Lesson and Activity on Coordinate Planes
Adriana Magallanes
This is a lesson, which introduces the Cartesian coordinate plane, its
parts, and “its origin,” and historical practical uses. The parts of the coordinate plane that are
introduced will be the axes, the origin, the quadrants, and the concept of
ordered pairs. We will also discuss the
French mathematician René Descartes development in the creation and
documentation of the coordinate plane, as stated by many. In contrast, the way this has been used by
Native Americans will also be introduced and explored. Next, students will create their own
Cartesian plane, label all parts, and create their own “bead” design as
discussed in uses of the coordinate plane by Native Americans. In conclusion, after all points plotted have
been recorded correctly they will reproduce their creations using a “virtual beadloom” website.
The anticipatory set for this lesson, the attention
grabber as it’s known, is an easy but fun warm-up
activity (see appendix A). The students
must be partnered up another and each has a piece of paper and a small sticker
of writing utensil of some sort. Have
the students stand back-to-back, one facing the front of the room. Instruct the students who are facing you to
put their stick, or make a dot on their paper.
Meanwhile those who are no doing this remain looking to the back
wall. Now, have the students with the
dot on their paper attempt to instruct their partner where to put their dot
without looking at one another. They
will find that this is a difficult task to do because a plain sheet of white
paper does not have any reference points; it is all based on the judgment and
interpretation of the individual.
Finally, have the students discover this through discussion.
Following the warm-up the history of Rene Descartes, the
French mathematician is introduced (see appendix A). The students will come to find out that it
has been documented in history that he thought up the idea of a coordinate
plane while lying sick in bed… and so forth.
The students will be interested if the delivery is good. This will also demonstrate that good ideas do
not always come to scholars in a classroom, or lab setting, it’s really quite
the contrary.
A transparency of ¼” graphing paper should be shown now,
students should also have their own sheet on which they will mimic what is
being modeled (notes). On here, we will
make a Cartesian coordinate plane.
First, the axes should be introduced along with the terms horizontal and
vertical, and the (0,0) coordinate called the
origin. Next, name and number the
sections created by the axes called quadrants. Following this, discuss the
concept of ordered pairs, including the order (x,y) and how they are determined. This can then be followed by discussion of
analogies such as an intersection, and how much easier the opening activity
would have been using this tool, the coordinate plane.
While asking the students what other uses that there may
be for this idea that was thought up so long ago, pose this question, “Do you
think that no one else had ever thought of this concept or used it until
Descartes came up with it one day? If so, why?” My students often struggle when I ask
questions of this nature because they have been taught to listen and write,
rather than listen, question, and learn.
Let the silence continue until there is a response, this may take a few
minutes. Then, present them with
pictures.
The pictures I presented were from the website of Dr.
Eglash of RPI. There were four pictures,
consisting of Native American embroidery, a sand painting, a buffalo hide drum,
and beadwork. I enlarged the images and
put one on transparency and asked if anyone could see a coordinate plane being
used in the picture. The first student
to raise their hand was asked to come up and draw in on the picture. This leads into the historical discussion of
the coordinate plane and it use by the Native Americans, before Descartes had
“thought of it.” Though the Native
American had not named it and shared it with the rest of the world,
The activity portion of this lesson is based on the
historical, Native American use of the coordinate plane within their culture
and characteristics of the drawings and designs that were used. There were samples of student works passed
around, from elaborate and time consuming designs to simple ones. Each student is then given a piece of white
construction paper, 11 ½” by 18” to make their coordinate plane on. This is done as a class with a ruler so that
everyone has 1”X1” squares (or close to that) on their paper. Students then have the option to use the
paper vertically or horizontal and to create a labeled coordinate plane, on
which they will be creating their beadwork.
Putting their design together has no boundaries, using
their imagination, individuality, and being creative is encouraged. The only requirement is that it had to be
symmetrical or on four fold symmetry. I
provided four different colors of paper, and gave each student a template to
cut out 1” circles. Also, it should be
explained that the circles need to be placed on a coordinate where two lines
were intersecting, not inside the squares to make determining the ordered pairs
easier. Now, any last questions should
be answered and at that point the activity should begin.
As students finish, they will be instructed to write down
the coordinates for each point they have plotted with a colored dot and group
them either by color or by color and quadrant, on a separate sheet of
paper. Once they have documented all
ordered pair, students will go to the virtual beadloom
(http://www.rpi.edu/~eglash/csdt.html)
webpage and recreate their beadwork there.
