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Pi by the Slice
Our curriculum project entitled “Pi by the slice” will be
taught over the course of
five days. The curriculum consists of learning the radius and diameter of a
circle, finding
pi and circumference, applying these discoveries to find the area of a circle
and creating a
final presentation. We will have activities planned each day that allow the
children to put
abstract ideas into concrete examples to prove mathematical formulas . We will
also
incorporate group work and get the parents involved in the school life of their
children.
On Monday we will teach about the diameter and radius of a circle. Since
Hartford is known for its bike cops, we will have them ride in for visual
reference and
give the students the chance to measure the wheels. Since this is the first part
of the
curriculum, we just want to define terms and have the students recognize them.
This
corresponds to the “Knowledge” level in Bloom’s Taxonomy. We will have one
volunteer measure across the wheel and we will introduce that as the diameter.
We will
have another volunteer measure only halfway across the wheel and introduce that
as
radius. The students will then figure out the relationship they share and how to
fit that
into their respective formulas. They will realize that radius is half the
diameter and that
conversely, diameter is twice the radius. They can apply this information to
everyday life.
They will examine the tire of the police officer’s bike
and see the number 65 centimeters
x 1.5 centimeters. We will explain to them that the number 65 corresponds to the
diameter of the bicycle wheel. This illustrates that Linguistic Intelligence is
being used,
which is one of Howard Gardner’s Multiple Intelligences . During this first
lesson, the
students are merely learning the language of math. They will need to know what
those
terms mean and how to use them appropriately in the context of the class.
On Tuesday, the students will discover pi and use it to find circumference. They
will be moving up a level in Bloom’s Taxonomy to “application”. The students
will
apply what they learned about diameter to finding pi and the circumference of a
circle.
Since it is better for the kids to see then just to tell them, we will use
another activity to
show this.
The class will be brought out into the hallway where a big red circle is on the
floor. We will ask for two volunteers. To explain this activity we will use two
hypothetical students, Elsie and Jose. The students will be asked to stand on
the same
side of the circle. When they are given the go, Jose will walk around the
circle, which is
the circumference, and Elsie will walk across the circle, which is the diameter.
Elsie will
keep walking back and forth until Jose has walked around the circle one full
time. The
students must be walking the same speed for this to work and must also be around
the
same height so that one student does not take bigger steps than the other . If
this is done
correctly, the students will see that Elsie walked the diameter, three and a
little bit more
times while Jose walked the circumference once. This activity is a great example
of one
of Howard Gardner’s Multiple Intelligences, Bodily/Kinesthetic Intelligence. The
volunteers are using their bodies to figure out how to find circumference by
walking
around and across the circle.
From here, the students will learn that three and a little bit more means pi
which
more specifically is 3.14.By analyzing the situation and moving up another level
in
Bloom’s Taxonomy, the class will make the inference that pi times the diameter
equals
the circumference. Some students may not be able to comprehend the example
shown, or
may not believe that this is possible and that it was just a fluke. So to
reinforce this
example we will have the nonbelievers demonstrate the same lesson, while using
different tools .
The next example will deal with using a belt to discover radius and pi. A belt
will
be tied so that it is in a circular shape and will be put on the floor. A piece
of yarn will be
used to measure the circumference of the belt and cut. Then using that length of
yarn, we
will show that it can go across the diameter three and a little bit more times,
which we
just established as pi. Both of the examples used for this lesson are a good
example for
Gardner’s Logical/ Mathematical Intelligence . The students will understand the
central
principles of these activities while using logic and math .
To wrap up Tuesday’s lesson, we will give the class a list of diameters. They
will
use them to find the circumference and plot diameter vs. circumference on graph
paper .
They will realize that there is a linear relationship between diameter and
circumference.
They will then be asked to calculate the slope and see that the slope of the
line is 3.14,
which is pi. This incorporates CMT preparation into the curriculum. In order to
do well
on the CMT’s, the students need to be able to read graphs and determine the
relationships
between them. Another attribute this curriculum provides is the ability to view
the same
problem in different ways and come up with different methods in which to solve
them,
using important critical thinking skills.
On Wednesday, we will introduce a project that the kids will construct
throughout
the rest of the week. Since no group work occurs in the math room at Hartford
Magnet
Middle School, students will work in groups to complete this project. They will
have to
create their own methods of discovering pi and circumference. This project will
turn into
a competition. Different groups will be awarded different prizes based on
different
standards. They will have Wednesday to work on the project and are encouraged to
continue and finish it after school or at home. They will have until Friday to
complete it.
We will give them a rubric on how they will be graded on the project (Appendix
A).
They can use this as a guide to ensure they receive the best grade possible.
This project
will help the students move up yet another level in Bloom’s Taxonomy, synthesis.
In this
level, students will create their own methods and propose it to the judges.
