1. Open GeoGebra and create triangle A ( -3, -1 ) B (-10, -1) C ( -5, -6). Reflect the triangle
over the x axis.
2. Click “File” and then “New Window” and create quadrilateral DEFG using the following
coordinates: D ( 3, 4) E (3, 8) F (6, 4) G (6,8). Reflect the quadrilateral over the y axis.
3. On the same page as question #2, create triangle JKL using the following coordinates: J
(1, -3) K (6, -3) L (5, -4). Reflect this triangle over x = 1.
4. Click “File” then “New Window”. Create triangle PQR using coordinates of your choice.
Rotate that triangle 90° clockwise and 90° counterclockwise.
5. On your own paper, write down a description of the relationship between rotating
90°clockwise and 270°counterclockwise.
The students will leave their GeoGebra windows open after the quiz, and I’ll print their
answers. Each question will be worth 20 points and I’ll look to see if they plotted the original
figure correctly and if they completed the correct transformation.
Appendix A
Initial Shape
Coordinates of Image
Coordinates of Image
After Reflection Over X Axis
After Reflections Over Y Axis
A ( 3, -2 )
A'
A'
B ( 7, -2 )
B'
B'
C ( 5, -6 )
C'
C'
Do you notice any patterns between the original coordinates and the coordinates of each image?
______________________________________________________________________________
J ( -1, -1 )
J'
J'
K (-7, -1 )
K'
K'
L ( -7, -7 )
L'
L'
M ( -1, -7 )
M'
M'
R ( 3, 4 )
R'
R'
S ( 3, 8 )
S'
S'
T ( 6, 10 )
T'
T'
Write down the rule you think each coordinate follows when reflected over the x axis _________
______________________________________________________________________________
Write down the rule you think each coordinate follows when reflected over the y axis _________
______________________________________________________________________________
Appendix B
Initial Shape
Coordinates of Image
Coordinates of Image
After Reflection Over X Axis
After Reflections Over Y Axis
A ( 1, -2 )
A'
A'
B ( 0, -2 )
B'
B'
C ( 1, -10 )
C'
C'
J ( -1, -2 )
J'
J'
K (-6, -2 )
K'
K'
L ( -6, -6 )
L'
L'
M ( -1, -6 )
M'
M'
R ( 0, 4 )
R'
R'
S ( 2, 6 )
S'
S'
T ( 6, 10 )
T'
T'
Appendix C
Coordinates
Line of Reflection
A ( -3, 6 )
B ( -3, 10 )
Coordinates After Reflection
A'
X = -1
B'
C (-8, 1 )
C'
D ( 1, 1 )
D'
E ( 1, 8 )
Y=3
E'
F ( 6, 6 )
F'
P ( 0, 6 )
P'
Q ( 0, 0 )
X = 10
Q'
R (6, 0 )
R'
L ( 1, 1 )
L'
M ( 1, 8 )
N ( 6, 6 )
Y = -2
M'
N'