During this last stage of the lesson/activity, the ability to properly
write down the ordered pairs and reproduce it on the website will demonstrate
understanding of the coordinate plane.
Those students who have not identified the correct ordered pair will be
deliberated with one-on-one to determine where the misunderstanding has taken
place. Lastly, after the entire lesson
was completed, students were given a short quiz to demonstrate cognizance and
comprehension of the coordinate plane.
I, Adriana Magallanes, gave this
lesson to a seventh grade pre-algebra class. Students were excited to learn a concept without
the book but seemed to feel awkward with the posing of questions. However, it was expected because this is the
typical reaction when this practice is employed. This exercise offered another point of view
to the student frame of mind and they were able to relate to it. They also came up with their own examples of
where they thought the idea of the coordinate plane could have been used such
as the Aztec calendar, serapès, and even pottery
designs they have seen. Although not
all of the examples were correct, it was enlightening to see students attempting
to make their own connection to the curriculum.
There were many who became highly engaged in making their “beadwork,”
as well as those who did not want to go to too much trouble. However, on the whole this was a successful
lesson for another outlook on the “Cartesian” coordinate plane, not dominated
by Eurocentrism.
This was a lesson that addressed the standards, but was adapted for
the student population that it is serving.
At no time was it ever watered down but rather enriched.
Appendix A
Introduction
Anticipatory Set- Activity with a partner, related to ambiguity without
the use of a coordinate plane.
Objective- Represent quantitative relationships graphically,
interpret meaning of specific parts of a graph, understand the use of graphs to
plot simple figures, and understand how to graph and read ordered pairs.
CA Standards: Algebra and Functions 1.5,
represent quantitative relationships graphically and interpret the meaning of a
specific part of a graph in the situation represented by the graph and
Measurement and Geometry 3.2, understand and use coordinate planes to plot
simple figures, determine lengths and areas related to them, and determine
their image under translations and reflections.
Objective- students will be able to identify specific parts of a
coordinate plane and is origin, as well as plot point and identify ordered
pairs of specific points that have been plotted.
Activity- Each student will be creating his or her own
Cartesian coordinate plane and points to be plotted, as related to historical
lecture.
Strategy- Begin with partner activity followed by explanation
of Rene Descartes creation of the Cartesian coordinate plane, as he named it. Next,
question to students, “Do you think he was the first to come up with the idea
just because it has been written down that way?” discussion. Introduction of Native
American works that used this same idea but also incorporated symmetry, show
examples.
Title 1- This population will be able to create a meaningful
application and related to the subject.
They will also be required to label all parts of the plane, list all
coordinates by quadrant, and reproduce this on the “virtual beadloom”
to show success of documentation.
GATE- These students will be expected to develop intricate designs and also
label the plane correctly, and provide guidance at their table as
necessary.
RSP and LEP- This activity will provide a hands-on application and
connection to the concepts presented. In
addition, allowing them to develop their own design and name it will also
encourage “ownership” of the concept.
Modeling- Show a sample of what a finished product could look
like. Encourage creativity and
individuality.
Guided Practice- This will essentially be taking place during the
entire activity, as students demonstrate acquired knowledge.
Do you now agree that Native Americans have used the
Cartesian coordinate plain effectively?
Should they be given credit for the idea even though it was not formally
document as such? Small quiz on parts of
a Cartesian coordinate plane, plotting points, and identifying ordered pairs of
points that have been plotted previously.
Warm Up Activity
Have the students get into
pairs with their backs to each other.
Give one of the students of each pair a sheet of plain paper with a dot,
a sticker, or some other kind of mark on it, (you may decide to have the student
mark the paper on their own). Have one
student describe to the other one the location of the mark, dot, or sticker
while the second student tries to put a mark on the location on the plain sheet
of paper from the description given.
Repeat the activity with the other student doing the describing
(optional). When completed discuss how
easy or difficult the task was. Point
out that a way of doing it with more success and accuracy would be to use some
kind of mapping system, such as the one developed by Descartes.
Brief History of Descartes
René
Descartes was a French mathematician and philosopher who lived from
1596-1650. A story goes that once, while
sick in bed, he noticed a fly on the ceiling.
It came to him that there should be a way to accurately describe the
location of the fly to someone else, like his nurse. Supposedly, from this thought he developed
the Cartesian coordinate plane with coordinates to determine the exact location
of points on a plane.