On Thursday we will introduce finding the area of a circle and the different
methods to acquire the correct answer. The students would return to the same
circle in the
hallway that we had used to discover Pi in Tuesday’s lesson and be able to apply
their
previous knowledge of radius to discover the area. In Bloom’s Taxonomy, this is
the
“Application” level. According to this level the students should be able to
apply Pi and
achieve a hypothesis on how to solve for the area . Once the students formed a
hypothesis we would explain the formula, area is equal to Pi times the radius,
squared.
We are going to implement this by having another two volunteers go to the
circle, one of
whom will estimate how many squares are inside the circle. They would be told
that each
square’s side is measured at one square foot, making it easy for them to count
the tiles
and acquire the area in feet. The other volunteer would measure how many squares
it
takes to get to the radius. We would then ask them to think about ways in which
they
could get the area from the radius. The students would be using logical and
spatial
intelligences to solve for the area, as described by Gardner’s Multiple
Intelligences
Theory. They will be applying logic in order to determine which method is most
efficient. They will be using spatial intelligence because they will be figuring
out how
much room the circle takes up.
The second exercise and group activity would begin as soon as we return to the
classroom. We once again decided to do a group activity because it encourages
the
students to work with and teach other students, which would reap better learning
instead
of individual activities. The classroom would be set up with groups of about
four
students, sitting at the tables with a different size circle already predrawn
on the
graphing paper. Each group would use the “counting squares” method and the
formula
method to discover the area. They would then evaluate both methods and determine
which one is the most efficient and precise. After that we would instruct them
to rotate to
a different table and check the previous group’s work. They would rotate until
they
reached their original table. Through this activity, the students have moved
again to the
second to last level in Bloom’s Taxonomy, Analysis. In this activity, students
will
analyze which method works better for finding the area of a circle. With these
lesson
plans and activities, we would be fulfilling the Connecticut standards in the
geometry and
measurement sections, 3.1 to 3.3.
On Friday, the presentations of the project will take
place. It is very important for
a school to get the parents involved in the school life of their child.
Therefore, we will
invite a few parents to be judges of the competition while the rest of the
parents will be
asked to come and support their children. Prizes will be awarded based on
various
qualities of the method such as most creative method, most mathematically
precise
method, easiest method and also group qualities such as most organized, most
energetic,
and best teamwork. This project will wrap up the pi and circumference unit.
Through this
competition, the students have reached the final level in Bloom’s Taxonomy,
Evaluation The students will have to argue and defend their method for finding
pi and
circumference.
Through this five day curriculum students will learn what the diameter and
radius
of a circle is, discover pi and use it to find circumference, apply pi to
finding the area of a
circle and finally, use what they’ve seen and learned to design their own
methods for
discovering pi and finding circumference. On Monday, we will have a Hartford
Bike Cop
ride into the classroom. We will use one of the tires on the bike to show the
students what
diameter and radius is and their relationship. On Tuesday the students will use
what they
learned about diameter to find pi and circumference. We will do an activity
where they
see firsthand that Pi times the diameter equals the circumference. On
Wednesday, the
students will begin their projects on finding their own methods to discover
circumference
and Pi. On Thursday the students will apply what they have learned from previous
lessons to find the area of a circle. They will partake in an activity where
they count the
number of tiles in the hallway circle to find area and learn the formula. Then
they will get
into groups and where they will find the area of a circle using the counting
squares
method and the formula. The groups will also rotate to different groups and
check the
work of that group. On Friday, the presentations will be presented. Groups will
be .
awarded prizes based on certain qualities. This will conclude the unit on pi.
The whole
lesson incorporated four of Gardner’s Intelligences; Linguistic,
Logical/Mathematical,
Spatial and Bodily/Kinesthetic Intelligence. It also incorporated the six levels
in Bloom’s
Taxonomy; Knowledge, Understanding, Application, Analysis, Synthesis and
Evaluation.
Appendix A
Methods for Discovering Pi and Circumference Rubric
CATEGORY  4  3  2  1 
Preparedness  Group is completely prepared and has obviously rehearsed. 
Group seems pretty prepared but might have needed a couple more rehearsals 
Group is somewhat prepared, but it is clear that rehearsal was lacking. 
Group does not seem at all prepared to present. 
Speech  Speaks clearly and distinctly all the time, and mispronounces no words. 
Speaks clearly and distinctly all the time, but mispronounces one word. 
Speaks clearly and distinctly most of the time.
Mispronounces no more than one word. 
Often mumbles or can not be understood OR mispronounces more than one word. 
Comprehension  Group is able to accurately answer almost all questions posed by classmates about the topic. 
Group is able to accurately answer most questions
posed by classmates about the topic. 
Group is able to accurately answer a few questions posed by classmates about the topic. 
Group is unable to accurately answer questions
posed by classmates about the topic. 
Content  Shows a full understanding of the topic.  Shows a good understanding of the topic.  Shows a good understanding of parts of the topic.  Does not seem to understand the topic very well. 
Props  Group uses several props that show considerable work/creativity and which make the presentation better. 
Group uses 1 prop that shows considerable work/creativity and which make the presentation better. 
Group uses 1 prop which makes the presentation
better. 
Group uses no props OR the props chosen detract from the presentation. 
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